Subgroups — C₃₁×#19360 #21476119

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3252	C₂	1	2	yes	maximal
2	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1696	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1695	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1693	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1672	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1671	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1559	C₂	1	2	yes	maximal, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₃₁×#529	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#499	—	2	1	no	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#479	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#470	—	2	2	no	—
23	C₃₁×#468	—	2	2	no	—
24	C₃₁×#464	—	2	2	no	—
25	C₃₁×#462	C₂²	1	2	yes	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#436	C₂²	1	1	yes	characteristic
30	C₃₁×#435	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#433	—	2	1	no	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#427	—	2	1	no	—
36	C₃₁×#427	—	2	1	no	—
37	C₃₁×#423	C₂²	1	2	yes	—
38	C₃₁×#405	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	1	yes	characteristic
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	1	no	—
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×#126	—	2	1	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#125	D₄	1	1	yes	characteristic
59	C₃₁×#124	D₄	1	1	yes	characteristic
60	C₃₁×#123	—	2	1	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
64	C₃₁×#121	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	1	no	—
76	C₃₁×#107	—	2	1	no	—
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₃₁×#97	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	Q₁₆×C₃₁	—	8	2	no	—
112	SD₁₆×C₃₁	—	8	2	no	—
113	M₁₆×C₃₁	—	4	2	no	—
114	M₁₆×C₃₁	—	8	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₃₁	#41	1	1	yes	characteristic
149	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
150	#19360	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	#121	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	2	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	8	2	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
171	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
172	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
176	C₈×C₃₁	—	2	1	no	—
177	C₈×C₃₁	—	2	1	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	1	no	—
180	C₈×C₃₁	—	4	2	no	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	#137	1	2	yes	—
183	C₈×C₃₁	#137	1	2	yes	—
184	#3252	C₂×C₃₁	1	2	yes	—
185	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#2966	C₂×C₃₁	1	1	yes	characteristic
187	#1696	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1695	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1694	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1693	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1672	C₂×C₃₁	1	2	yes	complemented
192	#1671	C₂×C₃₁	1	2	yes	complemented
193	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1559	C₂×C₃₁	1	2	yes	complemented
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	8	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
212	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
213	C₄×C₃₁	#577	1	2	yes	—
214	#533	C₂²×C₃₁	1	2	yes	—
215	#529	C₂²×C₃₁	1	1	yes	characteristic
216	#500	C₂²×C₃₁	1	1	yes	characteristic
217	#500	C₂²×C₃₁	1	1	yes	characteristic
218	#499	—	2	1	no	—
219	#499	—	2	1	no	—
220	#498	C₂²×C₃₁	1	2	yes	—
221	#479	—	2	2	no	—
222	#477	C₂²×C₃₁	1	2	yes	—
223	#476	C₂²×C₃₁	1	2	yes	—
224	#470	—	2	2	no	—
225	#468	—	2	2	no	—
226	#464	—	2	2	no	—
227	#462	C₂²×C₃₁	1	2	yes	—
228	#461	C₂²×C₃₁	1	2	yes	—
229	#446	C₂²×C₃₁	1	1	yes	characteristic
230	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
231	#436	C₂²×C₃₁	1	1	yes	characteristic
232	#435	—	2	1	no	—
233	C₈×D₄	—	2	1	no	—
234	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
235	#433	—	2	1	no	—
236	#432	C₂²×C₃₁	1	1	yes	characteristic
237	#427	—	2	1	no	—
238	#427	—	2	1	no	—
239	#423	C₂²×C₃₁	1	2	yes	—
240	#405	—	2	1	no	—
241	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	#334	C₂²×C₃₁	1	2	yes	—
243	#334	C₂²×C₃₁	1	1	yes	characteristic
244	#334	C₂²×C₃₁	1	1	yes	characteristic
245	#333	C₂²×C₃₁	1	2	yes	—
246	#333	C₂²×C₃₁	1	2	yes	—
247	#332	C₂²×C₃₁	1	2	yes	—
248	#332	C₂²×C₃₁	1	2	yes	—
249	C₂×C₃₁	—	4	1	no	—
250	C₂×C₃₁	—	8	1	no	—
251	C₂×C₃₁	#3039	1	1	yes	characteristic
252	C₂×C₃₁	#3060	1	1	yes	characteristic
253	C₂×C₃₁	#3067	1	1	yes	characteristic
254	C₂×Q₁₆	—	2	2	no	—
255	C₂×SD₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	1	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
261	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
262	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
263	#126	—	2	1	no	—
264	#125	—	2	2	no	—
265	#125	D₄×C₃₁	1	1	yes	characteristic
266	#124	D₄×C₃₁	1	1	yes	characteristic
267	#123	—	2	1	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	2	2	no	—
270	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
271	#121	D₄×C₃₁	1	1	yes	characteristic
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
277	#117	—	2	1	no	—
278	#117	—	2	2	no	—
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	#107	—	2	1	no	—
281	#107	—	2	1	no	—
282	#107	—	2	1	no	—
283	#107	—	2	1	no	—
284	#107	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	—	2	2	no	—
287	#106	—	2	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
294	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
295	#103	—	2	2	no	—
296	#103	—	2	2	no	—
297	#103	—	2	2	no	—
298	#102	—	2	2	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	2	1	no	—
301	C₂²⋊C₈	—	2	1	no	—
302	#97	—	2	1	no	—
303	#97	—	2	1	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
306	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₃₁	#19360	1	1	yes	characteristic, direct product factor, Sylow
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
319	Q₁₆	—	8	2	no	—
320	SD₁₆	—	8	2	no	—
321	M₁₆	—	4	2	no	—
322	M₁₆	—	8	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	4	2	no	—
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₄²	#41×C₃₁	1	1	yes	characteristic
357	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
358	C₂³	—	2	1	no	—
359	C₂³	C₃₁×#121	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
378	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
379	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	2	no	—
389	C₈	C₃₁×#137	1	2	yes	—
390	C₈	C₃₁×#137	1	2	yes	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	2	no	—
407	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
408	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
409	C₄	C₃₁×#577	1	2	yes	—
410	C₂	—	4	1	no	—
411	C₂	—	8	1	no	—
412	C₂	C₃₁×#3039	1	1	yes	characteristic
413	C₂	C₃₁×#3060	1	1	yes	characteristic
414	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#121	D₄
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×#3160	C₂	semidirect
4. C₃₁×#1696	C₂	semidirect
5. C₃₁×#1695	C₂	semidirect
6. C₃₁×#1694	C₂	semidirect
7. C₃₁×#1693	C₂	semidirect
8. C₃₁×#1672	C₂	semidirect
9. C₃₁×#1671	C₂	semidirect
10. C₃₁×#1566	C₂	semidirect
11. C₃₁×#1559	C₂	semidirect
150. #19360	C₃₁	direct (N × H)
185. #3160	C₂×C₃₁	semidirect
187. #1696	C₂×C₃₁	semidirect
188. #1695	C₂×C₃₁	semidirect
189. #1694	C₂×C₃₁	semidirect
190. #1693	C₂×C₃₁	semidirect
191. #1672	C₂×C₃₁	semidirect
192. #1671	C₂×C₃₁	semidirect
193. #1566	C₂×C₃₁	semidirect
194. #1559	C₂×C₃₁	semidirect
309. C₃₁	#19360	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×#3252	2
C₃₁×#3160	2
C₃₁×#2966	2
C₃₁×#1696	2
C₃₁×#1695	2
C₃₁×#1694	2
C₃₁×#1693	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1566	2
C₃₁×#1559	2
#19360	31

Subgroups — C31×#19360 #21476119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19360 #21476119