Subgroups — C₃₁×#22983 #21479742

Order 7936.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 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×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1946	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1866	C₂	1	2	yes	maximal
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1765	C₂	1	2	yes	maximal
11	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#131	C₂²	1	2	yes	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#479	C₄	1	2	yes	complemented
16	C₃₁×#478	C₄	1	2	yes	complemented
17	C₃₁×#477	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#476	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#475	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#474	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#446	—	2	2	no	—
22	C₃₁×#446	—	2	1	no	—
23	C₈×Q₈×C₃₁	—	2	1	no	—
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₃₁×#427	C₂²	1	2	yes	—
26	C₃₁×#424	—	2	2	no	—
27	C₃₁×#424	C₂²	1	2	yes	—
28	C₃₁×#424	C₂²	1	2	yes	—
29	C₃₁×#423	—	2	1	no	—
30	C₃₁×#423	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
36	C₃₁×#416	C₂²	1	2	yes	—
37	C₃₁×#405	C₂²	1	2	yes	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#97	C₂²	1	2	yes	—
40	C₄³×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#341	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₃₁×#339	C₂²	1	2	yes	—
46	C₃₁×#336	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#131	—	2	4	no	—
49	C₃₁×#131	—	4	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	C₃₁×#125	C₂×C₄	1	2	yes	—
60	C₃₁×C₂²⋊Q₈	—	2	1	no	—
61	C₃₁×C₂²⋊Q₈	—	2	1	no	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×Q₈×C₃₁	—	2	1	no	—
64	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	C₂³	1	2	yes	—
72	C₃₁×#117	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#108	—	4	2	no	—
80	C₃₁×#107	C₂×C₄	1	2	yes	—
81	C₃₁×#107	C₂×C₄	1	2	yes	—
82	C₃₁×#106	C₂×C₄	1	2	yes	—
83	C₃₁×#106	C₂×C₄	1	2	yes	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
92	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
93	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
96	C₃₁×#104	—	2	2	no	—
97	C₃₁×#104	Q₈	1	4	yes	complemented
98	C₃₁×#103	C₂×C₄	1	2	yes	—
99	C₃₁×#103	C₂×C₄	1	2	yes	—
100	C₃₁×#102	C₂×C₄	1	2	yes	—
101	C₃₁×#102	C₂×C₄	1	2	yes	—
102	C₃₁×#97	—	2	2	no	—
103	C₃₁×#97	—	2	1	no	—
104	C₃₁×#97	—	4	2	no	—
105	C₄×C₈×C₃₁	—	2	2	no	—
106	C₄×C₈×C₃₁	—	2	1	no	—
107	#41×C₃₁	—	2	1	no	—
108	#41×C₃₁	—	2	1	no	—
109	#41×C₃₁	C₂×Q₈	1	2	yes	—
110	#41×C₃₁	#41	1	2	yes	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	M₁₆×C₃₁	—	2	1	no	—
125	M₁₆×C₃₁	—	2	1	no	—
126	M₁₆×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	—	2	1	no	—
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	4	4	no	—
130	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	4	4	no	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	1	no	—
156	C₂²⋊C₄×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	4	2	no	—
165	C₄²×C₃₁	—	4	2	no	—
166	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
171	#22983	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
174	Q₈×C₃₁	—	2	1	no	—
175	Q₈×C₃₁	—	4	2	no	—
176	Q₈×C₃₁	—	8	1	no	—
177	Q₈×C₃₁	C₄×Q₈	1	2	yes	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	1	no	—
181	D₄×C₃₁	C₄×Q₈	1	2	yes	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	8	1	no	—
199	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₃₁	—	2	2	no	—
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	2	2	no	—
212	C₈×C₃₁	—	2	2	no	—
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	1	no	—
216	#3076	C₂×C₃₁	1	1	yes	characteristic
217	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2906	C₂×C₃₁	1	1	yes	characteristic
219	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1946	C₂×C₃₁	1	2	yes	complemented
222	#1922	C₂×C₃₁	1	2	yes	complemented
223	#1866	C₂×C₃₁	1	2	yes	—
224	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1765	C₂×C₃₁	1	2	yes	—
226	C₄×#104	C₂×C₃₁	1	1	yes	characteristic
227	C₂²×C₃₁	—	2	1	no	—
228	C₂²×C₃₁	—	2	1	no	—
229	C₂²×C₃₁	—	2	2	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂²×C₃₁	#552	1	1	yes	characteristic
234	C₂²×C₃₁	#554	1	1	yes	characteristic
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	2	1	no	—
237	C₄×C₃₁	—	2	1	no	—
238	C₄×C₃₁	—	2	1	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	#444	1	2	yes	—
246	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
247	C₄×C₃₁	#520	1	1	yes	characteristic
248	C₄×C₃₁	#546	1	1	yes	characteristic
249	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
250	C₂×#131	C₂²×C₃₁	1	2	yes	—
251	#533	C₂²×C₃₁	1	1	yes	characteristic
252	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
253	#479	C₄×C₃₁	1	2	yes	complemented
254	#478	C₄×C₃₁	1	2	yes	complemented
255	#477	C₄×C₃₁	1	1	yes	characteristic, complemented
256	#476	C₄×C₃₁	1	1	yes	characteristic, complemented
257	#475	C₄×C₃₁	1	1	yes	characteristic, complemented
258	#474	C₄×C₃₁	1	1	yes	characteristic, complemented
259	#446	—	2	2	no	—
260	#446	—	2	1	no	—
261	C₈×Q₈	—	2	1	no	—
262	C₂×#108	C₂²×C₃₁	1	2	yes	—
263	#427	C₂²×C₃₁	1	2	yes	—
264	#424	—	2	2	no	—
265	#424	C₂²×C₃₁	1	2	yes	—
266	#424	C₂²×C₃₁	1	2	yes	—
267	#423	—	2	1	no	—
268	#423	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₈	—	2	1	no	—
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
274	#416	C₂²×C₃₁	1	2	yes	—
275	#405	C₂²×C₃₁	1	2	yes	—
276	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#97	C₂²×C₃₁	1	2	yes	—
278	C₄³	C₂²×C₃₁	1	1	yes	characteristic
279	#341	C₂²×C₃₁	1	2	yes	—
280	#339	C₂²×C₃₁	1	2	yes	—
281	#339	C₂²×C₃₁	1	1	yes	characteristic
282	#339	C₂²×C₃₁	1	1	yes	characteristic
283	#339	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	C₂×C₃₁	—	2	1	no	—
286	C₂×C₃₁	—	2	1	no	—
287	C₂×C₃₁	—	4	2	no	—
288	C₂×C₃₁	#2360	1	1	yes	characteristic
289	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
290	C₂×C₃₁	#2956	1	1	yes	characteristic
291	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
292	#131	—	2	4	no	—
293	#131	—	4	2	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
296	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
300	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
301	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
303	#125	C₂×C₄×C₃₁	1	2	yes	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
314	#117	—	2	1	no	—
315	#117	C₂³×C₃₁	1	1	yes	characteristic
316	#117	C₂³×C₃₁	1	2	yes	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	#108	—	4	2	no	—
324	#107	C₂×C₄×C₃₁	1	2	yes	—
325	#107	C₂×C₄×C₃₁	1	2	yes	—
326	#106	C₂×C₄×C₃₁	1	2	yes	—
327	#106	C₂×C₄×C₃₁	1	2	yes	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
337	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
339	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
340	#104	—	2	2	no	—
341	#104	Q₈×C₃₁	1	4	yes	complemented
342	#103	C₂×C₄×C₃₁	1	2	yes	—
343	#103	C₂×C₄×C₃₁	1	2	yes	—
344	#102	C₂×C₄×C₃₁	1	2	yes	—
345	#102	C₂×C₄×C₃₁	1	2	yes	—
346	#97	—	2	2	no	—
347	#97	—	2	1	no	—
348	#97	—	4	2	no	—
349	C₄×C₈	—	2	2	no	—
350	C₄×C₈	—	2	1	no	—
351	C₃₁	#22983	1	1	yes	characteristic, direct product factor, Sylow
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	C₂×Q₈×C₃₁	1	2	yes	—
355	#41	#41×C₃₁	1	2	yes	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	1	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	4	4	no	—
375	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	4	4	no	—
384	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
387	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
395	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
396	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
397	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	4	2	no	—
410	C₄²	—	4	2	no	—
411	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
412	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
413	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	8	1	no	—
421	Q₈	C₄×Q₈×C₃₁	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	C₄×Q₈×C₃₁	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	8	1	no	—
443	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
451	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
452	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
466	C₂²	C₃₁×#552	1	1	yes	characteristic
467	C₂²	C₃₁×#554	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	2	no	—
478	C₄	C₃₁×#444	1	2	yes	—
479	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
480	C₄	C₃₁×#520	1	1	yes	characteristic
481	C₄	C₃₁×#546	1	1	yes	characteristic
482	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	C₃₁×#2360	1	1	yes	characteristic
487	C₂	C₂×C₃₁×#444	1	1	yes	characteristic
488	C₂	C₃₁×#2956	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×#1994	C₂	semidirect
5. C₃₁×#1950	C₂	semidirect
6. C₃₁×#1946	C₂	semidirect
7. C₃₁×#1922	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
15. C₃₁×#479	C₄	semidirect
16. C₃₁×#478	C₄	semidirect
17. C₃₁×#477	C₄	semidirect
18. C₃₁×#476	C₄	semidirect
19. C₃₁×#475	C₄	semidirect
20. C₃₁×#474	C₄	semidirect
97. C₃₁×#104	Q₈	semidirect
171. #22983	C₃₁	direct (N × H)
217. #2962	C₂×C₃₁	semidirect
219. #1994	C₂×C₃₁	semidirect
220. #1950	C₂×C₃₁	semidirect
221. #1946	C₂×C₃₁	semidirect
222. #1922	C₂×C₃₁	semidirect
224. #1767	C₂×C₃₁	semidirect
253. #479	C₄×C₃₁	semidirect
254. #478	C₄×C₃₁	semidirect
255. #477	C₄×C₃₁	semidirect
256. #476	C₄×C₃₁	semidirect
257. #475	C₄×C₃₁	semidirect
258. #474	C₄×C₃₁	semidirect
341. #104	Q₈×C₃₁	semidirect
351. C₃₁	#22983	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×#1994	2
C₃₁×#1950	2
C₃₁×#1946	2
C₃₁×#1922	2
C₃₁×#1866	2
C₃₁×#1767	2
C₃₁×#1765	2
C₄×C₃₁×#104	2
#22983	31

Subgroups — C31×#22983 #21479742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22983 #21479742