Group #24917839 — subgroups

Subgroups — #24917839

Order 10624.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581789	C₂	1	1	yes	maximal, characteristic
2	#19128067	C₂	1	4	yes	maximal, complemented
3	#19128049	C₂	1	4	yes	maximal, complemented
4	#19127705	C₂	1	4	yes	maximal, complemented
5	C₄×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₃₂	—	2	2	no	—
8	C₂²×D₃₃₂	C₂²	1	2	yes	—
9	C₂×#13050574	C₂²	1	2	yes	—
10	#14581789	—	2	2	no	—
11	#14581789	C₂²	1	2	yes	—
12	#14581789	C₂²	1	2	yes	—
13	#14581712	C₂²	1	16	yes	complemented
14	#14581703	C₂²	1	2	yes	complemented
15	C₄×C₈₃⋊C₈	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	complemented
17	#125×C₈₃	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₈₃	—	2	2	no	—
19	C₄×Q₈×C₈₃	—	2	2	no	—
20	#117×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
22	C₂³×D₈₃	—	2	2	no	—
23	C₂×D₃₃₂	—	2	8	no	—
24	C₂×D₃₃₂	—	2	2	no	—
25	C₂×D₃₃₂	—	2	4	no	—
26	C₂×D₃₃₂	—	2	4	no	—
27	C₂×D₃₃₂	—	4	8	no	—
28	C₂×D₃₃₂	C₂³	1	4	yes	—
29	#13050574	—	2	4	no	—
30	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
31	C₂×Q₈×C₈₃	—	2	2	no	—
32	C₂²×C₄×C₈₃	D₄	1	2	yes	—
33	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₈₃	—	2	4	no	—
35	C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₈₃	—	2	2	no	—
38	C₄²×C₈₃	—	2	2	no	—
39	C₄²×C₈₃	D₄	1	2	yes	—
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₄²×C₈₃	C₂³	1	1	yes	characteristic
42	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₈₃	—	2	4	no	—
44	C₂²×D₈₃	—	4	8	no	—
45	C₂²×D₈₃	—	4	4	no	—
46	Q₈×C₈₃	—	4	4	no	—
47	C₂×C₄×C₈₃	—	2	4	no	—
48	C₂×C₄×C₈₃	—	2	4	no	—
49	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₃	#41	1	4	yes	—
53	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
54	D₃₃₂	—	2	8	no	—
55	D₃₃₂	—	4	8	no	—
56	D₃₃₂	—	4	4	no	—
57	D₃₃₂	—	4	8	no	—
58	C₈₃⋊C₈	—	2	4	no	—
59	C₂²×C₈₃	—	2	2	no	—
60	C₂²×C₈₃	#124	1	2	yes	—
61	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₈₃	—	4	4	no	—
63	C₂×D₈₃	—	4	8	no	—
64	C₂×D₈₃	—	8	8	no	—
65	C₄×C₈₃	—	2	2	no	—
66	C₄×C₈₃	—	4	4	no	—
67	C₄×C₈₃	#124	1	2	yes	—
68	C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₈₃	C₂×#41	1	2	yes	—
70	C₂×C₈₃	—	2	2	no	—
71	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
72	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₈₃	—	8	4	no	—
74	#3138	—	83	1	no	maximal, Sylow
75	C₈₃	#3138	1	1	yes	characteristic, complemented, Sylow
76	#533	D₈₃	1	1	yes	characteristic, complemented
77	C₂×#127	—	83	1	no	—
78	#490	—	83	4	no	—
79	#486	—	83	4	no	—
80	#418	—	83	4	no	—
81	C₄×M₁₆	—	83	1	no	—
82	C₂²×D₄	—	83	2	no	—
83	C₂²×D₄	—	166	2	no	—
84	C₂×M₁₆	—	83	2	no	—
85	C₄⋊Q₈	C₂×D₈₃	1	2	yes	complemented
86	#127	—	83	2	no	—
87	#127	—	83	2	no	—
88	#127	—	166	2	no	—
89	#125	C₂×D₈₃	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₈₃	1	2	yes	—
93	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
94	#102	—	83	16	no	—
95	#97	—	83	2	no	—
96	C₄×C₈	—	83	2	no	—
97	C₂⁴	—	166	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	332	8	no	—
100	C₂×D₄	—	83	4	no	—
101	C₂×D₄	—	166	4	no	—
102	C₂×D₄	—	166	4	no	—
103	C₂×D₄	—	166	2	no	—
104	C₂×D₄	—	166	8	no	—
105	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
107	M₁₆	—	166	4	no	—
108	C₂×C₈	—	83	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₈₃	1	2	yes	—
115	C₄²	C₂²×D₈₃	1	1	yes	characteristic
116	C₄²	C₂²×D₈₃	1	1	yes	characteristic
117	C₂³	—	332	8	no	—
118	C₂³	—	332	4	no	—
119	C₂³	—	166	4	no	—
120	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	332	8	no	—
123	D₄	—	332	4	no	—
124	D₄	—	332	8	no	—
125	D₄	—	166	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
131	C₂×C₄	#13050591	1	4	yes	—
132	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
133	C₈	—	166	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	332	4	no	—
136	C₂²	—	332	8	no	—
137	C₂²	—	664	8	no	—
138	C₂²	#14581781	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#14581781	1	2	yes	—
143	C₄	#14581817	1	4	yes	—
144	C₄	C₂×#13050591	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	664	4	no	—
147	C₂	C₂×#14581781	1	1	yes	characteristic
148	C₂	C₂×#14581817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581789

C₂

yes

maximal, characteristic

#19128067

C₂

yes

maximal, complemented

#19128049

C₂

yes

maximal, complemented

#19127705

C₂

yes

maximal, complemented

C₄×#13050574

C₂

yes

maximal, characteristic, complemented

#533×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₃₂

—

C₂²×D₃₃₂

C₂²

yes

—

C₂×#13050574

C₂²

yes

—

#14581789

—

#14581789

C₂²

yes

—

#14581789

C₂²

yes

—

#14581712

C₂²

yes

complemented

#14581703

C₂²

yes

complemented

C₄×C₈₃⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₈₃

C₂²

yes

complemented

#125×C₈₃

C₂²

yes

complemented

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂³×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

C₂³

yes

—

#13050574

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#124

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

C₂×#41

yes

—

C₂×C₈₃

—

C₂×C₈₃

C₂×#124

yes

characteristic

C₂×C₈₃

C₂×AGL(1,ℤ/8)

yes

—

D₈₃

—

#3138

—

maximal, Sylow

C₈₃

#3138

yes

characteristic, complemented, Sylow

#533

D₈₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂²×D₄

—

C₂²×D₄

—

166

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₈₃

yes

—

—

—

166

—

#125

C₂×D₈₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

166

—

C₂×Q₈

—

C₂×D₄

—

332

—

100

C₂×D₄

—

101

C₂×D₄

—

166

—

102

C₂×D₄

—

166

—

103

C₂×D₄

—

166

—

104

C₂×D₄

—

166

—

105

C₂²×C₄

C₂²⋊D₈₃

yes

—

106

C₂²×C₄

C₂²×D₈₃

yes

characteristic

107

M₁₆

—

166

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₈₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₈₃

yes

—

115

C₄²

C₂²×D₈₃

yes

characteristic

116

C₄²

C₂²×D₈₃

yes

characteristic

117

C₂³

—

332

—

118

C₂³

—

332

—

119

C₂³

—

166

—

120

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

121

Q₈

—

122

D₄

—

332

—

123

D₄

—

332

—

124

D₄

—

332

—

125

D₄

—

166

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

131

C₂×C₄

#13050591

yes

—

132

C₂×C₄

C₂³×D₈₃

yes

characteristic

133

C₈

—

166

—

134

C₂²

—

135

C₂²

—

332

—

136

C₂²

—

332

—

137

C₂²

—

664

—

138

C₂²

#14581781

yes

—

139

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#14581781

yes

—

143

C₄

#14581817

yes

—

144

C₄

C₂×#13050591

yes

—

145

C₂

—

146

C₂

—

664

—

147

C₂

C₂×#14581781

yes

characteristic

148

C₂

C₂×#14581817

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#533×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#14581789	C₂
hypercenter	#533	D₈₃
hypocenter	C₈₃	#3138
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#533×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

C₂

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19128067	C₂	semidirect
3. #19128049	C₂	semidirect
4. #19127705	C₂	semidirect
5. C₄×#13050574	C₂	semidirect
6. #533×C₈₃	C₂	semidirect
13. #14581712	C₂²	semidirect
14. #14581703	C₂²	semidirect
15. C₄×C₈₃⋊C₈	C₂²	semidirect
16. C₄⋊Q₈×C₈₃	C₂²	semidirect
17. #125×C₈₃	C₂²	semidirect
75. C₈₃	#3138	semidirect
76. #533	D₈₃	semidirect
85. C₄⋊Q₈	C₂×D₈₃	semidirect
89. #125	C₂×D₈₃	semidirect

H ≅ G/N

type

2. #19128067

C₂

semidirect

3. #19128049

C₂

semidirect

4. #19127705

C₂

semidirect

5. C₄×#13050574

C₂

semidirect

6. #533×C₈₃

C₂

semidirect

13. #14581712

C₂²

semidirect

14. #14581703

C₂²

semidirect

15. C₄×C₈₃⋊C₈

C₂²

semidirect

16. C₄⋊Q₈×C₈₃

C₂²

semidirect

17. #125×C₈₃

C₂²

semidirect

75. C₈₃

#3138

semidirect

76. #533

D₈₃

semidirect

85. C₄⋊Q₈

C₂×D₈₃

semidirect

89. #125

C₂×D₈₃

semidirect

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

subgroup	index
C₂×#14581789	2
#19128067	2
#19128049	2
#19127705	2
C₄×#13050574	2
#533×C₈₃	2
#3138	83

subgroup

index

#533×C₈₃

Subgroups — #24917839

Conjugacy classes of subgroups

Complements

Maximal subgroups