Subgroups — C₃₁×#2461 #16842394

Order 3968.

646 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#396	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#394	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#392	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#388	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#386	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#385	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#124	—	2	4	no	—
23	C₃₁×#123	—	2	4	no	—
24	C₃₁×C₂²⋊Q₈	—	2	4	no	—
25	C₄×D₄×C₃₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
49	C₃₁×#95	C₂²	1	1	yes	characteristic
50	C₃₁×#95	C₂²	1	1	yes	characteristic
51	C₃₁×#95	C₂²	1	1	yes	characteristic
52	C₃₁×#95	C₂²	1	1	yes	characteristic
53	C₃₁×#95	C₂²	1	1	yes	characteristic
54	C₃₁×#95	C₂²	1	1	yes	characteristic
55	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
56	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
57	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	D₄	1	4	yes	complemented
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₃₁	—	2	2	no	—
99	C₂²⋊C₄×C₃₁	—	2	2	no	—
100	C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂²⋊C₄×C₃₁	—	2	2	no	—
102	C₂²⋊C₄×C₃₁	—	2	2	no	—
103	C₂²⋊C₄×C₃₁	—	2	2	no	—
104	C₂²⋊C₄×C₃₁	—	2	2	no	—
105	C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂²⋊C₄×C₃₁	D₄	1	4	yes	complemented
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₂³×C₃₁	—	2	1	no	—
112	C₂³×C₃₁	—	2	1	no	—
113	C₂³×C₃₁	—	2	1	no	—
114	C₂³×C₃₁	—	2	1	no	—
115	C₂³×C₃₁	—	2	1	no	—
116	C₂³×C₃₁	—	2	1	no	—
117	C₂³×C₃₁	—	2	1	no	—
118	C₂³×C₃₁	—	2	1	no	—
119	C₂³×C₃₁	—	2	1	no	—
120	C₂³×C₃₁	—	2	1	no	—
121	C₂³×C₃₁	—	2	1	no	—
122	C₂³×C₃₁	—	2	1	no	—
123	C₂³×C₃₁	—	2	1	no	—
124	C₂³×C₃₁	—	2	1	no	—
125	C₂³×C₃₁	—	2	1	no	—
126	C₂³×C₃₁	—	2	1	no	—
127	C₂³×C₃₁	—	2	1	no	—
128	C₂³×C₃₁	—	2	1	no	—
129	C₂³×C₃₁	—	2	1	no	—
130	C₂³×C₃₁	C₂×D₄	1	2	yes	—
131	C₂³×C₃₁	#41	1	2	yes	—
132	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
133	Q₈×C₃₁	—	4	4	no	—
134	D₄×C₃₁	—	2	8	no	—
135	D₄×C₃₁	—	4	4	no	—
136	D₄×C₃₁	—	4	4	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	4	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×C₄×C₃₁	#41	1	2	yes	—
174	C₂×C₄×C₃₁	#41	1	2	yes	—
175	#2461	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	2	2	no	—
179	C₂²×C₃₁	—	2	2	no	—
180	C₂²×C₃₁	—	2	2	no	—
181	C₂²×C₃₁	—	2	2	no	—
182	C₂²×C₃₁	—	4	1	no	—
183	C₂²×C₃₁	—	4	1	no	—
184	C₂²×C₃₁	—	4	1	no	—
185	C₂²×C₃₁	—	4	1	no	—
186	C₂²×C₃₁	—	4	1	no	—
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	4	1	no	—
216	C₂×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
217	C₂×#123	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#396	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#395	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#394	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#392	C₂×C₃₁	1	1	yes	characteristic
224	#390	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#388	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#386	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#385	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×C₃₁	—	4	2	no	—
232	C₂×C₃₁	—	4	2	no	—
233	C₂×C₃₁	—	8	1	no	—
234	C₂×C₃₁	#525	1	1	yes	characteristic
235	C₂×C₃₁	#532	1	1	yes	characteristic
236	C₂×C₃₁	#542	1	1	yes	characteristic
237	C₂×C₃₁	D₄²	1	1	yes	characteristic
238	C₂×C₃₁	#547	1	1	yes	characteristic
239	C₂×C₃₁	#548	1	1	yes	characteristic
240	C₂×C₃₁	#551	1	1	yes	characteristic
241	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
244	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
245	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
247	#124	—	2	4	no	—
248	#123	—	2	4	no	—
249	C₂²⋊Q₈	—	2	4	no	—
250	C₄×D₄	—	2	4	no	—
251	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
274	#95	C₂²×C₃₁	1	1	yes	characteristic
275	#95	C₂²×C₃₁	1	1	yes	characteristic
276	#95	C₂²×C₃₁	1	1	yes	characteristic
277	#95	C₂²×C₃₁	1	1	yes	characteristic
278	#95	C₂²×C₃₁	1	1	yes	characteristic
279	#95	C₂²×C₃₁	1	1	yes	characteristic
280	C₃₁	#2461	1	1	yes	characteristic, direct product factor, Sylow
281	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
282	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
283	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	4	1	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	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	D₄×C₃₁	1	4	yes	complemented
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₂²⋊C₄	D₄×C₃₁	1	4	yes	complemented
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	1	no	—
347	C₂³	—	2	1	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	2	1	no	—
351	C₂³	—	2	1	no	—
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	C₂×D₄×C₃₁	1	2	yes	—
357	C₂³	#41×C₃₁	1	2	yes	—
358	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
359	Q₈	—	4	4	no	—
360	D₄	—	2	8	no	—
361	D₄	—	4	4	no	—
362	D₄	—	4	4	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
396	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
397	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂×C₄	#41×C₃₁	1	2	yes	—
400	C₂×C₄	#41×C₃₁	1	2	yes	—
401	C₂²	—	2	2	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	4	1	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	—	4	1	no	—
417	C₂²	—	4	1	no	—
418	C₂²	—	4	1	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	4	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
424	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
427	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
428	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
429	C₄	—	2	2	no	—
430	C₄	—	2	2	no	—
431	C₄	—	2	2	no	—
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₂	—	4	2	no	—
442	C₂	—	4	2	no	—
443	C₂	—	8	1	no	—
444	C₂	C₃₁×#525	1	1	yes	characteristic
445	C₂	C₃₁×#532	1	1	yes	characteristic
446	C₂	C₃₁×#542	1	1	yes	characteristic
447	C₂	D₄²×C₃₁	1	1	yes	characteristic
448	C₂	C₃₁×#547	1	1	yes	characteristic
449	C₂	C₃₁×#548	1	1	yes	characteristic
450	C₂	C₃₁×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₃₁×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#124	C₂	semidirect
2. C₂×C₃₁×#123	C₂	semidirect
3. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₄×D₄×C₃₁	C₂	semidirect
5. C₃₁×#396	C₂	semidirect
6. C₃₁×#395	C₂	semidirect
7. C₃₁×#394	C₂	semidirect
9. C₃₁×#390	C₂	semidirect
10. C₃₁×#388	C₂	semidirect
11. C₃₁×#388	C₂	semidirect
12. C₃₁×#388	C₂	semidirect
13. C₃₁×#386	C₂	semidirect
14. C₃₁×#385	C₂	semidirect
15. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
94. C₄⋊C₄×C₃₁	D₄	semidirect
107. C₂²⋊C₄×C₃₁	D₄	semidirect
175. #2461	C₃₁	direct (N × H)
216. C₂×#124	C₂×C₃₁	semidirect
217. C₂×#123	C₂×C₃₁	semidirect
218. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
219. C₂×C₄×D₄	C₂×C₃₁	semidirect
220. #396	C₂×C₃₁	semidirect
221. #395	C₂×C₃₁	semidirect
222. #394	C₂×C₃₁	semidirect
224. #390	C₂×C₃₁	semidirect
225. #388	C₂×C₃₁	semidirect
226. #388	C₂×C₃₁	semidirect
227. #388	C₂×C₃₁	semidirect
228. #386	C₂×C₃₁	semidirect
229. #385	C₂×C₃₁	semidirect
230. C₄×C₄⋊C₄	C₂×C₃₁	semidirect
280. C₃₁	#2461	direct (N × H)
320. C₄⋊C₄	D₄×C₃₁	semidirect
333. C₂²⋊C₄	D₄×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×#124	2
C₂×C₃₁×#123	2
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₄×D₄×C₃₁	2
C₃₁×#396	2
C₃₁×#395	2
C₃₁×#394	2
C₃₁×#392	2
C₃₁×#390	2
C₃₁×#388	2
C₃₁×#386	2
C₃₁×#385	2
C₄×C₄⋊C₄×C₃₁	2
#2461	31

Subgroups — C31×#2461 #16842394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2461 #16842394