Group #24920142 — subgroups

Subgroups — #24920142

Order 10624.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#19128430	C₂	1	1	yes	maximal, characteristic
3	#19128052	C₂	1	3	yes	maximal
4	#19127997	C₂	1	3	yes	maximal
5	#19127912	C₂	1	3	yes	maximal
6	C₄×Dic₃₃₂	C₂	1	3	yes	maximal
7	C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₃₂	C₂²	1	3	yes	—
9	#14581792	C₂²	1	3	yes	—
10	#14581792	C₂²	1	3	yes	—
11	#14581791	—	2	3	no	—
12	#14581783	—	2	3	no	—
13	Q₈×Dic₈₃	—	2	1	no	—
14	Q₈×Dic₈₃	—	2	1	no	—
15	C₄×Dic₁₆₆	C₂²	1	3	yes	—
16	C₄×Dic₁₆₆	C₂²	1	3	yes	—
17	#14581726	C₂²	1	3	yes	—
18	#14581717	C₂²	1	6	yes	—
19	#14581717	C₂²	1	3	yes	—
20	#14581717	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	3	yes	—
23	C₄×C₈×C₈₃	C₂²	1	3	yes	—
24	C₂×Dic₁₆₆	—	2	3	no	—
25	C₂×Dic₁₆₆	C₂³	1	3	yes	—
26	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
27	Dic₃₃₂	—	2	6	no	—
28	C₄⋊Dic₈₃	—	2	3	no	—
29	C₄⋊Dic₈₃	—	2	3	no	—
30	C₄⋊Dic₈₃	—	2	3	no	—
31	C₄⋊Dic₈₃	C₂³	1	3	yes	—
32	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
33	Dic₈₃⋊C₄	—	2	3	no	—
34	Dic₈₃⋊C₄	—	2	3	no	—
35	C₄×Dic₈₃	—	2	3	no	—
36	C₄×Dic₈₃	—	2	3	no	—
37	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂³	1	3	yes	—
39	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₃	D₄	1	3	yes	—
41	C₄²×C₈₃	C₂³	1	3	yes	—
42	Q₈×C₈₃	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
45	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₈₃	—	2	1	no	—
47	C₂×Dic₈₃	—	2	1	no	—
48	C₂×Dic₈₃	—	2	3	no	—
49	C₂×Dic₈₃	—	2	3	no	—
50	Dic₁₆₆	—	2	1	no	—
51	Dic₁₆₆	—	2	3	no	—
52	Dic₁₆₆	—	2	3	no	—
53	Dic₁₆₆	—	8	3	no	—
54	Dic₁₆₆	#41	1	2	yes	—
55	C₈×C₈₃	—	2	3	no	—
56	C₈×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₈₃	—	2	3	no	—
59	C₄×C₈₃	C₂×Q₁₆	1	6	yes	—
60	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
61	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₈₃	—	2	2	no	—
63	Dic₈₃	—	4	3	no	—
64	Dic₈₃	—	4	3	no	—
65	Dic₈₃	—	4	1	no	—
66	C₂×C₈₃	#548	1	1	yes	characteristic
67	C₂×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₈₃	#577	1	1	yes	characteristic
69	#3363	—	83	1	no	maximal, Sylow
70	C₈₃	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	—	83	1	no	—
72	#557	—	83	1	no	—
73	#487	—	83	3	no	—
74	#477	—	83	3	no	—
75	#462	—	83	3	no	—
76	C₈×Q₈	D₈₃	1	1	yes	characteristic
77	C₄×Q₁₆	—	83	3	no	—
78	C₂×Q₁₆	—	83	3	no	—
79	C₄⋊Q₈	—	83	3	no	—
80	C₄⋊Q₈	—	83	3	no	—
81	C₄⋊Q₈	—	166	3	no	—
82	#125	—	166	3	no	—
83	C₄×Q₈	—	83	3	no	—
84	C₄×Q₈	—	83	3	no	—
85	C₄×Q₈	—	166	1	no	—
86	C₄×Q₈	—	166	1	no	—
87	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
88	#107	—	83	3	no	—
89	C₄⋊C₈	C₂×D₈₃	1	3	yes	—
90	#103	—	83	1	no	—
91	#103	—	83	3	no	—
92	#103	—	83	6	no	—
93	C₄×C₈	C₂×D₈₃	1	3	yes	—
94	C₂×Q₈	—	83	1	no	—
95	C₂×Q₈	—	83	3	no	—
96	C₂×Q₈	—	166	3	no	—
97	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
98	Q₁₆	—	166	6	no	—
99	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₈₃	1	3	yes	—
101	C₄⋊C₄	—	83	3	no	—
102	C₄⋊C₄	—	83	1	no	—
103	C₄⋊C₄	—	166	3	no	—
104	C₄⋊C₄	—	166	3	no	—
105	C₄⋊C₄	—	166	3	no	—
106	C₄⋊C₄	—	166	3	no	—
107	C₄⋊C₄	—	166	3	no	—
108	C₄⋊C₄	D₃₃₂	1	3	yes	—
109	C₄²	—	166	3	no	—
110	C₄²	—	166	3	no	—
111	C₄²	C₂²×D₈₃	1	3	yes	—
112	Q₈	—	83	2	no	—
113	Q₈	—	664	3	no	—
114	Q₈	—	166	3	no	—
115	Q₈	—	166	3	no	—
116	Q₈	—	166	1	no	—
117	Q₈	Dic₃₃₂	1	4	yes	complemented
118	C₂×C₄	—	166	1	no	—
119	C₂×C₄	—	166	3	no	—
120	C₂×C₄	—	166	3	no	—
121	C₂×C₄	—	166	1	no	—
122	C₂×C₄	C₂×D₃₃₂	1	3	yes	—
123	C₂×C₄	C₂×D₃₃₂	1	3	yes	—
124	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
125	C₈	—	2	3	no	—
126	C₈	—	2	1	no	—
127	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	332	3	no	—
130	C₄	—	332	3	no	—
131	C₄	—	332	1	no	—
132	C₄	—	166	2	no	—
133	C₄	C₂×Dic₃₃₂	1	6	yes	—
134	C₄	#41×D₈₃	1	1	yes	characteristic
135	C₄	#14581840	1	1	yes	characteristic
136	C₂	#19128363	1	1	yes	characteristic
137	C₂	C₂²×Dic₃₃₂	1	1	yes	characteristic
138	C₂	#19128513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₆₆

C₂

yes

maximal, characteristic

#19128430

C₂

yes

maximal, characteristic

#19128052

C₂

yes

maximal

#19127997

C₂

yes

maximal

#19127912

C₂

yes

maximal

C₄×Dic₃₃₂

C₂

yes

maximal

C₈×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

—

#14581792

C₂²

yes

—

#14581792

C₂²

yes

—

#14581791

—

#14581783

—

Q₈×Dic₈₃

—

Q₈×Dic₈₃

—

C₄×Dic₁₆₆

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581726

C₂²

yes

—

#14581717

C₂²

yes

—

#14581717

C₂²

yes

—

#14581717

C₂²

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

C₂×Dic₁₆₆

C₂³

yes

characteristic

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

C₂³

yes

characteristic

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

—

Q₈×C₈₃

Q₁₆

yes

complemented

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

#41

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

C₂×#41

yes

characteristic

C₄×C₈₃

2₋¹⁺⁴

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#548

yes

characteristic

C₂×C₈₃

C₂²×Q₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

C₈×Q₈

D₈₃

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

#125

—

166

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

C₄×Q₈

C₂×D₈₃

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

#103

—

#103

—

C₄×C₈

C₂×D₈₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

D₃₃₂

yes

characteristic

Q₁₆

—

166

—

C₂×C₈

C₂²×D₈₃

yes

characteristic

100

C₂×C₈

C₂²×D₈₃

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

166

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

—

166

—

108

C₄⋊C₄

D₃₃₂

yes

—

109

C₄²

—

166

—

110

C₄²

—

166

—

111

C₄²

C₂²×D₈₃

yes

—

112

Q₈

—

113

Q₈

—

664

—

114

Q₈

—

166

—

115

Q₈

—

166

—

116

Q₈

—

166

—

117

Q₈

Dic₃₃₂

yes

complemented

118

C₂×C₄

—

166

—

119

C₂×C₄

—

166

—

120

C₂×C₄

—

166

—

121

C₂×C₄

—

166

—

122

C₂×C₄

C₂×D₃₃₂

yes

—

123

C₂×C₄

C₂×D₃₃₂

yes

—

124

C₂×C₄

C₂³×D₈₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×D₃₃₂

yes

characteristic

128

C₄

—

129

C₄

—

332

—

130

C₄

—

332

—

131

C₄

—

332

—

132

C₄

—

166

—

133

C₄

C₂×Dic₃₃₂

yes

—

134

C₄

#41×D₈₃

yes

characteristic

135

C₄

#14581840

yes

characteristic

136

C₂

#19128363

yes

characteristic

137

C₂

C₂²×Dic₃₃₂

yes

characteristic

138

C₂

#19128513

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₈×Q₈×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂²×D₄
hypercenter	C₈×Q₈	D₈₃
hypocenter	C₈₃	#3363
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₃₂

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₈×Q₈×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂²×D₄

hypercenter

C₈×Q₈

D₈₃

hypocenter

C₈₃

#3363

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
42. Q₈×C₈₃	Q₁₆	semidirect
70. C₈₃	#3363	semidirect
117. Q₈	Dic₃₃₂	semidirect

H ≅ G/N

type

42. Q₈×C₈₃

Q₁₆

semidirect

70. C₈₃

#3363

semidirect

117. Q₈

Dic₃₃₂

semidirect

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

subgroup	index
Q₈×Dic₁₆₆	2
#19128430	2
#19128052	2
#19127997	2
#19127912	2
C₄×Dic₃₃₂	2
C₈×Q₈×C₈₃	2
#3363	83

subgroup

index

Q₈×Dic₁₆₆

C₄×Dic₃₃₂

C₈×Q₈×C₈₃

#3363

Subgroups — #24920142

Conjugacy classes of subgroups

Complements

Maximal subgroups