Subgroups — C₃₁×#61224 #21517983

Order 7936.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3320	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3315	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3312	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3307	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3063	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3013	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2932	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2898	C₂	1	2	yes	maximal, complemented
16	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
17	C₂×C₃₁×#137	C₂²	1	2	yes	—
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#131	C₂²	1	2	yes	—
20	C₃₁×#553	—	2	4	no	—
21	C₃₁×#553	C₂²	1	4	yes	—
22	C₃₁×#552	C₂²	1	4	yes	—
23	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
24	C₃₁×#547	—	2	4	no	—
25	C₃₁×#547	C₂²	1	4	yes	—
26	C₃₁×#541	—	2	1	no	—
27	C₃₁×#540	—	2	1	no	—
28	C₃₁×#537	C₂²	1	2	yes	—
29	C₃₁×#536	—	2	2	no	—
30	C₃₁×#536	C₂²	1	2	yes	—
31	C₃₁×#532	—	2	1	no	—
32	C₃₁×#532	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
34	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
35	C₃₁×#529	—	2	2	no	—
36	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
37	C₃₁×#520	C₂²	1	2	yes	—
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#491	C₂²	1	4	yes	—
41	C₃₁×#491	C₂²	1	4	yes	—
42	C₃₁×#485	C₂²	1	4	yes	—
43	C₃₁×#485	C₂²	1	4	yes	—
44	C₃₁×#484	C₂²	1	4	yes	—
45	C₃₁×#483	C₂²	1	4	yes	—
46	C₃₁×#479	C₂²	1	4	yes	—
47	C₃₁×#478	C₂²	1	4	yes	—
48	C₃₁×#470	C₂²	1	4	yes	—
49	C₃₁×#468	C₂²	1	4	yes	—
50	C₃₁×#467	C₂²	1	4	yes	—
51	C₃₁×#465	C₂²	1	4	yes	—
52	C₃₁×#462	C₂²	1	4	yes	—
53	C₃₁×#461	C₂²	1	4	yes	—
54	C₃₁×#452	C₂²	1	4	yes	—
55	C₃₁×#451	C₂²	1	4	yes	—
56	C₃₁×#449	C₂²	1	4	yes	—
57	C₃₁×#448	C₂²	1	4	yes	—
58	C₃₁×#442	C₂²	1	2	yes	—
59	C₃₁×#441	C₂²	1	2	yes	—
60	C₃₁×#440	C₂²	1	2	yes	—
61	C₃₁×#440	C₂²	1	2	yes	—
62	C₃₁×#435	C₂²	1	4	yes	—
63	C₃₁×#435	C₂²	1	4	yes	—
64	C₃₁×#428	C₂²	1	2	yes	—
65	C₃₁×#427	C₂²	1	1	yes	characteristic
66	C₃₁×#423	C₂²	1	1	yes	characteristic
67	C₃₁×#419	C₂²	1	2	yes	—
68	C₃₁×#419	C₂²	1	4	yes	—
69	C₃₁×#417	C₂²	1	2	yes	—
70	C₃₁×#417	C₂²	1	4	yes	—
71	C₃₁×#416	C₂²	1	1	yes	characteristic
72	C₃₁×#416	C₂²	1	4	yes	—
73	C₃₁×#416	C₂²	1	1	yes	characteristic
74	C₂×C₃₁×#103	C₂²	1	2	yes	—
75	C₂×C₃₁×#103	C₂²	1	2	yes	—
76	C₂×C₃₁×#103	C₂²	1	4	yes	—
77	C₃₁×#408	C₂²	1	2	yes	—
78	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
79	C₃₁×2₋¹⁺⁴	—	2	8	no	—
80	C₃₁×2₋¹⁺⁴	—	4	4	no	—
81	C₂×#41×C₃₁	—	2	2	no	—
82	C₂×#41×C₃₁	—	2	4	no	—
83	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×#41×C₃₁	C₂³	1	4	yes	—
85	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×#41×C₃₁	C₂³	1	2	yes	—
87	C₂²×Q₈×C₃₁	—	2	2	no	—
88	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
89	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₃₁×#137	—	4	4	no	—
91	C₃₁×#135	—	2	4	no	—
92	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
97	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×#131	—	4	4	no	—
99	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
100	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
101	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
104	C₃₁×C₄⋊Q₈	—	2	2	no	—
105	C₃₁×C₄⋊Q₈	—	2	4	no	—
106	C₃₁×C₄⋊Q₈	—	2	1	no	—
107	C₃₁×C₄⋊Q₈	—	2	1	no	—
108	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
109	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
110	C₃₁×#127	D₄	1	1	yes	characteristic
111	C₃₁×#126	—	2	4	no	—
112	C₃₁×#126	—	2	4	no	—
113	C₃₁×#125	—	2	4	no	—
114	C₃₁×#125	—	2	1	no	—
115	C₃₁×#125	—	2	1	no	—
116	C₃₁×#125	C₂³	1	4	yes	—
117	C₃₁×#124	—	2	2	no	—
118	C₃₁×#124	D₄	1	2	yes	—
119	C₃₁×#123	—	2	4	no	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×#123	—	2	4	no	—
123	C₃₁×#123	—	2	4	no	—
124	C₃₁×C₂²⋊Q₈	—	2	2	no	—
125	C₃₁×C₂²⋊Q₈	—	2	4	no	—
126	C₃₁×C₂²⋊Q₈	—	2	4	no	—
127	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
128	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
129	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
130	C₃₁×#121	—	2	2	no	—
131	C₃₁×#121	—	2	4	no	—
132	C₃₁×#121	—	2	4	no	—
133	C₃₁×#121	—	2	4	no	—
134	C₃₁×#121	D₄	1	4	yes	—
135	C₃₁×#121	C₂³	1	4	yes	—
136	C₄×Q₈×C₃₁	—	2	2	no	—
137	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
138	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
139	C₄×D₄×C₃₁	—	2	2	no	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	—	2	4	no	—
142	C₄×D₄×C₃₁	—	2	2	no	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	—	2	4	no	—
145	C₄×D₄×C₃₁	C₂³	1	2	yes	—
146	C₄×D₄×C₃₁	C₂³	1	2	yes	—
147	C₄×D₄×C₃₁	C₂³	1	4	yes	—
148	C₃₁×#117	—	2	4	no	—
149	C₃₁×#117	—	2	1	no	—
150	C₃₁×#117	—	2	1	no	—
151	C₃₁×#117	C₂³	1	1	yes	characteristic
152	C₃₁×#117	C₂³	1	4	yes	—
153	C₃₁×#117	C₂³	1	1	yes	characteristic
154	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
159	C₂×C₄²×C₃₁	—	2	1	no	—
160	C₂×C₄²×C₃₁	—	2	1	no	—
161	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
162	C₃₁×#107	C₂³	1	2	yes	—
163	C₃₁×#107	C₂³	1	4	yes	—
164	C₃₁×#106	C₂³	1	2	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
167	C₃₁×#103	C₂³	1	2	yes	—
168	C₃₁×#103	C₂³	1	4	yes	—
169	C₃₁×#103	C₂³	1	4	yes	—
170	C₃₁×#103	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	4	yes	—
172	C₃₁×#103	C₂³	1	4	yes	—
173	C₃₁×#103	C₂³	1	4	yes	—
174	C₃₁×#103	C₂³	1	4	yes	—
175	C₃₁×#103	C₂³	1	2	yes	—
176	C₃₁×#102	C₂³	1	2	yes	—
177	C₃₁×#102	C₂³	1	4	yes	—
178	C₃₁×#102	C₂³	1	2	yes	—
179	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
180	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
181	C₃₁×#97	C₂³	1	2	yes	—
182	C₃₁×#97	C₂³	1	2	yes	—
183	#41×C₃₁	—	2	4	no	—
184	#41×C₃₁	—	2	4	no	—
185	#41×C₃₁	—	2	2	no	—
186	#41×C₃₁	—	2	8	no	—
187	#41×C₃₁	—	2	2	no	—
188	#41×C₃₁	—	2	2	no	—
189	#41×C₃₁	—	4	4	no	—
190	#41×C₃₁	—	4	8	no	—
191	#41×C₃₁	—	4	16	no	—
192	#41×C₃₁	#41	1	4	yes	—
193	C₂×Q₈×C₃₁	—	2	4	no	—
194	C₂×Q₈×C₃₁	—	2	2	no	—
195	C₂×Q₈×C₃₁	—	2	8	no	—
196	C₂×Q₈×C₃₁	—	2	4	no	—
197	C₂×Q₈×C₃₁	—	2	4	no	—
198	C₂×Q₈×C₃₁	—	4	4	no	—
199	C₂×Q₈×C₃₁	—	4	8	no	—
200	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
201	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
202	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
203	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
204	C₂×D₄×C₃₁	—	2	1	no	—
205	C₂×D₄×C₃₁	—	2	1	no	—
206	C₂×D₄×C₃₁	—	2	4	no	—
207	C₂×D₄×C₃₁	—	2	4	no	—
208	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
209	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
211	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₃₁	—	2	1	no	—
214	C₂²×C₄×C₃₁	—	2	2	no	—
215	C₂²×C₄×C₃₁	—	2	4	no	—
216	C₂²×C₄×C₃₁	—	2	2	no	—
217	C₂²×C₄×C₃₁	—	2	4	no	—
218	C₂²×C₄×C₃₁	—	2	2	no	—
219	C₂²×C₄×C₃₁	—	2	1	no	—
220	C₂²×C₄×C₃₁	—	2	4	no	—
221	C₂²×C₄×C₃₁	—	2	2	no	—
222	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
224	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
225	Q₁₆×C₃₁	—	2	2	no	—
226	Q₁₆×C₃₁	—	4	4	no	—
227	SD₁₆×C₃₁	—	2	2	no	—
228	SD₁₆×C₃₁	—	2	2	no	—
229	SD₁₆×C₃₁	—	4	4	no	—
230	D₈×C₃₁	—	2	2	no	—
231	M₁₆×C₃₁	—	2	4	no	—
232	M₁₆×C₃₁	—	4	4	no	—
233	C₂×C₈×C₃₁	—	2	2	no	—
234	C₂×C₈×C₃₁	—	2	2	no	—
235	C₂×C₈×C₃₁	—	4	4	no	—
236	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
237	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
238	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
239	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
240	C₄⋊C₄×C₃₁	—	2	4	no	—
241	C₄⋊C₄×C₃₁	—	2	4	no	—
242	C₄⋊C₄×C₃₁	—	2	4	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	4	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	—	2	2	no	—
247	C₄⋊C₄×C₃₁	—	2	4	no	—
248	C₄⋊C₄×C₃₁	—	2	2	no	—
249	C₄⋊C₄×C₃₁	—	2	2	no	—
250	C₄⋊C₄×C₃₁	—	2	4	no	—
251	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
252	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
253	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
254	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
255	C₂²⋊C₄×C₃₁	—	2	4	no	—
256	C₂²⋊C₄×C₃₁	—	2	1	no	—
257	C₂²⋊C₄×C₃₁	—	2	4	no	—
258	C₂²⋊C₄×C₃₁	—	2	1	no	—
259	C₂²⋊C₄×C₃₁	—	2	4	no	—
260	C₂²⋊C₄×C₃₁	—	2	4	no	—
261	C₂²⋊C₄×C₃₁	—	2	2	no	—
262	C₂²⋊C₄×C₃₁	—	2	2	no	—
263	C₂²⋊C₄×C₃₁	—	2	4	no	—
264	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
265	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
266	C₄²×C₃₁	—	2	2	no	—
267	C₄²×C₃₁	—	2	1	no	—
268	C₄²×C₃₁	—	2	1	no	—
269	C₄²×C₃₁	—	2	2	no	—
270	C₄²×C₃₁	—	2	4	no	—
271	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₃₁	C₂⁴	1	2	yes	—
274	#61224	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂³×C₃₁	—	2	1	no	—
276	C₂³×C₃₁	—	2	1	no	—
277	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
278	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
279	Q₈×C₃₁	—	2	4	no	—
280	Q₈×C₃₁	—	2	1	no	—
281	Q₈×C₃₁	—	2	1	no	—
282	Q₈×C₃₁	—	2	4	no	—
283	Q₈×C₃₁	—	2	4	no	—
284	Q₈×C₃₁	—	2	1	no	—
285	Q₈×C₃₁	—	4	8	no	—
286	Q₈×C₃₁	—	4	4	no	—
287	Q₈×C₃₁	C₂×#41	1	2	yes	—
288	D₄×C₃₁	—	2	4	no	—
289	D₄×C₃₁	—	2	8	no	—
290	D₄×C₃₁	—	2	4	no	—
291	D₄×C₃₁	—	2	1	no	—
292	D₄×C₃₁	—	2	1	no	—
293	D₄×C₃₁	—	2	1	no	—
294	D₄×C₃₁	—	4	4	no	—
295	D₄×C₃₁	—	4	8	no	—
296	D₄×C₃₁	—	4	2	no	—
297	D₄×C₃₁	—	8	4	no	—
298	D₄×C₃₁	C₂×#41	1	2	yes	—
299	C₂×C₄×C₃₁	—	2	1	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	4	no	—
302	C₂×C₄×C₃₁	—	2	4	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	1	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	2	4	no	—
308	C₂×C₄×C₃₁	—	4	2	no	—
309	C₂×C₄×C₃₁	—	4	8	no	—
310	C₂×C₄×C₃₁	—	4	8	no	—
311	C₂×C₄×C₃₁	—	4	4	no	—
312	C₂×C₄×C₃₁	—	4	4	no	—
313	C₂×C₄×C₃₁	—	4	2	no	—
314	C₂×C₄×C₃₁	—	8	2	no	—
315	C₂×C₄×C₃₁	—	8	4	no	—
316	C₂×C₄×C₃₁	—	8	2	no	—
317	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
319	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
320	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
322	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
323	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
324	C₈×C₃₁	—	2	1	no	—
325	C₈×C₃₁	—	2	1	no	—
326	C₈×C₃₁	—	2	2	no	—
327	C₈×C₃₁	—	2	4	no	—
328	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#3320	C₂×C₃₁	1	4	yes	complemented
331	#3315	C₂×C₃₁	1	4	yes	complemented
332	#3312	C₂×C₃₁	1	4	yes	complemented
333	#3307	C₂×C₃₁	1	4	yes	complemented
334	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#3097	C₂×C₃₁	1	2	yes	complemented
336	#3080	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#3063	C₂×C₃₁	1	2	yes	complemented
338	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#3013	C₂×C₃₁	1	2	yes	complemented
340	#2943	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#2932	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#2898	C₂×C₃₁	1	2	yes	complemented
343	C₂²×C₃₁	—	2	2	no	—
344	C₂²×C₃₁	—	2	4	no	—
345	C₂²×C₃₁	—	2	4	no	—
346	C₂²×C₃₁	—	2	2	no	—
347	C₂²×C₃₁	—	2	4	no	—
348	C₂²×C₃₁	—	4	2	no	—
349	C₂²×C₃₁	—	4	2	no	—
350	C₂²×C₃₁	—	4	1	no	—
351	C₂²×C₃₁	#546	1	2	yes	—
352	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
353	C₄×C₃₁	—	2	2	no	—
354	C₄×C₃₁	—	2	2	no	—
355	C₄×C₃₁	—	2	2	no	—
356	C₄×C₃₁	—	2	4	no	—
357	C₄×C₃₁	—	4	2	no	—
358	C₄×C₃₁	—	4	2	no	—
359	C₄×C₃₁	—	4	4	no	—
360	C₄×C₃₁	—	4	4	no	—
361	C₄×C₃₁	—	4	1	no	—
362	C₄×C₃₁	#546	1	2	yes	—
363	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
364	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
365	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
366	C₂×#137	C₂²×C₃₁	1	2	yes	—
367	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#131	C₂²×C₃₁	1	2	yes	—
369	#553	—	2	4	no	—
370	#553	C₂²×C₃₁	1	4	yes	—
371	#552	C₂²×C₃₁	1	4	yes	—
372	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
373	#547	—	2	4	no	—
374	#547	C₂²×C₃₁	1	4	yes	—
375	#541	—	2	1	no	—
376	#540	—	2	1	no	—
377	#537	C₂²×C₃₁	1	2	yes	—
378	#536	—	2	2	no	—
379	#536	C₂²×C₃₁	1	2	yes	—
380	#532	—	2	1	no	—
381	#532	C₂²×C₃₁	1	1	yes	characteristic
382	C₂×C₄⋊Q₈	—	2	1	no	—
383	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
384	#529	—	2	2	no	—
385	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
386	#520	C₂²×C₃₁	1	2	yes	—
387	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
388	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
389	#491	C₂²×C₃₁	1	4	yes	—
390	#491	C₂²×C₃₁	1	4	yes	—
391	#485	C₂²×C₃₁	1	4	yes	—
392	#485	C₂²×C₃₁	1	4	yes	—
393	#484	C₂²×C₃₁	1	4	yes	—
394	#483	C₂²×C₃₁	1	4	yes	—
395	#479	C₂²×C₃₁	1	4	yes	—
396	#478	C₂²×C₃₁	1	4	yes	—
397	#470	C₂²×C₃₁	1	4	yes	—
398	#468	C₂²×C₃₁	1	4	yes	—
399	#467	C₂²×C₃₁	1	4	yes	—
400	#465	C₂²×C₃₁	1	4	yes	—
401	#462	C₂²×C₃₁	1	4	yes	—
402	#461	C₂²×C₃₁	1	4	yes	—
403	#452	C₂²×C₃₁	1	4	yes	—
404	#451	C₂²×C₃₁	1	4	yes	—
405	#449	C₂²×C₃₁	1	4	yes	—
406	#448	C₂²×C₃₁	1	4	yes	—
407	#442	C₂²×C₃₁	1	2	yes	—
408	#441	C₂²×C₃₁	1	2	yes	—
409	#440	C₂²×C₃₁	1	2	yes	—
410	#440	C₂²×C₃₁	1	2	yes	—
411	#435	C₂²×C₃₁	1	4	yes	—
412	#435	C₂²×C₃₁	1	4	yes	—
413	#428	C₂²×C₃₁	1	2	yes	—
414	#427	C₂²×C₃₁	1	1	yes	characteristic
415	#423	C₂²×C₃₁	1	1	yes	characteristic
416	#419	C₂²×C₃₁	1	2	yes	—
417	#419	C₂²×C₃₁	1	4	yes	—
418	#417	C₂²×C₃₁	1	2	yes	—
419	#417	C₂²×C₃₁	1	4	yes	—
420	#416	C₂²×C₃₁	1	1	yes	characteristic
421	#416	C₂²×C₃₁	1	4	yes	—
422	#416	C₂²×C₃₁	1	1	yes	characteristic
423	C₂×#103	C₂²×C₃₁	1	2	yes	—
424	C₂×#103	C₂²×C₃₁	1	2	yes	—
425	C₂×#103	C₂²×C₃₁	1	4	yes	—
426	#408	C₂²×C₃₁	1	2	yes	—
427	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
428	C₂×C₃₁	—	2	2	no	—
429	C₂×C₃₁	—	4	4	no	—
430	C₂×C₃₁	—	4	2	no	—
431	C₂×C₃₁	—	8	1	no	—
432	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
433	C₂×C₃₁	#3586	1	2	yes	—
434	2₋¹⁺⁴	—	2	8	no	—
435	2₋¹⁺⁴	—	4	4	no	—
436	C₂×#41	—	2	2	no	—
437	C₂×#41	—	2	4	no	—
438	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×#41	C₂³×C₃₁	1	4	yes	—
440	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×#41	C₂³×C₃₁	1	2	yes	—
442	C₂²×Q₈	—	2	2	no	—
443	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
444	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
445	#137	—	4	4	no	—
446	#135	—	2	4	no	—
447	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
448	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
449	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
452	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
453	#131	—	4	4	no	—
454	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
455	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
456	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
457	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
458	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
459	C₄⋊Q₈	—	2	2	no	—
460	C₄⋊Q₈	—	2	4	no	—
461	C₄⋊Q₈	—	2	1	no	—
462	C₄⋊Q₈	—	2	1	no	—
463	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
465	#127	D₄×C₃₁	1	1	yes	characteristic
466	#126	—	2	4	no	—
467	#126	—	2	4	no	—
468	#125	—	2	4	no	—
469	#125	—	2	1	no	—
470	#125	—	2	1	no	—
471	#125	C₂³×C₃₁	1	4	yes	—
472	#124	—	2	2	no	—
473	#124	D₄×C₃₁	1	2	yes	—
474	#123	—	2	4	no	—
475	#123	—	2	4	no	—
476	#123	—	2	4	no	—
477	#123	—	2	4	no	—
478	#123	—	2	4	no	—
479	C₂²⋊Q₈	—	2	2	no	—
480	C₂²⋊Q₈	—	2	4	no	—
481	C₂²⋊Q₈	—	2	4	no	—
482	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
483	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
484	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
485	#121	—	2	2	no	—
486	#121	—	2	4	no	—
487	#121	—	2	4	no	—
488	#121	—	2	4	no	—
489	#121	D₄×C₃₁	1	4	yes	—
490	#121	C₂³×C₃₁	1	4	yes	—
491	C₄×Q₈	—	2	2	no	—
492	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
493	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
494	C₄×D₄	—	2	2	no	—
495	C₄×D₄	—	2	4	no	—
496	C₄×D₄	—	2	4	no	—
497	C₄×D₄	—	2	2	no	—
498	C₄×D₄	—	2	2	no	—
499	C₄×D₄	—	2	4	no	—
500	C₄×D₄	C₂³×C₃₁	1	2	yes	—
501	C₄×D₄	C₂³×C₃₁	1	2	yes	—
502	C₄×D₄	C₂³×C₃₁	1	4	yes	—
503	#117	—	2	4	no	—
504	#117	—	2	1	no	—
505	#117	—	2	1	no	—
506	#117	C₂³×C₃₁	1	1	yes	characteristic
507	#117	C₂³×C₃₁	1	1	yes	characteristic
508	#117	C₂³×C₃₁	1	4	yes	—
509	C₂×C₄⋊C₄	—	2	2	no	—
510	C₂×C₄⋊C₄	—	2	2	no	—
511	C₂×C₄⋊C₄	—	2	4	no	—
512	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
513	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
514	C₂×C₄²	—	2	1	no	—
515	C₂×C₄²	—	2	1	no	—
516	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
517	#107	C₂³×C₃₁	1	2	yes	—
518	#107	C₂³×C₃₁	1	4	yes	—
519	#106	C₂³×C₃₁	1	2	yes	—
520	#106	C₂³×C₃₁	1	4	yes	—
521	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
522	#103	C₂³×C₃₁	1	2	yes	—
523	#103	C₂³×C₃₁	1	4	yes	—
524	#103	C₂³×C₃₁	1	4	yes	—
525	#103	C₂³×C₃₁	1	4	yes	—
526	#103	C₂³×C₃₁	1	4	yes	—
527	#103	C₂³×C₃₁	1	4	yes	—
528	#103	C₂³×C₃₁	1	4	yes	—
529	#103	C₂³×C₃₁	1	4	yes	—
530	#103	C₂³×C₃₁	1	2	yes	—
531	#102	C₂³×C₃₁	1	2	yes	—
532	#102	C₂³×C₃₁	1	4	yes	—
533	#102	C₂³×C₃₁	1	2	yes	—
534	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
535	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
536	#97	C₂³×C₃₁	1	2	yes	—
537	#97	C₂³×C₃₁	1	2	yes	—
538	C₃₁	#61224	1	1	yes	characteristic, direct product factor, Sylow
539	#41	—	2	4	no	—
540	#41	—	2	4	no	—
541	#41	—	2	2	no	—
542	#41	—	2	8	no	—
543	#41	—	2	2	no	—
544	#41	—	2	2	no	—
545	#41	—	4	4	no	—
546	#41	—	4	8	no	—
547	#41	—	4	16	no	—
548	#41	#41×C₃₁	1	4	yes	—
549	C₂×Q₈	—	2	4	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	2	8	no	—
552	C₂×Q₈	—	2	4	no	—
553	C₂×Q₈	—	2	4	no	—
554	C₂×Q₈	—	4	4	no	—
555	C₂×Q₈	—	4	8	no	—
556	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
557	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
558	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
559	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	4	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
565	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
566	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
567	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
568	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	4	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	—	2	4	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	4	no	—
577	C₂²×C₄	—	2	2	no	—
578	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
579	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
580	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
581	Q₁₆	—	2	2	no	—
582	Q₁₆	—	4	4	no	—
583	SD₁₆	—	2	2	no	—
584	SD₁₆	—	2	2	no	—
585	SD₁₆	—	4	4	no	—
586	D₈	—	2	2	no	—
587	M₁₆	—	2	4	no	—
588	M₁₆	—	4	4	no	—
589	C₂×C₈	—	2	2	no	—
590	C₂×C₈	—	2	2	no	—
591	C₂×C₈	—	4	4	no	—
592	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
593	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
594	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
595	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	4	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
608	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
610	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
611	C₂²⋊C₄	—	2	4	no	—
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	2	2	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
621	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
622	C₄²	—	2	2	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	2	2	no	—
626	C₄²	—	2	4	no	—
627	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
628	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
629	C₄²	C₂⁴×C₃₁	1	2	yes	—
630	C₂³	—	2	1	no	—
631	C₂³	—	2	1	no	—
632	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
633	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
634	Q₈	—	2	4	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	2	1	no	—
637	Q₈	—	2	4	no	—
638	Q₈	—	2	4	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	4	8	no	—
641	Q₈	—	4	4	no	—
642	Q₈	C₂×#41×C₃₁	1	2	yes	—
643	D₄	—	2	4	no	—
644	D₄	—	2	8	no	—
645	D₄	—	2	4	no	—
646	D₄	—	2	1	no	—
647	D₄	—	2	1	no	—
648	D₄	—	2	1	no	—
649	D₄	—	4	4	no	—
650	D₄	—	4	8	no	—
651	D₄	—	4	2	no	—
652	D₄	—	8	4	no	—
653	D₄	C₂×#41×C₃₁	1	2	yes	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	4	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	4	no	—
663	C₂×C₄	—	4	2	no	—
664	C₂×C₄	—	4	8	no	—
665	C₂×C₄	—	4	8	no	—
666	C₂×C₄	—	4	4	no	—
667	C₂×C₄	—	4	4	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	8	2	no	—
670	C₂×C₄	—	8	4	no	—
671	C₂×C₄	—	8	2	no	—
672	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
673	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
674	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
675	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
676	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
677	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
678	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
679	C₈	—	2	1	no	—
680	C₈	—	2	1	no	—
681	C₈	—	2	2	no	—
682	C₈	—	2	4	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	2	2	no	—
685	C₂²	—	2	4	no	—
686	C₂²	—	2	4	no	—
687	C₂²	—	2	4	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	1	no	—
690	C₂²	—	4	2	no	—
691	C₂²	C₃₁×#546	1	2	yes	—
692	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
693	C₄	—	2	2	no	—
694	C₄	—	2	4	no	—
695	C₄	—	2	2	no	—
696	C₄	—	2	2	no	—
697	C₄	—	4	4	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	4	no	—
702	C₄	C₃₁×#546	1	2	yes	—
703	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
704	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
705	C₂	—	2	2	no	—
706	C₂	—	4	4	no	—
707	C₂	—	4	2	no	—
708	C₂	—	8	1	no	—
709	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
710	C₂	C₃₁×#3586	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×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3499	C₂	semidirect
3. C₃₁×#3320	C₂	semidirect
4. C₃₁×#3315	C₂	semidirect
5. C₃₁×#3312	C₂	semidirect
6. C₃₁×#3307	C₂	semidirect
7. C₂×C₃₁×#491	C₂	semidirect
8. C₃₁×#3097	C₂	semidirect
9. C₃₁×#3080	C₂	semidirect
10. C₃₁×#3063	C₂	semidirect
11. C₃₁×#3039	C₂	semidirect
12. C₃₁×#3013	C₂	semidirect
13. C₃₁×#2943	C₂	semidirect
14. C₃₁×#2932	C₂	semidirect
15. C₃₁×#2898	C₂	semidirect
274. #61224	C₃₁	direct (N × H)
328. #3512	C₂×C₃₁	semidirect
329. #3499	C₂×C₃₁	semidirect
330. #3320	C₂×C₃₁	semidirect
331. #3315	C₂×C₃₁	semidirect
332. #3312	C₂×C₃₁	semidirect
333. #3307	C₂×C₃₁	semidirect
334. C₂×#491	C₂×C₃₁	semidirect
335. #3097	C₂×C₃₁	semidirect
336. #3080	C₂×C₃₁	semidirect
337. #3063	C₂×C₃₁	semidirect
338. #3039	C₂×C₃₁	semidirect
339. #3013	C₂×C₃₁	semidirect
340. #2943	C₂×C₃₁	semidirect
341. #2932	C₂×C₃₁	semidirect
342. #2898	C₂×C₃₁	semidirect
538. C₃₁	#61224	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×#3512	2
C₃₁×#3499	2
C₃₁×#3320	2
C₃₁×#3315	2
C₃₁×#3312	2
C₃₁×#3307	2
C₂×C₃₁×#491	2
C₃₁×#3097	2
C₃₁×#3080	2
C₃₁×#3063	2
C₃₁×#3039	2
C₃₁×#3013	2
C₃₁×#2943	2
C₃₁×#2932	2
C₃₁×#2898	2
#61224	31

Subgroups — C31×#61224 #21517983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61224 #21517983