Subgroups — C₃₁×#23130 #21479889

Order 7936.

978 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2363	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×#102	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#121	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₃₁×#466	C₄	1	8	yes	complemented
16	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#390	—	2	2	no	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#388	—	2	2	no	—
26	C₃₁×#386	—	2	2	no	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×D₈×C₃₁	—	4	2	no	—
41	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₃₁	—	4	2	no	—
43	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#121	—	2	4	no	—
48	C₃₁×#121	C₂×C₄	1	8	yes	complemented
49	C₄×D₄×C₃₁	—	2	4	no	—
50	C₄×D₄×C₃₁	—	4	2	no	—
51	C₃₁×#117	—	2	4	no	—
52	C₃₁×#117	—	4	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
66	C₃₁×#107	C₂×C₄	1	4	yes	complemented
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#102	—	4	2	no	—
69	C₃₁×#102	C₂×C₄	1	8	yes	complemented
70	C₃₁×C₂²⋊C₈	—	2	2	no	—
71	C₄×C₈×C₃₁	—	4	2	no	—
72	C₃₁×#95	—	2	2	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₃₁×#95	D₄	1	1	yes	characteristic
75	C₃₁×#95	D₄	1	1	yes	characteristic
76	C₂⁴×C₃₁	—	2	2	no	—
77	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×D₄×C₃₁	—	2	8	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	8	2	no	—
85	C₂×D₄×C₃₁	—	8	2	no	—
86	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	4	4	no	—
93	C₂²×C₄×C₃₁	—	4	4	no	—
94	C₂²×C₄×C₃₁	—	8	2	no	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	#41	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
101	D₈×C₃₁	—	4	4	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	4	4	no	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	8	2	no	—
115	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
116	C₄⋊C₄×C₃₁	#41	1	4	yes	—
117	C₂²⋊C₄×C₃₁	—	2	8	no	—
118	C₂²⋊C₄×C₃₁	—	2	4	no	—
119	C₂²⋊C₄×C₃₁	—	4	4	no	—
120	C₂²⋊C₄×C₃₁	—	4	4	no	—
121	C₂²⋊C₄×C₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₃₁	—	8	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	4	4	no	—
126	C₄²×C₃₁	—	8	2	no	—
127	#23130	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	2	2	no	—
129	C₂³×C₃₁	—	2	2	no	—
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	2	1	no	—
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	4	2	no	—
136	C₂³×C₃₁	—	4	2	no	—
137	C₂³×C₃₁	—	4	2	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	C₄×D₄	1	2	yes	—
140	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	D₄×C₃₁	—	2	4	no	—
142	D₄×C₃₁	—	2	4	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	8	8	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	8	8	no	—
165	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₃₁	#135	1	4	yes	—
167	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
171	C₈×C₃₁	—	2	2	no	—
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
175	#2363	C₂×C₃₁	1	2	yes	complemented
176	#1934	C₂×C₃₁	1	2	yes	complemented
177	#1921	C₂×C₃₁	1	2	yes	complemented
178	#1879	C₂×C₃₁	1	2	yes	complemented
179	#1875	C₂×C₃₁	1	2	yes	complemented
180	C₄×#102	C₂×C₃₁	1	2	yes	complemented
181	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1735	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂²×C₃₁	—	2	2	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	—	2	2	no	—
186	C₂²×C₃₁	—	4	2	no	—
187	C₂²×C₃₁	—	4	2	no	—
188	C₂²×C₃₁	—	4	4	no	—
189	C₂²×C₃₁	—	4	4	no	—
190	C₂²×C₃₁	—	4	4	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	—	8	2	no	—
195	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	#546	1	2	yes	—
197	C₂²×C₃₁	C₂×#135	1	2	yes	—
198	C₂²×C₃₁	#576	1	1	yes	characteristic
199	C₂²×C₃₁	#576	1	1	yes	characteristic
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	4	4	no	—
203	C₄×C₃₁	—	4	4	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	8	2	no	—
206	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
207	C₄×C₃₁	#518	1	1	yes	characteristic
208	C₄×C₃₁	#553	1	2	yes	—
209	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
210	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#121	C₂²×C₃₁	1	2	yes	—
212	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
213	C₂×#117	C₂²×C₃₁	1	2	yes	—
214	#466	C₄×C₃₁	1	8	yes	complemented
215	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#102	C₂²×C₃₁	1	2	yes	—
217	C₂×#102	C₂²×C₃₁	1	2	yes	—
218	C₂×#102	C₂²×C₃₁	1	2	yes	—
219	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
221	#390	—	2	2	no	—
222	#390	C₂²×C₃₁	1	2	yes	—
223	#389	C₂²×C₃₁	1	2	yes	—
224	#388	—	2	2	no	—
225	#386	—	2	2	no	—
226	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
227	C₄×C₂²⋊C₄	—	2	2	no	—
228	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
229	#340	C₂²×C₃₁	1	2	yes	—
230	#340	C₂²×C₃₁	1	2	yes	—
231	#340	C₂²×C₃₁	1	2	yes	—
232	#340	C₂²×C₃₁	1	1	yes	characteristic
233	#340	C₂²×C₃₁	1	1	yes	characteristic
234	#336	C₂²×C₃₁	1	2	yes	—
235	C₂×C₃₁	—	4	2	no	—
236	C₂×C₃₁	—	8	4	no	—
237	C₂×C₃₁	#2364	1	1	yes	characteristic
238	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
239	C₂×C₃₁	#2948	1	1	yes	characteristic
240	C₂×C₃₁	#3297	1	2	yes	—
241	C₂×C₃₁	#3305	1	2	yes	—
242	C₂²×D₄	—	2	2	no	—
243	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
244	C₂³×C₄	—	2	2	no	—
245	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
246	C₂×D₈	—	4	2	no	—
247	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
248	C₂²×C₈	—	4	2	no	—
249	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	#121	—	2	4	no	—
254	#121	C₂×C₄×C₃₁	1	8	yes	complemented
255	C₄×D₄	—	2	4	no	—
256	C₄×D₄	—	4	2	no	—
257	#117	—	2	4	no	—
258	#117	—	4	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
262	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
272	#107	C₂×C₄×C₃₁	1	4	yes	complemented
273	#102	—	2	4	no	—
274	#102	—	4	2	no	—
275	#102	C₂×C₄×C₃₁	1	8	yes	complemented
276	C₂²⋊C₈	—	2	2	no	—
277	C₄×C₈	—	4	2	no	—
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	#95	D₄×C₃₁	1	1	yes	characteristic
281	#95	D₄×C₃₁	1	1	yes	characteristic
282	C₃₁	#23130	1	1	yes	characteristic, direct product factor, Sylow
283	C₂⁴	—	2	2	no	—
284	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₂×D₄	—	2	8	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	8	2	no	—
292	C₂×D₄	—	8	2	no	—
293	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	4	4	no	—
300	C₂²×C₄	—	4	4	no	—
301	C₂²×C₄	—	8	2	no	—
302	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
306	C₂²×C₄	#41×C₃₁	1	2	yes	—
307	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
308	D₈	—	4	4	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	4	4	no	—
315	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	8	2	no	—
322	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
323	C₄⋊C₄	#41×C₃₁	1	4	yes	—
324	C₂²⋊C₄	—	2	8	no	—
325	C₂²⋊C₄	—	2	4	no	—
326	C₂²⋊C₄	—	4	4	no	—
327	C₂²⋊C₄	—	4	4	no	—
328	C₂²⋊C₄	—	4	4	no	—
329	C₂²⋊C₄	—	8	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	4	4	no	—
333	C₄²	—	8	2	no	—
334	C₂³	—	2	2	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	4	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	C₄×D₄×C₃₁	1	2	yes	—
346	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
347	D₄	—	2	4	no	—
348	D₄	—	2	4	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	D₄	—	8	8	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	8	no	—
371	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
372	C₂×C₄	C₃₁×#135	1	4	yes	—
373	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
375	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
377	C₈	—	2	2	no	—
378	C₈	—	2	2	no	—
379	C₈	—	4	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	4	no	—
386	C₂²	—	4	4	no	—
387	C₂²	—	4	4	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	2	no	—
390	C₂²	—	8	2	no	—
391	C₂²	—	8	2	no	—
392	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
393	C₂²	C₃₁×#546	1	2	yes	—
394	C₂²	C₂×C₃₁×#135	1	2	yes	—
395	C₂²	C₃₁×#576	1	1	yes	characteristic
396	C₂²	C₃₁×#576	1	1	yes	characteristic
397	C₄	—	2	2	no	—
398	C₄	—	2	4	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	4	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
404	C₄	C₃₁×#518	1	1	yes	characteristic
405	C₄	C₃₁×#553	1	2	yes	—
406	C₂	—	4	2	no	—
407	C₂	—	8	4	no	—
408	C₂	C₃₁×#2364	1	1	yes	characteristic
409	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
410	C₂	C₃₁×#2948	1	1	yes	characteristic
411	C₂	C₃₁×#3297	1	2	yes	—
412	C₂	C₃₁×#3305	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2363	C₂	semidirect
3. C₃₁×#1934	C₂	semidirect
4. C₃₁×#1921	C₂	semidirect
5. C₃₁×#1879	C₂	semidirect
6. C₃₁×#1875	C₂	semidirect
7. C₄×C₃₁×#102	C₂	semidirect
8. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#1735	C₂	semidirect
15. C₃₁×#466	C₄	semidirect
43. C₂²×C₈×C₃₁	C₂×C₄	semidirect
48. C₃₁×#121	C₂×C₄	semidirect
66. C₃₁×#107	C₂×C₄	semidirect
69. C₃₁×#102	C₂×C₄	semidirect
127. #23130	C₃₁	direct (N × H)
175. #2363	C₂×C₃₁	semidirect
176. #1934	C₂×C₃₁	semidirect
177. #1921	C₂×C₃₁	semidirect
178. #1879	C₂×C₃₁	semidirect
179. #1875	C₂×C₃₁	semidirect
180. C₄×#102	C₂×C₃₁	semidirect
181. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
182. #1735	C₂×C₃₁	semidirect
214. #466	C₄×C₃₁	semidirect
249. C₂²×C₈	C₂×C₄×C₃₁	semidirect
254. #121	C₂×C₄×C₃₁	semidirect
272. #107	C₂×C₄×C₃₁	semidirect
275. #102	C₂×C₄×C₃₁	semidirect
282. C₃₁	#23130	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#466	2
C₃₁×#2363	2
C₃₁×#1934	2
C₃₁×#1921	2
C₃₁×#1879	2
C₃₁×#1875	2
C₄×C₃₁×#102	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1735	2
#23130	31

Subgroups — C31×#23130 #21479889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23130 #21479889