Subgroups — C₃₁×#26081 #21482840

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3084	C₂	1	2	yes	maximal
3	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2090	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2060	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1859	C₂	1	2	yes	maximal
9	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
13	C₃₁×#544	C₂²	1	2	yes	—
14	C₃₁×#520	C₂²	1	2	yes	—
15	C₃₁×#501	—	2	2	no	—
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#498	—	2	1	no	—
18	C₃₁×#479	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#476	—	2	2	no	—
21	C₃₁×#475	—	2	2	no	—
22	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
23	C₃₁×#428	—	2	2	no	—
24	C₃₁×#428	C₂²	1	2	yes	—
25	C₃₁×#428	C₂²	1	2	yes	—
26	C₃₁×#428	C₂²	1	2	yes	—
27	C₃₁×#427	—	2	1	no	—
28	C₃₁×#427	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#106	—	2	1	no	—
30	C₂×C₃₁×#106	C₂²	1	2	yes	—
31	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
33	C₃₁×#424	C₂²	1	2	yes	—
34	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
35	C₃₁×#417	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	1	yes	characteristic
37	C₃₁×#405	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#384	C₂²	1	2	yes	—
40	C₃₁×#343	C₂²	1	2	yes	—
41	C₃₁×#343	C₂²	1	2	yes	—
42	C₃₁×#341	C₂²	1	1	yes	characteristic
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#131	—	2	2	no	—
47	C₃₁×#131	Q₈	1	4	yes	complemented
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	—	2	1	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#125	—	2	2	no	—
59	C₃₁×C₂²⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	4	2	no	—
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	C₂³	1	1	yes	characteristic
65	C₃₁×#117	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#108	—	2	2	no	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	D₄	1	2	yes	—
76	C₃₁×#106	—	2	1	no	—
77	C₃₁×#106	—	2	1	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×#106	—	2	1	no	—
81	C₃₁×#106	—	2	1	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#106	D₄	1	1	yes	characteristic
86	C₃₁×#106	D₄	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×#104	Q₈	1	4	yes	complemented
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	D₄	1	2	yes	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	D₄	1	2	yes	—
94	C₃₁×#97	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	#41×C₃₁	—	2	1	no	—
98	#41×C₃₁	—	2	1	no	—
99	#41×C₃₁	C₂×Q₈	1	2	yes	—
100	#41×C₃₁	C₂×Q₈	1	2	yes	—
101	C₂×Q₈×C₃₁	—	4	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₃₁	—	2	1	no	—
112	M₁₆×C₃₁	—	2	1	no	—
113	M₁₆×C₃₁	—	2	4	no	—
114	M₁₆×C₃₁	—	2	2	no	—
115	M₁₆×C₃₁	—	2	2	no	—
116	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
117	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	4	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	2	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	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	8	4	no	—
138	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²⋊C₄×C₃₁	—	2	2	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
149	#26081	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	1	no	—
153	Q₈×C₃₁	—	16	2	no	—
154	Q₈×C₃₁	C₄⋊Q₈	1	2	yes	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	1	no	—
158	D₄×C₃₁	C₄⋊Q₈	1	2	yes	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	8	2	no	—
170	C₂×C₄×C₃₁	—	8	2	no	—
171	C₂×C₄×C₃₁	—	16	2	no	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈×C₃₁	—	2	2	no	—
179	C₈×C₃₁	—	2	2	no	—
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
183	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
184	#3163	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#3084	C₂×C₃₁	1	2	yes	—
186	#2912	C₂×C₃₁	1	1	yes	characteristic
187	#2090	C₂×C₃₁	1	2	yes	complemented
188	#2060	C₂×C₃₁	1	2	yes	complemented
189	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1859	C₂×C₃₁	1	2	yes	—
192	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1815	C₂×C₃₁	1	1	yes	characteristic
194	#1764	C₂×C₃₁	1	1	yes	characteristic
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	#544	1	1	yes	characteristic
199	C₂²×C₃₁	D₄²	1	1	yes	characteristic
200	C₂²×C₃₁	#554	1	1	yes	characteristic
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
209	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
210	C₄×C₃₁	#547	1	1	yes	characteristic
211	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
212	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
213	#544	C₂²×C₃₁	1	2	yes	—
214	#520	C₂²×C₃₁	1	2	yes	—
215	#501	—	2	2	no	—
216	#499	—	2	1	no	—
217	#498	—	2	1	no	—
218	#479	—	2	2	no	—
219	#478	—	2	2	no	—
220	#476	—	2	2	no	—
221	#475	—	2	2	no	—
222	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
223	#428	—	2	2	no	—
224	#428	C₂²×C₃₁	1	2	yes	—
225	#428	C₂²×C₃₁	1	2	yes	—
226	#428	C₂²×C₃₁	1	2	yes	—
227	#427	—	2	1	no	—
228	#427	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#106	—	2	1	no	—
230	C₂×#106	C₂²×C₃₁	1	2	yes	—
231	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
233	#424	C₂²×C₃₁	1	2	yes	—
234	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
235	#417	C₂²×C₃₁	1	2	yes	—
236	#416	C₂²×C₃₁	1	1	yes	characteristic
237	#405	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
239	#384	C₂²×C₃₁	1	2	yes	—
240	#343	C₂²×C₃₁	1	2	yes	—
241	#343	C₂²×C₃₁	1	2	yes	—
242	#341	C₂²×C₃₁	1	1	yes	characteristic
243	#339	C₂²×C₃₁	1	1	yes	characteristic
244	#339	C₂²×C₃₁	1	2	yes	—
245	C₂×C₃₁	—	2	1	no	—
246	C₂×C₃₁	—	2	1	no	—
247	C₂×C₃₁	—	4	2	no	—
248	C₂×C₃₁	#2440	1	1	yes	characteristic
249	C₂×C₃₁	#3291	1	2	yes	—
250	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
251	#131	—	2	2	no	—
252	#131	Q₈×C₃₁	1	4	yes	complemented
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
255	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
256	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	#125	—	2	2	no	—
263	#125	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	—	4	2	no	—
268	#117	—	2	2	no	—
269	#117	C₂³×C₃₁	1	2	yes	—
270	#117	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	4	2	no	—
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	#108	—	2	2	no	—
277	#107	—	2	2	no	—
278	#107	—	2	2	no	—
279	#107	—	2	2	no	—
280	#107	D₄×C₃₁	1	2	yes	—
281	#106	—	2	1	no	—
282	#106	—	2	1	no	—
283	#106	—	2	2	no	—
284	#106	—	2	2	no	—
285	#106	—	2	1	no	—
286	#106	—	2	1	no	—
287	#106	—	2	2	no	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	#106	D₄×C₃₁	1	1	yes	characteristic
291	#106	D₄×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	2	no	—
294	#104	Q₈×C₃₁	1	4	yes	complemented
295	#103	—	2	2	no	—
296	#103	D₄×C₃₁	1	2	yes	—
297	#102	—	2	2	no	—
298	#102	D₄×C₃₁	1	2	yes	—
299	#97	—	2	1	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	2	2	no	—
302	C₃₁	#26081	1	1	yes	characteristic, direct product factor, Sylow
303	#41	—	2	1	no	—
304	#41	—	2	1	no	—
305	#41	C₂×Q₈×C₃₁	1	2	yes	—
306	#41	C₂×Q₈×C₃₁	1	2	yes	—
307	C₂×Q₈	—	4	2	no	—
308	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
317	M₁₆	—	2	1	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	4	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
323	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
334	C₂×C₈	#41×C₃₁	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	8	4	no	—
344	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
346	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	4	2	no	—
354	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
355	C₂³	—	2	1	no	—
356	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
357	Q₈	—	2	1	no	—
358	Q₈	—	16	2	no	—
359	Q₈	C₃₁×C₄⋊Q₈	1	2	yes	—
360	D₄	—	2	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	2	1	no	—
363	D₄	C₃₁×C₄⋊Q₈	1	2	yes	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	—	16	2	no	—
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	2	no	—
385	C₈	—	2	2	no	—
386	C₈	—	4	2	no	—
387	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
388	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	C₃₁×#544	1	1	yes	characteristic
393	C₂²	D₄²×C₃₁	1	1	yes	characteristic
394	C₂²	C₃₁×#554	1	1	yes	characteristic
395	C₄	—	2	4	no	—
396	C₄	—	2	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	2	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	2	no	—
402	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
403	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
404	C₄	C₃₁×#547	1	1	yes	characteristic
405	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	4	2	no	—
409	C₂	C₃₁×#2440	1	1	yes	characteristic
410	C₂	C₃₁×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3163	C₂	semidirect
4. C₃₁×#2090	C₂	semidirect
5. C₃₁×#2060	C₂	semidirect
6. C₃₁×#1951	C₂	semidirect
7. C₃₁×#1945	C₂	semidirect
9. C₃₁×#1844	C₂	semidirect
47. C₃₁×#131	Q₈	semidirect
89. C₃₁×#104	Q₈	semidirect
149. #26081	C₃₁	direct (N × H)
184. #3163	C₂×C₃₁	semidirect
187. #2090	C₂×C₃₁	semidirect
188. #2060	C₂×C₃₁	semidirect
189. #1951	C₂×C₃₁	semidirect
190. #1945	C₂×C₃₁	semidirect
192. #1844	C₂×C₃₁	semidirect
252. #131	Q₈×C₃₁	semidirect
294. #104	Q₈×C₃₁	semidirect
302. C₃₁	#26081	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×#3163	2
C₃₁×#3084	2
C₃₁×#2912	2
C₃₁×#2090	2
C₃₁×#2060	2
C₃₁×#1951	2
C₃₁×#1945	2
C₃₁×#1859	2
C₃₁×#1844	2
C₃₁×#1815	2
C₃₁×#1764	2
#26081	31

Subgroups — C31×#26081 #21482840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26081 #21482840