Subgroups — C₃₁×#26083 #21482842

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3083	C₂	1	2	yes	maximal
3	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2089	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2060	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1952	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×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1816	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×#500	—	2	1	no	—
17	C₃₁×#499	—	2	1	no	—
18	C₃₁×#479	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#477	—	2	2	no	—
21	C₃₁×#474	—	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×#107	—	2	1	no	—
30	C₂×C₃₁×#107	C₂²	1	2	yes	—
31	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#107	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	1	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	1	no	—
77	C₃₁×#107	—	2	1	no	—
78	C₃₁×#107	—	2	2	no	—
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#107	—	2	2	no	—
81	C₃₁×#107	D₄	1	1	yes	characteristic
82	C₃₁×#107	D₄	1	1	yes	characteristic
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#106	—	2	2	no	—
86	C₃₁×#106	D₄	1	2	yes	—
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	1	yes	characteristic
92	C₃₁×#103	D₄	1	1	yes	characteristic
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	D₄	1	1	yes	characteristic
95	C₃₁×#102	D₄	1	1	yes	characteristic
96	C₃₁×#97	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	1	no	—
100	#41×C₃₁	—	2	1	no	—
101	#41×C₃₁	C₂×Q₈	1	2	yes	—
102	#41×C₃₁	C₂×Q₈	1	2	yes	—
103	C₂×Q₈×C₃₁	—	4	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	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₃₁	—	2	1	no	—
114	M₁₆×C₃₁	—	2	1	no	—
115	M₁₆×C₃₁	—	2	4	no	—
116	M₁₆×C₃₁	—	2	2	no	—
117	M₁₆×C₃₁	—	2	2	no	—
118	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
119	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	—	8	4	no	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²⋊C₄×C₃₁	—	2	2	no	—
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₂²⋊C₄×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	4	2	no	—
152	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
153	#26083	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
156	Q₈×C₃₁	—	2	1	no	—
157	Q₈×C₃₁	—	16	2	no	—
158	Q₈×C₃₁	C₄⋊Q₈	1	2	yes	—
159	D₄×C₃₁	—	2	2	no	—
160	D₄×C₃₁	—	2	2	no	—
161	D₄×C₃₁	—	2	1	no	—
162	D₄×C₃₁	C₄⋊Q₈	1	2	yes	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	8	2	no	—
177	C₂×C₄×C₃₁	—	16	2	no	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₃₁	—	2	2	no	—
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	2	1	no	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
190	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
191	#3164	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#3083	C₂×C₃₁	1	2	yes	—
193	#2913	C₂×C₃₁	1	1	yes	characteristic
194	#2089	C₂×C₃₁	1	2	yes	complemented
195	#2060	C₂×C₃₁	1	2	yes	complemented
196	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1859	C₂×C₃₁	1	2	yes	—
199	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1816	C₂×C₃₁	1	1	yes	characteristic
201	#1764	C₂×C₃₁	1	1	yes	characteristic
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	#544	1	1	yes	characteristic
208	C₂²×C₃₁	D₄²	1	1	yes	characteristic
209	C₂²×C₃₁	#554	1	1	yes	characteristic
210	C₄×C₃₁	—	2	4	no	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	4	1	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
219	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
220	C₄×C₃₁	#547	1	1	yes	characteristic
221	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
222	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
223	#544	C₂²×C₃₁	1	2	yes	—
224	#520	C₂²×C₃₁	1	2	yes	—
225	#501	—	2	2	no	—
226	#500	—	2	1	no	—
227	#499	—	2	1	no	—
228	#479	—	2	2	no	—
229	#478	—	2	2	no	—
230	#477	—	2	2	no	—
231	#474	—	2	2	no	—
232	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
233	#428	—	2	2	no	—
234	#428	C₂²×C₃₁	1	2	yes	—
235	#428	C₂²×C₃₁	1	2	yes	—
236	#428	C₂²×C₃₁	1	2	yes	—
237	#427	—	2	1	no	—
238	#427	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#107	—	2	1	no	—
240	C₂×#107	C₂²×C₃₁	1	2	yes	—
241	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
243	#424	C₂²×C₃₁	1	2	yes	—
244	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
245	#417	C₂²×C₃₁	1	2	yes	—
246	#416	C₂²×C₃₁	1	1	yes	characteristic
247	#405	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
249	#384	C₂²×C₃₁	1	2	yes	—
250	#343	C₂²×C₃₁	1	2	yes	—
251	#343	C₂²×C₃₁	1	2	yes	—
252	#341	C₂²×C₃₁	1	1	yes	characteristic
253	#339	C₂²×C₃₁	1	1	yes	characteristic
254	#339	C₂²×C₃₁	1	2	yes	—
255	C₂×C₃₁	—	2	1	no	—
256	C₂×C₃₁	—	2	1	no	—
257	C₂×C₃₁	—	4	2	no	—
258	C₂×C₃₁	#2440	1	1	yes	characteristic
259	C₂×C₃₁	#3292	1	1	yes	characteristic
260	C₂×C₃₁	#3293	1	1	yes	characteristic
261	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
262	#131	—	2	2	no	—
263	#131	Q₈×C₃₁	1	4	yes	complemented
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	2	no	—
273	#125	—	2	2	no	—
274	#125	—	2	2	no	—
275	C₂²⋊Q₈	—	4	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	—	4	2	no	—
279	#117	—	2	2	no	—
280	#117	C₂³×C₃₁	1	2	yes	—
281	#117	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	4	2	no	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
287	#108	—	2	2	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	2	no	—
291	#107	—	2	2	no	—
292	#107	—	2	1	no	—
293	#107	—	2	1	no	—
294	#107	—	2	2	no	—
295	#107	—	2	2	no	—
296	#107	—	2	2	no	—
297	#107	D₄×C₃₁	1	1	yes	characteristic
298	#107	D₄×C₃₁	1	1	yes	characteristic
299	#106	—	2	2	no	—
300	#106	—	2	2	no	—
301	#106	—	2	2	no	—
302	#106	D₄×C₃₁	1	2	yes	—
303	C₄⋊C₈	—	2	2	no	—
304	C₄⋊C₈	—	2	2	no	—
305	#104	Q₈×C₃₁	1	4	yes	complemented
306	#103	—	2	2	no	—
307	#103	D₄×C₃₁	1	1	yes	characteristic
308	#103	D₄×C₃₁	1	1	yes	characteristic
309	#102	—	2	2	no	—
310	#102	D₄×C₃₁	1	1	yes	characteristic
311	#102	D₄×C₃₁	1	1	yes	characteristic
312	#97	—	2	1	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	—	2	2	no	—
315	C₃₁	#26083	1	1	yes	characteristic, direct product factor, Sylow
316	#41	—	2	1	no	—
317	#41	—	2	1	no	—
318	#41	C₂×Q₈×C₃₁	1	2	yes	—
319	#41	C₂×Q₈×C₃₁	1	2	yes	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
330	M₁₆	—	2	1	no	—
331	M₁₆	—	2	1	no	—
332	M₁₆	—	2	4	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
336	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	4	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
348	C₂×C₈	#41×C₃₁	1	2	yes	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	8	4	no	—
358	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
361	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	1	no	—
368	C₄²	—	4	2	no	—
369	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
372	Q₈	—	2	1	no	—
373	Q₈	—	16	2	no	—
374	Q₈	C₃₁×C₄⋊Q₈	1	2	yes	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	—	2	1	no	—
378	D₄	C₃₁×C₄⋊Q₈	1	2	yes	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	16	2	no	—
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	1	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	2	no	—
405	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
406	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	C₃₁×#544	1	1	yes	characteristic
413	C₂²	D₄²×C₃₁	1	1	yes	characteristic
414	C₂²	C₃₁×#554	1	1	yes	characteristic
415	C₄	—	2	4	no	—
416	C₄	—	2	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	1	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
424	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
425	C₄	C₃₁×#547	1	1	yes	characteristic
426	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	4	2	no	—
430	C₂	C₃₁×#2440	1	1	yes	characteristic
431	C₂	C₃₁×#3292	1	1	yes	characteristic
432	C₂	C₃₁×#3293	1	1	yes	characteristic

	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×#3164	C₂	semidirect
4. C₃₁×#2089	C₂	semidirect
5. C₃₁×#2060	C₂	semidirect
6. C₃₁×#1952	C₂	semidirect
7. C₃₁×#1945	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
47. C₃₁×#131	Q₈	semidirect
89. C₃₁×#104	Q₈	semidirect
153. #26083	C₃₁	direct (N × H)
191. #3164	C₂×C₃₁	semidirect
194. #2089	C₂×C₃₁	semidirect
195. #2060	C₂×C₃₁	semidirect
196. #1952	C₂×C₃₁	semidirect
197. #1945	C₂×C₃₁	semidirect
199. #1845	C₂×C₃₁	semidirect
263. #131	Q₈×C₃₁	semidirect
305. #104	Q₈×C₃₁	semidirect
315. C₃₁	#26083	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×#3164	2
C₃₁×#3083	2
C₃₁×#2913	2
C₃₁×#2089	2
C₃₁×#2060	2
C₃₁×#1952	2
C₃₁×#1945	2
C₃₁×#1859	2
C₃₁×#1845	2
C₃₁×#1816	2
C₃₁×#1764	2
#26083	31

Subgroups — C31×#26083 #21482842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26083 #21482842