Subgroups — C₃₁×#7258 #21464017

Order 7936.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	—	2	1	no	—
7	C₂×C₃₁×#107	C₄	1	2	yes	complemented
8	C₃₁×#380	—	2	1	no	—
9	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
10	C₃₁×C₂²⋊C₁₆	C₄	1	2	yes	complemented
11	C₃₁×#340	—	2	1	no	—
12	C₃₁×#340	—	2	1	no	—
13	C₃₁×#340	C₄	1	2	yes	complemented
14	C₂³×C₄×C₃₁	—	4	1	no	—
15	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	—	4	2	no	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
28	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
29	C₂×C₁₆×C₃₁	—	2	2	no	—
30	C₂×C₁₆×C₃₁	—	4	2	no	—
31	C₃₁×#107	—	4	2	no	—
32	C₃₁×#95	—	4	1	no	—
33	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	—	2	2	no	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	—	4	4	no	—
37	C₂²×C₄×C₃₁	—	4	2	no	—
38	C₂²×C₄×C₃₁	—	4	2	no	—
39	C₂²×C₄×C₃₁	—	8	2	no	—
40	C₂²×C₄×C₃₁	—	8	1	no	—
41	C₂²×C₄×C₃₁	—	8	1	no	—
42	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
48	C₂×C₈×C₃₁	—	2	2	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	4	1	no	—
51	C₂×C₈×C₃₁	—	4	1	no	—
52	C₂×C₈×C₃₁	—	4	1	no	—
53	C₂×C₈×C₃₁	—	4	1	no	—
54	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	4	4	no	—
60	C₄⋊C₄×C₃₁	—	4	2	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₂²⋊C₄×C₃₁	—	8	1	no	—
64	C₂²⋊C₄×C₃₁	—	8	1	no	—
65	C₁₆×C₃₁	—	4	2	no	—
66	#7258	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	2	2	no	—
69	C₂³×C₃₁	#102	1	1	yes	characteristic
70	C₂³×C₃₁	#102	1	1	yes	characteristic
71	C₂³×C₃₁	#107	1	1	yes	characteristic
72	C₂³×C₃₁	D₁₆	1	1	yes	characteristic
73	C₂³×C₃₁	SD₃₂	1	1	yes	characteristic
74	C₂³×C₃₁	SD₃₂	1	1	yes	characteristic
75	C₂³×C₃₁	Q₃₂	1	1	yes	characteristic
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	8	4	no	—
85	C₂×C₄×C₃₁	—	8	4	no	—
86	C₂×C₄×C₃₁	—	8	2	no	—
87	C₂×C₄×C₃₁	—	8	2	no	—
88	C₂×C₄×C₃₁	—	8	2	no	—
89	C₂×C₄×C₃₁	—	8	2	no	—
90	C₂×C₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#99	1	2	yes	—
93	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
94	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#104	1	2	yes	—
96	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	1	no	—
101	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
102	#1808	C₂×C₃₁	1	1	yes	characteristic
103	C₂×#340	C₂×C₃₁	1	1	yes	characteristic
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	2	1	no	—
112	C₂²×C₃₁	#327	1	2	yes	—
113	C₂²×C₃₁	#340	1	1	yes	characteristic
114	C₂²×C₃₁	#357	1	1	yes	characteristic
115	C₂²×C₃₁	#357	1	1	yes	characteristic
116	C₂²×C₃₁	#358	1	1	yes	characteristic
117	C₂²×C₃₁	#358	1	1	yes	characteristic
118	C₂²×C₃₁	#360	1	2	yes	—
119	C₂²×C₃₁	#366	1	1	yes	characteristic
120	C₂²×C₃₁	#367	1	1	yes	characteristic
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	8	4	no	—
124	C₄×C₃₁	—	16	2	no	—
125	C₄×C₃₁	#328	1	2	yes	—
126	C₄×C₃₁	#339	1	1	yes	characteristic
127	C₄×C₃₁	#342	1	1	yes	characteristic
128	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
129	C₂²×C₄⋊C₄	—	2	1	no	—
130	C₂²×C₁₆	—	2	1	no	—
131	C₂×#107	C₄×C₃₁	1	2	yes	complemented
132	#380	—	2	1	no	—
133	C₂²⋊C₁₆	—	2	1	no	—
134	C₂²⋊C₁₆	C₄×C₃₁	1	2	yes	complemented
135	#340	—	2	1	no	—
136	#340	—	2	1	no	—
137	#340	C₄×C₃₁	1	2	yes	complemented
138	C₂×C₃₁	—	2	1	no	—
139	C₂×C₃₁	—	2	1	no	—
140	C₂×C₃₁	—	2	1	no	—
141	C₂×C₃₁	—	2	1	no	—
142	C₂×C₃₁	#1294	1	1	yes	characteristic
143	C₂×C₃₁	#1347	1	2	yes	—
144	C₂×C₃₁	#1348	1	2	yes	—
145	C₂×C₃₁	#1380	1	1	yes	characteristic
146	C₂×C₃₁	#1386	1	1	yes	characteristic
147	C₂³×C₄	—	4	1	no	—
148	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	—	4	2	no	—
150	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
152	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	2	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₂²⋊C₄	—	4	1	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	4	2	no	—
164	#107	—	4	2	no	—
165	#95	—	4	1	no	—
166	C₃₁	#7258	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	8	2	no	—
174	C₂²×C₄	—	8	1	no	—
175	C₂²×C₄	—	8	1	no	—
176	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	8	1	no	—
198	C₂²⋊C₄	—	8	1	no	—
199	C₁₆	—	4	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	C₃₁×#102	1	1	yes	characteristic
203	C₂³	C₃₁×#102	1	1	yes	characteristic
204	C₂³	C₃₁×#107	1	1	yes	characteristic
205	C₂³	C₃₁×D₁₆	1	1	yes	characteristic
206	C₂³	C₃₁×SD₃₂	1	1	yes	characteristic
207	C₂³	C₃₁×SD₃₂	1	1	yes	characteristic
208	C₂³	C₃₁×Q₃₂	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
225	C₂×C₄	C₃₁×#99	1	2	yes	—
226	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
227	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
228	C₂×C₄	C₃₁×#104	1	2	yes	—
229	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	C₃₁×#327	1	2	yes	—
243	C₂²	C₃₁×#340	1	1	yes	characteristic
244	C₂²	C₃₁×#357	1	1	yes	characteristic
245	C₂²	C₃₁×#357	1	1	yes	characteristic
246	C₂²	C₃₁×#358	1	1	yes	characteristic
247	C₂²	C₃₁×#358	1	1	yes	characteristic
248	C₂²	C₃₁×#360	1	2	yes	—
249	C₂²	C₃₁×#366	1	1	yes	characteristic
250	C₂²	C₃₁×#367	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	16	2	no	—
255	C₄	C₃₁×#328	1	2	yes	—
256	C₄	C₃₁×#339	1	1	yes	characteristic
257	C₄	C₃₁×#342	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	C₃₁×#1294	1	1	yes	characteristic
263	C₂	C₃₁×#1347	1	2	yes	—
264	C₂	C₃₁×#1348	1	2	yes	—
265	C₂	C₃₁×#1380	1	1	yes	characteristic
266	C₂	C₃₁×#1386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#102
commutator	C₂×C₈	C₄²×C₃₁
Frattini, cosocle	C₂³×C₈	C₂²×C₃₁
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂×C₃₁×#107	C₄	semidirect
10. C₃₁×C₂²⋊C₁₆	C₄	semidirect
13. C₃₁×#340	C₄	semidirect
66. #7258	C₃₁	direct (N × H)
131. C₂×#107	C₄×C₃₁	semidirect
134. C₂²⋊C₁₆	C₄×C₃₁	semidirect
137. #340	C₄×C₃₁	semidirect
166. C₃₁	#7258	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×C₂²⋊C₁₆	2
C₃₁×#1808	2
C₂×C₃₁×#340	2
#7258	31

Subgroups — C31×#7258 #21464017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7258 #21464017