Subgroups — C₂×C₄×C₃₁×#95 #21470504

Order 7936.

4462 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₃₁	C₂	1	3	yes	maximal
2	C₂²×C₃₁×#95	C₂	1	4	yes	maximal
3	C₄×C₃₁×#95	C₂	1	8	yes	maximal, direct product factor
4	C₂⁴×C₄×C₃₁	C₂²	1	6	yes	—
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₃₁	—	2	24	no	—
7	C₂²×C₄²×C₃₁	C₄	1	24	yes	complemented
8	C₂²×C₄²×C₃₁	C₂²	1	12	yes	—
9	C₂×C₃₁×#95	C₄	1	32	yes	direct product factor
10	C₂×C₃₁×#95	C₂²	1	16	yes	—
11	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	—	2	24	no	—
13	C₂³×C₄×C₃₁	—	2	24	no	—
14	C₂³×C₄×C₃₁	C₂×C₄	1	24	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	48	yes	—
16	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₃₁	D₄	1	12	yes	—
18	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
21	C₂×C₄²×C₃₁	—	2	96	no	—
22	C₂×C₄²×C₃₁	—	2	48	no	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	48	yes	complemented
24	C₃₁×#95	C₂×C₄	1	64	yes	direct product factor
25	C₂⁴×C₃₁	C₂²×C₄	1	6	yes	—
26	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
27	C₂⁴×C₃₁	C₂×D₄	1	6	yes	—
28	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
29	C₂⁴×C₃₁	#41	1	8	yes	—
30	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	—	2	48	no	—
32	C₂²×C₄×C₃₁	—	2	48	no	—
33	C₂²×C₄×C₃₁	—	2	192	no	—
34	C₂²×C₄×C₃₁	—	2	24	no	—
35	C₂²×C₄×C₃₁	—	2	24	no	—
36	C₂²×C₄×C₃₁	C₄²	1	16	yes	—
37	C₂²×C₄×C₃₁	C₄²	1	96	yes	complemented
38	C₂²×C₄×C₃₁	C₂²⋊C₄	1	48	yes	—
39	C₂²×C₄×C₃₁	C₄⋊C₄	1	48	yes	—
40	C₂²×C₄×C₃₁	C₂²×C₄	1	48	yes	—
41	C₂²×C₄×C₃₁	C₂²×C₄	1	24	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
44	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
45	C₄²×C₃₁	—	2	96	no	—
46	C₂×C₄×#95	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₃₁	C₂×C₄²	1	4	yes	—
48	C₂³×C₃₁	C₂×C₄²	1	24	yes	—
49	C₂³×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
50	C₂³×C₃₁	C₂×C₄⋊C₄	1	12	yes	—
51	C₂³×C₃₁	#117	1	24	yes	—
52	C₂³×C₃₁	C₄×D₄	1	48	yes	—
53	C₂³×C₃₁	C₄×Q₈	1	16	yes	—
54	C₂³×C₃₁	C₂³×C₄	1	3	yes	—
55	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
56	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
57	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
58	C₂³×C₃₁	C₂×#41	1	4	yes	—
59	C₂×C₄×C₃₁	—	2	24	no	—
60	C₂×C₄×C₃₁	—	2	24	no	—
61	C₂×C₄×C₃₁	—	2	96	no	—
62	C₂×C₄×C₃₁	—	2	192	no	—
63	C₂×C₄×C₃₁	#95	1	64	yes	direct product factor
64	C₂×C₄×C₃₁	C₂×C₄²	1	8	yes	—
65	C₂×C₄×C₃₁	C₂×C₄²	1	48	yes	complemented
66	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	24	yes	—
67	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	24	yes	—
68	C₂³×C₄²	C₂×C₃₁	1	3	yes	—
69	C₂²×#95	C₂×C₃₁	1	4	yes	—
70	C₄×#95	C₂×C₃₁	1	8	yes	direct product factor
71	C₂²×C₃₁	C₄³	1	8	yes	—
72	C₂²×C₃₁	C₂×#95	1	8	yes	—
73	C₂²×C₃₁	#376	1	8	yes	—
74	C₂²×C₃₁	C₄×C₂²⋊C₄	1	48	yes	—
75	C₂²×C₃₁	C₄×C₄⋊C₄	1	48	yes	—
76	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
77	C₂²×C₃₁	C₂²×C₄²	1	6	yes	—
78	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
79	C₂²×C₃₁	C₂²×C₄⋊C₄	1	3	yes	—
80	C₂²×C₃₁	C₂×#117	1	6	yes	—
81	C₂²×C₃₁	C₂×C₄×D₄	1	12	yes	—
82	C₂²×C₃₁	C₂×C₄×Q₈	1	4	yes	—
83	C₄×C₃₁	—	2	48	no	—
84	C₄×C₃₁	C₂×#95	1	16	yes	direct product factor
85	C₂⁴×C₄	C₂²×C₃₁	1	6	yes	—
86	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
87	C₂²×C₄²	—	2	24	no	—
88	C₂²×C₄²	C₄×C₃₁	1	24	yes	complemented
89	C₂²×C₄²	C₂²×C₃₁	1	12	yes	—
90	C₂×#95	C₄×C₃₁	1	32	yes	direct product factor
91	C₂×#95	C₂²×C₃₁	1	16	yes	—
92	C₂×C₃₁	C₄×#95	1	16	yes	direct product factor
93	C₂×C₃₁	C₂×C₄³	1	1	yes	characteristic
94	C₂×C₃₁	C₂²×#95	1	1	yes	characteristic
95	C₂×C₃₁	C₂×#376	1	1	yes	characteristic
96	C₂×C₃₁	C₂×C₄×C₂²⋊C₄	1	6	yes	—
97	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	6	yes	—
98	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
99	C₂³×C₄	—	2	24	no	—
100	C₂³×C₄	—	2	24	no	—
101	C₂³×C₄	C₂×C₄×C₃₁	1	48	yes	—
102	C₂³×C₄	C₂×C₄×C₃₁	1	24	yes	—
103	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
104	C₂³×C₄	D₄×C₃₁	1	12	yes	—
105	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
106	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
107	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
108	C₂×C₄²	—	2	96	no	—
109	C₂×C₄²	—	2	48	no	—
110	C₂×C₄²	C₂×C₄×C₃₁	1	48	yes	complemented
111	#95	C₂×C₄×C₃₁	1	64	yes	direct product factor
112	C₃₁	C₂×C₄×#95	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴	C₂²×C₄×C₃₁	1	6	yes	—
114	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
115	C₂⁴	C₂×D₄×C₃₁	1	6	yes	—
116	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
117	C₂⁴	#41×C₃₁	1	8	yes	—
118	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
119	C₂²×C₄	—	2	48	no	—
120	C₂²×C₄	—	2	48	no	—
121	C₂²×C₄	—	2	192	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	—	2	24	no	—
124	C₂²×C₄	C₄²×C₃₁	1	16	yes	—
125	C₂²×C₄	C₄²×C₃₁	1	96	yes	complemented
126	C₂²×C₄	C₂²⋊C₄×C₃₁	1	48	yes	—
127	C₂²×C₄	C₄⋊C₄×C₃₁	1	48	yes	—
128	C₂²×C₄	C₂²×C₄×C₃₁	1	48	yes	—
129	C₂²×C₄	C₂²×C₄×C₃₁	1	24	yes	—
130	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
131	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
132	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
133	C₄²	—	2	96	no	—
134	C₂³	C₂×C₄²×C₃₁	1	4	yes	—
135	C₂³	C₂×C₄²×C₃₁	1	24	yes	—
136	C₂³	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
137	C₂³	C₂×C₄⋊C₄×C₃₁	1	12	yes	—
138	C₂³	C₃₁×#117	1	24	yes	—
139	C₂³	C₄×D₄×C₃₁	1	48	yes	—
140	C₂³	C₄×Q₈×C₃₁	1	16	yes	—
141	C₂³	C₂³×C₄×C₃₁	1	3	yes	—
142	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
143	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
144	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
145	C₂³	C₂×#41×C₃₁	1	4	yes	—
146	C₂×C₄	—	2	24	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	—	2	96	no	—
149	C₂×C₄	—	2	192	no	—
150	C₂×C₄	C₃₁×#95	1	64	yes	direct product factor
151	C₂×C₄	C₂×C₄²×C₃₁	1	8	yes	—
152	C₂×C₄	C₂×C₄²×C₃₁	1	48	yes	complemented
153	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	24	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	24	yes	—
155	C₂²	C₄³×C₃₁	1	8	yes	—
156	C₂²	C₂×C₃₁×#95	1	8	yes	—
157	C₂²	C₃₁×#376	1	8	yes	—
158	C₂²	C₄×C₂²⋊C₄×C₃₁	1	48	yes	—
159	C₂²	C₄×C₄⋊C₄×C₃₁	1	48	yes	—
160	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
161	C₂²	C₂²×C₄²×C₃₁	1	6	yes	—
162	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
163	C₂²	C₂²×C₄⋊C₄×C₃₁	1	3	yes	—
164	C₂²	C₂×C₃₁×#117	1	6	yes	—
165	C₂²	C₂×C₄×D₄×C₃₁	1	12	yes	—
166	C₂²	C₂×C₄×Q₈×C₃₁	1	4	yes	—
167	C₄	—	2	48	no	—
168	C₄	C₂×C₃₁×#95	1	16	yes	direct product factor
169	C₂	C₄×C₃₁×#95	1	16	yes	direct product factor
170	C₂	C₂×C₄³×C₃₁	1	1	yes	characteristic
171	C₂	C₂²×C₃₁×#95	1	1	yes	characteristic
172	C₂	C₂×C₃₁×#376	1	1	yes	characteristic
173	C₂	C₂×C₄×C₂²⋊C₄×C₃₁	1	6	yes	—
174	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	6	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₃₁	C₂²
commutator	C₂	C₂×C₄³×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
socle, prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₃₁×#95	C₂	direct (N × H)
7. C₂²×C₄²×C₃₁	C₄	semidirect
9. C₂×C₃₁×#95	C₄	direct (N × H)
23. C₂×C₄²×C₃₁	C₂×C₄	semidirect
24. C₃₁×#95	C₂×C₄	direct (N × H)
37. C₂²×C₄×C₃₁	C₄²	semidirect
46. C₂×C₄×#95	C₃₁	direct (N × H)
63. C₂×C₄×C₃₁	#95	direct (N × H)
65. C₂×C₄×C₃₁	C₂×C₄²	semidirect
70. C₄×#95	C₂×C₃₁	direct (N × H)
84. C₄×C₃₁	C₂×#95	direct (N × H)
88. C₂²×C₄²	C₄×C₃₁	semidirect
90. C₂×#95	C₄×C₃₁	direct (N × H)
92. C₂×C₃₁	C₄×#95	direct (N × H)
110. C₂×C₄²	C₂×C₄×C₃₁	semidirect
111. #95	C₂×C₄×C₃₁	direct (N × H)
112. C₃₁	C₂×C₄×#95	direct (N × H)
125. C₂²×C₄	C₄²×C₃₁	semidirect
150. C₂×C₄	C₃₁×#95	direct (N × H)
152. C₂×C₄	C₂×C₄²×C₃₁	semidirect
168. C₄	C₂×C₃₁×#95	direct (N × H)
169. C₂	C₄×C₃₁×#95	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₃₁	2
C₂²×C₃₁×#95	2
C₄×C₃₁×#95	2
C₂×C₄×#95	31

Subgroups — C2×C4×C31×#95 #21470504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₁×#95 #21470504