Group #24906174 — subgroups

Subgroups — #24906174

Order 10624.

198 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19128258	C₂	1	1	yes	maximal, characteristic
2	#19127754	C₂	1	1	yes	maximal, characteristic
3	#19127449	C₂	1	2	yes	maximal, complemented
4	#19127448	C₂	1	2	yes	maximal, complemented
5	#432×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581792	C₂²	1	1	yes	characteristic
7	#14581792	C₂²	1	1	yes	characteristic
8	#14581784	—	2	1	no	—
9	#14581771	—	2	1	no	—
10	#14581771	C₄	1	2	yes	—
11	C₄×Dic₁₆₆	—	2	1	no	—
12	C₄×Dic₁₆₆	C₄	1	2	yes	—
13	#14581751	—	2	1	no	—
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
15	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
16	C₄×C₈₃⋊C₈	—	2	1	no	—
17	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₈₃	—	2	1	no	—
20	#97×C₈₃	—	2	1	no	—
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
23	C₂×C₈₃⋊C₈	—	2	2	no	—
24	C₄⋊Dic₈₃	—	2	1	no	—
25	C₄⋊Dic₈₃	—	2	1	no	—
26	C₄⋊Dic₈₃	—	2	1	no	—
27	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
28	Dic₈₃⋊C₄	—	2	2	no	—
29	Dic₈₃⋊C₄	—	2	2	no	—
30	C₂²⋊Dic₈₃	—	2	2	no	—
31	C₄×Dic₈₃	—	2	2	no	—
32	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₃	—	2	2	no	—
36	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	—	2	2	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	2	no	—
51	C₂×Dic₈₃	—	2	2	no	—
52	Dic₁₆₆	—	4	2	no	—
53	Dic₁₆₆	—	4	2	no	—
54	C₈×C₈₃	—	4	2	no	—
55	C₈₃⋊C₈	—	4	2	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	—	2	1	no	—
59	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	2	1	no	—
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	#135	1	2	yes	—
65	C₄×C₈₃	#137	1	2	yes	—
66	Dic₈₃	—	4	2	no	—
67	Dic₈₃	—	4	2	no	—
68	C₂×C₈₃	—	4	1	no	—
69	C₂×C₈₃	#413	1	1	yes	characteristic
70	C₂×C₈₃	#416	1	1	yes	characteristic
71	C₂×C₈₃	#419	1	1	yes	characteristic
72	#1554	—	83	1	no	maximal, Sylow
73	C₈₃	#1554	1	1	yes	characteristic, complemented, Sylow
74	#533	—	83	1	no	—
75	#432	D₈₃	1	1	yes	characteristic
76	#431	—	83	1	no	—
77	#333	—	83	2	no	—
78	#332	—	83	2	no	—
79	C₄⋊Q₈	—	83	1	no	—
80	C₄⋊Q₈	—	83	1	no	—
81	#125	—	166	1	no	—
82	C₂²⋊Q₈	—	83	2	no	—
83	C₂²⋊Q₈	—	166	1	no	—
84	C₄×Q₈	—	83	2	no	—
85	C₄×Q₈	—	166	1	no	—
86	#117	—	166	1	no	—
87	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
88	C₄⋊C₈	—	83	2	no	—
89	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	—	166	1	no	—
92	#97	—	2	1	no	—
93	C₄×C₈	—	166	1	no	—
94	C₂×Q₈	—	83	2	no	—
95	C₂×Q₈	—	166	1	no	—
96	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
97	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	166	2	no	—
101	C₄⋊C₄	—	83	2	no	—
102	C₄⋊C₄	—	166	2	no	—
103	C₄⋊C₄	—	166	2	no	—
104	C₄⋊C₄	—	166	1	no	—
105	C₄⋊C₄	—	166	1	no	—
106	C₄⋊C₄	—	166	1	no	—
107	C₂²⋊C₄	—	166	2	no	—
108	C₄²	—	166	2	no	—
109	C₄²	C₄×D₈₃	1	1	yes	characteristic
110	C₄²	D₃₃₂	1	1	yes	characteristic
111	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
112	C₄²	C₂²×D₈₃	1	1	yes	characteristic
113	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
114	Q₈	—	332	2	no	—
115	Q₈	—	332	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	166	2	no	—
120	C₂×C₄	—	166	2	no	—
121	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
126	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	332	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	332	2	no	—
138	C₄	—	332	2	no	—
139	C₄	#14581812	1	2	yes	—
140	C₄	#14581825	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#19127683	1	1	yes	characteristic
143	C₂	#19127695	1	1	yes	characteristic
144	C₂	#19127711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

#19127754

C₂

yes

maximal, characteristic

#19127449

C₂

yes

maximal, complemented

#19127448

C₂

yes

maximal, complemented

#432×C₈₃

C₂

yes

maximal, characteristic

#14581792

C₂²

yes

characteristic

#14581792

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

C₄

yes

—

#14581751

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#135

yes

—

C₄×C₈₃

#137

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#533

—

#432

D₈₃

yes

—

—

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

166

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

C₄×Q₈

—

166

—

#117

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#97

—

C₄×C₈

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

166

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

166

—

103

C₄⋊C₄

—

166

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₂²⋊C₄

—

166

—

108

C₄²

—

166

—

109

C₄²

C₄×D₈₃

yes

characteristic

110

C₄²

D₃₃₂

yes

characteristic

111

C₄²

C₂²⋊D₈₃

yes

characteristic

112

C₄²

C₂²×D₈₃

yes

characteristic

113

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

114

Q₈

—

332

—

115

Q₈

—

332

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

166

—

120

C₂×C₄

—

166

—

121

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

124

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

126

C₂×C₄

C₂×D₃₃₂

yes

characteristic

127

C₈

—

128

C₈

—

332

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

332

—

138

C₄

—

332

—

139

C₄

#14581812

yes

—

140

C₄

#14581825

yes

—

141

C₂

—

142

C₂

#19127683

yes

characteristic

143

C₂

#19127695

yes

characteristic

144

C₂

#19127711

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#432×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#432	D₈₃
hypocenter	C₈₃	#1554
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#432×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #19127449	C₂	semidirect
4. #19127448	C₂	semidirect
73. C₈₃	#1554	semidirect

H ≅ G/N

type

3. #19127449

C₂

semidirect

4. #19127448

C₂

semidirect

73. C₈₃

#1554

semidirect

Subgroups — #24906174

Conjugacy classes of subgroups

Complements