Subgroups — C₃₁×#18941 #21475700

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3548	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1644	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1641	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1551	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#562	—	2	2	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₃₁×#554	C₂²	1	2	yes	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₂²	1	2	yes	—
17	C₃₁×#547	—	2	1	no	—
18	C₃₁×#547	—	2	1	no	—
19	C₃₁×#540	—	2	2	no	—
20	C₃₁×#537	—	2	2	no	—
21	C₃₁×#537	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#501	—	2	1	no	—
27	C₃₁×#494	C₂²	1	1	yes	characteristic
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#490	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#482	—	2	2	no	—
33	C₃₁×#477	—	2	1	no	—
34	C₃₁×#477	C₂²	1	2	yes	—
35	C₃₁×#476	C₂²	1	2	yes	—
36	C₃₁×#474	C₂²	1	2	yes	—
37	C₃₁×#467	—	2	2	no	—
38	C₃₁×#464	—	2	2	no	—
39	C₃₁×#462	—	2	1	no	—
40	C₃₁×#462	C₂²	1	2	yes	—
41	C₃₁×#452	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	D₄	1	2	yes	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	D₄	1	1	yes	characteristic
67	C₃₁×#127	C₂³	1	1	yes	characteristic
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×#124	D₄	1	2	yes	—
73	C₃₁×#123	—	2	4	no	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
83	C₃₁×#121	—	2	4	no	—
84	C₃₁×#121	—	2	4	no	—
85	C₃₁×#121	—	2	2	no	—
86	C₃₁×#121	—	2	4	no	—
87	C₃₁×#121	—	2	4	no	—
88	C₃₁×#121	—	2	1	no	—
89	C₃₁×#121	—	2	1	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	—	2	2	no	—
92	C₃₁×#121	D₄	1	2	yes	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	C₂³	1	2	yes	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	—	2	1	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	—	2	1	no	—
112	C₃₁×C₄⋊C₈	—	2	1	no	—
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#103	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	1	no	—
122	C₃₁×C₂²⋊C₈	—	2	1	no	—
123	C₃₁×#97	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	8	1	no	—
127	#41×C₃₁	—	8	1	no	—
128	#41×C₃₁	—	8	2	no	—
129	#41×C₃₁	—	8	2	no	—
130	#41×C₃₁	—	8	2	no	—
131	#41×C₃₁	—	8	4	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	Q₁₆×C₃₁	—	8	2	no	—
156	Q₁₆×C₃₁	—	8	2	no	—
157	SD₁₆×C₃₁	—	8	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	2	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	8	1	no	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	4	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	4	1	no	—
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
195	#18941	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	#120	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	1	no	—
205	Q₈×C₃₁	—	4	1	no	—
206	Q₈×C₃₁	—	8	1	no	—
207	D₄×C₃₁	—	2	4	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	1	no	—
211	D₄×C₃₁	—	4	1	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	8	1	no	—
214	D₄×C₃₁	—	8	2	no	—
215	D₄×C₃₁	—	8	2	no	—
216	D₄×C₃₁	—	8	4	no	—
217	D₄×C₃₁	—	8	4	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	4	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	8	4	no	—
232	C₂×C₄×C₃₁	—	8	4	no	—
233	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
234	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
235	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₃₁	—	4	1	no	—
240	C₈×C₃₁	—	4	1	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	2	no	—
243	#3548	C₂×C₃₁	1	1	yes	characteristic
244	#3249	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3235	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1644	C₂×C₃₁	1	2	yes	complemented
248	#1641	C₂×C₃₁	1	2	yes	complemented
249	#1632	C₂×C₃₁	1	2	yes	complemented
250	#1631	C₂×C₃₁	1	2	yes	complemented
251	#1551	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1480	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	2	1	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	8	1	no	—
273	C₄×C₃₁	—	8	1	no	—
274	C₄×C₃₁	#576	1	1	yes	characteristic
275	C₄×C₃₁	#577	1	1	yes	characteristic
276	C₄×C₃₁	#578	1	2	yes	—
277	#562	—	2	2	no	—
278	#554	—	2	1	no	—
279	#554	C₂²×C₃₁	1	2	yes	—
280	#550	—	2	1	no	—
281	#550	C₂²×C₃₁	1	2	yes	—
282	#547	—	2	1	no	—
283	#547	—	2	1	no	—
284	#540	—	2	2	no	—
285	#537	—	2	2	no	—
286	#537	—	2	2	no	—
287	#533	C₂²×C₃₁	1	1	yes	characteristic
288	#533	C₂²×C₃₁	1	1	yes	characteristic
289	#532	C₂²×C₃₁	1	1	yes	characteristic
290	#532	C₂²×C₃₁	1	1	yes	characteristic
291	#501	—	2	1	no	—
292	#494	C₂²×C₃₁	1	1	yes	characteristic
293	#492	C₂²×C₃₁	1	1	yes	characteristic
294	#490	—	2	1	no	—
295	#487	C₂²×C₃₁	1	1	yes	characteristic
296	#486	C₂²×C₃₁	1	1	yes	characteristic
297	#482	—	2	2	no	—
298	#477	—	2	1	no	—
299	#477	C₂²×C₃₁	1	2	yes	—
300	#476	C₂²×C₃₁	1	2	yes	—
301	#474	C₂²×C₃₁	1	2	yes	—
302	#467	—	2	2	no	—
303	#464	—	2	2	no	—
304	#462	—	2	1	no	—
305	#462	C₂²×C₃₁	1	2	yes	—
306	#452	—	2	2	no	—
307	#432	C₂²×C₃₁	1	1	yes	characteristic
308	#431	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	#333	C₂²×C₃₁	1	2	yes	—
311	#331	C₂²×C₃₁	1	2	yes	—
312	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
313	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
314	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
315	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
316	C₂×C₃₁	—	4	1	no	—
317	C₂×C₃₁	—	8	2	no	—
318	C₂×C₃₁	—	8	2	no	—
319	C₂×C₃₁	#3011	1	1	yes	characteristic
320	C₂×C₃₁	#3012	1	1	yes	characteristic
321	C₂×C₃₁	#3019	1	1	yes	characteristic
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	D₄×C₃₁	1	2	yes	—
326	C₂×Q₁₆	—	2	2	no	—
327	C₂×Q₁₆	—	2	2	no	—
328	C₂×SD₁₆	—	2	2	no	—
329	C₂²×C₈	—	2	1	no	—
330	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
332	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
336	#127	—	2	2	no	—
337	#127	D₄×C₃₁	1	1	yes	characteristic
338	#127	C₂³×C₃₁	1	1	yes	characteristic
339	#125	—	2	2	no	—
340	#125	—	2	1	no	—
341	#125	—	2	1	no	—
342	#124	—	2	1	no	—
343	#124	D₄×C₃₁	1	2	yes	—
344	#123	—	2	4	no	—
345	#123	—	2	4	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
354	#121	—	2	4	no	—
355	#121	—	2	4	no	—
356	#121	—	2	2	no	—
357	#121	—	2	4	no	—
358	#121	—	2	4	no	—
359	#121	—	2	1	no	—
360	#121	—	2	1	no	—
361	#121	—	2	1	no	—
362	#121	—	2	2	no	—
363	#121	D₄×C₃₁	1	2	yes	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	C₂³×C₃₁	1	2	yes	—
374	#117	—	2	1	no	—
375	#117	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	#107	—	2	2	no	—
380	#107	—	2	2	no	—
381	#106	—	2	2	no	—
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
385	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	C₂²⋊C₈	—	2	1	no	—
393	C₂²⋊C₈	—	2	1	no	—
394	#97	—	2	1	no	—
395	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₃₁	#18941	1	1	yes	characteristic, direct product factor, Sylow
398	#41	—	8	1	no	—
399	#41	—	8	1	no	—
400	#41	—	8	2	no	—
401	#41	—	8	2	no	—
402	#41	—	8	2	no	—
403	#41	—	8	4	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
409	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
427	Q₁₆	—	8	2	no	—
428	Q₁₆	—	8	2	no	—
429	SD₁₆	—	8	2	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	8	1	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	4	1	no	—
450	C₄⋊C₄	—	4	1	no	—
451	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	2	no	—
459	C₂²⋊C₄	—	2	2	no	—
460	C₂²⋊C₄	—	4	1	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	4	1	no	—
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	C₃₁×#120	1	1	yes	characteristic
470	Q₈	—	2	2	no	—
471	Q₈	—	2	2	no	—
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	1	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	8	1	no	—
478	D₄	—	2	4	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	1	no	—
482	D₄	—	4	1	no	—
483	D₄	—	4	2	no	—
484	D₄	—	8	1	no	—
485	D₄	—	8	2	no	—
486	D₄	—	8	2	no	—
487	D₄	—	8	4	no	—
488	D₄	—	8	4	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	8	4	no	—
503	C₂×C₄	—	8	4	no	—
504	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
505	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
506	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
510	C₈	—	4	1	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	2	no	—
513	C₈	—	4	2	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	8	1	no	—
533	C₄	—	8	1	no	—
534	C₄	C₃₁×#576	1	1	yes	characteristic
535	C₄	C₃₁×#577	1	1	yes	characteristic
536	C₄	C₃₁×#578	1	2	yes	—
537	C₂	—	4	1	no	—
538	C₂	—	8	2	no	—
539	C₂	—	8	2	no	—
540	C₂	C₃₁×#3011	1	1	yes	characteristic
541	C₂	C₃₁×#3012	1	1	yes	characteristic
542	C₂	C₃₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3249	C₂	semidirect
3. C₃₁×#3235	C₂	semidirect
4. C₃₁×#3210	C₂	semidirect
5. C₃₁×#1644	C₂	semidirect
6. C₃₁×#1641	C₂	semidirect
7. C₃₁×#1632	C₂	semidirect
8. C₃₁×#1631	C₂	semidirect
9. C₃₁×#1551	C₂	semidirect
10. C₃₁×#1497	C₂	semidirect
11. C₃₁×#1480	C₂	semidirect
195. #18941	C₃₁	direct (N × H)
244. #3249	C₂×C₃₁	semidirect
245. #3235	C₂×C₃₁	semidirect
246. #3210	C₂×C₃₁	semidirect
247. #1644	C₂×C₃₁	semidirect
248. #1641	C₂×C₃₁	semidirect
249. #1632	C₂×C₃₁	semidirect
250. #1631	C₂×C₃₁	semidirect
251. #1551	C₂×C₃₁	semidirect
252. #1497	C₂×C₃₁	semidirect
253. #1480	C₂×C₃₁	semidirect
397. C₃₁	#18941	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×#3548	2
C₃₁×#3249	2
C₃₁×#3235	2
C₃₁×#3210	2
C₃₁×#1644	2
C₃₁×#1641	2
C₃₁×#1632	2
C₃₁×#1631	2
C₃₁×#1551	2
C₃₁×#1497	2
C₃₁×#1480	2
#18941	31

Subgroups — C31×#18941 #21475700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18941 #21475700