Subgroups — C₃₁×#22984 #21479743

Order 7936.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1922	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1922	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1866	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1866	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic
14	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1759	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₃₁×#533	C₂²	1	1	yes	characteristic
19	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#479	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#479	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#478	C₄	1	1	yes	characteristic, complemented
23	C₃₁×#478	C₄	1	1	yes	characteristic, complemented
24	C₃₁×#477	C₄	1	1	yes	characteristic, complemented
25	C₃₁×#476	C₄	1	1	yes	characteristic, complemented
26	C₃₁×#475	C₄	1	1	yes	characteristic, complemented
27	C₃₁×#474	C₄	1	1	yes	characteristic, complemented
28	C₃₁×#446	—	2	1	no	—
29	C₃₁×#446	—	2	1	no	—
30	C₃₁×#446	—	2	1	no	—
31	C₈×Q₈×C₃₁	—	2	1	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
34	C₃₁×#427	C₂²	1	1	yes	characteristic
35	C₃₁×#427	C₂²	1	1	yes	characteristic
36	C₃₁×#424	—	2	1	no	—
37	C₃₁×#424	—	2	1	no	—
38	C₃₁×#424	C₂²	1	1	yes	characteristic
39	C₃₁×#424	C₂²	1	1	yes	characteristic
40	C₃₁×#424	C₂²	1	1	yes	characteristic
41	C₃₁×#424	C₂²	1	1	yes	characteristic
42	C₃₁×#423	—	2	1	no	—
43	C₃₁×#423	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
49	C₃₁×#416	C₂²	1	1	yes	characteristic
50	C₃₁×#416	C₂²	1	1	yes	characteristic
51	C₃₁×#405	C₂²	1	1	yes	characteristic
52	C₃₁×#405	C₂²	1	1	yes	characteristic
53	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
55	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
56	C₃₁×#376	C₂²	1	1	yes	characteristic
57	C₃₁×#341	C₂²	1	1	yes	characteristic
58	C₃₁×#341	C₂²	1	1	yes	characteristic
59	C₃₁×#339	C₂²	1	1	yes	characteristic
60	C₃₁×#339	C₂²	1	1	yes	characteristic
61	C₃₁×#339	C₂²	1	1	yes	characteristic
62	C₃₁×#339	C₂²	1	1	yes	characteristic
63	C₃₁×#339	C₂²	1	1	yes	characteristic
64	C₃₁×#339	C₂²	1	1	yes	characteristic
65	C₃₁×#336	C₂²	1	1	yes	characteristic
66	C₃₁×#336	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#131	—	2	2	no	—
69	C₃₁×#131	—	2	2	no	—
70	C₃₁×#131	—	4	1	no	—
71	C₃₁×#131	—	4	1	no	—
72	C₂×M₁₆×C₃₁	—	2	1	no	—
73	C₂×M₁₆×C₃₁	—	2	1	no	—
74	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	—	2	1	no	—
79	C₂²×C₈×C₃₁	—	2	1	no	—
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
87	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	C₂³	1	1	yes	characteristic
100	C₃₁×#117	C₂³	1	1	yes	characteristic
101	C₃₁×#117	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	—	2	1	no	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₃₁×#108	—	4	1	no	—
107	C₃₁×#108	—	4	1	no	—
108	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
109	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
110	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
111	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
112	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
113	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
114	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
115	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	—	2	1	no	—
118	C₃₁×C₄⋊C₈	—	2	1	no	—
119	C₃₁×C₄⋊C₈	—	2	1	no	—
120	C₃₁×C₄⋊C₈	—	2	1	no	—
121	C₃₁×C₄⋊C₈	—	2	1	no	—
122	C₃₁×C₄⋊C₈	—	2	1	no	—
123	C₃₁×C₄⋊C₈	—	2	1	no	—
124	C₃₁×C₄⋊C₈	—	2	1	no	—
125	C₃₁×C₄⋊C₈	—	2	1	no	—
126	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
127	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
128	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
129	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
130	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
131	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
132	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
133	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
134	C₃₁×#104	—	2	2	no	—
135	C₃₁×#104	Q₈	1	4	yes	complemented
136	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
137	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
138	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
139	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
140	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
141	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
142	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
143	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
144	C₃₁×#97	—	2	1	no	—
145	C₃₁×#97	—	2	1	no	—
146	C₃₁×#97	—	2	1	no	—
147	C₃₁×#97	—	4	1	no	—
148	C₄×C₈×C₃₁	—	2	1	no	—
149	C₄×C₈×C₃₁	—	2	1	no	—
150	C₄×C₈×C₃₁	—	2	1	no	—
151	C₄×C₈×C₃₁	—	4	1	no	—
152	C₃₁×#95	—	2	1	no	—
153	C₃₁×#95	—	2	1	no	—
154	#41×C₃₁	—	2	1	no	—
155	#41×C₃₁	—	2	1	no	—
156	#41×C₃₁	C₂×Q₈	1	2	yes	—
157	#41×C₃₁	#41	1	2	yes	—
158	C₂×Q₈×C₃₁	—	2	1	no	—
159	C₂×Q₈×C₃₁	—	2	1	no	—
160	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	1	no	—
168	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	M₁₆×C₃₁	—	2	1	no	—
172	M₁₆×C₃₁	—	2	1	no	—
173	M₁₆×C₃₁	—	2	1	no	—
174	M₁₆×C₃₁	—	2	1	no	—
175	M₁₆×C₃₁	—	2	1	no	—
176	M₁₆×C₃₁	—	2	1	no	—
177	M₁₆×C₃₁	—	2	1	no	—
178	M₁₆×C₃₁	—	4	2	no	—
179	M₁₆×C₃₁	—	4	2	no	—
180	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	1	no	—
186	C₂×C₈×C₃₁	—	2	1	no	—
187	C₂×C₈×C₃₁	—	2	1	no	—
188	C₂×C₈×C₃₁	—	2	1	no	—
189	C₂×C₈×C₃₁	—	2	1	no	—
190	C₂×C₈×C₃₁	—	2	1	no	—
191	C₂×C₈×C₃₁	—	2	1	no	—
192	C₂×C₈×C₃₁	—	2	1	no	—
193	C₂×C₈×C₃₁	—	4	2	no	—
194	C₂×C₈×C₃₁	—	4	2	no	—
195	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
196	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
197	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
198	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
199	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₃₁	—	2	1	no	—
204	C₄⋊C₄×C₃₁	—	2	1	no	—
205	C₄⋊C₄×C₃₁	—	2	1	no	—
206	C₄⋊C₄×C₃₁	—	2	1	no	—
207	C₄⋊C₄×C₃₁	—	2	1	no	—
208	C₄⋊C₄×C₃₁	—	2	1	no	—
209	C₄⋊C₄×C₃₁	—	2	1	no	—
210	C₄⋊C₄×C₃₁	—	2	1	no	—
211	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
213	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
214	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
215	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
216	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
217	C₂²⋊C₄×C₃₁	—	2	1	no	—
218	C₂²⋊C₄×C₃₁	—	2	1	no	—
219	C₂²⋊C₄×C₃₁	—	2	1	no	—
220	C₂²⋊C₄×C₃₁	—	2	1	no	—
221	C₂²⋊C₄×C₃₁	—	2	1	no	—
222	C₄²×C₃₁	—	2	1	no	—
223	C₄²×C₃₁	—	2	1	no	—
224	C₄²×C₃₁	—	2	1	no	—
225	C₄²×C₃₁	—	2	1	no	—
226	C₄²×C₃₁	—	2	1	no	—
227	C₄²×C₃₁	—	2	1	no	—
228	C₄²×C₃₁	—	2	1	no	—
229	C₄²×C₃₁	—	2	1	no	—
230	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
231	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
232	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
233	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
235	#22984	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₃₁	—	2	1	no	—
237	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
238	Q₈×C₃₁	—	2	1	no	—
239	Q₈×C₃₁	—	4	2	no	—
240	Q₈×C₃₁	—	8	1	no	—
241	Q₈×C₃₁	C₄×Q₈	1	2	yes	—
242	D₄×C₃₁	—	2	2	no	—
243	D₄×C₃₁	—	2	2	no	—
244	D₄×C₃₁	—	2	1	no	—
245	D₄×C₃₁	C₄×Q₈	1	2	yes	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	4	1	no	—
257	C₂×C₄×C₃₁	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	2	no	—
259	C₂×C₄×C₃₁	—	4	1	no	—
260	C₂×C₄×C₃₁	—	4	1	no	—
261	C₂×C₄×C₃₁	—	4	1	no	—
262	C₂×C₄×C₃₁	—	4	1	no	—
263	C₂×C₄×C₃₁	—	4	1	no	—
264	C₂×C₄×C₃₁	—	8	1	no	—
265	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
275	C₈×C₃₁	—	2	1	no	—
276	C₈×C₃₁	—	2	1	no	—
277	C₈×C₃₁	—	2	1	no	—
278	C₈×C₃₁	—	2	1	no	—
279	C₈×C₃₁	—	2	1	no	—
280	C₈×C₃₁	—	2	1	no	—
281	C₈×C₃₁	—	2	1	no	—
282	C₈×C₃₁	—	2	1	no	—
283	C₈×C₃₁	—	2	2	no	—
284	C₈×C₃₁	—	4	1	no	—
285	C₈×C₃₁	—	4	1	no	—
286	C₈×C₃₁	—	4	1	no	—
287	#3076	C₂×C₃₁	1	1	yes	characteristic
288	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2906	C₂×C₃₁	1	1	yes	characteristic
290	#1995	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#1922	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1922	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#1866	C₂×C₃₁	1	1	yes	characteristic
297	#1866	C₂×C₃₁	1	1	yes	characteristic
298	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#1765	C₂×C₃₁	1	1	yes	characteristic
300	#1764	C₂×C₃₁	1	1	yes	characteristic
301	#1759	C₂×C₃₁	1	1	yes	characteristic
302	C₂²×C₃₁	—	2	1	no	—
303	C₂²×C₃₁	—	2	1	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	4	1	no	—
306	C₂²×C₃₁	—	4	1	no	—
307	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
308	C₂²×C₃₁	#552	1	1	yes	characteristic
309	C₂²×C₃₁	#554	1	1	yes	characteristic
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	2	1	no	—
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	1	no	—
317	C₄×C₃₁	—	4	1	no	—
318	C₄×C₃₁	—	4	1	no	—
319	C₄×C₃₁	—	4	1	no	—
320	C₄×C₃₁	—	4	1	no	—
321	C₄×C₃₁	—	4	1	no	—
322	C₄×C₃₁	—	8	1	no	—
323	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
324	C₄×C₃₁	#520	1	1	yes	characteristic
325	C₄×C₃₁	#546	1	1	yes	characteristic
326	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
327	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
329	#533	C₂²×C₃₁	1	1	yes	characteristic
330	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
331	#479	C₄×C₃₁	1	1	yes	characteristic, complemented
332	#479	C₄×C₃₁	1	1	yes	characteristic, complemented
333	#478	C₄×C₃₁	1	1	yes	characteristic, complemented
334	#478	C₄×C₃₁	1	1	yes	characteristic, complemented
335	#477	C₄×C₃₁	1	1	yes	characteristic, complemented
336	#476	C₄×C₃₁	1	1	yes	characteristic, complemented
337	#475	C₄×C₃₁	1	1	yes	characteristic, complemented
338	#474	C₄×C₃₁	1	1	yes	characteristic, complemented
339	#446	—	2	1	no	—
340	#446	—	2	1	no	—
341	#446	—	2	1	no	—
342	C₈×Q₈	—	2	1	no	—
343	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
345	#427	C₂²×C₃₁	1	1	yes	characteristic
346	#427	C₂²×C₃₁	1	1	yes	characteristic
347	#424	—	2	1	no	—
348	#424	—	2	1	no	—
349	#424	C₂²×C₃₁	1	1	yes	characteristic
350	#424	C₂²×C₃₁	1	1	yes	characteristic
351	#424	C₂²×C₃₁	1	1	yes	characteristic
352	#424	C₂²×C₃₁	1	1	yes	characteristic
353	#423	—	2	1	no	—
354	#423	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₈	—	2	1	no	—
356	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
360	#416	C₂²×C₃₁	1	1	yes	characteristic
361	#416	C₂²×C₃₁	1	1	yes	characteristic
362	#405	C₂²×C₃₁	1	1	yes	characteristic
363	#405	C₂²×C₃₁	1	1	yes	characteristic
364	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
367	#376	C₂²×C₃₁	1	1	yes	characteristic
368	#341	C₂²×C₃₁	1	1	yes	characteristic
369	#341	C₂²×C₃₁	1	1	yes	characteristic
370	#339	C₂²×C₃₁	1	1	yes	characteristic
371	#339	C₂²×C₃₁	1	1	yes	characteristic
372	#339	C₂²×C₃₁	1	1	yes	characteristic
373	#339	C₂²×C₃₁	1	1	yes	characteristic
374	#339	C₂²×C₃₁	1	1	yes	characteristic
375	#339	C₂²×C₃₁	1	1	yes	characteristic
376	#336	C₂²×C₃₁	1	1	yes	characteristic
377	#336	C₂²×C₃₁	1	1	yes	characteristic
378	C₂×C₃₁	—	2	1	no	—
379	C₂×C₃₁	—	2	1	no	—
380	C₂×C₃₁	—	4	2	no	—
381	C₂×C₃₁	#2360	1	1	yes	characteristic
382	C₂×C₃₁	#2955	1	1	yes	characteristic
383	C₂×C₃₁	#2957	1	1	yes	characteristic
384	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
385	#131	—	2	2	no	—
386	#131	—	2	2	no	—
387	#131	—	4	1	no	—
388	#131	—	4	1	no	—
389	C₂×M₁₆	—	2	1	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
395	C₂²×C₈	—	2	1	no	—
396	C₂²×C₈	—	2	1	no	—
397	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
402	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
403	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
404	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₂²⋊Q₈	—	2	1	no	—
406	C₂²⋊Q₈	—	2	1	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
410	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
414	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
415	#117	—	2	1	no	—
416	#117	C₂³×C₃₁	1	1	yes	characteristic
417	#117	C₂³×C₃₁	1	1	yes	characteristic
418	#117	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
423	#108	—	4	1	no	—
424	#108	—	4	1	no	—
425	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
426	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
427	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
428	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
429	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
430	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
431	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
432	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
433	C₄⋊C₈	—	2	1	no	—
434	C₄⋊C₈	—	2	1	no	—
435	C₄⋊C₈	—	2	1	no	—
436	C₄⋊C₈	—	2	1	no	—
437	C₄⋊C₈	—	2	1	no	—
438	C₄⋊C₈	—	2	1	no	—
439	C₄⋊C₈	—	2	1	no	—
440	C₄⋊C₈	—	2	1	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	1	no	—
443	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
444	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
445	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
446	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
447	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
448	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
449	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
450	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
451	#104	—	2	2	no	—
452	#104	Q₈×C₃₁	1	4	yes	complemented
453	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
454	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
455	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
456	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
457	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
458	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
459	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
460	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
461	#97	—	2	1	no	—
462	#97	—	2	1	no	—
463	#97	—	2	1	no	—
464	#97	—	4	1	no	—
465	C₄×C₈	—	2	1	no	—
466	C₄×C₈	—	2	1	no	—
467	C₄×C₈	—	2	1	no	—
468	C₄×C₈	—	4	1	no	—
469	#95	—	2	1	no	—
470	#95	—	2	1	no	—
471	C₃₁	#22984	1	1	yes	characteristic, direct product factor, Sylow
472	#41	—	2	1	no	—
473	#41	—	2	1	no	—
474	#41	C₂×Q₈×C₃₁	1	2	yes	—
475	#41	#41×C₃₁	1	2	yes	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
489	M₁₆	—	2	1	no	—
490	M₁₆	—	2	1	no	—
491	M₁₆	—	2	1	no	—
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	1	no	—
494	M₁₆	—	2	1	no	—
495	M₁₆	—	2	1	no	—
496	M₁₆	—	4	2	no	—
497	M₁₆	—	4	2	no	—
498	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	4	2	no	—
512	C₂×C₈	—	4	2	no	—
513	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
514	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
515	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
516	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
517	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
518	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
530	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
531	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
532	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
533	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
534	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
549	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
550	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
551	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
553	C₂³	—	2	1	no	—
554	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
555	Q₈	—	2	1	no	—
556	Q₈	—	4	2	no	—
557	Q₈	—	8	1	no	—
558	Q₈	C₄×Q₈×C₃₁	1	2	yes	—
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	2	1	no	—
562	D₄	C₄×Q₈×C₃₁	1	2	yes	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	8	1	no	—
582	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
590	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
591	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
592	C₈	—	2	1	no	—
593	C₈	—	2	1	no	—
594	C₈	—	2	1	no	—
595	C₈	—	2	1	no	—
596	C₈	—	2	1	no	—
597	C₈	—	2	1	no	—
598	C₈	—	2	1	no	—
599	C₈	—	2	1	no	—
600	C₈	—	2	2	no	—
601	C₈	—	4	1	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
610	C₂²	C₃₁×#552	1	1	yes	characteristic
611	C₂²	C₃₁×#554	1	1	yes	characteristic
612	C₄	—	2	2	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	1	no	—
619	C₄	—	4	1	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	1	no	—
623	C₄	—	4	1	no	—
624	C₄	—	8	1	no	—
625	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
626	C₄	C₃₁×#520	1	1	yes	characteristic
627	C₄	C₃₁×#546	1	1	yes	characteristic
628	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	4	2	no	—
632	C₂	C₃₁×#2360	1	1	yes	characteristic
633	C₂	C₃₁×#2955	1	1	yes	characteristic
634	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₄⋊C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2962	C₂	semidirect
4. C₃₁×#1995	C₂	semidirect
5. C₃₁×#1950	C₂	semidirect
6. C₃₁×#1946	C₂	semidirect
7. C₃₁×#1945	C₂	semidirect
8. C₃₁×#1922	C₂	semidirect
9. C₃₁×#1922	C₂	semidirect
12. C₃₁×#1768	C₂	semidirect
20. C₃₁×#479	C₄	semidirect
21. C₃₁×#479	C₄	semidirect
22. C₃₁×#478	C₄	semidirect
23. C₃₁×#478	C₄	semidirect
24. C₃₁×#477	C₄	semidirect
25. C₃₁×#476	C₄	semidirect
26. C₃₁×#475	C₄	semidirect
27. C₃₁×#474	C₄	semidirect
135. C₃₁×#104	Q₈	semidirect
235. #22984	C₃₁	direct (N × H)
288. #2962	C₂×C₃₁	semidirect
290. #1995	C₂×C₃₁	semidirect
291. #1950	C₂×C₃₁	semidirect
292. #1946	C₂×C₃₁	semidirect
293. #1945	C₂×C₃₁	semidirect
294. #1922	C₂×C₃₁	semidirect
295. #1922	C₂×C₃₁	semidirect
298. #1768	C₂×C₃₁	semidirect
331. #479	C₄×C₃₁	semidirect
332. #479	C₄×C₃₁	semidirect
333. #478	C₄×C₃₁	semidirect
334. #478	C₄×C₃₁	semidirect
335. #477	C₄×C₃₁	semidirect
336. #476	C₄×C₃₁	semidirect
337. #475	C₄×C₃₁	semidirect
338. #474	C₄×C₃₁	semidirect
452. #104	Q₈×C₃₁	semidirect
471. C₃₁	#22984	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3076	2
C₃₁×#2962	2
C₃₁×#2906	2
C₃₁×#1995	2
C₃₁×#1950	2
C₃₁×#1946	2
C₃₁×#1945	2
C₃₁×#1922	2
C₃₁×#1866	2
C₃₁×#1768	2
C₃₁×#1765	2
C₃₁×#1764	2
C₃₁×#1759	2
#22984	31

Subgroups — C31×#22984 #21479743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22984 #21479743