Group #24906136 — subgroups

Subgroups — #24906136

Order 10624.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19127754	C₂	1	1	yes	maximal, characteristic
3	#19127448	C₂	1	4	yes	maximal, complemented
4	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	—	2	1	no	—
6	#14581792	—	2	1	no	—
7	#14581792	—	2	1	no	—
8	#14581792	C₄	1	2	yes	—
9	#14581792	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₃	—	2	1	no	—
11	C₂×C₄⋊Dic₈₃	C₄	1	2	yes	—
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
14	C₄×C₈₃⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₈₃	—	2	1	no	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₃	—	2	1	no	—
18	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
19	C₂×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	—	4	2	no	—
22	C₂×Dic₁₆₆	—	4	2	no	—
23	C₂²×Dic₈₃	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₄⋊Dic₈₃	—	2	2	no	—
26	C₄⋊Dic₈₃	—	2	2	no	—
27	C₄⋊Dic₈₃	—	2	2	no	—
28	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
32	M₁₆×C₈₃	—	4	2	no	—
33	C₂×C₈×C₈₃	—	2	2	no	—
34	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	D₈	1	2	yes	—
45	C₂×C₄×C₈₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₈₃	—	2	4	no	—
50	C₂×Dic₈₃	—	4	4	no	—
51	Dic₁₆₆	—	4	4	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	#102	1	2	yes	—
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	#101	1	2	yes	—
63	C₄×C₈₃	#102	1	2	yes	—
64	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
65	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₈₃	#137	1	2	yes	—
67	Dic₈₃	—	4	4	no	—
68	C₂×C₈₃	—	2	2	no	—
69	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
70	C₂×C₈₃	C₂×#102	1	1	yes	characteristic
71	C₂×C₈₃	#419	1	1	yes	characteristic
72	#1546	—	83	1	no	maximal, Sylow
73	C₈₃	#1546	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	83	1	no	—
75	#431	—	83	1	no	—
76	#423	D₈₃	1	1	yes	characteristic
77	#332	—	83	4	no	—
78	C₂²×Q₈	—	166	1	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	83	2	no	—
81	C₄⋊Q₈	—	83	2	no	—
82	C₄⋊Q₈	—	166	1	no	—
83	C₄⋊Q₈	—	166	1	no	—
84	C₂×C₄⋊C₄	—	83	2	no	—
85	C₂×C₄⋊C₄	—	166	1	no	—
86	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	83	2	no	—
89	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
90	C₂²⋊C₈	—	166	1	no	—
91	C₄×C₈	—	166	1	no	—
92	C₂×Q₈	—	332	2	no	—
93	C₂×Q₈	—	332	2	no	—
94	C₂×Q₈	—	166	2	no	—
95	C₂×Q₈	—	166	2	no	—
96	C₂²×C₄	—	166	2	no	—
97	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
98	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	166	2	no	—
103	C₄⋊C₄	—	83	4	no	—
104	C₄⋊C₄	—	166	2	no	—
105	C₄⋊C₄	—	166	2	no	—
106	C₄⋊C₄	—	166	2	no	—
107	C₄²	C₄×D₈₃	1	1	yes	characteristic
108	C₄²	D₃₃₂	1	1	yes	characteristic
109	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
110	C₄²	C₂²×D₈₃	1	1	yes	characteristic
111	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
112	Q₈	—	332	2	no	—
113	Q₈	—	332	2	no	—
114	Q₈	—	332	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	332	4	no	—
118	C₂×C₄	—	166	4	no	—
119	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	#13050575	1	2	yes	—
123	C₂×C₄	#13050577	1	2	yes	—
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	#14581712	1	2	yes	—
132	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	332	4	no	—
136	C₄	#14581710	1	2	yes	—
137	C₄	#14581712	1	2	yes	—
138	C₄	C₂×#13050575	1	1	yes	characteristic
139	C₄	C₂×#13050577	1	1	yes	characteristic
140	C₄	#14581825	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#14581710	1	1	yes	characteristic
143	C₂	C₂×#14581712	1	1	yes	characteristic
144	C₂	#19127711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

#19127754

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal, complemented

#423×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

—

#14581792

—

#14581792

—

#14581792

C₄

yes

—

#14581792

C₂²

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

M₁₆×C₈₃

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

D₈

yes

—

C₂×C₄×C₈₃

SD₁₆

yes

—

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

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#102

yes

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#101

yes

—

C₄×C₈₃

#102

yes

—

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

#137

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

—

#423

D₈₃

yes

characteristic

#332

—

C₂²×Q₈

—

166

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

166

—

C₂×Q₈

—

332

—

C₂×Q₈

—

332

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

166

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₄²

C₄×D₈₃

yes

characteristic

108

C₄²

D₃₃₂

yes

characteristic

109

C₄²

C₂²⋊D₈₃

yes

characteristic

110

C₄²

C₂²×D₈₃

yes

characteristic

111

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

112

Q₈

—

332

—

113

Q₈

—

332

—

114

Q₈

—

332

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

332

—

118

C₂×C₄

—

166

—

119

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

122

C₂×C₄

#13050575

yes

—

123

C₂×C₄

#13050577

yes

—

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

#14581712

yes

—

132

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

332

—

136

C₄

#14581710

yes

—

137

C₄

#14581712

yes

—

138

C₄

C₂×#13050575

yes

characteristic

139

C₄

C₂×#13050577

yes

characteristic

140

C₄

#14581825

yes

—

141

C₂

—

142

C₂

C₂×#14581710

yes

characteristic

143

C₂

C₂×#14581712

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	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1546
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#423×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 ◁ #423

N	H ≅ G/N	type
3. #19127448	C₂	semidirect
73. C₈₃	#1546	semidirect

H ≅ G/N

type

3. #19127448

C₂

semidirect

73. C₈₃

#1546

semidirect

Subgroups — #24906136

Conjugacy classes of subgroups

Complements