Group #24908403 — subgroups

Subgroups — #24908403

Order 10624.

280 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19127455	C₂	1	4	yes	maximal
4	C₂×#97×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	C₄	1	1	yes	characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₁₆₆	—	2	2	no	—
8	C₄×Dic₁₆₆	—	2	2	no	—
9	C₂×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
10	C₂×Dic₈₃⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
12	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
13	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	#97×C₈₃	—	2	2	no	—
16	C₂×Dic₁₆₆	—	2	2	no	—
17	C₂×Dic₁₆₆	—	2	4	no	—
18	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
19	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	4	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	Dic₈₃⋊C₄	—	2	4	no	—
24	C₄×Dic₈₃	—	2	2	no	—
25	C₄×Dic₈₃	—	2	2	no	—
26	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₈₃	—	2	2	no	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	Q₈	1	4	yes	complemented
32	C₄²×C₈₃	D₄	1	2	yes	—
33	C₄²×C₈₃	D₄	1	2	yes	—
34	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	—	2	2	no	—
36	C₂×C₄×C₈₃	—	2	2	no	—
37	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₈₃	#41	1	2	yes	—
43	C₂×Dic₈₃	—	2	2	no	—
44	C₂×Dic₈₃	—	2	2	no	—
45	C₂×Dic₈₃	—	2	2	no	—
46	C₂×Dic₈₃	—	2	4	no	—
47	C₂×Dic₈₃	M₁₆	1	4	yes	—
48	Dic₁₆₆	—	2	8	no	—
49	C₈×C₈₃	—	2	4	no	—
50	C₈₃⋊C₈	—	4	2	no	—
51	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
54	C₂²×C₈₃	#131	1	2	yes	—
55	C₄×C₈₃	—	2	2	no	—
56	C₄×C₈₃	—	2	2	no	—
57	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₈₃	#124	1	1	yes	characteristic
59	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₈₃	—	2	4	no	—
61	Dic₈₃	—	4	2	no	—
62	C₂×C₈₃	#391	1	1	yes	characteristic
63	C₂×C₈₃	#407	1	1	yes	characteristic
64	C₂×C₈₃	#408	1	1	yes	characteristic
65	C₂×C₈₃	#446	1	4	yes	—
66	#1985	—	83	1	no	maximal, Sylow
67	C₈₃	#1985	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	—	83	1	no	—
69	C₂×C₄⋊C₈	—	83	1	no	—
70	C₂×#97	D₈₃	1	1	yes	characteristic
71	#336	—	83	4	no	—
72	C₂²×Q₈	—	83	1	no	—
73	C₂²×C₈	—	83	2	no	—
74	C₂²×C₈	C₂×D₈₃	1	2	yes	—
75	C₄×Q₈	—	166	2	no	—
76	C₄×Q₈	—	166	2	no	—
77	C₂×C₄⋊C₄	—	83	2	no	—
78	C₂×C₄⋊C₄	—	83	1	no	—
79	C₂×C₄²	—	83	2	no	—
80	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
81	C₄⋊C₈	—	166	2	no	—
82	#97	—	2	2	no	—
83	C₂×Q₈	—	166	4	no	—
84	C₂×Q₈	—	166	2	no	—
85	C₂²×C₄	—	83	2	no	—
86	C₂²×C₄	—	83	2	no	—
87	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	166	2	no	—
93	C₂×C₈	—	166	4	no	—
94	C₂×C₈	Dic₁₆₆	1	4	yes	complemented
95	C₄⋊C₄	—	166	4	no	—
96	C₄⋊C₄	—	166	2	no	—
97	C₄²	—	166	2	no	—
98	C₄²	—	166	2	no	—
99	C₄²	D₃₃₂	1	2	yes	—
100	C₄²	C₂²⋊D₈₃	1	2	yes	—
101	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
102	Q₈	—	166	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	83	4	no	—
106	C₂×C₄	—	166	4	no	—
107	C₂×C₄	—	166	2	no	—
108	C₂×C₄	—	166	2	no	—
109	C₂×C₄	—	166	2	no	—
110	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
111	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
113	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
115	C₂×C₄	#13050590	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	332	2	no	—
118	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
119	C₂²	C₄×Dic₁₆₆	1	2	yes	—
120	C₂²	M₁₆×D₈₃	1	2	yes	—
121	C₂²	#14581800	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	332	2	no	—
125	C₄	—	166	4	no	—
126	C₄	#14581771	1	2	yes	—
127	C₄	#14581780	1	1	yes	characteristic
128	C₄	#14581792	1	1	yes	characteristic
129	C₂	#19127607	1	1	yes	characteristic
130	C₂	#19127664	1	1	yes	characteristic
131	C₂	#19127669	1	1	yes	characteristic
132	C₂	#19127829	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₆₆

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127455

C₂

yes

maximal

C₂×#97×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₄

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

—

C₂×C₄⋊Dic₈₃

C₄

yes

characteristic

C₂×Dic₈₃⋊C₄

C₄

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂²×C₈×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#97×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

Q₈

yes

complemented

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

M₁₆

yes

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₄×Q₈

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₂²×C₈₃

#131

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#124

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

#1985

—

maximal, Sylow

C₈₃

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₈₃

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

—

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

#97

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

Dic₁₆₆

yes

complemented

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

D₃₃₂

yes

—

100

C₄²

C₂²⋊D₈₃

yes

—

101

C₂³

C₂×C₄×D₈₃

yes

characteristic

102

Q₈

—

166

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

166

—

107

C₂×C₄

—

166

—

108

C₂×C₄

—

166

—

109

C₂×C₄

—

166

—

110

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

111

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

112

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

113

C₂×C₄

C₂×D₃₃₂

yes

characteristic

114

C₂×C₄

C₂×Dic₁₆₆

yes

—

115

C₂×C₄

#13050590

yes

—

116

C₈

—

117

C₈

—

332

—

118

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

119

C₂²

C₄×Dic₁₆₆

yes

—

120

C₂²

M₁₆×D₈₃

yes

—

121

C₂²

#14581800

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

332

—

125

C₄

—

166

—

126

C₄

#14581771

yes

—

127

C₄

#14581780

yes

characteristic

128

C₄

#14581792

yes

characteristic

129

C₂

#19127607

yes

characteristic

130

C₂

#19127664

yes

characteristic

131

C₂

#19127669

yes

characteristic

132

C₂

#19127829

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator, socle, prime radical	C₂³×C₈₃	C₂²×C₄
Fitting	C₂×#97×C₈₃	C₂
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×#97	D₈₃
hypocenter	C₈₃	#1985
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator, socle, prime radical

C₂³×C₈₃

C₂²×C₄

Fitting

C₂×#97×C₈₃

C₂

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×#97

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
31. C₂×C₈×C₈₃	Q₈	semidirect
67. C₈₃	#1985	semidirect
94. C₂×C₈	Dic₁₆₆	semidirect

H ≅ G/N

type

31. C₂×C₈×C₈₃

Q₈

semidirect

67. C₈₃

#1985

semidirect

94. C₂×C₈

Dic₁₆₆

semidirect

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

subgroup	index
C₂×C₄×Dic₁₆₆	2
C₂×C₄⋊(C₈₃⋊C₈)	2
#19127455	2
C₂×#97×C₈₃	2
#1985	83

subgroup

index

C₂×C₄×Dic₁₆₆

C₂×C₄⋊(C₈₃⋊C₈)

#19127455

C₂×#97×C₈₃

#1985

Subgroups — #24908403

Conjugacy classes of subgroups

Complements

Maximal subgroups