Group #24916534 — subgroups

Subgroups — #24916534

Order 10624.

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

normal subgroups only

This group has 394 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	#19127773	C₂	1	8	yes	maximal, complemented
2	#19127758	C₂	1	2	yes	maximal, complemented
3	#19127668	C₂	1	4	yes	maximal, complemented
4	#532×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13050574	C₂²	1	4	yes	—
6	C₂²×C₈₃⋊C₈	C₂²	1	4	yes	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
8	C₂²⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
9	C₂²⋊(C₈₃⋊C₈)	C₂²	1	8	yes	—
10	#14581703	C₂²	1	4	yes	—
11	C₂×#41×C₈₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₈₃	C₄	1	1	yes	characteristic
13	#127×C₈₃	C₄	1	1	yes	characteristic
14	#124×C₈₃	C₄	1	2	yes	—
15	#121×C₈₃	C₄	1	4	yes	—
16	C₄×D₄×C₈₃	C₂²	1	4	yes	—
17	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
18	#13050574	—	4	4	no	—
19	C₂×C₈₃⋊C₈	—	4	4	no	—
20	C₂×C₈₃⋊C₈	C₂³	1	8	yes	—
21	#41×C₈₃	—	4	2	no	—
22	#41×C₈₃	—	4	2	no	—
23	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
26	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₈₃	C₂×C₄	1	8	yes	—
32	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₈₃	C₂³	1	2	yes	—
35	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
36	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₈₃	—	4	2	no	—
38	D₄×C₈₃	—	4	4	no	—
39	D₄×C₈₃	—	4	2	no	—
40	D₄×C₈₃	—	4	4	no	—
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	—	2	4	no	—
43	C₂×C₄×C₈₃	—	2	4	no	—
44	C₂×C₄×C₈₃	—	4	2	no	—
45	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₈₃⋊C₈	—	2	8	no	—
51	C₂²×C₈₃	—	2	1	no	—
52	C₂²×C₈₃	—	2	4	no	—
53	C₂²×C₈₃	—	2	2	no	—
54	C₂²×C₈₃	—	2	8	no	—
55	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₈₃	—	2	1	no	—
57	C₄×C₈₃	—	2	4	no	—
58	C₄×C₈₃	—	2	2	no	—
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₈₃	—	4	1	no	—
62	C₂×C₈₃	—	4	4	no	—
63	C₂×C₈₃	#518	1	1	yes	characteristic
64	C₂×C₈₃	#568	1	2	yes	—
65	#2980	—	83	1	no	maximal, Sylow
66	C₈₃	#2980	1	1	yes	characteristic, complemented, Sylow
67	#532	D₈₃	1	1	yes	characteristic
68	#435	—	83	8	no	—
69	#432	—	83	2	no	—
70	#408	—	83	4	no	—
71	C₂×#41	C₂×D₈₃	1	2	yes	—
72	C₂×M₁₆	—	83	4	no	—
73	C₂²×C₈	—	83	4	no	—
74	C₄⋊Q₈	Dic₈₃	1	1	yes	characteristic
75	#127	Dic₈₃	1	1	yes	characteristic
76	#124	Dic₈₃	1	2	yes	—
77	#121	Dic₈₃	1	4	yes	—
78	C₄×D₄	C₂×D₈₃	1	4	yes	—
79	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
80	C₄⋊C₈	—	83	4	no	—
81	C₂²⋊C₈	—	83	4	no	—
82	C₂²⋊C₈	—	83	8	no	—
83	#97	—	83	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₈₃	1	2	yes	—
87	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
88	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
89	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
90	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
93	M₁₆	—	332	4	no	—
94	C₂×C₈	—	332	4	no	—
95	C₂×C₈	—	83	8	no	—
96	C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₈₃	1	8	yes	—
98	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
100	C₄²	C₂²×D₈₃	1	2	yes	—
101	C₂³	C₂²×Dic₈₃	1	4	yes	—
102	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
113	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
114	C₂×C₄	C₂²×Dic₈₃	1	4	yes	—
115	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
116	C₈	—	166	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₈₃	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#14581837	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#19128176	1	1	yes	characteristic
130	C₂	#19128468	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127773

C₂

yes

maximal, complemented

#19127758

C₂

yes

maximal, complemented

#19127668

C₂

yes

maximal, complemented

#532×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581703

C₂²

yes

—

C₂×#41×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₄

yes

characteristic

#127×C₈₃

C₄

yes

characteristic

#124×C₈₃

C₄

yes

—

#121×C₈₃

C₄

yes

—

C₄×D₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂³

yes

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂²×C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

H(5,𝔽₂)

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#518

yes

characteristic

C₂×C₈₃

#568

yes

—

#2980

—

maximal, Sylow

C₈₃

#2980

yes

characteristic, complemented, Sylow

#532

D₈₃

yes

—

—

—

C₂×#41

C₂×D₈₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₈₃

yes

characteristic

#127

Dic₈₃

yes

characteristic

#124

Dic₈₃

yes

—

#121

Dic₈₃

yes

—

C₄×D₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₈₃

yes

—

C₂×D₄

C₂×Dic₈₃

yes

—

C₂×D₄

C₂×Dic₈₃

yes

—

C₂×D₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

—

M₁₆

—

332

—

C₂×C₈

—

332

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₂²⋊C₄

C₂×Dic₈₃

yes

—

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

100

C₄²

C₂²×D₈₃

yes

—

101

C₂³

C₂²×Dic₈₃

yes

—

102

C₂³

C₂²×Dic₈₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

112

C₂×C₄

C₂²×Dic₈₃

yes

—

113

C₂×C₄

C₂²×Dic₈₃

yes

—

114

C₂×C₄

C₂²×Dic₈₃

yes

—

115

C₂×C₄

C₂³×D₈₃

yes

characteristic

116

C₈

—

166

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₈₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#14581837

yes

—

127

C₂

—

128

C₂

—

129

C₂

#19128176

yes

characteristic

130

C₂

#19128468

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting, prime radical	#532×C₈₃	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#532	D₈₃
hypocenter	C₈₃	#2980
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₃

commutator, socle

C₂²×C₈₃

C₂³×C₄

Fitting, prime radical

#532×C₈₃

C₂

cosocle

C₂×C₄×C₈₃

C₂⁴

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #19127773	C₂	semidirect
2. #19127758	C₂	semidirect
3. #19127668	C₂	semidirect
66. C₈₃	#2980	semidirect

H ≅ G/N

type

1. #19127773

C₂

semidirect

2. #19127758

C₂

semidirect

3. #19127668

C₂

semidirect

66. C₈₃

#2980

semidirect

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

subgroup	index
#19127773	2
#19127758	2
#19127668	2
#532×C₈₃	2
#2980	83

subgroup

index

#532×C₈₃

Subgroups — #24916534

Conjugacy classes of subgroups

Complements

Maximal subgroups