Group #24907314 — subgroups

Subgroups — C₄⋊C₈×Dic₈₃ #24907314

Order 10624.

316 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₈₃	C₂	1	2	yes	maximal
3	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127455	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₄	1	4	yes	complemented
12	C₈×Dic₈₃	—	2	4	no	—
13	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₈₃	C₄	1	4	yes	complemented
16	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
17	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₈₃	—	2	1	no	—
22	C₄×Dic₈₃	—	2	1	no	—
23	C₄×Dic₈₃	—	2	4	no	—
24	C₄×Dic₈₃	C₈	1	8	yes	complemented
25	C₄×Dic₈₃	D₄	1	2	yes	—
26	C₄×Dic₈₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
33	C₄²×C₈₃	C₂×C₄	1	2	yes	—
34	C₄²×C₈₃	C₂×C₄	1	2	yes	—
35	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	—	2	2	no	—
37	C₂×C₄×C₈₃	C₄²	1	2	yes	—
38	C₂×C₄×C₈₃	C₄²	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×C₈	1	4	yes	—
40	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
46	C₂×Dic₈₃	—	2	2	no	—
47	C₂×Dic₈₃	—	2	2	no	—
48	C₂×Dic₈₃	C₄⋊C₄	1	4	yes	—
49	C₂×Dic₈₃	C₂×C₈	1	4	yes	—
50	C₂×Dic₈₃	C₂×C₈	1	4	yes	—
51	C₂×Dic₈₃	M₁₆	1	4	yes	—
52	C₈×C₈₃	—	2	4	no	—
53	C₈₃⋊C₈	—	2	4	no	—
54	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈₃	#117	1	1	yes	characteristic
57	C₂²×C₈₃	C₂²×C₈	1	2	yes	—
58	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	C₄×C₈	1	4	yes	complemented
61	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
62	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
63	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
64	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
65	Dic₈₃	—	2	4	no	—
66	Dic₈₃	C₄⋊C₈	1	8	yes	direct product factor
67	C₂×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂×C₈₃	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
71	C₂×C₈₃	#431	1	2	yes	—
72	C₄×C₄⋊C₈	—	83	1	no	maximal, Sylow
73	C₈₃	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊C₈	—	83	1	no	—
75	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
76	C₂×C₄×C₈	—	83	2	no	—
77	C₄³	—	83	1	no	—
78	#336	—	83	2	no	—
79	C₂²×C₈	—	83	2	no	—
80	C₂²×C₈	C₂×D₈₃	1	2	yes	—
81	C₂×C₄²	—	83	2	no	—
82	C₂×C₄²	—	83	1	no	—
83	C₂×C₄²	—	83	2	no	—
84	C₂×C₄²	—	83	1	no	—
85	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
86	C₄⋊C₈	—	83	4	no	—
87	C₄⋊C₈	Dic₈₃	1	4	yes	direct product factor
88	C₄×C₈	—	166	4	no	—
89	C₂²×C₄	—	83	2	no	—
90	C₂²×C₄	—	83	2	no	—
91	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	83	4	no	—
97	C₂×C₈	—	166	2	no	—
98	C₂×C₈	—	166	2	no	—
99	C₂×C₈	C₂×Dic₈₃	1	4	yes	—
100	C₄²	—	83	8	no	—
101	C₄²	—	83	2	no	—
102	C₄²	—	83	2	no	—
103	C₄²	—	166	4	no	—
104	C₄²	—	166	1	no	—
105	C₄²	—	166	1	no	—
106	C₄²	C₄×D₈₃	1	2	yes	—
107	C₄²	C₂×Dic₈₃	1	2	yes	—
108	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	83	4	no	—
111	C₂×C₄	—	83	4	no	—
112	C₂×C₄	—	83	4	no	—
113	C₂×C₄	—	83	4	no	—
114	C₂×C₄	—	166	2	no	—
115	C₂×C₄	—	166	2	no	—
116	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
117	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
118	C₂×C₄	C₈×D₈₃	1	4	yes	—
119	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
123	C₂×C₄	#13050589	1	1	yes	characteristic
124	C₂×C₄	#13050591	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	166	4	no	—
127	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
128	C₂²	C₄⋊C₄×D₈₃	1	1	yes	characteristic
129	C₂²	#14581750	1	1	yes	characteristic
130	C₂²	C₂×C₈×D₈₃	1	2	yes	—
131	C₂²	M₁₆×D₈₃	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	83	8	no	—
134	C₄	—	166	4	no	—
135	C₄	C₈×Dic₈₃	1	4	yes	complemented
136	C₄	#14581754	1	1	yes	characteristic
137	C₄	D₄×Dic₈₃	1	1	yes	characteristic
138	C₄	#14581757	1	1	yes	characteristic
139	C₄	Q₈×Dic₈₃	1	1	yes	characteristic
140	C₂	C₄⋊C₄×Dic₈₃	1	1	yes	characteristic
141	C₂	C₂×C₈×Dic₈₃	1	1	yes	characteristic
142	C₂	M₁₆×Dic₈₃	1	1	yes	characteristic
143	C₂	C₄⋊C₈×D₈₃	1	2	yes	—
144	C₂	#19127753	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₃

C₂

yes

maximal

C₄²×Dic₈₃

C₂

yes

maximal, characteristic

#19127455

C₂

yes

maximal

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₄

yes

complemented

C₈×Dic₈₃

—

C₂²×C₈×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₄

yes

complemented

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₈

yes

complemented

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

C₂×Dic₈₃

C₂×C₈

yes

—

C₂×Dic₈₃

C₂×C₈

yes

—

C₂×Dic₈₃

M₁₆

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

#117

yes

characteristic

C₂²×C₈₃

C₂²×C₈

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₄×C₈

yes

complemented

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

C₄⋊C₈

yes

direct product factor

C₂×C₈₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₃

C₂×C₄×C₈

yes

characteristic

C₂×C₈₃

C₄×M₁₆

yes

characteristic

C₂×C₈₃

C₂×C₄⋊C₈

yes

—

C₂×C₈₃

#431

yes

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₈₃

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

C₂×C₄×C₈

—

C₄³

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₈₃

yes

direct product factor

C₄×C₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

166

—

104

C₄²

—

166

—

105

C₄²

—

166

—

106

C₄²

C₄×D₈₃

yes

—

107

C₄²

C₂×Dic₈₃

yes

—

108

C₂³

C₂×C₄×D₈₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

166

—

115

C₂×C₄

—

166

—

116

C₂×C₄

C₄×Dic₈₃

yes

—

117

C₂×C₄

C₄×Dic₈₃

yes

—

118

C₂×C₄

C₈×D₈₃

yes

—

119

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

120

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

121

C₂×C₄

D₄×D₈₃

yes

characteristic

122

C₂×C₄

Q₈×D₈₃

yes

characteristic

123

C₂×C₄

#13050589

yes

characteristic

124

C₂×C₄

#13050591

yes

characteristic

125

C₈

—

126

C₈

—

166

—

127

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

128

C₂²

C₄⋊C₄×D₈₃

yes

characteristic

129

C₂²

#14581750

yes

characteristic

130

C₂²

C₂×C₈×D₈₃

yes

—

131

C₂²

M₁₆×D₈₃

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

166

—

135

C₄

C₈×Dic₈₃

yes

complemented

136

C₄

#14581754

yes

characteristic

137

C₄

D₄×Dic₈₃

yes

characteristic

138

C₄

#14581757

yes

characteristic

139

C₄

Q₈×Dic₈₃

yes

characteristic

140

C₂

C₄⋊C₄×Dic₈₃

yes

characteristic

141

C₂

C₂×C₈×Dic₈₃

yes

characteristic

142

C₂

M₁₆×Dic₈₃

yes

characteristic

143

C₂

C₄⋊C₈×D₈₃

yes

—

144

C₂

#19127753

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator

C₂×C₈₃

C₂×C₄×C₈

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₃

hypocenter

C₈₃

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
11. C₄⋊(C₈₃⋊C₈)	C₄	semidirect
15. C₄⋊C₈×C₈₃	C₄	semidirect
24. C₄×Dic₈₃	C₈	semidirect
60. C₄×C₈₃	C₄×C₈	semidirect
66. Dic₈₃	C₄⋊C₈	direct (N × H)
73. C₈₃	C₄×C₄⋊C₈	semidirect
87. C₄⋊C₈	Dic₈₃	direct (N × H)
135. C₄	C₈×Dic₈₃	semidirect

H ≅ G/N

type

11. C₄⋊(C₈₃⋊C₈)

C₄

semidirect

15. C₄⋊C₈×C₈₃

C₄

semidirect

24. C₄×Dic₈₃

C₈

semidirect

60. C₄×C₈₃

C₄×C₈

semidirect

66. Dic₈₃

C₄⋊C₈

direct (N × H)

73. C₈₃

C₄×C₄⋊C₈

semidirect

87. C₄⋊C₈

Dic₈₃

direct (N × H)

135. C₄

C₈×Dic₈₃

semidirect

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

subgroup	index
C₂×C₄⋊(C₈₃⋊C₈)	2
C₂×C₈×Dic₈₃	2
C₄²×Dic₈₃	2
#19127455	2
C₂×C₄⋊C₈×C₈₃	2
C₄×C₄⋊C₈	83

subgroup

index

C₂×C₄⋊(C₈₃⋊C₈)

C₂×C₈×Dic₈₃

C₄²×Dic₈₃

#19127455

C₂×C₄⋊C₈×C₈₃

C₄×C₄⋊C₈

Subgroups — C₄⋊C₈×Dic₈₃ #24907314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4⋊C8×Dic83 #24907314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₈×Dic₈₃ #24907314