Subgroups — C₃₁×#23730 #21480489

Order 7936.

1126 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#416	C₂	1	2	yes	maximal
3	C₃₁×#2297	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2097	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2096	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2094	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2093	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#427	—	2	8	no	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₃₁×#416	—	2	8	no	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₃₁×#391	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#390	C₂²	1	1	yes	characteristic
30	C₃₁×#390	C₂²	1	1	yes	characteristic
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#382	C₂²	1	2	yes	—
35	C₃₁×#380	—	2	2	no	—
36	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	4	no	—
43	C₂×#41×C₃₁	—	2	4	no	—
44	C₂×#41×C₃₁	Q₈	1	4	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	D₄	1	2	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	—	2	4	no	—
52	C₂²×C₈×C₃₁	—	2	4	no	—
53	C₂²×C₈×C₃₁	D₄	1	8	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#117	—	2	4	no	—
57	C₃₁×#117	—	2	4	no	—
58	C₃₁×#117	—	2	4	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	4	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	D₄	1	2	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	2	4	no	—
76	C₃₁×#107	—	2	4	no	—
77	C₃₁×#106	—	2	4	no	—
78	C₃₁×#106	—	2	4	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	4	no	—
86	C₂⁴×C₃₁	—	2	1	no	—
87	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
88	#41×C₃₁	—	2	16	no	—
89	#41×C₃₁	—	4	8	no	—
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
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	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
117	C₂²×C₄×C₃₁	#41	1	4	yes	—
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₃₁	—	2	16	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
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	4	no	—
134	C₄⋊C₄×C₃₁	—	4	4	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₄²×C₃₁	—	2	4	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	4	4	no	—
144	#23730	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
153	C₂³×C₃₁	#123	1	2	yes	—
154	C₂³×C₃₁	#127	1	2	yes	—
155	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₃₁	—	2	4	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	—	4	8	no	—
160	D₄×C₃₁	—	4	2	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	8	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	8	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	4	no	—
179	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
180	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
181	C₂×C₄×C₃₁	#123	1	8	yes	—
182	C₂×C₄×C₃₁	#123	1	2	yes	—
183	C₂×C₄×C₃₁	#127	1	2	yes	—
184	C₂×C₄×C₃₁	#135	1	4	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₈×C₃₁	—	2	4	no	—
190	C₈×C₃₁	—	2	4	no	—
191	C₂×#427	C₂×C₃₁	1	2	yes	complemented
192	C₂×#416	C₂×C₃₁	1	2	yes	—
193	#2297	C₂×C₃₁	1	1	yes	characteristic
194	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2097	C₂×C₃₁	1	2	yes	complemented
196	#2096	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2094	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#2093	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#2092	C₂×C₃₁	1	2	yes	complemented
201	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	1	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	#399	1	4	yes	—
212	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
214	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×#135	1	2	yes	—
216	C₂²×C₃₁	#575	1	2	yes	—
217	C₄×C₃₁	—	2	4	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	8	1	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#399	1	4	yes	—
225	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
226	C₄×C₃₁	C₂×#123	1	2	yes	—
227	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
228	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
229	C₂×#117	C₂²×C₃₁	1	2	yes	—
230	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
231	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
232	#427	—	2	8	no	—
233	C₂×#107	C₂²×C₃₁	1	2	yes	—
234	C₂×#107	C₂²×C₃₁	1	2	yes	—
235	C₂×#106	C₂²×C₃₁	1	2	yes	—
236	C₂×#106	C₂²×C₃₁	1	2	yes	—
237	#416	—	2	8	no	—
238	C₂×#103	C₂²×C₃₁	1	2	yes	—
239	C₂×#103	C₂²×C₃₁	1	2	yes	—
240	C₂×#102	C₂²×C₃₁	1	2	yes	—
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	#391	C₂²×C₃₁	1	1	yes	characteristic
243	#391	C₂²×C₃₁	1	1	yes	characteristic
244	#390	C₂²×C₃₁	1	1	yes	characteristic
245	#390	C₂²×C₃₁	1	1	yes	characteristic
246	#386	—	2	2	no	—
247	#384	C₂²×C₃₁	1	1	yes	characteristic
248	#384	C₂²×C₃₁	1	1	yes	characteristic
249	#382	C₂²×C₃₁	1	2	yes	—
250	#380	—	2	2	no	—
251	C₄×C₄⋊C₄	—	2	2	no	—
252	C₄×C₂²⋊C₄	—	2	2	no	—
253	#336	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	1	yes	characteristic
255	#336	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₃₁	—	2	4	no	—
257	C₂×C₃₁	—	8	2	no	—
258	C₂×C₃₁	C₂×#399	1	1	yes	characteristic
259	C₂×C₃₁	#3076	1	2	yes	—
260	C₂×C₃₁	#3091	1	2	yes	—
261	C₂×C₃₁	#3147	1	1	yes	characteristic
262	C₂×C₃₁	#3150	1	1	yes	characteristic
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	Q₈×C₃₁	1	4	yes	—
267	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
270	C₂³×C₄	—	2	1	no	—
271	C₂³×C₄	D₄×C₃₁	1	2	yes	—
272	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	—	2	4	no	—
274	C₂²×C₈	—	2	4	no	—
275	C₂²×C₈	D₄×C₃₁	1	8	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
278	#117	—	2	4	no	—
279	#117	—	2	4	no	—
280	#117	—	2	4	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₄²	—	2	4	no	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	D₄×C₃₁	1	2	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	—	2	4	no	—
298	#107	—	2	4	no	—
299	#106	—	2	4	no	—
300	#106	—	2	4	no	—
301	#103	—	2	4	no	—
302	#103	—	2	4	no	—
303	#102	—	2	4	no	—
304	#102	—	2	4	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	4	no	—
308	C₃₁	#23730	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	—	2	1	no	—
310	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
311	#41	—	2	16	no	—
312	#41	—	4	8	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
339	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
340	C₂²×C₄	#41×C₃₁	1	4	yes	—
341	C₂²×C₄	#41×C₃₁	1	2	yes	—
342	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₂×C₈	—	2	16	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
349	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	4	no	—
357	C₄⋊C₄	—	4	4	no	—
358	C₂²⋊C₄	—	4	4	no	—
359	C₂²⋊C₄	—	4	4	no	—
360	C₂²⋊C₄	—	4	2	no	—
361	C₂²⋊C₄	—	4	4	no	—
362	C₄²	—	2	4	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	4	2	no	—
366	C₄²	—	4	4	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
375	C₂³	C₃₁×#123	1	2	yes	—
376	C₂³	C₃₁×#127	1	2	yes	—
377	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	Q₈	—	2	4	no	—
379	Q₈	—	4	2	no	—
380	D₄	—	2	4	no	—
381	D₄	—	4	8	no	—
382	D₄	—	4	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	4	8	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	8	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
402	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
403	C₂×C₄	C₃₁×#123	1	2	yes	—
404	C₂×C₄	C₃₁×#123	1	8	yes	—
405	C₂×C₄	C₃₁×#127	1	2	yes	—
406	C₂×C₄	C₃₁×#135	1	4	yes	—
407	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
408	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
410	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
411	C₈	—	2	4	no	—
412	C₈	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	2	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	1	no	—
421	C₂²	—	8	1	no	—
422	C₂²	C₃₁×#399	1	4	yes	—
423	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
424	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
425	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
426	C₂²	C₂×C₃₁×#135	1	2	yes	—
427	C₂²	C₃₁×#575	1	2	yes	—
428	C₄	—	2	4	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	2	no	—
433	C₄	—	8	1	no	—
434	C₄	—	8	2	no	—
435	C₄	C₃₁×#399	1	4	yes	—
436	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
437	C₄	C₂×C₃₁×#123	1	2	yes	—
438	C₂	—	2	4	no	—
439	C₂	—	8	2	no	—
440	C₂	C₂×C₃₁×#399	1	1	yes	characteristic
441	C₂	C₃₁×#3076	1	2	yes	—
442	C₂	C₃₁×#3091	1	2	yes	—
443	C₂	C₃₁×#3147	1	1	yes	characteristic
444	C₂	C₃₁×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2285	C₂	semidirect
5. C₃₁×#2097	C₂	semidirect
6. C₃₁×#2096	C₂	semidirect
7. C₃₁×#2095	C₂	semidirect
8. C₃₁×#2094	C₂	semidirect
9. C₃₁×#2093	C₂	semidirect
10. C₃₁×#2092	C₂	semidirect
11. C₂×C₃₁×#336	C₂	semidirect
144. #23730	C₃₁	direct (N × H)
191. C₂×#427	C₂×C₃₁	semidirect
194. #2285	C₂×C₃₁	semidirect
195. #2097	C₂×C₃₁	semidirect
196. #2096	C₂×C₃₁	semidirect
197. #2095	C₂×C₃₁	semidirect
198. #2094	C₂×C₃₁	semidirect
199. #2093	C₂×C₃₁	semidirect
200. #2092	C₂×C₃₁	semidirect
201. C₂×#336	C₂×C₃₁	semidirect
308. C₃₁	#23730	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×#427	2
C₂×C₃₁×#416	2
C₃₁×#2297	2
C₃₁×#2285	2
C₃₁×#2097	2
C₃₁×#2096	2
C₃₁×#2095	2
C₃₁×#2094	2
C₃₁×#2093	2
C₃₁×#2092	2
C₂×C₃₁×#336	2
#23730	31

Subgroups — C31×#23730 #21480489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23730 #21480489