Subgroups — C₃₁×#61410 #21518169

Order 7936.

1958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3400	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3397	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3394	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3392	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3154	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3043	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3042	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2949	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	C₂²	1	2	yes	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#559	C₂²	1	4	yes	—
22	C₃₁×#557	C₂²	1	4	yes	complemented
23	C₃₁×#556	C₂²	1	4	yes	complemented
24	C₃₁×#552	—	2	4	no	—
25	C₃₁×#550	C₂²	1	4	yes	—
26	C₃₁×#546	—	2	4	no	—
27	C₃₁×#541	—	2	2	no	—
28	C₃₁×#540	—	2	2	no	—
29	C₃₁×#536	C₂²	1	2	yes	—
30	C₃₁×#535	C₂²	1	2	yes	—
31	C₃₁×#533	—	2	2	no	—
32	C₃₁×#533	C₂²	1	1	yes	characteristic
33	C₃₁×#532	—	2	2	no	—
34	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
36	C₃₁×#519	C₂²	1	2	yes	—
37	C₃₁×#518	C₂²	1	2	yes	—
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#497	C₂²	1	4	yes	complemented
41	C₃₁×#496	C₂²	1	4	yes	complemented
42	C₃₁×#495	C₂²	1	4	yes	complemented
43	C₃₁×#492	C₂²	1	4	yes	—
44	C₃₁×#492	C₂²	1	4	yes	—
45	C₃₁×#492	C₂²	1	4	yes	complemented
46	C₃₁×#464	C₂²	1	8	yes	complemented
47	C₃₁×#463	C₂²	1	8	yes	complemented
48	C₃₁×#462	C₂²	1	4	yes	complemented
49	C₃₁×#461	C₂²	1	4	yes	complemented
50	C₃₁×#461	C₂²	1	8	yes	complemented
51	C₃₁×#460	C₂²	1	4	yes	—
52	C₃₁×#460	C₂²	1	8	yes	complemented
53	C₃₁×#459	C₂²	1	4	yes	—
54	C₃₁×#446	C₂²	1	4	yes	complemented
55	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
56	C₃₁×#440	C₂²	1	8	yes	complemented
57	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
58	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
59	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
60	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
61	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
62	C₃₁×#427	C₂²	1	1	yes	characteristic
63	C₃₁×#424	C₂²	1	2	yes	—
64	C₃₁×#423	C₂²	1	1	yes	characteristic
65	C₃₁×#419	C₂²	1	2	yes	—
66	C₃₁×#418	C₂²	1	2	yes	—
67	C₃₁×#416	C₂²	1	1	yes	characteristic
68	C₃₁×#416	C₂²	1	1	yes	characteristic
69	C₃₁×#405	C₂²	1	2	yes	—
70	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
71	C₂×#41×C₃₁	—	2	4	no	—
72	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×D₄×C₃₁	—	2	2	no	—
76	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#137	—	4	4	no	—
78	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
79	C₃₁×#135	—	2	4	no	—
80	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂×SD₁₆×C₃₁	—	2	8	no	—
83	C₂×SD₁₆×C₃₁	—	4	4	no	—
84	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
85	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
91	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×D₈×C₃₁	C₂³	1	4	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊Q₈	—	2	1	no	—
97	C₃₁×C₄⋊Q₈	—	2	1	no	—
98	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
99	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
100	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
101	C₃₁×#127	—	2	1	no	—
102	C₃₁×#127	—	2	1	no	—
103	C₃₁×#127	C₂³	1	4	yes	—
104	C₃₁×#127	C₂³	1	4	yes	—
105	C₃₁×#126	—	2	8	no	—
106	C₃₁×#125	—	2	4	no	—
107	C₃₁×#125	—	2	8	no	—
108	C₃₁×#125	D₄	1	2	yes	—
109	C₃₁×#124	—	2	4	no	—
110	C₃₁×#124	C₂³	1	4	yes	—
111	C₃₁×#124	C₂³	1	4	yes	—
112	C₃₁×#123	—	2	8	no	—
113	C₃₁×#123	—	2	4	no	—
114	C₃₁×C₂²⋊Q₈	—	2	4	no	—
115	C₃₁×C₂²⋊Q₈	—	2	4	no	—
116	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
117	C₃₁×#121	—	2	4	no	—
118	C₃₁×#121	—	2	4	no	—
119	C₃₁×#121	—	2	8	no	—
120	C₃₁×#120	—	2	4	no	—
121	C₄×Q₈×C₃₁	—	2	4	no	—
122	C₄×Q₈×C₃₁	—	2	4	no	—
123	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
125	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
126	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	2	no	—
129	C₄×D₄×C₃₁	—	2	4	no	—
130	C₄×D₄×C₃₁	—	2	4	no	—
131	C₄×D₄×C₃₁	C₂³	1	2	yes	—
132	C₄×D₄×C₃₁	C₂³	1	2	yes	—
133	C₄×D₄×C₃₁	C₂³	1	8	yes	—
134	C₃₁×#117	—	2	4	no	—
135	C₃₁×#117	—	2	4	no	—
136	C₃₁×#117	—	2	1	no	—
137	C₃₁×#117	—	2	1	no	—
138	C₃₁×#117	C₂³	1	1	yes	characteristic
139	C₃₁×#117	C₂³	1	1	yes	characteristic
140	C₃₁×#117	C₂³	1	2	yes	—
141	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂×C₄²×C₃₁	—	2	1	no	—
144	C₂×C₄²×C₃₁	—	2	1	no	—
145	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₃₁×#107	C₂³	1	2	yes	—
147	C₃₁×#106	C₂³	1	2	yes	—
148	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
150	C₃₁×#103	C₂³	1	2	yes	—
151	C₃₁×#103	C₂³	1	2	yes	—
152	C₃₁×#103	C₂³	1	8	yes	—
153	C₃₁×#102	C₂³	1	2	yes	—
154	C₃₁×#102	C₂³	1	2	yes	—
155	C₃₁×#102	C₂³	1	8	yes	—
156	C₃₁×#97	C₂³	1	4	yes	—
157	C₄×C₈×C₃₁	C₂³	1	2	yes	—
158	C₄×C₈×C₃₁	C₂³	1	2	yes	—
159	C₄×C₈×C₃₁	C₂³	1	4	yes	—
160	C₂⁴×C₃₁	—	2	2	no	—
161	#41×C₃₁	—	2	2	no	—
162	#41×C₃₁	—	2	2	no	—
163	#41×C₃₁	—	2	2	no	—
164	#41×C₃₁	—	2	2	no	—
165	#41×C₃₁	—	8	8	no	—
166	C₂×Q₈×C₃₁	—	2	2	no	—
167	C₂×Q₈×C₃₁	—	2	4	no	—
168	C₂×Q₈×C₃₁	—	2	4	no	—
169	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
170	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
172	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂×D₄×C₃₁	—	2	1	no	—
174	C₂×D₄×C₃₁	—	2	1	no	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	2	8	no	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	2	4	no	—
180	C₂×D₄×C₃₁	—	2	4	no	—
181	C₂×D₄×C₃₁	—	8	4	no	—
182	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	—	2	2	no	—
186	C₂²×C₄×C₃₁	—	2	2	no	—
187	C₂²×C₄×C₃₁	—	2	1	no	—
188	C₂²×C₄×C₃₁	—	2	1	no	—
189	C₂²×C₄×C₃₁	—	2	2	no	—
190	C₂²×C₄×C₃₁	—	2	8	no	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₃₁	—	2	2	no	—
194	Q₁₆×C₃₁	—	4	4	no	—
195	SD₁₆×C₃₁	—	2	8	no	—
196	SD₁₆×C₃₁	—	2	8	no	—
197	SD₁₆×C₃₁	—	2	2	no	—
198	SD₁₆×C₃₁	—	2	2	no	—
199	SD₁₆×C₃₁	—	4	8	no	—
200	SD₁₆×C₃₁	—	4	4	no	—
201	SD₁₆×C₃₁	—	4	4	no	—
202	D₈×C₃₁	—	2	2	no	—
203	D₈×C₃₁	—	4	4	no	—
204	M₁₆×C₃₁	—	2	4	no	—
205	C₂×C₈×C₃₁	—	2	2	no	—
206	C₂×C₈×C₃₁	#41	1	4	yes	complemented
207	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
208	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
209	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	—	2	4	no	—
213	C₄⋊C₄×C₃₁	—	2	2	no	—
214	C₄⋊C₄×C₃₁	—	2	2	no	—
215	C₄⋊C₄×C₃₁	—	2	8	no	—
216	C₄⋊C₄×C₃₁	—	2	8	no	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
219	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₂²⋊C₄×C₃₁	—	2	1	no	—
223	C₂²⋊C₄×C₃₁	—	2	1	no	—
224	C₂²⋊C₄×C₃₁	—	2	8	no	—
225	C₂²⋊C₄×C₃₁	—	2	2	no	—
226	C₂²⋊C₄×C₃₁	—	2	2	no	—
227	C₂²⋊C₄×C₃₁	—	2	8	no	—
228	C₂²⋊C₄×C₃₁	—	2	8	no	—
229	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
230	C₄²×C₃₁	—	2	1	no	—
231	C₄²×C₃₁	—	2	1	no	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	—	2	2	no	—
234	C₄²×C₃₁	—	2	8	no	—
235	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₃₁	C₂⁴	1	2	yes	—
238	C₄²×C₃₁	C₂⁴	1	4	yes	—
239	#61410	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₃₁	—	2	1	no	—
241	C₂³×C₃₁	—	2	4	no	—
242	C₂³×C₃₁	—	2	1	no	—
243	C₂³×C₃₁	—	4	4	no	—
244	C₂³×C₃₁	—	8	4	no	—
245	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₃₁	—	2	8	no	—
247	Q₈×C₃₁	—	2	4	no	—
248	Q₈×C₃₁	—	2	1	no	—
249	Q₈×C₃₁	—	2	1	no	—
250	Q₈×C₃₁	—	2	1	no	—
251	Q₈×C₃₁	—	2	4	no	—
252	Q₈×C₃₁	—	2	1	no	—
253	Q₈×C₃₁	—	4	4	no	—
254	D₄×C₃₁	—	2	1	no	—
255	D₄×C₃₁	—	2	1	no	—
256	D₄×C₃₁	—	2	4	no	—
257	D₄×C₃₁	—	2	1	no	—
258	D₄×C₃₁	—	2	4	no	—
259	D₄×C₃₁	—	2	1	no	—
260	D₄×C₃₁	—	4	2	no	—
261	D₄×C₃₁	—	4	2	no	—
262	D₄×C₃₁	—	4	4	no	—
263	D₄×C₃₁	—	4	8	no	—
264	D₄×C₃₁	—	8	4	no	—
265	D₄×C₃₁	—	8	8	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	2	no	—
268	C₂×C₄×C₃₁	—	2	1	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	1	no	—
272	C₂×C₄×C₃₁	—	4	4	no	—
273	C₂×C₄×C₃₁	—	4	2	no	—
274	C₂×C₄×C₃₁	—	4	2	no	—
275	C₂×C₄×C₃₁	—	4	4	no	—
276	C₂×C₄×C₃₁	—	4	4	no	—
277	C₂×C₄×C₃₁	—	8	8	no	—
278	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
283	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
284	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
285	C₈×C₃₁	—	2	2	no	—
286	C₈×C₃₁	—	2	4	no	—
287	C₈×C₃₁	C₂×#41	1	2	yes	—
288	C₈×C₃₁	C₂×#41	1	2	yes	—
289	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#3400	C₂×C₃₁	1	4	yes	complemented
292	#3397	C₂×C₃₁	1	4	yes	complemented
293	#3394	C₂×C₃₁	1	4	yes	complemented
294	#3392	C₂×C₃₁	1	4	yes	complemented
295	#3154	C₂×C₃₁	1	2	yes	complemented
296	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#3043	C₂×C₃₁	1	2	yes	complemented
298	#3042	C₂×C₃₁	1	2	yes	complemented
299	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#2949	C₂×C₃₁	1	2	yes	complemented
303	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	4	1	no	—
306	C₂²×C₃₁	—	4	1	no	—
307	C₂²×C₃₁	—	4	4	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	4	8	no	—
311	C₂²×C₃₁	—	16	4	no	—
312	C₂²×C₃₁	#550	1	2	yes	—
313	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	2	8	no	—
316	C₄×C₃₁	—	4	1	no	—
317	C₄×C₃₁	—	4	1	no	—
318	C₄×C₃₁	—	4	4	no	—
319	C₄×C₃₁	—	4	2	no	—
320	C₄×C₃₁	—	4	2	no	—
321	C₄×C₃₁	#550	1	2	yes	—
322	C₄×C₃₁	#577	1	4	yes	—
323	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
324	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂×#137	C₂²×C₃₁	1	2	yes	—
326	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
327	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
328	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
329	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
330	#559	C₂²×C₃₁	1	4	yes	—
331	#557	C₂²×C₃₁	1	4	yes	complemented
332	#556	C₂²×C₃₁	1	4	yes	complemented
333	#552	—	2	4	no	—
334	#550	C₂²×C₃₁	1	4	yes	—
335	#546	—	2	4	no	—
336	#541	—	2	2	no	—
337	#540	—	2	2	no	—
338	#536	C₂²×C₃₁	1	2	yes	—
339	#535	C₂²×C₃₁	1	2	yes	—
340	#533	—	2	2	no	—
341	#533	C₂²×C₃₁	1	1	yes	characteristic
342	#532	—	2	2	no	—
343	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
345	#519	C₂²×C₃₁	1	2	yes	—
346	#518	C₂²×C₃₁	1	2	yes	—
347	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
348	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
349	#497	C₂²×C₃₁	1	4	yes	complemented
350	#496	C₂²×C₃₁	1	4	yes	complemented
351	#495	C₂²×C₃₁	1	4	yes	complemented
352	#492	C₂²×C₃₁	1	4	yes	—
353	#492	C₂²×C₃₁	1	4	yes	—
354	#492	C₂²×C₃₁	1	4	yes	complemented
355	#464	C₂²×C₃₁	1	8	yes	complemented
356	#463	C₂²×C₃₁	1	8	yes	complemented
357	#462	C₂²×C₃₁	1	4	yes	complemented
358	#461	C₂²×C₃₁	1	4	yes	complemented
359	#461	C₂²×C₃₁	1	8	yes	complemented
360	#460	C₂²×C₃₁	1	4	yes	—
361	#460	C₂²×C₃₁	1	8	yes	complemented
362	#459	C₂²×C₃₁	1	4	yes	—
363	#446	C₂²×C₃₁	1	4	yes	complemented
364	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
365	#440	C₂²×C₃₁	1	8	yes	complemented
366	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
367	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
368	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
369	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
370	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
371	#427	C₂²×C₃₁	1	1	yes	characteristic
372	#424	C₂²×C₃₁	1	2	yes	—
373	#423	C₂²×C₃₁	1	1	yes	characteristic
374	#419	C₂²×C₃₁	1	2	yes	—
375	#418	C₂²×C₃₁	1	2	yes	—
376	#416	C₂²×C₃₁	1	1	yes	characteristic
377	#416	C₂²×C₃₁	1	1	yes	characteristic
378	#405	C₂²×C₃₁	1	2	yes	—
379	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
380	C₂×C₃₁	—	2	2	no	—
381	C₂×C₃₁	—	8	1	no	—
382	C₂×C₃₁	—	8	1	no	—
383	C₂×C₃₁	—	8	4	no	—
384	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
385	C₂×C₃₁	C₂×#577	1	2	yes	—
386	C₂×#41	—	2	4	no	—
387	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
389	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
390	C₂²×D₄	—	2	2	no	—
391	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
392	#137	—	4	4	no	—
393	AGL(1,ℤ/8)	—	4	4	no	—
394	#135	—	2	4	no	—
395	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
397	C₂×SD₁₆	—	2	8	no	—
398	C₂×SD₁₆	—	4	4	no	—
399	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
400	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
401	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
404	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
405	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
406	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₂×D₈	C₂³×C₃₁	1	4	yes	—
408	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
409	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
414	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
415	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
416	#127	—	2	1	no	—
417	#127	—	2	1	no	—
418	#127	C₂³×C₃₁	1	4	yes	—
419	#127	C₂³×C₃₁	1	4	yes	—
420	#126	—	2	8	no	—
421	#125	—	2	4	no	—
422	#125	—	2	8	no	—
423	#125	D₄×C₃₁	1	2	yes	—
424	#124	—	2	4	no	—
425	#124	C₂³×C₃₁	1	4	yes	—
426	#124	C₂³×C₃₁	1	4	yes	—
427	#123	—	2	8	no	—
428	#123	—	2	4	no	—
429	C₂²⋊Q₈	—	2	4	no	—
430	C₂²⋊Q₈	—	2	4	no	—
431	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
432	#121	—	2	4	no	—
433	#121	—	2	4	no	—
434	#121	—	2	8	no	—
435	#120	—	2	4	no	—
436	C₄×Q₈	—	2	4	no	—
437	C₄×Q₈	—	2	4	no	—
438	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
439	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
440	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
441	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
442	C₄×D₄	—	2	2	no	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	4	no	—
445	C₄×D₄	—	2	4	no	—
446	C₄×D₄	C₂³×C₃₁	1	2	yes	—
447	C₄×D₄	C₂³×C₃₁	1	2	yes	—
448	C₄×D₄	C₂³×C₃₁	1	8	yes	—
449	#117	—	2	4	no	—
450	#117	—	2	4	no	—
451	#117	—	2	1	no	—
452	#117	—	2	1	no	—
453	#117	C₂³×C₃₁	1	1	yes	characteristic
454	#117	C₂³×C₃₁	1	1	yes	characteristic
455	#117	C₂³×C₃₁	1	2	yes	—
456	C₂×C₄⋊C₄	—	2	2	no	—
457	C₂×C₂²⋊C₄	—	2	4	no	—
458	C₂×C₄²	—	2	1	no	—
459	C₂×C₄²	—	2	1	no	—
460	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
461	#107	C₂³×C₃₁	1	2	yes	—
462	#106	C₂³×C₃₁	1	2	yes	—
463	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
464	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
465	#103	C₂³×C₃₁	1	2	yes	—
466	#103	C₂³×C₃₁	1	2	yes	—
467	#103	C₂³×C₃₁	1	8	yes	—
468	#102	C₂³×C₃₁	1	2	yes	—
469	#102	C₂³×C₃₁	1	2	yes	—
470	#102	C₂³×C₃₁	1	8	yes	—
471	#97	C₂³×C₃₁	1	4	yes	—
472	C₄×C₈	C₂³×C₃₁	1	2	yes	—
473	C₄×C₈	C₂³×C₃₁	1	2	yes	—
474	C₄×C₈	C₂³×C₃₁	1	4	yes	—
475	C₃₁	#61410	1	1	yes	characteristic, direct product factor, Sylow
476	C₂⁴	—	2	2	no	—
477	#41	—	2	2	no	—
478	#41	—	2	2	no	—
479	#41	—	2	2	no	—
480	#41	—	2	2	no	—
481	#41	—	8	8	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	4	no	—
485	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
486	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
487	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
488	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	8	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	8	4	no	—
498	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
499	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
500	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	8	no	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
509	Q₁₆	—	2	2	no	—
510	Q₁₆	—	4	4	no	—
511	SD₁₆	—	2	8	no	—
512	SD₁₆	—	2	8	no	—
513	SD₁₆	—	2	2	no	—
514	SD₁₆	—	2	2	no	—
515	SD₁₆	—	4	8	no	—
516	SD₁₆	—	4	4	no	—
517	SD₁₆	—	4	4	no	—
518	D₈	—	2	2	no	—
519	D₈	—	4	4	no	—
520	M₁₆	—	2	4	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	#41×C₃₁	1	4	yes	complemented
523	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
524	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
525	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
526	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	8	no	—
532	C₄⋊C₄	—	2	8	no	—
533	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
535	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
536	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
537	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	—	2	8	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	8	no	—
544	C₂²⋊C₄	—	2	8	no	—
545	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	2	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	8	no	—
551	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
553	C₄²	C₂⁴×C₃₁	1	2	yes	—
554	C₄²	C₂⁴×C₃₁	1	4	yes	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	4	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	4	4	no	—
559	C₂³	—	8	4	no	—
560	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
561	Q₈	—	2	8	no	—
562	Q₈	—	2	4	no	—
563	Q₈	—	2	1	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	2	4	no	—
567	Q₈	—	2	1	no	—
568	Q₈	—	4	4	no	—
569	D₄	—	2	1	no	—
570	D₄	—	2	1	no	—
571	D₄	—	2	4	no	—
572	D₄	—	2	1	no	—
573	D₄	—	2	4	no	—
574	D₄	—	2	1	no	—
575	D₄	—	4	2	no	—
576	D₄	—	4	2	no	—
577	D₄	—	4	4	no	—
578	D₄	—	4	8	no	—
579	D₄	—	8	4	no	—
580	D₄	—	8	8	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	8	8	no	—
593	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
594	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
596	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
597	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
598	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
599	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
600	C₈	—	2	2	no	—
601	C₈	—	2	4	no	—
602	C₈	C₂×#41×C₃₁	1	2	yes	—
603	C₈	C₂×#41×C₃₁	1	2	yes	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	8	no	—
611	C₂²	—	16	4	no	—
612	C₂²	C₃₁×#550	1	2	yes	—
613	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
614	C₄	—	2	2	no	—
615	C₄	—	2	8	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	1	no	—
618	C₄	—	4	4	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	C₃₁×#550	1	2	yes	—
622	C₄	C₃₁×#577	1	4	yes	—
623	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
624	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
625	C₂	—	2	2	no	—
626	C₂	—	8	4	no	—
627	C₂	—	8	1	no	—
628	C₂	—	8	1	no	—
629	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
630	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3509	C₂	semidirect
2. C₃₁×#3508	C₂	semidirect
3. C₃₁×#3400	C₂	semidirect
4. C₃₁×#3397	C₂	semidirect
5. C₃₁×#3394	C₂	semidirect
6. C₃₁×#3392	C₂	semidirect
7. C₃₁×#3154	C₂	semidirect
8. C₂×C₃₁×#492	C₂	semidirect
9. C₃₁×#3043	C₂	semidirect
10. C₃₁×#3042	C₂	semidirect
11. C₃₁×#3039	C₂	semidirect
12. C₃₁×#3038	C₂	semidirect
13. C₃₁×#2960	C₂	semidirect
14. C₃₁×#2949	C₂	semidirect
15. C₄×C₃₁×#135	C₂	semidirect
22. C₃₁×#557	C₂²	semidirect
23. C₃₁×#556	C₂²	semidirect
40. C₃₁×#497	C₂²	semidirect
41. C₃₁×#496	C₂²	semidirect
42. C₃₁×#495	C₂²	semidirect
45. C₃₁×#492	C₂²	semidirect
46. C₃₁×#464	C₂²	semidirect
47. C₃₁×#463	C₂²	semidirect
48. C₃₁×#462	C₂²	semidirect
49. C₃₁×#461	C₂²	semidirect
50. C₃₁×#461	C₂²	semidirect
52. C₃₁×#460	C₂²	semidirect
54. C₃₁×#446	C₂²	semidirect
55. C₈×Q₈×C₃₁	C₂²	semidirect
56. C₃₁×#440	C₂²	semidirect
57. C₄×Q₁₆×C₃₁	C₂²	semidirect
58. C₄×SD₁₆×C₃₁	C₂²	semidirect
59. C₄×SD₁₆×C₃₁	C₂²	semidirect
60. C₄×SD₁₆×C₃₁	C₂²	semidirect
61. C₄×D₈×C₃₁	C₂²	semidirect
206. C₂×C₈×C₃₁	#41	semidirect
239. #61410	C₃₁	direct (N × H)
289. #3509	C₂×C₃₁	semidirect
290. #3508	C₂×C₃₁	semidirect
291. #3400	C₂×C₃₁	semidirect
292. #3397	C₂×C₃₁	semidirect
293. #3394	C₂×C₃₁	semidirect
294. #3392	C₂×C₃₁	semidirect
295. #3154	C₂×C₃₁	semidirect
296. C₂×#492	C₂×C₃₁	semidirect
297. #3043	C₂×C₃₁	semidirect
298. #3042	C₂×C₃₁	semidirect
299. #3039	C₂×C₃₁	semidirect
300. #3038	C₂×C₃₁	semidirect
301. #2960	C₂×C₃₁	semidirect
302. #2949	C₂×C₃₁	semidirect
303. C₄×#135	C₂×C₃₁	semidirect
331. #557	C₂²×C₃₁	semidirect
332. #556	C₂²×C₃₁	semidirect
349. #497	C₂²×C₃₁	semidirect
350. #496	C₂²×C₃₁	semidirect
351. #495	C₂²×C₃₁	semidirect
354. #492	C₂²×C₃₁	semidirect
355. #464	C₂²×C₃₁	semidirect
356. #463	C₂²×C₃₁	semidirect
357. #462	C₂²×C₃₁	semidirect
358. #461	C₂²×C₃₁	semidirect
359. #461	C₂²×C₃₁	semidirect
361. #460	C₂²×C₃₁	semidirect
363. #446	C₂²×C₃₁	semidirect
364. C₈×Q₈	C₂²×C₃₁	semidirect
365. #440	C₂²×C₃₁	semidirect
366. C₄×Q₁₆	C₂²×C₃₁	semidirect
367. C₄×SD₁₆	C₂²×C₃₁	semidirect
368. C₄×SD₁₆	C₂²×C₃₁	semidirect
369. C₄×SD₁₆	C₂²×C₃₁	semidirect
370. C₄×D₈	C₂²×C₃₁	semidirect
475. C₃₁	#61410	direct (N × H)
522. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3509	2
C₃₁×#3508	2
C₃₁×#3400	2
C₃₁×#3397	2
C₃₁×#3394	2
C₃₁×#3392	2
C₃₁×#3154	2
C₂×C₃₁×#492	2
C₃₁×#3043	2
C₃₁×#3042	2
C₃₁×#3039	2
C₃₁×#3038	2
C₃₁×#2960	2
C₃₁×#2949	2
C₄×C₃₁×#135	2
#61410	31

Subgroups — C31×#61410 #21518169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61410 #21518169