Group #24916397 — subgroups

Subgroups — #24916397

Order 10624.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#19128258	C₂	1	1	yes	maximal, characteristic
2	#19127831	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₁₆₆	C₂	1	4	yes	maximal, complemented
4	#19127727	C₂	1	2	yes	maximal
5	#19127725	C₂	1	1	yes	maximal, characteristic
6	#19127656	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	2	yes	—
9	#14581792	C₄	1	2	yes	—
10	#14581784	C₄	1	2	yes	—
11	#14581771	C₄	1	4	yes	—
12	C₄×Dic₁₆₆	C₂²	1	4	yes	—
13	#14581751	C₂²	1	2	yes	—
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
15	Dic₈₃⋊C₈	C₂²	1	8	yes	—
16	#14581702	C₂²	1	4	yes	—
17	C₈×Dic₈₃	C₂²	1	4	yes	—
18	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₈₃	C₂²	1	2	yes	—
22	C₄×C₈×C₈₃	C₂²	1	2	yes	—
23	C₂×Dic₁₆₆	C₂×C₄	1	4	yes	—
24	#13050574	—	2	4	no	—
25	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
28	Dic₈₃⋊C₄	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
30	C₄×Dic₈₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₈₃	—	2	2	no	—
34	C₂×C₈×C₈₃	Q₈	1	4	yes	complemented
35	C₂×C₈×C₈₃	C₂³	1	2	yes	—
36	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₈₃	C₂³	1	2	yes	—
41	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	—	2	4	no	—
43	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₈₃	#41	1	2	yes	—
48	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₈₃	C₂²×C₄	1	8	yes	—
50	Dic₁₆₆	—	2	8	no	—
51	C₈×C₈₃	—	2	2	no	—
52	C₈×C₈₃	C₂×Q₈	1	2	yes	—
53	C₈×C₈₃	C₂×Q₈	1	2	yes	—
54	C₈₃⋊C₈	—	2	4	no	—
55	C₂²×C₈₃	—	2	2	no	—
56	C₂²×C₈₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₈₃	—	2	2	no	—
59	C₄×C₈₃	C₄×Q₈	1	2	yes	—
60	C₄×C₈₃	#131	1	4	yes	—
61	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
63	Dic₈₃	—	2	8	no	—
64	C₂×C₈₃	—	2	2	no	—
65	C₂×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₈₃	C₂×#131	1	2	yes	—
67	#2960	—	83	1	no	maximal, Sylow
68	C₈₃	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	83	1	no	—
70	#446	—	83	4	no	—
71	C₈×Q₈	—	83	4	no	—
72	#424	—	83	2	no	—
73	#423	—	83	1	no	—
74	#405	—	83	2	no	—
75	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
76	C₂×M₁₆	—	83	2	no	—
77	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
79	C₄⋊Q₈	—	83	2	no	—
80	#125	—	83	2	no	—
81	C₂²⋊Q₈	—	83	4	no	—
82	C₄×Q₈	—	83	4	no	—
83	#117	—	83	2	no	—
84	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
85	C₄⋊C₈	—	83	8	no	—
86	C₄⋊C₈	—	83	4	no	—
87	#97	—	83	4	no	—
88	C₄×C₈	—	83	4	no	—
89	C₄×C₈	C₂×D₈₃	1	2	yes	—
90	C₄×C₈	C₂×D₈₃	1	2	yes	—
91	C₂×Q₈	—	83	4	no	—
92	C₂²×C₄	C₄×D₈₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
94	M₁₆	—	166	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	83	4	no	—
97	C₂×C₈	Dic₁₆₆	1	4	yes	complemented
98	C₂×C₈	C₂²×D₈₃	1	2	yes	—
99	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
101	C₄⋊C₄	—	83	4	no	—
102	C₄⋊C₄	—	83	8	no	—
103	C₄⋊C₄	—	83	4	no	—
104	C₂²⋊C₄	—	83	4	no	—
105	C₄²	—	83	4	no	—
106	C₄²	C₄×D₈₃	1	1	yes	characteristic
107	C₄²	C₄×D₈₃	1	1	yes	characteristic
108	C₄²	C₂²×D₈₃	1	2	yes	—
109	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
110	Q₈	—	166	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	83	8	no	—
113	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
117	C₂×C₄	#13050590	1	2	yes	—
118	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	166	4	no	—
121	C₈	C₂×Dic₁₆₆	1	2	yes	—
122	C₈	C₂×Dic₁₆₆	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₁₆₆	1	2	yes	—
125	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	166	8	no	—
128	C₄	C₄×Dic₁₆₆	1	2	yes	—
129	C₄	#14581798	1	4	yes	—
130	C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
131	C₄	C₂×#13050590	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
134	C₂	C₂×#14581798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

#19127831

C₂

yes

maximal, complemented

C₈×Dic₁₆₆

C₂

yes

maximal, complemented

#19127727

C₂

yes

maximal

#19127725

C₂

yes

maximal, characteristic

#19127656

C₂

yes

maximal

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

#14581792

C₄

yes

—

#14581784

C₄

yes

—

#14581771

C₄

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581751

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

Dic₈₃⋊C₈

C₂²

yes

—

#14581702

C₂²

yes

—

C₈×Dic₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

#13050574

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

Q₈

yes

complemented

C₂×C₈×C₈₃

C₂³

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

C₂²×C₄

yes

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×Q₈

yes

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₄×Q₈

yes

—

C₄×C₈₃

#131

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

C₂×#41

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₈₃

C₂×#131

yes

—

#2960

—

maximal, Sylow

C₈₃

#2960

yes

characteristic, complemented, Sylow

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₈₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

—

C₄×C₈

C₂×D₈₃

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₆₆

yes

complemented

C₂×C₈

C₂²×D₈₃

yes

—

C₂×C₈

C₂²×D₈₃

yes

characteristic

100

C₂×C₈

C₂²×D₈₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₈₃

yes

characteristic

107

C₄²

C₄×D₈₃

yes

characteristic

108

C₄²

C₂²×D₈₃

yes

—

109

C₂³

C₂×C₄×D₈₃

yes

characteristic

110

Q₈

—

166

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₈₃

yes

—

114

C₂×C₄

C₂×C₄×D₈₃

yes

—

115

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

116

C₂×C₄

C₂×Dic₁₆₆

yes

—

117

C₂×C₄

#13050590

yes

—

118

C₂×C₄

C₂³×D₈₃

yes

characteristic

119

C₈

—

120

C₈

—

166

—

121

C₈

C₂×Dic₁₆₆

yes

—

122

C₈

C₂×Dic₁₆₆

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₁₆₆

yes

—

125

C₂²

C₂²×C₄×D₈₃

yes

characteristic

126

C₄

—

127

C₄

—

166

—

128

C₄

C₄×Dic₁₆₆

yes

—

129

C₄

#14581798

yes

—

130

C₄

C₂²×Dic₁₆₆

yes

characteristic

131

C₄

C₂×#13050590

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₁₆₆

yes

characteristic

134

C₂

C₂×#14581798

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#2960
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₈₃

commutator, socle

C₂²×C₈₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₂×C₄×C₈×C₈₃

C₂

prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂×C₄×C₈₃

C₂⁴

hypercenter

C₂×C₄×C₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19127831	C₂	semidirect
3. C₈×Dic₁₆₆	C₂	semidirect
34. C₂×C₈×C₈₃	Q₈	semidirect
68. C₈₃	#2960	semidirect
97. C₂×C₈	Dic₁₆₆	semidirect

H ≅ G/N

type

2. #19127831

C₂

semidirect

3. C₈×Dic₁₆₆

C₂

semidirect

34. C₂×C₈×C₈₃

Q₈

semidirect

68. C₈₃

#2960

semidirect

97. C₂×C₈

Dic₁₆₆

semidirect

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

subgroup	index
#19128258	2
#19127831	2
C₈×Dic₁₆₆	2
#19127727	2
#19127725	2
#19127656	2
C₂×C₄×C₈×C₈₃	2
#2960	83

subgroup

index

C₈×Dic₁₆₆

C₂×C₄×C₈×C₈₃

Subgroups — #24916397

Conjugacy classes of subgroups

Complements

Maximal subgroups