Subgroups — C₃₁×#9595 #21466354

Order 7936.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1939	C₂	1	2	yes	maximal
2	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#340	C₄	1	2	yes	—
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×#334	—	2	2	no	—
17	C₃₁×#322	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	—	2	2	no	—
26	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×#107	—	2	2	no	—
28	C₃₁×#107	C₈	1	4	yes	complemented
29	C₃₁×C₄⋊C₈	—	2	2	no	—
30	C₃₁×C₄⋊C₈	—	2	2	no	—
31	C₃₁×C₄⋊C₈	—	4	4	no	—
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	—	2	1	no	—
35	C₄×C₈×C₃₁	D₄	1	2	yes	—
36	C₄×C₈×C₃₁	Q₈	1	2	yes	—
37	C₄×C₈×C₃₁	Q₈	1	2	yes	—
38	C₃₁×#95	—	2	2	no	—
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
44	C₂×C₈×C₃₁	—	2	1	no	—
45	C₂×C₈×C₃₁	—	2	1	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
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	2	no	—
53	C₂×C₈×C₃₁	—	4	2	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
63	C₄²×C₃₁	—	4	2	no	—
64	C₄²×C₃₁	—	4	2	no	—
65	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₃₁	#41	1	1	yes	characteristic
69	#9595	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	1	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	4	2	no	—
75	C₂×C₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	—	4	2	no	—
77	C₂×C₄×C₃₁	—	4	2	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
89	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₃₁	#137	1	2	yes	—
91	C₈×C₃₁	—	2	2	no	—
92	C₈×C₃₁	—	4	2	no	—
93	C₈×C₃₁	—	4	2	no	—
94	C₈×C₃₁	—	8	2	no	—
95	C₈×C₃₁	C₄⋊C₈	1	4	yes	complemented
96	#1939	C₂×C₃₁	1	2	yes	—
97	C₄×#107	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
99	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
100	#1279	C₂×C₃₁	1	2	yes	—
101	C₂²×C₃₁	#389	1	1	yes	characteristic
102	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
103	C₂²×C₃₁	#436	1	1	yes	characteristic
104	C₂²×C₃₁	#441	1	1	yes	characteristic
105	C₂²×C₃₁	#442	1	1	yes	characteristic
106	C₂²×C₃₁	#444	1	1	yes	characteristic
107	C₂²×C₃₁	#444	1	1	yes	characteristic
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	4	no	—
111	C₄×C₃₁	—	8	2	no	—
112	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₃₁	#424	1	1	yes	characteristic
114	C₄×C₃₁	C₈×Q₈	1	1	yes	characteristic
115	C₄×C₃₁	#446	1	1	yes	characteristic
116	C₄×C₃₁	#468	1	2	yes	—
117	C₄×C₃₁	#501	1	2	yes	—
118	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
119	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	2	2	no	—
121	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
126	#340	C₄×C₃₁	1	2	yes	—
127	#336	—	2	2	no	—
128	#334	—	2	2	no	—
129	#322	—	2	2	no	—
130	C₂×C₃₁	#1579	1	1	yes	characteristic
131	C₂×C₃₁	#1580	1	1	yes	characteristic
132	C₂×C₃₁	#1593	1	1	yes	characteristic
133	C₂×C₃₁	#1594	1	1	yes	characteristic
134	C₂×C₃₁	#1939	1	1	yes	characteristic
135	C₂×C₃₁	#1943	1	1	yes	characteristic
136	C₂×C₃₁	#1946	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₈	D₄×C₃₁	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
146	#107	—	2	2	no	—
147	#107	C₈×C₃₁	1	4	yes	complemented
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	4	4	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	D₄×C₃₁	1	2	yes	—
155	C₄×C₈	Q₈×C₃₁	1	2	yes	—
156	C₄×C₈	Q₈×C₃₁	1	2	yes	—
157	#95	—	2	2	no	—
158	C₃₁	#9595	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
176	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
177	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
178	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
187	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
188	C₄²	#41×C₃₁	1	1	yes	characteristic
189	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
201	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
203	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
207	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
208	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
209	C₂×C₄	C₃₁×#137	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	2	no	—
213	C₈	—	8	2	no	—
214	C₈	C₃₁×C₄⋊C₈	1	4	yes	complemented
215	C₂²	C₃₁×#389	1	1	yes	characteristic
216	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
217	C₂²	C₃₁×#436	1	1	yes	characteristic
218	C₂²	C₃₁×#441	1	1	yes	characteristic
219	C₂²	C₃₁×#442	1	1	yes	characteristic
220	C₂²	C₃₁×#444	1	1	yes	characteristic
221	C₂²	C₃₁×#444	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
227	C₄	C₃₁×#424	1	1	yes	characteristic
228	C₄	C₈×Q₈×C₃₁	1	1	yes	characteristic
229	C₄	C₃₁×#446	1	1	yes	characteristic
230	C₄	C₃₁×#468	1	2	yes	—
231	C₄	C₃₁×#501	1	2	yes	—
232	C₂	C₃₁×#1579	1	1	yes	characteristic
233	C₂	C₃₁×#1580	1	1	yes	characteristic
234	C₂	C₃₁×#1593	1	1	yes	characteristic
235	C₂	C₃₁×#1594	1	1	yes	characteristic
236	C₂	C₃₁×#1939	1	1	yes	characteristic
237	C₂	C₃₁×#1943	1	1	yes	characteristic
238	C₂	C₃₁×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
28. C₃₁×#107	C₈	semidirect
69. #9595	C₃₁	direct (N × H)
95. C₈×C₃₁	C₄⋊C₈	semidirect
147. #107	C₈×C₃₁	semidirect
158. C₃₁	#9595	direct (N × H)
214. C₈	C₃₁×C₄⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#1939	2
C₄×C₃₁×#107	2
C₂×C₃₁×#334	2
C₂×C₃₁×#322	2
C₃₁×#1279	2
#9595	31

Subgroups — C31×#9595 #21466354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9595 #21466354