Subgroups — C₃₁×#23127 #21479886

Order 7936.

1078 conjugacy classes of subgroups.

The subgroup lattice has 424 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×#2361	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1874	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	C₂×C₃₁×#340	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	—	2	2	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#466	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
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	C₃₁×#385	—	2	2	no	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
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	—	2	2	no	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×D₈×C₃₁	—	4	2	no	—
42	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
43	C₂²×C₈×C₃₁	—	4	2	no	—
44	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#121	—	2	4	no	—
49	C₃₁×#121	C₂×C₄	1	8	yes	complemented
50	C₄×D₄×C₃₁	—	2	4	no	—
51	C₄×D₄×C₃₁	—	4	2	no	—
52	C₄×D₄×C₃₁	—	4	8	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#107	C₂×C₄	1	4	yes	complemented
69	C₃₁×#102	—	2	4	no	—
70	C₃₁×#102	—	4	2	no	—
71	C₃₁×#102	C₂×C₄	1	8	yes	complemented
72	C₃₁×C₂²⋊C₈	—	2	2	no	—
73	C₄×C₈×C₃₁	—	4	2	no	—
74	C₃₁×#95	—	2	2	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	D₄	1	2	yes	—
77	C₂⁴×C₃₁	—	2	2	no	—
78	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×D₄×C₃₁	—	2	8	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	—	2	2	no	—
85	C₂×D₄×C₃₁	—	4	4	no	—
86	C₂×D₄×C₃₁	—	4	4	no	—
87	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	4	4	no	—
94	C₂²×C₄×C₃₁	—	4	4	no	—
95	C₂²×C₄×C₃₁	—	4	4	no	—
96	C₂²×C₄×C₃₁	—	4	4	no	—
97	C₂²×C₄×C₃₁	—	8	2	no	—
98	C₂²×C₄×C₃₁	—	8	2	no	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	#41	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	D₈×C₃₁	—	4	4	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	4	4	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
121	C₄⋊C₄×C₃₁	#41	1	4	yes	—
122	C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂²⋊C₄×C₃₁	—	2	8	no	—
124	C₂²⋊C₄×C₃₁	—	4	4	no	—
125	C₂²⋊C₄×C₃₁	—	4	4	no	—
126	C₂²⋊C₄×C₃₁	—	8	2	no	—
127	C₂²⋊C₄×C₃₁	D₈	1	4	yes	complemented
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	4	4	no	—
131	#23127	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	4	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	—	4	2	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	C₄×D₄	1	2	yes	—
142	C₂³×C₃₁	C₂×D₈	1	2	yes	—
143	C₂³×C₃₁	#135	1	2	yes	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	2	4	no	—
147	D₄×C₃₁	—	4	16	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	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	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	8	8	no	—
169	C₂×C₄×C₃₁	—	8	8	no	—
170	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
172	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
176	C₈×C₃₁	—	2	2	no	—
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	—	4	2	no	—
179	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
180	#2361	C₂×C₃₁	1	2	yes	complemented
181	#1934	C₂×C₃₁	1	2	yes	complemented
182	#1918	C₂×C₃₁	1	2	yes	complemented
183	#1879	C₂×C₃₁	1	2	yes	complemented
184	#1874	C₂×C₃₁	1	2	yes	complemented
185	C₄×#102	C₂×C₃₁	1	2	yes	complemented
186	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂×#340	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	4	4	no	—
194	C₂²×C₃₁	—	4	4	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	C₄×D₈	1	4	yes	—
200	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₃₁	#546	1	2	yes	—
202	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
203	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
204	C₂²×C₃₁	#577	1	2	yes	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	4	no	—
207	C₄×C₃₁	—	4	4	no	—
208	C₄×C₃₁	—	4	4	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
212	C₄×C₃₁	#518	1	1	yes	characteristic
213	C₄×C₃₁	#553	1	2	yes	—
214	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#121	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄×D₄	—	2	2	no	—
218	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
219	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
220	#466	C₄×C₃₁	1	8	yes	complemented
221	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×#102	C₂²×C₃₁	1	2	yes	—
223	C₂×#102	C₂²×C₃₁	1	2	yes	—
224	C₂×#102	C₂²×C₃₁	1	2	yes	—
225	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
227	#390	C₂²×C₃₁	1	2	yes	—
228	#389	C₂²×C₃₁	1	2	yes	—
229	#388	—	2	2	no	—
230	#386	—	2	2	no	—
231	#385	—	2	2	no	—
232	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
233	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
234	#340	C₂²×C₃₁	1	2	yes	—
235	#340	C₂²×C₃₁	1	2	yes	—
236	#340	C₂²×C₃₁	1	2	yes	—
237	#340	C₂²×C₃₁	1	1	yes	characteristic
238	#340	C₂²×C₃₁	1	1	yes	characteristic
239	#336	C₂²×C₃₁	1	2	yes	—
240	C₂×C₃₁	—	2	4	no	—
241	C₂×C₃₁	—	8	4	no	—
242	C₂×C₃₁	#2364	1	1	yes	characteristic
243	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
244	C₂×C₃₁	#2952	1	1	yes	characteristic
245	C₂×C₃₁	#3294	1	2	yes	—
246	C₂×C₃₁	#3301	1	2	yes	—
247	C₂²×D₄	—	2	2	no	—
248	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	—	2	2	no	—
251	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
252	C₂×D₈	—	4	2	no	—
253	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
254	C₂²×C₈	—	4	2	no	—
255	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
256	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
257	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	#121	—	2	4	no	—
260	#121	C₂×C₄×C₃₁	1	8	yes	complemented
261	C₄×D₄	—	2	4	no	—
262	C₄×D₄	—	4	2	no	—
263	C₄×D₄	—	4	8	no	—
264	C₂×C₄⋊C₄	—	2	4	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	4	2	no	—
268	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
279	#107	C₂×C₄×C₃₁	1	4	yes	complemented
280	#102	—	2	4	no	—
281	#102	—	4	2	no	—
282	#102	C₂×C₄×C₃₁	1	8	yes	complemented
283	C₂²⋊C₈	—	2	2	no	—
284	C₄×C₈	—	4	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	#95	D₄×C₃₁	1	2	yes	—
288	C₃₁	#23127	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	—	2	2	no	—
290	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	4	4	no	—
298	C₂×D₄	—	4	4	no	—
299	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	—	4	4	no	—
307	C₂²×C₄	—	4	4	no	—
308	C₂²×C₄	—	4	4	no	—
309	C₂²×C₄	—	8	2	no	—
310	C₂²×C₄	—	8	2	no	—
311	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	C₂²×C₄	#41×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
317	D₈	—	4	4	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	4	4	no	—
324	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	4	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
333	C₄⋊C₄	#41×C₃₁	1	4	yes	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	2	8	no	—
336	C₂²⋊C₄	—	4	4	no	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₂²⋊C₄	—	8	2	no	—
339	C₂²⋊C₄	D₈×C₃₁	1	4	yes	complemented
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	4	4	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	4	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	C₄×D₄×C₃₁	1	2	yes	—
353	C₂³	C₂×D₈×C₃₁	1	2	yes	—
354	C₂³	C₃₁×#135	1	2	yes	—
355	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	D₄	—	2	4	no	—
357	D₄	—	2	4	no	—
358	D₄	—	4	16	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	8	8	no	—
380	C₂×C₄	—	8	8	no	—
381	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
382	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
383	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
385	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
387	C₈	—	2	2	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	C₄×D₈×C₃₁	1	4	yes	—
402	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₃₁×#546	1	2	yes	—
404	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
405	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
406	C₂²	C₃₁×#577	1	2	yes	—
407	C₄	—	2	2	no	—
408	C₄	—	2	4	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	2	no	—
412	C₄	—	8	2	no	—
413	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
414	C₄	C₃₁×#518	1	1	yes	characteristic
415	C₄	C₃₁×#553	1	2	yes	—
416	C₂	—	2	4	no	—
417	C₂	—	8	4	no	—
418	C₂	C₃₁×#2364	1	1	yes	characteristic
419	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
420	C₂	C₃₁×#2952	1	1	yes	characteristic
421	C₂	C₃₁×#3294	1	2	yes	—
422	C₂	C₃₁×#3301	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×#2361	C₂	semidirect
3. C₃₁×#1934	C₂	semidirect
4. C₃₁×#1918	C₂	semidirect
5. C₃₁×#1879	C₂	semidirect
6. C₃₁×#1874	C₂	semidirect
7. C₄×C₃₁×#102	C₂	semidirect
8. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
9. C₂×C₃₁×#340	C₂	semidirect
16. C₃₁×#466	C₄	semidirect
42. C₂×D₈×C₃₁	C₂×C₄	semidirect
44. C₂²×C₈×C₃₁	C₂×C₄	semidirect
49. C₃₁×#121	C₂×C₄	semidirect
68. C₃₁×#107	C₂×C₄	semidirect
71. C₃₁×#102	C₂×C₄	semidirect
127. C₂²⋊C₄×C₃₁	D₈	semidirect
131. #23127	C₃₁	direct (N × H)
180. #2361	C₂×C₃₁	semidirect
181. #1934	C₂×C₃₁	semidirect
182. #1918	C₂×C₃₁	semidirect
183. #1879	C₂×C₃₁	semidirect
184. #1874	C₂×C₃₁	semidirect
185. C₄×#102	C₂×C₃₁	semidirect
186. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
187. C₂×#340	C₂×C₃₁	semidirect
220. #466	C₄×C₃₁	semidirect
253. C₂×D₈	C₂×C₄×C₃₁	semidirect
255. C₂²×C₈	C₂×C₄×C₃₁	semidirect
260. #121	C₂×C₄×C₃₁	semidirect
279. #107	C₂×C₄×C₃₁	semidirect
282. #102	C₂×C₄×C₃₁	semidirect
288. C₃₁	#23127	direct (N × H)
339. C₂²⋊C₄	D₈×C₃₁	semidirect

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×#2361	2
C₃₁×#1934	2
C₃₁×#1918	2
C₃₁×#1879	2
C₃₁×#1874	2
C₄×C₃₁×#102	2
C₈×C₂²⋊C₄×C₃₁	2
C₂×C₃₁×#340	2
#23127	31

Subgroups — C31×#23127 #21479886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23127 #21479886