Group #24917775 — subgroups

Subgroups — C₂×#19128053 #24917775

Order 10624.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14581792	C₂	1	1	yes	maximal, characteristic
2	#19128053	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14581715	C₂	1	4	yes	maximal
4	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
8	#14581792	—	2	2	no	—
9	#14581792	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₈₃	—	2	1	no	—
11	C₂×C₄⋊Dic₈₃	—	2	1	no	—
12	#14581715	C₂²	1	16	yes	—
13	C₄×C₈₃⋊C₈	C₂²	1	4	yes	—
14	C₂²×Q₈×C₈₃	—	2	1	no	—
15	C₄⋊Q₈×C₈₃	—	2	2	no	—
16	C₄⋊Q₈×C₈₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₆₆	—	2	8	no	—
21	C₂×Dic₁₆₆	—	2	2	no	—
22	C₂×Dic₁₆₆	C₂³	1	4	yes	—
23	C₂²×Dic₈₃	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	2	4	no	—
25	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₈₃	—	2	4	no	—
27	C₄⋊Dic₈₃	—	2	4	no	—
28	C₂×Q₈×C₈₃	—	2	4	no	—
29	C₂×Q₈×C₈₃	—	4	4	no	—
30	C₂²×C₄×C₈₃	—	2	2	no	—
31	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
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	C₄²×C₈₃	D₄	1	2	yes	—
38	C₄²×C₈₃	C₂³	1	2	yes	—
39	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₈₃	—	4	8	no	—
41	C₂×C₄×C₈₃	—	2	4	no	—
42	C₂×C₄×C₈₃	—	4	4	no	—
43	C₂×C₄×C₈₃	SD₁₆	1	4	yes	—
44	C₂×C₄×C₈₃	Q₁₆	1	4	yes	—
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₈₃	#41	1	4	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	4	no	—
51	C₂×Dic₈₃	—	4	4	no	—
52	Dic₁₆₆	—	2	8	no	—
53	Dic₁₆₆	—	4	4	no	—
54	C₈₃⋊C₈	—	2	4	no	—
55	C₂²×C₈₃	#124	1	2	yes	—
56	C₂²×C₈₃	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₈₃	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₈₃	—	4	4	no	—
60	C₄×C₈₃	#124	1	2	yes	—
61	C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
62	C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
63	C₄×C₈₃	C₂×#41	1	2	yes	—
64	Dic₈₃	—	4	4	no	—
65	C₂×C₈₃	#487	1	4	yes	complemented
66	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
67	C₂×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	83	1	no	maximal, Sylow
70	C₈₃	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	83	1	no	—
72	C₂×C₄⋊Q₈	D₈₃	1	1	yes	characteristic
73	#487	—	83	8	no	—
74	C₂×#103	—	83	4	no	—
75	C₂×C₄×C₈	—	83	1	no	—
76	C₂²×Q₈	—	2	1	no	—
77	C₂²×Q₈	—	83	2	no	—
78	C₂²×C₈	—	83	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	83	4	no	—
81	C₄⋊Q₈	—	166	2	no	—
82	C₄⋊Q₈	C₂×D₈₃	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	166	1	no	—
85	C₂×C₄⋊C₄	—	166	1	no	—
86	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
87	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
88	#103	—	83	16	no	—
89	C₄×C₈	—	83	4	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	83	4	no	—
93	C₂×Q₈	—	166	8	no	—
94	C₂×Q₈	—	166	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	166	2	no	—
97	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
100	C₂×C₈	—	83	4	no	—
101	C₂×C₈	—	166	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	166	4	no	—
105	C₄⋊C₄	—	166	4	no	—
106	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
107	C₄²	C₂²⋊D₈₃	1	2	yes	—
108	C₄²	C₂²×D₈₃	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
110	Q₈	—	4	8	no	—
111	Q₈	—	332	4	no	—
112	Q₈	—	166	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	332	4	no	—
116	C₂×C₄	—	166	4	no	—
117	C₂×C₄	#13050578	1	4	yes	—
118	C₂×C₄	#13050580	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
122	C₂×C₄	#13050591	1	4	yes	—
123	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
124	C₈	—	166	4	no	—
125	C₂²	#14581781	1	2	yes	—
126	C₂²	C₂×#13050578	1	2	yes	—
127	C₂²	C₂×#13050580	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	332	4	no	—
131	C₄	#14581781	1	2	yes	—
132	C₄	C₂×#13050578	1	4	yes	—
133	C₄	C₂×#13050580	1	4	yes	—
134	C₄	C₂×#13050591	1	2	yes	—
135	C₂	#19128053	1	4	yes	direct product factor
136	C₂	C₂×#14581781	1	1	yes	characteristic
137	C₂	C₂²×#13050578	1	1	yes	characteristic
138	C₂	C₂²×#13050580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

#19128053

C₂

yes

maximal, direct product factor

C₂×#14581715

C₂

yes

maximal

C₂×C₄×C₈₃⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581792

—

#14581792

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581715

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂²×Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

Q₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

#124

yes

—

C₂²×C₈₃

C₂×SD₁₆

yes

—

C₂²×C₈₃

C₂×Q₁₆

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#124

yes

—

C₄×C₈₃

C₂×SD₁₆

yes

—

C₄×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

C₂×#41

yes

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

C₂²×SD₁₆

yes

characteristic

C₂×C₈₃

C₂²×Q₁₆

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₈₃

yes

characteristic

#487

—

C₂×#103

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

166

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

C₂²×D₈₃

yes

—

107

C₄²

C₂²⋊D₈₃

yes

—

108

C₄²

C₂²×D₈₃

yes

—

109

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

110

Q₈

—

111

Q₈

—

332

—

112

Q₈

—

166

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

332

—

116

C₂×C₄

—

166

—

117

C₂×C₄

#13050578

yes

—

118

C₂×C₄

#13050580

yes

—

119

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

122

C₂×C₄

#13050591

yes

—

123

C₂×C₄

C₂³×D₈₃

yes

characteristic

124

C₈

—

166

—

125

C₂²

#14581781

yes

—

126

C₂²

C₂×#13050578

yes

—

127

C₂²

C₂×#13050580

yes

—

128

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

129

C₄

—

130

C₄

—

332

—

131

C₄

#14581781

yes

—

132

C₄

C₂×#13050578

yes

—

133

C₄

C₂×#13050580

yes

—

134

C₄

C₂×#13050591

yes

—

135

C₂

#19128053

yes

direct product factor

136

C₂

C₂×#14581781

yes

characteristic

137

C₂

C₂²×#13050578

yes

characteristic

138

C₂

C₂²×#13050580

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂×C₄⋊Q₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₈₃
hypocenter	C₈₃	C₂×#487
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

C₂×C₄⋊Q₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₈₃

hypocenter

C₈₃

C₂×#487

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #19128053	C₂	direct (N × H)
65. C₂×C₈₃	#487	semidirect
70. C₈₃	C₂×#487	semidirect
135. C₂	#19128053	direct (N × H)

H ≅ G/N

type

2. #19128053

C₂

direct (N × H)

65. C₂×C₈₃

#487

semidirect

70. C₈₃

C₂×#487

semidirect

135. C₂

#19128053

direct (N × H)

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

subgroup	index
C₂×#14581792	2
#19128053	2
C₂×#14581715	2
C₂×C₄×C₈₃⋊C₈	2
C₂×C₄⋊Q₈×C₈₃	2
C₂×#487	83

subgroup

index

C₂×#14581792

#19128053

C₂×#14581715

C₂×C₄×C₈₃⋊C₈

C₂×C₄⋊Q₈×C₈₃

C₂×#487

Subgroups — C₂×#19128053 #24917775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19128053 #24917775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128053 #24917775