Subgroups — C₃₁×#21872 #21478631

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1874	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1804	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1802	C₂	1	2	yes	maximal, complemented
12	C₃₁×#540	—	2	4	no	—
13	C₃₁×#540	C₄	1	8	yes	complemented
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#123	—	2	2	no	—
17	C₂×C₃₁×#121	—	2	2	no	—
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#433	—	2	4	no	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#389	—	2	2	no	—
33	C₃₁×#385	—	2	2	no	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#382	C₂²	1	1	yes	characteristic
37	C₃₁×#382	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	—	2	2	no	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	C₂×C₄	1	4	yes	complemented
52	C₃₁×#125	—	2	2	no	—
53	C₃₁×#125	C₂×C₄	1	4	yes	complemented
54	C₃₁×#123	—	2	16	no	—
55	C₃₁×#121	—	2	4	no	—
56	C₃₁×#121	—	2	16	no	—
57	C₃₁×#121	C₂×C₄	1	8	yes	—
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
60	C₃₁×#117	—	4	2	no	—
61	C₃₁×#117	C₂×C₄	1	4	yes	—
62	C₃₁×#117	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	—	2	4	no	—
82	C₃₁×#102	—	4	2	no	—
83	C₃₁×#102	—	4	2	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#102	—	4	2	no	—
86	C₃₁×C₂²⋊C₈	—	2	4	no	—
87	C₃₁×#97	—	2	2	no	—
88	C₄×C₈×C₃₁	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₂⁴×C₃₁	—	2	2	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×D₄×C₃₁	—	2	8	no	—
94	C₂×D₄×C₃₁	—	2	8	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	8	2	no	—
101	C₂×D₄×C₃₁	—	8	2	no	—
102	C₂×D₄×C₃₁	—	8	2	no	—
103	C₂×D₄×C₃₁	—	8	2	no	—
104	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	4	2	no	—
114	C₂²×C₄×C₃₁	—	8	2	no	—
115	C₂²×C₄×C₃₁	—	8	2	no	—
116	C₂²×C₄×C₃₁	—	8	2	no	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	8	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	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	8	2	no	—
137	C₄⋊C₄×C₃₁	—	8	2	no	—
138	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²⋊C₄×C₃₁	—	2	8	no	—
141	C₂²⋊C₄×C₃₁	—	2	8	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
144	C₄²×C₃₁	—	4	2	no	—
145	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
146	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄²×C₃₁	C₂×D₄	1	2	yes	—
148	C₄²×C₃₁	C₂×D₄	1	2	yes	—
149	#21872	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	8	no	—
168	D₄×C₃₁	—	8	8	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	8	8	no	—
189	C₂×C₄×C₃₁	—	8	4	no	—
190	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
191	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
192	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
196	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
197	C₈×C₃₁	—	4	2	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₂×#540	C₂×C₃₁	1	1	yes	characteristic
200	C₂×#433	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1991	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1875	C₂×C₃₁	1	2	yes	complemented
207	#1874	C₂×C₃₁	1	2	yes	complemented
208	#1804	C₂×C₃₁	1	2	yes	complemented
209	#1802	C₂×C₃₁	1	2	yes	complemented
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	4	no	—
216	C₂²×C₃₁	—	4	4	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²×C₃₁	#535	1	1	yes	characteristic
224	C₂²×C₃₁	#536	1	1	yes	characteristic
225	C₂²×C₃₁	#576	1	2	yes	—
226	C₂²×C₃₁	#577	1	2	yes	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	8	1	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	#520	1	2	yes	—
236	C₄×C₃₁	#537	1	2	yes	—
237	#540	—	2	4	no	—
238	#540	C₄×C₃₁	1	8	yes	complemented
239	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#123	—	2	2	no	—
242	C₂×#121	—	2	2	no	—
243	C₂×#121	C₂²×C₃₁	1	2	yes	—
244	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
245	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	#433	—	2	4	no	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×#102	C₂²×C₃₁	1	2	yes	—
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₃₁	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
255	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#389	—	2	2	no	—
258	#385	—	2	2	no	—
259	#384	C₂²×C₃₁	1	1	yes	characteristic
260	#384	C₂²×C₃₁	1	1	yes	characteristic
261	#382	C₂²×C₃₁	1	1	yes	characteristic
262	#382	C₂²×C₃₁	1	1	yes	characteristic
263	#340	C₂²×C₃₁	1	2	yes	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	—	8	4	no	—
269	C₂×C₃₁	#2311	1	1	yes	characteristic
270	C₂×C₃₁	#2897	1	2	yes	—
271	C₂×C₃₁	#3220	1	1	yes	characteristic
272	C₂×C₃₁	#3221	1	1	yes	characteristic
273	C₂×C₃₁	#3241	1	2	yes	—
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
277	C₂³×C₄	—	2	2	no	—
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
281	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
282	#127	—	2	2	no	—
283	#127	C₂×C₄×C₃₁	1	4	yes	complemented
284	#125	—	2	2	no	—
285	#125	C₂×C₄×C₃₁	1	4	yes	complemented
286	#123	—	2	16	no	—
287	#121	—	2	4	no	—
288	#121	—	2	16	no	—
289	#121	C₂×C₄×C₃₁	1	8	yes	—
290	C₄×D₄	—	2	4	no	—
291	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
292	#117	—	4	2	no	—
293	#117	C₂×C₄×C₃₁	1	4	yes	—
294	#117	D₄×C₃₁	1	4	yes	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	4	2	no	—
300	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	C₄⋊C₈	—	2	4	no	—
314	#102	—	4	2	no	—
315	#102	—	4	2	no	—
316	#102	—	4	2	no	—
317	#102	—	4	2	no	—
318	C₂²⋊C₈	—	2	4	no	—
319	#97	—	2	2	no	—
320	C₄×C₈	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	C₃₁	#21872	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	—	2	2	no	—
325	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	8	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	—	8	2	no	—
336	C₂×D₄	—	8	2	no	—
337	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	8	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	8	2	no	—
348	C₂²×C₄	—	8	2	no	—
349	C₂²×C₄	—	8	2	no	—
350	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₄⋊C₄	—	8	2	no	—
371	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
372	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
373	C₂²⋊C₄	—	2	8	no	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
377	C₄²	—	4	2	no	—
378	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
379	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
380	C₄²	C₂×D₄×C₃₁	1	2	yes	—
381	C₄²	C₂×D₄×C₃₁	1	2	yes	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
394	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	D₄	—	2	4	no	—
396	D₄	—	2	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	8	no	—
400	D₄	—	8	8	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	8	8	no	—
421	C₂×C₄	—	8	4	no	—
422	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
423	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
424	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
425	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
428	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
429	C₈	—	4	2	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	1	no	—
434	C₂²	—	4	1	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂²	C₃₁×#535	1	1	yes	characteristic
445	C₂²	C₃₁×#536	1	1	yes	characteristic
446	C₂²	C₃₁×#576	1	2	yes	—
447	C₂²	C₃₁×#577	1	2	yes	—
448	C₄	—	2	2	no	—
449	C₄	—	2	4	no	—
450	C₄	—	2	4	no	—
451	C₄	—	4	4	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	1	no	—
455	C₄	—	8	1	no	—
456	C₄	C₃₁×#520	1	2	yes	—
457	C₄	C₃₁×#537	1	2	yes	—
458	C₂	—	4	2	no	—
459	C₂	—	8	4	no	—
460	C₂	C₃₁×#2311	1	1	yes	characteristic
461	C₂	C₃₁×#2897	1	2	yes	—
462	C₂	C₃₁×#3220	1	1	yes	characteristic
463	C₂	C₃₁×#3221	1	1	yes	characteristic
464	C₂	C₃₁×#3241	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×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#433	C₂	semidirect
3. C₃₁×#2317	C₂	semidirect
4. C₃₁×#1992	C₂	semidirect
5. C₃₁×#1991	C₂	semidirect
6. C₃₁×#1961	C₂	semidirect
7. C₃₁×#1957	C₂	semidirect
8. C₃₁×#1875	C₂	semidirect
9. C₃₁×#1874	C₂	semidirect
10. C₃₁×#1804	C₂	semidirect
11. C₃₁×#1802	C₂	semidirect
13. C₃₁×#540	C₄	semidirect
51. C₃₁×#127	C₂×C₄	semidirect
53. C₃₁×#125	C₂×C₄	semidirect
59. C₄×D₄×C₃₁	C₂×C₄	semidirect
149. #21872	C₃₁	direct (N × H)
200. C₂×#433	C₂×C₃₁	semidirect
201. #2317	C₂×C₃₁	semidirect
202. #1992	C₂×C₃₁	semidirect
203. #1991	C₂×C₃₁	semidirect
204. #1961	C₂×C₃₁	semidirect
205. #1957	C₂×C₃₁	semidirect
206. #1875	C₂×C₃₁	semidirect
207. #1874	C₂×C₃₁	semidirect
208. #1804	C₂×C₃₁	semidirect
209. #1802	C₂×C₃₁	semidirect
238. #540	C₄×C₃₁	semidirect
283. #127	C₂×C₄×C₃₁	semidirect
285. #125	C₂×C₄×C₃₁	semidirect
291. C₄×D₄	C₂×C₄×C₃₁	semidirect
323. C₃₁	#21872	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×#540	2
C₂×C₃₁×#433	2
C₃₁×#2317	2
C₃₁×#1992	2
C₃₁×#1991	2
C₃₁×#1961	2
C₃₁×#1957	2
C₃₁×#1875	2
C₃₁×#1874	2
C₃₁×#1804	2
C₃₁×#1802	2
#21872	31

Subgroups — C31×#21872 #21478631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21872 #21478631