Subgroups — C₃₁×#22783 #21479542

Order 7936.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1937	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1876	C₂	1	2	yes	maximal
7	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
17	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
18	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
19	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
20	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
21	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
22	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
23	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
24	C₃₁×#433	—	2	2	no	—
25	C₃₁×#432	—	2	1	no	—
26	C₃₁×#431	—	2	1	no	—
27	C₃₁×#427	C₂²	1	1	yes	characteristic
28	C₃₁×#424	—	2	2	no	—
29	C₃₁×#424	C₂²	1	2	yes	—
30	C₃₁×#423	—	2	1	no	—
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₂×C₃₁×#104	C₂²	1	2	yes	—
37	C₃₁×#416	C₂²	1	1	yes	characteristic
38	C₃₁×#416	C₂²	1	1	yes	characteristic
39	C₃₁×#408	C₂²	1	2	yes	—
40	C₃₁×#405	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₃₁×#341	C₂²	1	2	yes	—
45	C₃₁×#341	C₂²	1	2	yes	—
46	C₃₁×#339	C₂²	1	2	yes	—
47	C₃₁×#339	C₂²	1	2	yes	—
48	C₃₁×#336	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#135	—	2	2	no	—
52	C₃₁×#135	—	2	2	no	—
53	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₃₁×#131	—	2	4	no	—
58	C₃₁×#131	—	4	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
66	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₃₁×#117	—	2	1	no	—
79	C₃₁×#117	—	2	1	no	—
80	C₃₁×#117	C₂³	1	1	yes	characteristic
81	C₃₁×#117	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
88	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
89	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	—	2	2	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	1	no	—
94	C₃₁×C₄⋊C₈	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
98	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
99	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
100	C₃₁×#104	—	2	4	no	—
101	C₃₁×#104	—	4	2	no	—
102	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
103	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
104	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
105	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
106	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
107	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
108	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
109	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
110	C₃₁×C₂²⋊C₈	—	2	2	no	—
111	C₃₁×C₂²⋊C₈	—	2	2	no	—
112	C₃₁×#97	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	—	4	2	no	—
115	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#95	—	2	1	no	—
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	#41×C₃₁	—	2	1	no	—
125	#41×C₃₁	—	2	1	no	—
126	#41×C₃₁	#41	1	2	yes	—
127	#41×C₃₁	#41	1	2	yes	—
128	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₃₁	—	2	1	no	—
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
144	Q₁₆×C₃₁	—	2	2	no	—
145	SD₁₆×C₃₁	—	2	2	no	—
146	SD₁₆×C₃₁	—	2	2	no	—
147	D₈×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	2	no	—
151	M₁₆×C₃₁	—	2	2	no	—
152	M₁₆×C₃₁	—	4	4	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	2	no	—
161	C₂×C₈×C₃₁	—	4	4	no	—
162	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
176	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	2	1	no	—
179	C₂²⋊C₄×C₃₁	—	2	1	no	—
180	C₂²⋊C₄×C₃₁	—	2	1	no	—
181	C₂²⋊C₄×C₃₁	—	2	1	no	—
182	C₂²⋊C₄×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	—	2	1	no	—
184	C₄²×C₃₁	—	2	1	no	—
185	C₄²×C₃₁	—	2	1	no	—
186	C₄²×C₃₁	—	2	1	no	—
187	C₄²×C₃₁	—	2	1	no	—
188	C₄²×C₃₁	—	2	1	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	#22783	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	1	no	—
200	Q₈×C₃₁	—	2	1	no	—
201	Q₈×C₃₁	—	2	1	no	—
202	Q₈×C₃₁	#117	1	2	yes	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	—	2	1	no	—
206	D₄×C₃₁	—	2	1	no	—
207	D₄×C₃₁	—	2	1	no	—
208	D₄×C₃₁	—	4	1	no	—
209	D₄×C₃₁	—	4	1	no	—
210	D₄×C₃₁	#117	1	2	yes	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	8	1	no	—
229	C₂×C₄×C₃₁	—	8	1	no	—
230	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
239	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
240	C₈×C₃₁	—	2	1	no	—
241	C₈×C₃₁	—	2	1	no	—
242	C₈×C₃₁	—	2	1	no	—
243	C₈×C₃₁	—	2	1	no	—
244	C₈×C₃₁	—	2	2	no	—
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	4	2	no	—
247	C₈×C₃₁	—	4	2	no	—
248	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
249	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1937	C₂×C₃₁	1	2	yes	complemented
252	#1922	C₂×C₃₁	1	2	yes	complemented
253	#1876	C₂×C₃₁	1	2	yes	—
254	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1806	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1806	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1764	C₂×C₃₁	1	2	yes	complemented
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	1	no	—
267	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂²×C₃₁	#552	1	1	yes	characteristic
269	C₂²×C₃₁	#553	1	1	yes	characteristic
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	—	4	1	no	—
276	C₄×C₃₁	—	4	1	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	4	1	no	—
279	C₄×C₃₁	—	4	1	no	—
280	C₄×C₃₁	#443	1	2	yes	—
281	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
282	C₄×C₃₁	#520	1	1	yes	characteristic
283	C₄×C₃₁	#546	1	1	yes	characteristic
284	C₄×C₃₁	#548	1	1	yes	characteristic
285	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#131	C₂²×C₃₁	1	2	yes	—
287	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
288	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
289	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
290	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
291	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
292	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
293	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
294	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
295	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
296	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
297	#433	—	2	2	no	—
298	#432	—	2	1	no	—
299	#431	—	2	1	no	—
300	#427	C₂²×C₃₁	1	1	yes	characteristic
301	#424	—	2	2	no	—
302	#424	C₂²×C₃₁	1	2	yes	—
303	#423	—	2	1	no	—
304	#423	C₂²×C₃₁	1	1	yes	characteristic
305	#423	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₈	—	2	1	no	—
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#104	C₂²×C₃₁	1	2	yes	—
309	C₂×#104	C₂²×C₃₁	1	2	yes	—
310	#416	C₂²×C₃₁	1	1	yes	characteristic
311	#416	C₂²×C₃₁	1	1	yes	characteristic
312	#408	C₂²×C₃₁	1	2	yes	—
313	#405	C₂²×C₃₁	1	2	yes	—
314	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
316	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
317	#341	C₂²×C₃₁	1	2	yes	—
318	#341	C₂²×C₃₁	1	2	yes	—
319	#339	C₂²×C₃₁	1	2	yes	—
320	#339	C₂²×C₃₁	1	2	yes	—
321	#336	C₂²×C₃₁	1	2	yes	—
322	C₂×C₃₁	—	2	1	no	—
323	C₂×C₃₁	—	2	1	no	—
324	C₂×C₃₁	—	4	2	no	—
325	C₂×C₃₁	—	8	1	no	—
326	C₂×C₃₁	#2352	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
328	C₂×C₃₁	#2957	1	1	yes	characteristic
329	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
331	#135	—	2	2	no	—
332	#135	—	2	2	no	—
333	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
337	#131	—	2	4	no	—
338	#131	—	4	2	no	—
339	C₂×M₁₆	—	2	2	no	—
340	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
341	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
342	C₂²×C₈	—	2	2	no	—
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
346	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
347	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
348	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	—	2	1	no	—
354	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
358	#117	—	2	1	no	—
359	#117	—	2	1	no	—
360	#117	C₂³×C₃₁	1	1	yes	characteristic
361	#117	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
368	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
369	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
370	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₄⋊C₈	—	2	2	no	—
372	C₄⋊C₈	—	2	2	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
378	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
379	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
380	#104	—	2	4	no	—
381	#104	—	4	2	no	—
382	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
383	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
384	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
385	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
386	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
387	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
388	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
389	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
390	C₂²⋊C₈	—	2	2	no	—
391	C₂²⋊C₈	—	2	2	no	—
392	#97	—	2	2	no	—
393	C₄×C₈	—	2	2	no	—
394	C₄×C₈	—	4	2	no	—
395	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
397	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
398	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
399	#95	—	2	1	no	—
400	C₃₁	#22783	1	1	yes	characteristic, direct product factor, Sylow
401	#41	—	2	1	no	—
402	#41	—	2	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	2	1	no	—
406	#41	—	2	1	no	—
407	#41	#41×C₃₁	1	2	yes	—
408	#41	#41×C₃₁	1	2	yes	—
409	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
410	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
414	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	Q₁₆	—	2	2	no	—
426	SD₁₆	—	2	2	no	—
427	SD₁₆	—	2	2	no	—
428	D₈	—	2	2	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	4	4	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	4	4	no	—
443	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
444	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
448	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	2	1	no	—
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	2	1	no	—
482	Q₈	C₃₁×#117	1	2	yes	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	1	no	—
486	D₄	—	2	1	no	—
487	D₄	—	2	1	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	1	no	—
490	D₄	C₃₁×#117	1	2	yes	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	8	1	no	—
509	C₂×C₄	—	8	1	no	—
510	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
517	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
519	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
520	C₈	—	2	1	no	—
521	C₈	—	2	1	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	2	no	—
525	C₈	—	2	2	no	—
526	C₈	—	4	2	no	—
527	C₈	—	4	2	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
537	C₂²	C₃₁×#552	1	1	yes	characteristic
538	C₂²	C₃₁×#553	1	1	yes	characteristic
539	C₄	—	2	2	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	C₃₁×#443	1	2	yes	—
550	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
551	C₄	C₃₁×#520	1	1	yes	characteristic
552	C₄	C₃₁×#546	1	1	yes	characteristic
553	C₄	C₃₁×#548	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	1	no	—
558	C₂	C₃₁×#2352	1	1	yes	characteristic
559	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
560	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×C₃₁×#135	C₂²
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×#2925	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1937	C₂	semidirect
5. C₃₁×#1922	C₂	semidirect
7. C₃₁×#1837	C₂	semidirect
8. C₃₁×#1806	C₂	semidirect
9. C₃₁×#1806	C₂	semidirect
10. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
16. C₄×Q₁₆×C₃₁	C₄	semidirect
17. C₄×Q₁₆×C₃₁	C₄	semidirect
18. C₄×SD₁₆×C₃₁	C₄	semidirect
19. C₄×SD₁₆×C₃₁	C₄	semidirect
20. C₄×SD₁₆×C₃₁	C₄	semidirect
21. C₄×SD₁₆×C₃₁	C₄	semidirect
22. C₄×D₈×C₃₁	C₄	semidirect
23. C₄×D₈×C₃₁	C₄	semidirect
195. #22783	C₃₁	direct (N × H)
249. #2925	C₂×C₃₁	semidirect
250. #2906	C₂×C₃₁	semidirect
251. #1937	C₂×C₃₁	semidirect
252. #1922	C₂×C₃₁	semidirect
254. #1837	C₂×C₃₁	semidirect
255. #1806	C₂×C₃₁	semidirect
256. #1806	C₂×C₃₁	semidirect
257. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
258. #1764	C₂×C₃₁	semidirect
289. C₄×Q₁₆	C₄×C₃₁	semidirect
290. C₄×Q₁₆	C₄×C₃₁	semidirect
291. C₄×SD₁₆	C₄×C₃₁	semidirect
292. C₄×SD₁₆	C₄×C₃₁	semidirect
293. C₄×SD₁₆	C₄×C₃₁	semidirect
294. C₄×SD₁₆	C₄×C₃₁	semidirect
295. C₄×D₈	C₄×C₃₁	semidirect
296. C₄×D₈	C₄×C₃₁	semidirect
400. C₃₁	#22783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#135	2
C₃₁×#2925	2
C₃₁×#2906	2
C₃₁×#1937	2
C₃₁×#1922	2
C₃₁×#1876	2
C₃₁×#1837	2
C₃₁×#1806	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1764	2
#22783	31

Subgroups — C31×#22783 #21479542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22783 #21479542