Group #24916530 — subgroups

Subgroups — #24916530

Order 10624.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#19127779	C₂	1	4	yes	maximal, complemented
2	D₄×C₈₃⋊C₈	C₂	1	4	yes	maximal, complemented
3	#19127754	C₂	1	2	yes	maximal, complemented
4	#19127668	C₂	1	4	yes	maximal, complemented
5	#532×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	4	yes	—
7	C₂²×C₈₃⋊C₈	C₂²	1	4	yes	—
8	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
11	C₂²⋊(C₈₃⋊C₈)	C₂²	1	8	yes	—
12	C₄×C₈₃⋊C₈	C₂²	1	4	yes	—
13	C₂×#41×C₈₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₈₃	C₄	1	1	yes	characteristic
15	#127×C₈₃	C₄	1	1	yes	characteristic
16	#124×C₈₃	C₄	1	2	yes	—
17	#121×C₈₃	C₄	1	4	yes	—
18	C₄×D₄×C₈₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	#13050574	—	4	4	no	—
21	C₂×C₈₃⋊C₈	—	4	4	no	—
22	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
23	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
24	#41×C₈₃	—	4	2	no	—
25	#41×C₈₃	—	4	2	no	—
26	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₈₃	C₂×C₄	1	8	yes	—
35	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₃	C₂³	1	2	yes	—
38	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
39	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₈₃	—	4	2	no	—
41	D₄×C₈₃	—	2	8	no	—
42	D₄×C₈₃	—	4	2	no	—
43	D₄×C₈₃	—	4	4	no	—
44	C₂×C₄×C₈₃	—	2	4	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	2	4	no	—
48	C₂×C₄×C₈₃	—	2	4	no	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
54	C₈₃⋊C₈	—	2	4	no	—
55	C₈₃⋊C₈	—	2	4	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	4	no	—
58	C₂²×C₈₃	—	2	2	no	—
59	C₂²×C₈₃	—	2	8	no	—
60	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₈₃	—	2	1	no	—
62	C₄×C₈₃	—	2	4	no	—
63	C₄×C₈₃	—	2	2	no	—
64	C₄×C₈₃	#131	1	4	yes	—
65	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₈₃	—	4	1	no	—
68	C₂×C₈₃	—	4	4	no	—
69	C₂×C₈₃	#518	1	1	yes	characteristic
70	C₂×C₈₃	C₂×#131	1	2	yes	—
71	#2979	—	83	1	no	maximal, Sylow
72	C₈₃	#2979	1	1	yes	characteristic, complemented, Sylow
73	#532	D₈₃	1	1	yes	characteristic
74	#436	—	83	4	no	—
75	C₈×D₄	—	83	4	no	—
76	#431	—	83	2	no	—
77	#408	—	83	4	no	—
78	C₂×#41	C₂×D₈₃	1	2	yes	—
79	C₂×M₁₆	—	83	4	no	—
80	C₂²×C₈	—	83	4	no	—
81	C₄⋊Q₈	Dic₈₃	1	1	yes	characteristic
82	#127	Dic₈₃	1	1	yes	characteristic
83	#124	Dic₈₃	1	2	yes	—
84	#121	Dic₈₃	1	4	yes	—
85	C₄×D₄	C₂×D₈₃	1	4	yes	—
86	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
87	C₄⋊C₈	—	83	2	no	—
88	C₄⋊C₈	—	83	2	no	—
89	C₂²⋊C₈	—	83	4	no	—
90	C₂²⋊C₈	—	83	8	no	—
91	C₄×C₈	—	83	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₈₃	1	2	yes	—
95	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
96	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
97	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
98	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
101	M₁₆	—	332	4	no	—
102	C₂×C₈	—	332	4	no	—
103	C₂×C₈	—	83	4	no	—
104	C₂×C₈	—	83	4	no	—
105	C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₈₃	1	8	yes	—
107	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
108	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
109	C₄²	C₂²×D₈₃	1	2	yes	—
110	C₂³	C₂²×Dic₈₃	1	4	yes	—
111	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
123	C₂×C₄	C₂²×Dic₈₃	1	4	yes	—
124	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
125	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
126	C₈	—	166	4	no	—
127	C₈	—	166	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₈₃	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#14581799	1	4	yes	—
137	C₄	#14581837	1	1	yes	characteristic
138	C₄	#14581837	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#19128176	1	1	yes	characteristic
142	C₂	C₂×#14581799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127779

C₂

yes

maximal, complemented

D₄×C₈₃⋊C₈

C₂

yes

maximal, complemented

#19127754

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

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

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

—

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

—

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

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#131

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#518

yes

characteristic

C₂×C₈₃

C₂×#131

yes

—

#2979

—

maximal, Sylow

C₈₃

#2979

yes

characteristic, complemented, Sylow

#532

D₈₃

yes

characteristic

#436

—

C₈×D₄

—

#431

—

#408

—

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

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#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

100

C₂²×C₄

C₂²×D₈₃

yes

—

101

M₁₆

—

332

—

102

C₂×C₈

—

332

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

C₂×Dic₈₃

yes

—

106

C₂²⋊C₄

C₂×Dic₈₃

yes

—

107

C₄²

C₂×Dic₈₃

yes

characteristic

108

C₄²

C₂×Dic₈₃

yes

characteristic

109

C₄²

C₂²×D₈₃

yes

—

110

C₂³

C₂²×Dic₈₃

yes

—

111

C₂³

C₂²×Dic₈₃

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

122

C₂×C₄

C₂²×Dic₈₃

yes

—

123

C₂×C₄

C₂²×Dic₈₃

yes

—

124

C₂×C₄

C₂²×Dic₈₃

yes

—

125

C₂×C₄

C₂³×D₈₃

yes

characteristic

126

C₈

—

166

—

127

C₈

—

166

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₈₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#14581799

yes

—

137

C₄

#14581837

yes

characteristic

138

C₄

#14581837

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#19128176

yes

characteristic

142

C₂

C₂×#14581799

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	#2979
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. #19127779	C₂	semidirect
2. D₄×C₈₃⋊C₈	C₂	semidirect
3. #19127754	C₂	semidirect
4. #19127668	C₂	semidirect
72. C₈₃	#2979	semidirect

H ≅ G/N

type

1. #19127779

C₂

semidirect

2. D₄×C₈₃⋊C₈

C₂

semidirect

3. #19127754

C₂

semidirect

4. #19127668

C₂

semidirect

72. C₈₃

#2979

semidirect

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

subgroup	index
#19127779	2
D₄×C₈₃⋊C₈	2
#19127754	2
#19127668	2
#532×C₈₃	2
#2979	83

subgroup

index

D₄×C₈₃⋊C₈

#532×C₈₃

Subgroups — #24916530

Conjugacy classes of subgroups

Complements

Maximal subgroups