Subgroups — C₂×C₃₁×#2273 #21490753

Order 7936.

5790 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄×C₃₁	C₂	1	3	yes	maximal
2	C₃₁×#2273	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	12	yes	maximal, complemented
4	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
5	C₂⁴×C₄×C₃₁	C₂²	1	6	yes	—
6	C₂²×C₂²⋊C₄×C₃₁	C₄	1	48	yes	complemented
7	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	12	yes	—
8	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	24	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	8	yes	complemented
10	C₄×C₂²⋊C₄×C₃₁	C₂²	1	96	yes	complemented
11	C₂×C₃₁×#95	C₂²	1	8	yes	complemented
12	C₂⁵×C₃₁	—	4	8	no	—
13	C₂⁵×C₃₁	C₂×C₄	1	24	yes	—
14	C₂⁵×C₃₁	C₂³	1	3	yes	—
15	C₂⁵×C₃₁	C₂³	1	4	yes	—
16	C₂³×C₄×C₃₁	—	4	24	no	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	48	yes	complemented
18	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	96	yes	complemented
21	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	192	yes	complemented
22	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
23	C₂×C₄²×C₃₁	—	4	16	no	—
24	C₂×C₄²×C₃₁	C₂³	1	32	yes	complemented
25	C₃₁×#95	C₂³	1	32	yes	complemented
26	C₂⁴×C₃₁	—	2	24	no	—
27	C₂⁴×C₃₁	—	4	96	no	—
28	C₂⁴×C₃₁	—	4	16	no	—
29	C₂⁴×C₃₁	C₄²	1	64	yes	complemented
30	C₂⁴×C₃₁	C₂²×C₄	1	24	yes	—
31	C₂⁴×C₃₁	C₂²×C₄	1	12	yes	—
32	C₂⁴×C₃₁	C₂²×C₄	1	48	yes	—
33	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
34	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	—	2	48	no	—
36	C₂²×C₄×C₃₁	—	4	96	no	—
37	C₂²×C₄×C₃₁	—	4	48	no	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	48	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	96	yes	complemented
40	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
41	C₂²⋊C₄×C₃₁	C₂²×C₄	1	192	yes	complemented
42	C₄²×C₃₁	—	4	32	no	—
43	C₂×#2273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₃₁	—	2	48	no	—
45	C₂³×C₃₁	—	2	96	no	—
46	C₂³×C₃₁	—	2	24	no	—
47	C₂³×C₃₁	—	4	128	no	—
48	C₂³×C₃₁	—	4	96	no	—
49	C₂³×C₃₁	C₂×C₄²	1	48	yes	—
50	C₂³×C₃₁	C₂×C₄²	1	64	yes	complemented
51	C₂³×C₃₁	C₂³×C₄	1	24	yes	—
52	C₂³×C₃₁	C₂³×C₄	1	12	yes	—
53	C₂³×C₃₁	C₂³×C₄	1	6	yes	—
54	C₂³×C₃₁	H(5,𝔽₂)	1	8	yes	—
55	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
56	C₂×C₄×C₃₁	—	2	96	no	—
57	C₂×C₄×C₃₁	—	2	48	no	—
58	C₂×C₄×C₃₁	—	4	96	no	—
59	C₂×C₄×C₃₁	C₂³×C₄	1	48	yes	—
60	C₂³×C₂²⋊C₄	C₂×C₃₁	1	3	yes	—
61	#2273	C₂×C₃₁	1	16	yes	direct product factor
62	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	12	yes	complemented
63	C₂²×C₃₁	—	2	24	no	—
64	C₂²×C₃₁	—	2	48	no	—
65	C₂²×C₃₁	—	2	96	no	—
66	C₂²×C₃₁	—	4	64	no	—
67	C₂²×C₃₁	C₂²×C₄²	1	24	yes	—
68	C₂²×C₃₁	C₂²×C₄²	1	4	yes	—
69	C₂²×C₃₁	#518	1	24	yes	—
70	C₂²×C₃₁	C₂⁴×C₄	1	3	yes	—
71	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
72	C₄×C₃₁	—	2	48	no	—
73	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
74	C₂⁴×C₄	C₂²×C₃₁	1	6	yes	—
75	C₂²×C₂²⋊C₄	C₄×C₃₁	1	48	yes	complemented
76	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	12	yes	—
77	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	24	yes	—
78	C₂²×C₄²	C₂²×C₃₁	1	8	yes	complemented
79	C₄×C₂²⋊C₄	C₂²×C₃₁	1	96	yes	complemented
80	C₂×#95	C₂²×C₃₁	1	8	yes	complemented
81	C₂×C₃₁	—	2	24	no	—
82	C₂×C₃₁	#2273	1	8	yes	direct product factor
83	C₂×C₃₁	C₂³×C₄²	1	1	yes	characteristic
84	C₂×C₃₁	C₂×#518	1	6	yes	—
85	C₂⁵	—	4	8	no	—
86	C₂⁵	C₂×C₄×C₃₁	1	24	yes	—
87	C₂⁵	C₂³×C₃₁	1	3	yes	—
88	C₂⁵	C₂³×C₃₁	1	4	yes	—
89	C₂³×C₄	—	4	24	no	—
90	C₂³×C₄	C₂×C₄×C₃₁	1	48	yes	complemented
91	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
92	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
93	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	96	yes	complemented
94	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	192	yes	complemented
95	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
96	C₂×C₄²	—	4	16	no	—
97	C₂×C₄²	C₂³×C₃₁	1	32	yes	complemented
98	#95	C₂³×C₃₁	1	32	yes	complemented
99	C₃₁	C₂×#2273	1	1	yes	characteristic, direct product factor, Sylow
100	C₂⁴	—	2	24	no	—
101	C₂⁴	—	4	96	no	—
102	C₂⁴	—	4	16	no	—
103	C₂⁴	C₄²×C₃₁	1	64	yes	complemented
104	C₂⁴	C₂²×C₄×C₃₁	1	24	yes	—
105	C₂⁴	C₂²×C₄×C₃₁	1	12	yes	—
106	C₂⁴	C₂²×C₄×C₃₁	1	48	yes	—
107	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
108	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
109	C₂²×C₄	—	2	48	no	—
110	C₂²×C₄	—	4	96	no	—
111	C₂²×C₄	—	4	48	no	—
112	C₂²×C₄	C₂²×C₄×C₃₁	1	48	yes	—
113	C₂²×C₄	C₂²×C₄×C₃₁	1	96	yes	complemented
114	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
115	C₂²⋊C₄	C₂²×C₄×C₃₁	1	192	yes	complemented
116	C₄²	—	4	32	no	—
117	C₂³	—	2	48	no	—
118	C₂³	—	2	96	no	—
119	C₂³	—	2	24	no	—
120	C₂³	—	4	128	no	—
121	C₂³	—	4	96	no	—
122	C₂³	C₂×C₄²×C₃₁	1	48	yes	—
123	C₂³	C₂×C₄²×C₃₁	1	64	yes	complemented
124	C₂³	C₂³×C₄×C₃₁	1	24	yes	—
125	C₂³	C₂³×C₄×C₃₁	1	12	yes	—
126	C₂³	C₂³×C₄×C₃₁	1	6	yes	—
127	C₂³	C₃₁×H(5,𝔽₂)	1	8	yes	—
128	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
129	C₂×C₄	—	2	96	no	—
130	C₂×C₄	—	2	48	no	—
131	C₂×C₄	—	4	96	no	—
132	C₂×C₄	C₂³×C₄×C₃₁	1	48	yes	—
133	C₂²	—	2	24	no	—
134	C₂²	—	2	48	no	—
135	C₂²	—	2	96	no	—
136	C₂²	—	4	64	no	—
137	C₂²	C₂²×C₄²×C₃₁	1	24	yes	—
138	C₂²	C₂²×C₄²×C₃₁	1	4	yes	—
139	C₂²	C₃₁×#518	1	24	yes	—
140	C₂²	C₂⁴×C₄×C₃₁	1	3	yes	—
141	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
142	C₄	—	2	48	no	—
143	C₂	—	2	24	no	—
144	C₂	C₃₁×#2273	1	8	yes	direct product factor
145	C₂	C₂³×C₄²×C₃₁	1	1	yes	characteristic
146	C₂	C₂×C₃₁×#518	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂	C₂³×C₄²×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁶×C₃₁	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2273	C₂	direct (N × H)
3. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
6. C₂²×C₂²⋊C₄×C₃₁	C₄	semidirect
9. C₂²×C₄²×C₃₁	C₂²	semidirect
10. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
11. C₂×C₃₁×#95	C₂²	semidirect
17. C₂³×C₄×C₃₁	C₂×C₄	semidirect
20. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
21. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
24. C₂×C₄²×C₃₁	C₂³	semidirect
25. C₃₁×#95	C₂³	semidirect
29. C₂⁴×C₃₁	C₄²	semidirect
39. C₂²×C₄×C₃₁	C₂²×C₄	semidirect
41. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
43. C₂×#2273	C₃₁	direct (N × H)
50. C₂³×C₃₁	C₂×C₄²	semidirect
61. #2273	C₂×C₃₁	direct (N × H)
62. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
75. C₂²×C₂²⋊C₄	C₄×C₃₁	semidirect
78. C₂²×C₄²	C₂²×C₃₁	semidirect
79. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
80. C₂×#95	C₂²×C₃₁	semidirect
82. C₂×C₃₁	#2273	direct (N × H)
90. C₂³×C₄	C₂×C₄×C₃₁	semidirect
93. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
94. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
97. C₂×C₄²	C₂³×C₃₁	semidirect
98. #95	C₂³×C₃₁	semidirect
99. C₃₁	C₂×#2273	direct (N × H)
103. C₂⁴	C₄²×C₃₁	semidirect
113. C₂²×C₄	C₂²×C₄×C₃₁	semidirect
115. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
123. C₂³	C₂×C₄²×C₃₁	semidirect
144. C₂	C₃₁×#2273	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×C₂²⋊C₄×C₃₁	2
C₃₁×#2273	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×#2273	31

Subgroups — C2×C31×#2273 #21490753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2273 #21490753