Group #24906121 — subgroups

Subgroups — #24906121

Order 10624.

212 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19127446	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×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	C₂²×C₈×C₈₃	—	2	1	no	—
18	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₃	—	2	1	no	—
20	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
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	C₂×C₈×C₈₃	—	4	2	no	—
37	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
48	C₂×C₄×C₈₃	Q₁₆	1	2	yes	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	2	2	no	—
54	Dic₁₆₆	—	4	2	no	—
55	Dic₁₆₆	—	4	2	no	—
56	C₈×C₈₃	—	4	2	no	—
57	C₈₃⋊C₈	—	4	2	no	—
58	C₂²×C₈₃	—	2	1	no	—
59	C₂²×C₈₃	—	2	1	no	—
60	C₂²×C₈₃	#103	1	2	yes	—
61	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	#103	1	2	yes	—
66	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₄×C₈₃	#135	1	2	yes	—
69	Dic₈₃	—	4	2	no	—
70	Dic₈₃	—	4	2	no	—
71	C₂×C₈₃	—	2	2	no	—
72	C₂×C₈₃	#413	1	1	yes	characteristic
73	C₂×C₈₃	C₂×#103	1	1	yes	characteristic
74	C₂×C₈₃	#416	1	1	yes	characteristic
75	#1542	—	83	1	no	maximal, Sylow
76	C₈₃	#1542	1	1	yes	characteristic, complemented, Sylow
77	#533	—	83	1	no	—
78	#431	—	83	1	no	—
79	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
80	#333	—	83	2	no	—
81	#332	—	83	2	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₄⋊Q₈	—	83	1	no	—
84	C₄⋊Q₈	—	83	1	no	—
85	#125	—	166	1	no	—
86	C₂²⋊Q₈	—	83	2	no	—
87	C₂²⋊Q₈	—	166	1	no	—
88	C₄×Q₈	—	83	2	no	—
89	C₄×Q₈	—	166	1	no	—
90	#117	—	166	1	no	—
91	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	83	2	no	—
94	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
95	C₂²⋊C₈	—	166	1	no	—
96	C₄×C₈	—	166	1	no	—
97	C₂×Q₈	—	83	2	no	—
98	C₂×Q₈	—	166	1	no	—
99	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
100	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	166	2	no	—
105	C₄⋊C₄	—	83	2	no	—
106	C₄⋊C₄	—	166	2	no	—
107	C₄⋊C₄	—	166	2	no	—
108	C₄⋊C₄	—	166	1	no	—
109	C₄⋊C₄	—	166	1	no	—
110	C₄⋊C₄	—	166	1	no	—
111	C₂²⋊C₄	—	166	2	no	—
112	C₄²	—	166	2	no	—
113	C₄²	C₄×D₈₃	1	1	yes	characteristic
114	C₄²	D₃₃₂	1	1	yes	characteristic
115	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
116	C₄²	C₂²×D₈₃	1	1	yes	characteristic
117	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
118	Q₈	—	332	2	no	—
119	Q₈	—	332	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	166	2	no	—
123	C₂×C₄	—	166	2	no	—
124	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	#13050578	1	2	yes	—
128	C₂×C₄	#13050580	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	332	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#14581715	1	2	yes	—
137	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	332	2	no	—
142	C₄	—	332	2	no	—
143	C₄	#14581715	1	2	yes	—
144	C₄	C₂×#13050578	1	1	yes	characteristic
145	C₄	C₂×#13050580	1	1	yes	characteristic
146	C₄	#14581814	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	#19127683	1	1	yes	characteristic
149	C₂	C₂×#14581715	1	1	yes	characteristic
150	C₂	#19127698	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

#19127446

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×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

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₂²

yes

—

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

—

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

SD₁₆

yes

—

C₂×C₄×C₈₃

Q₁₆

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

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#103

yes

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#103

yes

—

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₄×C₈₃

#135

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#533

—

#431

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

#333

—

#332

—

C₂²×C₈

—

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

—

C₄⋊C₈

C₂×D₈₃

yes

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

100

C₂²×C₄

D₃₃₂

yes

characteristic

101

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

166

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

—

166

—

108

C₄⋊C₄

—

166

—

109

C₄⋊C₄

—

166

—

110

C₄⋊C₄

—

166

—

111

C₂²⋊C₄

—

166

—

112

C₄²

—

166

—

113

C₄²

C₄×D₈₃

yes

characteristic

114

C₄²

D₃₃₂

yes

characteristic

115

C₄²

C₂²⋊D₈₃

yes

characteristic

116

C₄²

C₂²×D₈₃

yes

characteristic

117

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

118

Q₈

—

332

—

119

Q₈

—

332

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

166

—

123

C₂×C₄

—

166

—

124

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

125

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

126

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

127

C₂×C₄

#13050578

yes

—

128

C₂×C₄

#13050580

yes

—

129

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

131

C₂×C₄

C₂×D₃₃₂

yes

characteristic

132

C₈

—

133

C₈

—

332

—

134

C₂²

—

135

C₂²

—

136

C₂²

#14581715

yes

—

137

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

332

—

142

C₄

—

332

—

143

C₄

#14581715

yes

—

144

C₄

C₂×#13050578

yes

characteristic

145

C₄

C₂×#13050580

yes

characteristic

146

C₄

#14581814

yes

—

147

C₂

—

148

C₂

#19127683

yes

characteristic

149

C₂

C₂×#14581715

yes

characteristic

150

C₂

#19127698

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#1542
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₃

hypocenter

C₈₃

#1542

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19127449	C₂	semidirect
4. #19127446	C₂	semidirect
76. C₈₃	#1542	semidirect

H ≅ G/N

type

3. #19127449

C₂

semidirect

4. #19127446

C₂

semidirect

76. C₈₃

#1542

semidirect

Subgroups — #24906121

Conjugacy classes of subgroups

Complements