Subgroups — C₃₁×#34208 #21490967

Order 7936.

3054 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2364	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2352	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2328	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2327	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2311	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#124	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#532	C₄	1	16	yes	complemented
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	C₂²	1	4	yes	—
17	C₃₁×#520	—	2	16	no	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
22	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	4	yes	—
24	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	4	yes	—
29	C₃₁×#390	C₂²	1	4	yes	—
30	C₃₁×#388	C₂²	1	16	yes	—
31	C₃₁×#387	C₂²	1	16	yes	—
32	C₃₁×#386	C₂²	1	8	yes	—
33	C₃₁×#386	C₂²	1	16	yes	—
34	C₃₁×#382	C₂²	1	2	yes	—
35	C₃₁×#382	C₂²	1	4	yes	—
36	C₃₁×#382	C₂²	1	2	yes	—
37	C₃₁×#380	C₂²	1	8	yes	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	—
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
41	C₃₁×#376	C₂²	1	2	yes	—
42	C₂×C₃₁×#95	C₂²	1	4	yes	—
43	C₄³×C₃₁	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	—	2	8	no	—
45	C₂×#41×C₃₁	—	2	8	no	—
46	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
47	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
57	C₃₁×#127	C₂×C₄	1	8	yes	complemented
58	C₃₁×#124	C₂×C₄	1	16	yes	complemented
59	C₃₁×#121	C₂×C₄	1	32	yes	complemented
60	C₄×Q₈×C₃₁	—	2	16	no	—
61	C₄×D₄×C₃₁	—	2	32	no	—
62	C₄×D₄×C₃₁	—	2	16	no	—
63	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	complemented
64	C₃₁×#117	—	2	4	no	—
65	C₃₁×#117	—	2	16	no	—
66	C₃₁×#117	D₄	1	8	yes	complemented
67	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
81	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
88	C₃₁×#95	C₂³	1	8	yes	—
89	C₃₁×#95	C₂³	1	8	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₃₁×#95	C₂³	1	16	yes	—
92	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
93	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
94	#41×C₃₁	—	2	64	no	—
95	C₂×Q₈×C₃₁	—	2	4	no	—
96	C₂×Q₈×C₃₁	—	2	4	no	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
98	C₂×D₄×C₃₁	—	2	8	no	—
99	C₂×D₄×C₃₁	—	2	8	no	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	4	8	no	—
103	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
104	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
105	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
106	C₂²×C₄×C₃₁	—	2	8	no	—
107	C₂²×C₄×C₃₁	—	2	8	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	16	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	—	4	16	no	—
112	C₂²×C₄×C₃₁	—	4	8	no	—
113	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
114	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₃₁	#41	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	8	no	—
127	C₄⋊C₄×C₃₁	—	2	16	no	—
128	C₄⋊C₄×C₃₁	—	2	8	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
131	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
132	C₂²⋊C₄×C₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₃₁	—	2	16	no	—
134	C₂²⋊C₄×C₃₁	—	4	16	no	—
135	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
136	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
137	C₄²×C₃₁	—	2	16	no	—
138	C₄²×C₃₁	—	2	4	no	—
139	C₄²×C₃₁	—	4	8	no	—
140	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
141	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
142	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
143	C₄²×C₃₁	C₂×D₄	1	8	yes	—
144	#34208	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	2	4	no	—
147	C₂³×C₃₁	—	2	4	no	—
148	C₂³×C₃₁	—	2	8	no	—
149	C₂³×C₃₁	—	2	8	no	—
150	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
151	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
152	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
153	C₂³×C₃₁	C₂×#41	1	2	yes	—
154	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	16	no	—
156	D₄×C₃₁	—	2	32	no	—
157	D₄×C₃₁	—	2	16	no	—
158	D₄×C₃₁	—	4	16	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	8	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	32	no	—
165	C₂×C₄×C₃₁	—	2	8	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	16	no	—
169	C₂×C₄×C₃₁	—	4	8	no	—
170	C₂×C₄×C₃₁	—	8	16	no	—
171	C₂×C₄×C₃₁	—	8	8	no	—
172	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	—
173	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
174	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
175	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
176	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
179	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
183	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
184	C₂×#520	C₂×C₃₁	1	2	yes	complemented
185	#2364	C₂×C₃₁	1	8	yes	complemented
186	#2352	C₂×C₃₁	1	8	yes	complemented
187	#2328	C₂×C₃₁	1	2	yes	complemented
188	#2327	C₂×C₃₁	1	2	yes	complemented
189	#2311	C₂×C₃₁	1	2	yes	complemented
190	C₄×#124	C₂×C₃₁	1	4	yes	complemented
191	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
192	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂²×C₃₁	—	2	4	no	—
194	C₂²×C₃₁	—	2	4	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	8	no	—
197	C₂²×C₃₁	—	2	16	no	—
198	C₂²×C₃₁	—	4	4	no	—
199	C₂²×C₃₁	—	4	4	no	—
200	C₂²×C₃₁	—	4	8	no	—
201	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
202	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
203	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
204	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
205	C₂²×C₃₁	#585	1	4	yes	—
206	C₄×C₃₁	—	2	8	no	—
207	C₄×C₃₁	—	2	4	no	—
208	C₄×C₃₁	—	2	4	no	—
209	C₄×C₃₁	—	2	8	no	—
210	C₄×C₃₁	—	4	8	no	—
211	C₄×C₃₁	—	4	4	no	—
212	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
213	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
214	C₂²×#41	C₂²×C₃₁	1	2	yes	—
215	#532	C₄×C₃₁	1	16	yes	complemented
216	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#124	C₂²×C₃₁	1	2	yes	—
219	C₂×#121	C₂²×C₃₁	1	4	yes	—
220	#520	—	2	16	no	—
221	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
222	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
223	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
224	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
225	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#117	C₂²×C₃₁	1	4	yes	—
227	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
228	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
230	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
231	#391	C₂²×C₃₁	1	4	yes	—
232	#390	C₂²×C₃₁	1	4	yes	—
233	#388	C₂²×C₃₁	1	16	yes	—
234	#387	C₂²×C₃₁	1	16	yes	—
235	#386	C₂²×C₃₁	1	8	yes	—
236	#386	C₂²×C₃₁	1	16	yes	—
237	#382	C₂²×C₃₁	1	2	yes	—
238	#382	C₂²×C₃₁	1	4	yes	—
239	#382	C₂²×C₃₁	1	2	yes	—
240	#380	C₂²×C₃₁	1	8	yes	—
241	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
242	C₄×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	—
243	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
244	#376	C₂²×C₃₁	1	2	yes	—
245	C₂×#95	C₂²×C₃₁	1	4	yes	—
246	C₄³	C₂²×C₃₁	1	2	yes	—
247	C₂×C₃₁	—	2	4	no	—
248	C₂×C₃₁	—	4	8	no	—
249	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
250	C₂×C₃₁	#3428	1	2	yes	—
251	C₂×C₃₁	#3500	1	2	yes	—
252	C₂×C₃₁	#3512	1	2	yes	—
253	C₂×#41	—	2	8	no	—
254	C₂×#41	—	2	8	no	—
255	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
256	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
257	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
258	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
259	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
260	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
261	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
262	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
263	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
264	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
265	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
266	#127	C₂×C₄×C₃₁	1	8	yes	complemented
267	#124	C₂×C₄×C₃₁	1	16	yes	complemented
268	#121	C₂×C₄×C₃₁	1	32	yes	complemented
269	C₄×Q₈	—	2	16	no	—
270	C₄×D₄	—	2	32	no	—
271	C₄×D₄	—	2	16	no	—
272	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	complemented
273	#117	—	2	4	no	—
274	#117	—	2	16	no	—
275	#117	D₄×C₃₁	1	8	yes	complemented
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	16	no	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
290	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
291	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
292	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
293	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
297	#95	C₂³×C₃₁	1	8	yes	—
298	#95	C₂³×C₃₁	1	8	yes	—
299	#95	C₂³×C₃₁	1	8	yes	—
300	#95	C₂³×C₃₁	1	16	yes	—
301	C₃₁	#34208	1	1	yes	characteristic, direct product factor, Sylow
302	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
303	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
304	#41	—	2	64	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
308	C₂×D₄	—	2	8	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	4	8	no	—
313	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
314	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
315	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	16	no	—
320	C₂²×C₄	—	4	4	no	—
321	C₂²×C₄	—	4	16	no	—
322	C₂²×C₄	—	4	8	no	—
323	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
324	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
325	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
327	C₂²×C₄	#41×C₃₁	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
330	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	16	no	—
338	C₄⋊C₄	—	2	8	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
341	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	16	no	—
344	C₂²⋊C₄	—	4	16	no	—
345	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
346	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
347	C₄²	—	2	16	no	—
348	C₄²	—	2	4	no	—
349	C₄²	—	4	8	no	—
350	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
351	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
352	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
353	C₄²	C₂×D₄×C₃₁	1	8	yes	—
354	C₂³	—	2	4	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	8	no	—
358	C₂³	—	2	8	no	—
359	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
360	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
361	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
362	C₂³	C₂×#41×C₃₁	1	2	yes	—
363	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
364	Q₈	—	2	16	no	—
365	D₄	—	2	32	no	—
366	D₄	—	2	16	no	—
367	D₄	—	4	16	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	32	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	16	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	8	16	no	—
380	C₂×C₄	—	8	8	no	—
381	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	—
382	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
383	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
384	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
385	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
386	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
388	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
391	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	8	no	—
396	C₂²	—	2	16	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	4	8	no	—
400	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
401	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
402	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
404	C₂²	C₃₁×#585	1	4	yes	—
405	C₄	—	2	8	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	4	no	—
408	C₄	—	2	8	no	—
409	C₄	—	4	8	no	—
410	C₄	—	4	4	no	—
411	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
412	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
413	C₂	—	2	4	no	—
414	C₂	—	4	8	no	—
415	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
416	C₂	C₃₁×#3428	1	2	yes	—
417	C₂	C₃₁×#3500	1	2	yes	—
418	C₂	C₃₁×#3512	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#520	C₂	semidirect
3. C₃₁×#2364	C₂	semidirect
4. C₃₁×#2352	C₂	semidirect
5. C₃₁×#2328	C₂	semidirect
6. C₃₁×#2327	C₂	semidirect
7. C₃₁×#2311	C₂	semidirect
8. C₄×C₃₁×#124	C₂	semidirect
9. C₂×C₃₁×#382	C₂	semidirect
10. C₄×C₃₁×#117	C₂	semidirect
12. C₃₁×#532	C₄	semidirect
46. C₂×#41×C₃₁	C₂×C₄	semidirect
56. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
57. C₃₁×#127	C₂×C₄	semidirect
58. C₃₁×#124	C₂×C₄	semidirect
59. C₃₁×#121	C₂×C₄	semidirect
63. C₄×D₄×C₃₁	C₂×C₄	semidirect
66. C₃₁×#117	D₄	semidirect
80. C₂×C₄²×C₃₁	C₂×C₄	semidirect
144. #34208	C₃₁	direct (N × H)
184. C₂×#520	C₂×C₃₁	semidirect
185. #2364	C₂×C₃₁	semidirect
186. #2352	C₂×C₃₁	semidirect
187. #2328	C₂×C₃₁	semidirect
188. #2327	C₂×C₃₁	semidirect
189. #2311	C₂×C₃₁	semidirect
190. C₄×#124	C₂×C₃₁	semidirect
191. C₂×#382	C₂×C₃₁	semidirect
192. C₄×#117	C₂×C₃₁	semidirect
215. #532	C₄×C₃₁	semidirect
255. C₂×#41	C₂×C₄×C₃₁	semidirect
265. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
266. #127	C₂×C₄×C₃₁	semidirect
267. #124	C₂×C₄×C₃₁	semidirect
268. #121	C₂×C₄×C₃₁	semidirect
272. C₄×D₄	C₂×C₄×C₃₁	semidirect
275. #117	D₄×C₃₁	semidirect
289. C₂×C₄²	C₂×C₄×C₃₁	semidirect
301. C₃₁	#34208	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×#532	2
C₂×C₃₁×#520	2
C₃₁×#2364	2
C₃₁×#2352	2
C₃₁×#2328	2
C₃₁×#2327	2
C₃₁×#2311	2
C₄×C₃₁×#124	2
C₂×C₃₁×#382	2
C₄×C₃₁×#117	2
#34208	31

Subgroups — C31×#34208 #21490967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34208 #21490967