Group #24907221 — subgroups

Subgroups — C₄×C₂²⋊(C₈₃⋊C₈) #24907221

Order 10624.

436 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₂²⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	#19127456	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₈	C₂²	1	4	yes	—
6	C₂²⋊(C₈₃⋊C₈)	C₄	1	8	yes	direct product factor
7	C₄×C₈₃⋊C₈	—	2	4	no	—
8	C₂³×C₄×C₈₃	C₄	1	2	yes	—
9	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₈₃	—	2	4	no	—
11	C₂×C₄²×C₈₃	C₄	1	2	yes	—
12	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₈₃⋊C₈	—	2	4	no	—
15	C₂×C₈₃⋊C₈	—	2	4	no	—
16	C₂×C₈₃⋊C₈	C₂×C₄	1	8	yes	complemented
17	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₈₃	—	2	4	no	—
19	C₂²×C₄×C₈₃	—	2	4	no	—
20	C₂²×C₄×C₈₃	—	2	2	no	—
21	C₂²×C₄×C₈₃	—	2	2	no	—
22	C₂²×C₄×C₈₃	C₈	1	8	yes	complemented
23	C₂²×C₄×C₈₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	C₄²×C₈₃	—	2	4	no	—
29	C₄²×C₈₃	—	2	2	no	—
30	C₄²×C₈₃	D₄	1	4	yes	—
31	C₂³×C₈₃	—	2	1	no	—
32	C₂³×C₈₃	—	2	1	no	—
33	C₂³×C₈₃	—	2	2	no	—
34	C₂³×C₈₃	C₄²	1	2	yes	—
35	C₂³×C₈₃	C₂×C₈	1	4	yes	—
36	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	—	2	1	no	—
38	C₂×C₄×C₈₃	—	2	2	no	—
39	C₂×C₄×C₈₃	—	2	1	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	2	16	no	—
42	C₂×C₄×C₈₃	—	2	8	no	—
43	C₂×C₄×C₈₃	C₄²	1	2	yes	—
44	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₈₃	C₂×C₈	1	4	yes	—
46	C₂×C₄×C₈₃	C₂×C₈	1	4	yes	—
47	C₂×C₄×C₈₃	M₁₆	1	4	yes	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₃	#41	1	2	yes	—
51	C₈₃⋊C₈	—	2	8	no	—
52	C₂²×C₈₃	—	2	2	no	—
53	C₂²×C₈₃	—	2	2	no	—
54	C₂²×C₈₃	—	2	4	no	—
55	C₂²×C₈₃	—	2	4	no	—
56	C₂²×C₈₃	C₄×C₈	1	4	yes	complemented
57	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈₃	#117	1	1	yes	characteristic
60	C₂²×C₈₃	C₂²×C₈	1	2	yes	—
61	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
62	C₄×C₈₃	—	2	2	no	—
63	C₄×C₈₃	—	2	4	no	—
64	C₄×C₈₃	C₂²⋊C₈	1	8	yes	complemented
65	C₄×C₈₃	C₄×D₄	1	4	yes	—
66	C₂×C₈₃	—	2	4	no	—
67	C₂×C₈₃	C₄×C₂²⋊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×C₂²⋊C₈	1	2	yes	—
71	C₂×C₈₃	#431	1	2	yes	—
72	C₄×C₂²⋊C₈	—	83	1	no	maximal, Sylow
73	C₈₃	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₄²	D₈₃	1	1	yes	characteristic
75	C₂×C₂²⋊C₈	—	83	2	no	—
76	C₂×C₄×C₈	—	83	2	no	—
77	#336	—	83	2	no	—
78	C₂³×C₄	Dic₈₃	1	2	yes	—
79	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
80	C₂²×C₈	—	83	4	no	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	Dic₈₃	1	2	yes	—
83	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
85	C₂²⋊C₈	—	83	8	no	—
86	C₄×C₈	—	166	4	no	—
87	C₂⁴	C₂×Dic₈₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	C₈₃⋊C₈	1	8	yes	complemented
93	C₂²×C₄	C₄×D₈₃	1	4	yes	—
94	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
96	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
97	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
98	C₂×C₈	—	83	8	no	—
99	C₂×C₈	—	166	4	no	—
100	C₂×C₈	—	166	4	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂²⋊D₈₃	1	4	yes	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	1	no	—
107	C₂³	C₄×Dic₈₃	1	2	yes	—
108	C₂³	C₂×C₈₃⋊C₈	1	4	yes	—
109	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	16	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₈₃	1	4	yes	—
118	C₂×C₄	C₂×C₈₃⋊C₈	1	4	yes	—
119	C₂×C₄	C₂×C₈₃⋊C₈	1	4	yes	—
120	C₂×C₄	#13050574	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
123	C₂×C₄	#13050590	1	2	yes	—
124	C₈	—	166	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	C₄×C₈₃⋊C₈	1	4	yes	complemented
130	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
132	C₂²	#14581751	1	1	yes	characteristic
133	C₂²	C₂²×C₈₃⋊C₈	1	2	yes	—
134	C₂²	C₂×#13050574	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	2	2	no	—
137	C₄	C₂²⋊(C₈₃⋊C₈)	1	8	yes	direct product factor
138	C₄	C₄×C₂²⋊D₈₃	1	4	yes	—
139	C₂	—	2	4	no	—
140	C₂	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
141	C₂	C₂×C₄×C₈₃⋊C₈	1	1	yes	characteristic
142	C₂	C₄×#13050574	1	1	yes	characteristic
143	C₂	C₂×C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
144	C₂	#19127754	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₈₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈₃⋊C₈

C₂

yes

maximal, complemented

#19127456

C₂

yes

maximal, complemented

C₂²×C₄²×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

direct product factor

C₄×C₈₃⋊C₈

—

C₂³×C₄×C₈₃

C₄

yes

—

C₂³×C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

complemented

C₂⁴×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₈

yes

complemented

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₄²

yes

—

C₂³×C₈₃

C₂×C₈

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×C₈

yes

complemented

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

#117

yes

characteristic

C₂²×C₈₃

C₂²×C₈

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊C₈

yes

complemented

C₄×C₈₃

C₄×D₄

yes

—

C₂×C₈₃

—

C₂×C₈₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₈₃

C₂×C₄×C₈

yes

characteristic

C₂×C₈₃

C₄×M₁₆

yes

characteristic

C₂×C₈₃

C₂×C₂²⋊C₈

yes

—

C₂×C₈₃

#431

yes

—

C₄×C₂²⋊C₈

—

maximal, Sylow

C₈₃

C₄×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₄²

D₈₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

#336

—

C₂³×C₄

Dic₈₃

yes

—

C₂³×C₄

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

166

—

C₂⁴

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₈₃⋊C₈

yes

complemented

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

166

—

100

C₂×C₈

—

166

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊D₈₃

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₄×Dic₈₃

yes

—

108

C₂³

C₂×C₈₃⋊C₈

yes

—

109

C₂³

C₂²×Dic₈₃

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄×Dic₈₃

yes

—

117

C₂×C₄

C₂²⋊Dic₈₃

yes

—

118

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

119

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

120

C₂×C₄

#13050574

yes

—

121

C₂×C₄

C₂×C₄×D₈₃

yes

—

122

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

123

C₂×C₄

#13050590

yes

—

124

C₈

—

166

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₄×C₈₃⋊C₈

yes

complemented

130

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

132

C₂²

#14581751

yes

characteristic

133

C₂²

C₂²×C₈₃⋊C₈

yes

—

134

C₂²

C₂×#13050574

yes

—

135

C₄

—

136

C₄

—

137

C₄

C₂²⋊(C₈₃⋊C₈)

yes

direct product factor

138

C₄

C₄×C₂²⋊D₈₃

yes

—

139

C₂

—

140

C₂

C₄×C₂²⋊Dic₈₃

yes

characteristic

141

C₂

C₂×C₄×C₈₃⋊C₈

yes

characteristic

142

C₂

C₄×#13050574

yes

characteristic

143

C₂

C₂×C₂²⋊(C₈₃⋊C₈)

yes

—

144

C₂

#19127754

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₈₃
commutator	C₂×C₈₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂²×C₄²×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₄
prime radical	C₂⁴×C₈₃	C₂×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂²×C₄²	D₈₃
hypocenter	C₈₃	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

subgroup

G/H

center

C₂×C₄²

C₂×D₈₃

commutator

C₂×C₈₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂²×C₄²×C₈₃

C₂

socle

C₂³×C₈₃

C₂²×C₄

prime radical

C₂⁴×C₈₃

C₂×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂²×C₄²

D₈₃

hypocenter

C₈₃

C₄×C₂²⋊C₈

LCS

G ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×C₄×C₈₃⋊C₈	C₂	semidirect
3. #19127456	C₂	semidirect
6. C₂²⋊(C₈₃⋊C₈)	C₄	direct (N × H)
16. C₂×C₈₃⋊C₈	C₂×C₄	semidirect
22. C₂²×C₄×C₈₃	C₈	semidirect
56. C₂²×C₈₃	C₄×C₈	semidirect
64. C₄×C₈₃	C₂²⋊C₈	semidirect
73. C₈₃	C₄×C₂²⋊C₈	semidirect
92. C₂²×C₄	C₈₃⋊C₈	semidirect
129. C₂²	C₄×C₈₃⋊C₈	semidirect
137. C₄	C₂²⋊(C₈₃⋊C₈)	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×C₈₃⋊C₈

C₂

semidirect

3. #19127456

C₂

semidirect

6. C₂²⋊(C₈₃⋊C₈)

C₄

direct (N × H)

16. C₂×C₈₃⋊C₈

C₂×C₄

semidirect

22. C₂²×C₄×C₈₃

C₈

semidirect

56. C₂²×C₈₃

C₄×C₈

semidirect

64. C₄×C₈₃

C₂²⋊C₈

semidirect

73. C₈₃

C₄×C₂²⋊C₈

semidirect

92. C₂²×C₄

C₈₃⋊C₈

semidirect

129. C₂²

C₄×C₈₃⋊C₈

semidirect

137. C₄

C₂²⋊(C₈₃⋊C₈)

direct (N × H)

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

subgroup	index
C₂×C₂²⋊(C₈₃⋊C₈)	2
C₂×C₄×C₈₃⋊C₈	2
#19127456	2
C₂²×C₄²×C₈₃	2
C₄×C₂²⋊C₈	83

subgroup

index

C₂×C₂²⋊(C₈₃⋊C₈)

C₂×C₄×C₈₃⋊C₈

#19127456

C₂²×C₄²×C₈₃

C₄×C₂²⋊C₈

Subgroups — C₄×C₂²⋊(C₈₃⋊C₈) #24907221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C22⋊(C83⋊C8) #24907221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊(C₈₃⋊C₈) #24907221