Group #16854275 — subgroups

Subgroups — #16854275

Order 3968.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92474	C₂	1	2	yes	maximal
2	#14521076	C₂	1	8	yes	maximal, complemented
3	#14521073	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#117	C₄	1	4	yes	—
7	C₃₁×#117	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
11	C₂×#92474	—	2	8	no	—
12	C₂×#92474	C₂²	1	4	yes	—
13	C₂×#92474	C₂²	1	4	yes	—
14	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
15	C₄⋊(C₃₁⋊C₈)	C₂²	1	16	yes	complemented
16	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	D₄	1	4	yes	—
21	C₂²×C₄×C₃₁	Q₈	1	4	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
25	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
26	C₄²×C₃₁	C₂×C₄	1	4	yes	—
27	C₄²×C₃₁	C₂³	1	4	yes	—
28	#92474	—	2	8	no	—
29	#92474	—	2	8	no	—
30	C₂×C₃₁⋊C₈	—	2	8	no	—
31	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
32	C₂³×C₃₁	—	4	2	no	—
33	C₂³×C₃₁	C₄⋊C₄	1	4	yes	—
34	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
35	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	4	4	no	—
37	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
45	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₃₁⋊C₈	—	2	8	no	—
48	#2905	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	2	4	no	—
51	C₂²×C₃₁	—	4	4	no	—
52	C₂²×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
53	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₃₁	—	2	8	no	—
56	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
58	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×M₁₆	—	31	2	no	—
61	C₂×#117	D₃₁	1	1	yes	characteristic
62	#424	—	31	8	no	—
63	#423	—	31	4	no	—
64	C₂×C₃₁	—	2	2	no	—
65	C₂×C₃₁	—	2	4	no	—
66	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃₁	#568	1	2	yes	—
68	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
69	C₂×M₁₆	—	31	4	no	—
70	C₂×M₁₆	—	31	4	no	—
71	C₂×M₁₆	—	62	8	no	—
72	C₂²×C₈	—	31	4	no	—
73	#117	Dic₃₁	1	4	yes	—
74	#117	C₂×D₃₁	1	4	yes	—
75	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
76	C₂×C₂²⋊C₄	Dic₃₁	1	2	yes	—
77	C₂×C₄²	C₂×D₃₁	1	2	yes	—
78	C₄⋊C₈	—	31	16	no	—
79	C₃₁	#2905	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
81	C₂²×C₄	Dic₆₂	1	4	yes	—
82	C₂²×C₄	D₁₂₄	1	4	yes	—
83	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
84	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
86	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
88	M₁₆	—	62	8	no	—
89	M₁₆	—	62	8	no	—
90	C₂×C₈	—	31	8	no	—
91	C₂×C₈	—	62	8	no	—
92	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	—
94	C₄²	C₂×Dic₃₁	1	4	yes	—
95	C₄²	C₂²×D₃₁	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₃₁	1	4	yes	—
98	C₂³	C₂²×Dic₃₁	1	2	yes	—
99	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₃₁	1	8	yes	—
103	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
104	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
105	C₂×C₄	C₂×D₁₂₄	1	4	yes	—
106	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
107	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
108	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
110	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
111	C₈	—	62	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₃₁	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₃₁	1	2	yes	—
117	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₃₁	1	4	yes	—
120	C₄	C₂×C₄⋊Dic₃₁	1	2	yes	—
121	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
122	C₄	C₂²×D₁₂₄	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₃₁	1	1	yes	characteristic
126	C₂	#14521816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#92474

C₂

yes

maximal

#14521076

C₂

yes

maximal, complemented

#14521073

C₂

yes

maximal, complemented

C₂×C₃₁×#117

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₄

yes

—

C₃₁×#117

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₂²⋊C₄×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₂×#92474

—

C₂×#92474

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

complemented

C₂⁴×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂³

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₃₁⋊C₈

—

#2905

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×M₁₆

—

C₂×#117

D₃₁

yes

characteristic

#424

—

#423

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#568

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

Dic₃₁

yes

—

#117

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₂²⋊C₄

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

—

C₄⋊C₈

—

C₃₁

#2905

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

Dic₆₂

yes

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₂²⋊C₄

C₂×Dic₃₁

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

—

C₂³

C₄⋊Dic₃₁

yes

—

C₂³

C₂²×Dic₃₁

yes

—

C₂³

C₂²×Dic₃₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₃₁

yes

—

102

C₂×C₄

C₄⋊Dic₃₁

yes

—

103

C₂×C₄

C₂×Dic₆₂

yes

—

104

C₂×C₄

C₂×Dic₆₂

yes

—

105

C₂×C₄

C₂×D₁₂₄

yes

—

106

C₂×C₄

C₂×D₁₂₄

yes

—

107

C₂×C₄

C₂²×Dic₃₁

yes

—

108

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

109

C₂×C₄

C₂²×Dic₃₁

yes

—

110

C₂×C₄

C₂³×D₃₁

yes

characteristic

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₃₁

yes

—

116

C₂²

C₂×C₄⋊Dic₃₁

yes

—

117

C₂²

C₂³×Dic₃₁

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₃₁

yes

—

120

C₄

C₂×C₄⋊Dic₃₁

yes

—

121

C₄

C₂²×Dic₆₂

yes

characteristic

122

C₄

C₂²×D₁₂₄

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₃₁

yes

characteristic

126

C₂

#14521816

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×#117	C₂
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	#2905
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₃₁×#117

C₂

prime radical

C₂⁴×C₃₁

C₂×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #14521076	C₂	semidirect
3. #14521073	C₂	semidirect
15. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
79. C₃₁	#2905	semidirect

H ≅ G/N

type

2. #14521076

C₂

semidirect

3. #14521073

C₂

semidirect

15. C₄⋊(C₃₁⋊C₈)

C₂²

semidirect

79. C₃₁

#2905

semidirect

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

subgroup	index
C₂²×#92474	2
#14521076	2
#14521073	2
C₂×C₃₁×#117	2
#2905	31

subgroup

index

C₂²×#92474

#14521076

#14521073

C₂×C₃₁×#117

#2905

Subgroups — #16854275

Conjugacy classes of subgroups

Complements

Maximal subgroups