Group #16854087 — subgroups

Subgroups — #16854087

Order 3968.

504 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#11758465	C₂	1	2	yes	maximal
2	#14520953	C₂	1	8	yes	maximal
3	#14520915	C₂	1	1	yes	maximal, characteristic
4	#14520914	C₂	1	2	yes	maximal
5	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₂	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	#11758465	—	2	8	no	—
16	C₃₁×#95	C₂²	1	4	yes	—
17	C₃₁⋊#95	C₂²	1	8	yes	—
18	C₃₁⋊#95	C₂²	1	4	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₁	C₂³	1	4	yes	—
23	C₂²×Dic₃₁	C₂³	1	2	yes	—
24	C₂²×Dic₃₁	C₂³	1	2	yes	—
25	C₂×Dic₆₂	—	2	8	no	—
26	C₂×Dic₆₂	—	2	16	no	—
27	C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₄²×C₃₁	Q₈	1	4	yes	complemented
29	C₄⋊Dic₃₁	—	2	4	no	—
30	Dic₃₁⋊C₄	—	2	8	no	—
31	C₄×Dic₃₁	—	2	8	no	—
32	C₄×Dic₃₁	D₄	1	4	yes	—
33	C₄×Dic₃₁	Q₈	1	4	yes	complemented
34	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	—	2	8	no	—
38	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
40	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
42	C₂×Dic₃₁	—	2	4	no	—
43	C₂×Dic₃₁	—	2	4	no	—
44	C₂×Dic₃₁	—	2	8	no	—
45	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
46	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
47	C₂×Dic₃₁	#41	1	8	yes	—
48	Dic₆₂	—	4	16	no	—
49	#2862	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×#41	1	4	yes	—
54	C₄×C₃₁	—	2	4	no	—
55	C₄×C₃₁	—	4	4	no	—
56	C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
57	Dic₃₁	—	2	4	no	—
58	Dic₃₁	—	4	4	no	—
59	Dic₃₁	#124	1	8	yes	—
60	C₂×C₄⋊Q₈	—	31	2	no	—
61	#391	—	31	8	no	—
62	#382	—	31	2	no	—
63	#382	—	31	1	no	—
64	#382	D₃₁	1	1	yes	characteristic
65	C₄³	—	31	1	no	—
66	C₂×C₃₁	C₂×#124	1	2	yes	—
67	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₃₁	#533	1	2	yes	—
69	C₂×C₃₁	#533	1	2	yes	—
70	C₂²×Q₈	—	31	4	no	—
71	C₄⋊Q₈	—	62	8	no	—
72	C₂×C₄⋊C₄	—	31	2	no	—
73	C₂×C₄⋊C₄	—	31	4	no	—
74	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
75	C₂×C₄²	—	31	1	no	—
76	C₂×C₄²	—	31	1	no	—
77	C₂×C₄²	—	31	4	no	—
78	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
79	#95	—	31	8	no	—
80	#95	—	31	4	no	—
81	#95	C₂×D₃₁	1	4	yes	—
82	C₃₁	#2862	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	62	8	no	—
84	C₂×Q₈	—	62	16	no	—
85	C₂²×C₄	—	31	2	no	—
86	C₂²×C₄	—	31	2	no	—
87	C₂²×C₄	—	31	4	no	—
88	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
89	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	62	8	no	—
93	C₄⋊C₄	—	62	4	no	—
94	C₄²	—	31	4	no	—
95	C₄²	—	31	4	no	—
96	C₄²	—	62	8	no	—
97	C₄²	Dic₆₂	1	4	yes	complemented
98	C₂³	C₂³×D₃₁	1	1	yes	characteristic
99	Q₈	—	124	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	31	4	no	—
104	C₂×C₄	—	31	8	no	—
105	C₂×C₄	—	31	4	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	—	62	4	no	—
108	C₂×C₄	—	62	8	no	—
109	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
110	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
111	C₂×C₄	D₄×D₃₁	1	2	yes	—
112	C₂×C₄	Q₈×D₃₁	1	2	yes	—
113	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
114	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
115	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
116	C₂²	#41×D₃₁	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	31	8	no	—
120	C₄	—	62	4	no	—
121	C₄	—	124	4	no	—
122	C₄	#11758465	1	4	yes	—
123	C₂	#124×D₃₁	1	2	yes	—
124	C₂	C₂×#11758465	1	1	yes	characteristic
125	C₂	#14521602	1	2	yes	—
126	C₂	#14521605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758465

C₂

yes

maximal

#14520953

C₂

yes

maximal

#14520915

C₂

yes

maximal, characteristic

#14520914

C₂

yes

maximal

C₄²×Dic₃₁

C₂

yes

maximal, characteristic

C₃₁×#382

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

#11758465

—

C₃₁×#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

Q₈

yes

complemented

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

Q₈

yes

complemented

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

#41

yes

—

Dic₆₂

—

#2862

—

maximal, Sylow

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#124

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₃₁

yes

characteristic

C₄³

—

C₂×C₃₁

C₂×#124

yes

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#533

yes

—

C₂×C₃₁

#533

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

C₂×D₃₁

yes

—

C₃₁

#2862

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆₂

yes

complemented

C₂³

C₂³×D₃₁

yes

characteristic

Q₈

—

124

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Dic₆₂

yes

—

110

C₂×C₄

C₂×Dic₆₂

yes

—

111

C₂×C₄

D₄×D₃₁

yes

—

112

C₂×C₄

Q₈×D₃₁

yes

—

113

C₂²

C₂²×Dic₆₂

yes

characteristic

114

C₂²

C₂×D₄×D₃₁

yes

characteristic

115

C₂²

C₂×Q₈×D₃₁

yes

characteristic

116

C₂²

#41×D₃₁

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

124

—

122

C₄

#11758465

yes

—

123

C₂

#124×D₃₁

yes

—

124

C₂

C₂×#11758465

yes

characteristic

125

C₂

#14521602

yes

—

126

C₂

#14521605

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₃₁×#382	C₂
hypercenter	#382	D₃₁
hypocenter	C₃₁	#2862
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₃₁×#382

C₂

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄²×C₃₁	Q₈	semidirect
33. C₄×Dic₃₁	Q₈	semidirect
82. C₃₁	#2862	semidirect
97. C₄²	Dic₆₂	semidirect

H ≅ G/N

type

28. C₄²×C₃₁

Q₈

semidirect

33. C₄×Dic₃₁

Q₈

semidirect

82. C₃₁

#2862

semidirect

97. C₄²

Dic₆₂

semidirect

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

subgroup	index
C₂×#11758465	2
#14520953	2
#14520915	2
#14520914	2
C₄²×Dic₃₁	2
C₃₁×#382	2
#2862	31

subgroup

index

C₄²×Dic₃₁

C₃₁×#382

#2862

Subgroups — #16854087

Conjugacy classes of subgroups

Complements

Maximal subgroups