Subgroups — C₃₁×#26082 #21482841

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 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×#3084	C₂	1	2	yes	maximal
3	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2090	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2061	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×#1860	C₂	1	2	yes	maximal
9	C₃₁×#1846	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×#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×#107	—	2	1	no	—
30	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#106	C₂²	1	2	yes	—
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×#382	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	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#108	—	2	2	no	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#107	D₄	1	1	yes	characteristic
79	C₃₁×#107	D₄	1	1	yes	characteristic
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#106	D₄	1	2	yes	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#104	Q₈	1	4	yes	complemented
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	D₄	1	1	yes	characteristic
91	C₃₁×#103	D₄	1	1	yes	characteristic
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	D₄	1	1	yes	characteristic
94	C₃₁×#102	D₄	1	1	yes	characteristic
95	C₃₁×#97	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₃₁×#95	—	4	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	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	4	2	no	—
151	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
152	#26082	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	1	no	—
156	Q₈×C₃₁	—	16	2	no	—
157	Q₈×C₃₁	C₄⋊Q₈	1	2	yes	—
158	D₄×C₃₁	—	2	2	no	—
159	D₄×C₃₁	—	2	2	no	—
160	D₄×C₃₁	—	2	1	no	—
161	D₄×C₃₁	C₄⋊Q₈	1	2	yes	—
162	C₂×C₄×C₃₁	—	2	1	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	1	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	16	2	no	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	2	1	no	—
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
189	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
190	#3164	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#3084	C₂×C₃₁	1	2	yes	—
192	#2913	C₂×C₃₁	1	1	yes	characteristic
193	#2090	C₂×C₃₁	1	2	yes	complemented
194	#2061	C₂×C₃₁	1	2	yes	complemented
195	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1860	C₂×C₃₁	1	2	yes	—
198	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1816	C₂×C₃₁	1	1	yes	characteristic
200	#1764	C₂×C₃₁	1	1	yes	characteristic
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	4	1	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	#544	1	1	yes	characteristic
207	C₂²×C₃₁	D₄²	1	1	yes	characteristic
208	C₂²×C₃₁	#554	1	1	yes	characteristic
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
219	C₄×C₃₁	#547	1	1	yes	characteristic
220	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
221	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
222	#544	C₂²×C₃₁	1	2	yes	—
223	#520	C₂²×C₃₁	1	2	yes	—
224	#501	—	2	2	no	—
225	#500	—	2	1	no	—
226	#499	—	2	1	no	—
227	#479	—	2	2	no	—
228	#478	—	2	2	no	—
229	#476	—	2	2	no	—
230	#475	—	2	2	no	—
231	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
232	#428	—	2	2	no	—
233	#428	C₂²×C₃₁	1	2	yes	—
234	#428	C₂²×C₃₁	1	2	yes	—
235	#428	C₂²×C₃₁	1	2	yes	—
236	#427	—	2	1	no	—
237	#427	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#107	—	2	1	no	—
239	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#106	C₂²×C₃₁	1	2	yes	—
242	#424	C₂²×C₃₁	1	2	yes	—
243	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
244	#417	C₂²×C₃₁	1	2	yes	—
245	#416	C₂²×C₃₁	1	1	yes	characteristic
246	#405	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	#382	C₂²×C₃₁	1	2	yes	—
249	#343	C₂²×C₃₁	1	2	yes	—
250	#343	C₂²×C₃₁	1	2	yes	—
251	#341	C₂²×C₃₁	1	1	yes	characteristic
252	#339	C₂²×C₃₁	1	1	yes	characteristic
253	#339	C₂²×C₃₁	1	2	yes	—
254	C₂×C₃₁	—	2	1	no	—
255	C₂×C₃₁	—	2	1	no	—
256	C₂×C₃₁	—	4	2	no	—
257	C₂×C₃₁	#2440	1	1	yes	characteristic
258	C₂×C₃₁	#3290	1	1	yes	characteristic
259	C₂×C₃₁	#3290	1	1	yes	characteristic
260	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
261	#131	—	2	2	no	—
262	#131	Q₈×C₃₁	1	4	yes	complemented
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
265	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₄⋊Q₈	—	2	2	no	—
271	C₄⋊Q₈	—	2	2	no	—
272	#125	—	2	2	no	—
273	#125	—	2	2	no	—
274	C₂²⋊Q₈	—	4	2	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×D₄	—	2	2	no	—
277	C₄×D₄	—	4	2	no	—
278	#117	—	2	2	no	—
279	#117	C₂³×C₃₁	1	2	yes	—
280	#117	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	#108	—	2	2	no	—
286	#107	—	2	1	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	2	no	—
291	#107	—	2	2	no	—
292	#107	—	2	2	no	—
293	#107	D₄×C₃₁	1	1	yes	characteristic
294	#107	D₄×C₃₁	1	1	yes	characteristic
295	#106	—	2	2	no	—
296	#106	—	2	2	no	—
297	#106	—	2	2	no	—
298	#106	—	2	2	no	—
299	#106	—	2	2	no	—
300	#106	D₄×C₃₁	1	2	yes	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	2	no	—
303	#104	Q₈×C₃₁	1	4	yes	complemented
304	#103	—	2	2	no	—
305	#103	D₄×C₃₁	1	1	yes	characteristic
306	#103	D₄×C₃₁	1	1	yes	characteristic
307	#102	—	2	2	no	—
308	#102	D₄×C₃₁	1	1	yes	characteristic
309	#102	D₄×C₃₁	1	1	yes	characteristic
310	#97	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	2	no	—
313	#95	—	4	2	no	—
314	C₃₁	#26082	1	1	yes	characteristic, direct product factor, Sylow
315	#41	—	2	1	no	—
316	#41	—	2	1	no	—
317	#41	C₂×Q₈×C₃₁	1	2	yes	—
318	#41	C₂×Q₈×C₃₁	1	2	yes	—
319	C₂×Q₈	—	4	2	no	—
320	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
329	M₁₆	—	2	1	no	—
330	M₁₆	—	2	1	no	—
331	M₁₆	—	2	4	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
335	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	4	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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×Q₈×C₃₁	1	2	yes	—
347	C₂×C₈	#41×C₃₁	1	2	yes	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
359	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	4	2	no	—
367	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
368	C₂³	—	2	1	no	—
369	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
370	Q₈	—	2	1	no	—
371	Q₈	—	16	2	no	—
372	Q₈	C₃₁×C₄⋊Q₈	1	2	yes	—
373	D₄	—	2	2	no	—
374	D₄	—	2	2	no	—
375	D₄	—	2	1	no	—
376	D₄	C₃₁×C₄⋊Q₈	1	2	yes	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
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	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	16	2	no	—
392	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	2	2	no	—
402	C₈	—	4	2	no	—
403	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
404	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	C₃₁×#544	1	1	yes	characteristic
411	C₂²	D₄²×C₃₁	1	1	yes	characteristic
412	C₂²	C₃₁×#554	1	1	yes	characteristic
413	C₄	—	2	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	2	no	—
421	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
422	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
423	C₄	C₃₁×#547	1	1	yes	characteristic
424	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
425	C₂	—	2	1	no	—
426	C₂	—	2	1	no	—
427	C₂	—	4	2	no	—
428	C₂	C₃₁×#2440	1	1	yes	characteristic
429	C₂	C₃₁×#3290	1	1	yes	characteristic
430	C₂	C₃₁×#3290	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×#2090	C₂	semidirect
5. C₃₁×#2061	C₂	semidirect
6. C₃₁×#1952	C₂	semidirect
7. C₃₁×#1945	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
47. C₃₁×#131	Q₈	semidirect
88. C₃₁×#104	Q₈	semidirect
152. #26082	C₃₁	direct (N × H)
190. #3164	C₂×C₃₁	semidirect
193. #2090	C₂×C₃₁	semidirect
194. #2061	C₂×C₃₁	semidirect
195. #1952	C₂×C₃₁	semidirect
196. #1945	C₂×C₃₁	semidirect
198. #1846	C₂×C₃₁	semidirect
262. #131	Q₈×C₃₁	semidirect
303. #104	Q₈×C₃₁	semidirect
314. C₃₁	#26082	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×#3084	2
C₃₁×#2913	2
C₃₁×#2090	2
C₃₁×#2061	2
C₃₁×#1952	2
C₃₁×#1945	2
C₃₁×#1860	2
C₃₁×#1846	2
C₃₁×#1816	2
C₃₁×#1764	2
#26082	31

Subgroups — C31×#26082 #21482841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26082 #21482841