Group #23776043 — subgroups

Subgroups — #23776043

Order 10112.

370 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 139 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	#19099292	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
3	#19098914	C₂	1	3	yes	maximal, complemented
4	#19098849	C₂	1	3	yes	maximal, complemented
5	#19098769	C₂	1	3	yes	maximal
6	C₄×#11929613	C₂	1	3	yes	maximal
7	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₉	—	2	1	no	—
9	C₂×#11929613	C₂²	1	3	yes	—
10	#14579161	C₂²	1	3	yes	—
11	#14579161	C₂²	1	3	yes	—
12	#14579152	—	2	3	no	—
13	#14579141	—	2	3	no	—
14	Q₈×Dic₇₉	—	2	1	no	—
15	C₄×Dic₁₅₈	C₂²	1	3	yes	—
16	C₄×D₃₁₆	C₂²	1	3	yes	—
17	#14579093	C₂²	1	3	yes	—
18	#14579086	C₂²	1	3	yes	—
19	#14579086	C₂²	1	6	yes	—
20	#14579083	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
23	C₄×C₈×C₇₉	C₂²	1	3	yes	—
24	C₂×Dic₁₅₈	—	2	3	no	—
25	C₂×Dic₁₅₈	C₂³	1	3	yes	—
26	Q₈×D₇₉	—	4	4	no	—
27	C₂×D₃₁₆	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₇₉	—	2	3	no	—
29	#11929613	—	2	6	no	—
30	C₄⋊Dic₇₉	—	2	3	no	—
31	C₄⋊Dic₇₉	—	2	3	no	—
32	C₄⋊Dic₇₉	—	2	3	no	—
33	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₇₉	C₂³	1	3	yes	—
35	Dic₇₉⋊C₄	—	2	3	no	—
36	(C₂×D₇₉)⋊C₄	—	2	3	no	—
37	C₄×Dic₇₉	—	2	3	no	—
38	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂³	1	3	yes	—
41	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
42	C₄²×C₇₉	C₂³	1	3	yes	—
43	C₂²×D₇₉	—	2	1	no	—
44	Q₈×C₇₉	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	1	no	—
49	C₂×Dic₇₉	—	2	3	no	—
50	C₂×Dic₇₉	—	2	3	no	—
51	D₃₁₆	—	2	1	no	—
52	D₃₁₆	#41	1	2	yes	—
53	C₄×D₇₉	—	4	6	no	—
54	Dic₁₅₈	—	2	3	no	—
55	Dic₁₅₈	—	2	3	no	—
56	Dic₁₅₈	—	4	6	no	—
57	C₈×C₇₉	—	2	3	no	—
58	C₈×C₇₉	—	2	1	no	—
59	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₇₉	—	2	2	no	—
61	C₂×D₇₉	—	4	1	no	—
62	C₂×D₇₉	—	4	1	no	—
63	C₄×C₇₉	—	2	3	no	—
64	C₄×C₇₉	C₂×SD₁₆	1	6	yes	—
65	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
66	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₇₉	—	4	3	no	—
68	Dic₇₉	—	4	3	no	—
69	Dic₇₉	—	4	1	no	—
70	C₂×C₇₉	#548	1	1	yes	characteristic
71	C₂×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
72	C₂×C₇₉	#578	1	1	yes	characteristic
73	D₇₉	—	4	2	no	—
74	#3359	—	79	1	no	maximal, Sylow
75	C₇₉	#3359	1	1	yes	characteristic, complemented, Sylow
76	#557	—	79	1	no	—
77	D₄×Q₈	—	79	1	no	—
78	#487	—	79	3	no	—
79	#475	—	79	3	no	—
80	#461	—	79	3	no	—
81	C₈×Q₈	D₇₉	1	1	yes	characteristic, complemented
82	C₄×SD₁₆	—	79	3	no	—
83	C₂²×Q₈	—	158	1	no	—
84	C₂×SD₁₆	—	79	3	no	—
85	C₄⋊Q₈	—	79	3	no	—
86	C₄⋊Q₈	—	79	3	no	—
87	#125	—	158	3	no	—
88	C₂²⋊Q₈	—	158	3	no	—
89	C₄×Q₈	—	79	3	no	—
90	C₄×Q₈	—	158	1	no	—
91	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
92	C₄×D₄	—	79	3	no	—
93	#106	—	79	3	no	—
94	C₄⋊C₈	C₂×D₇₉	1	3	yes	—
95	#103	—	79	3	no	—
96	#103	—	79	6	no	—
97	#102	—	79	1	no	—
98	C₄×C₈	C₂×D₇₉	1	3	yes	—
99	C₂×Q₈	—	316	4	no	—
100	C₂×Q₈	—	79	3	no	—
101	C₂×Q₈	—	158	3	no	—
102	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
103	C₂×D₄	—	79	1	no	—
104	C₂²×C₄	—	158	3	no	—
105	SD₁₆	—	158	6	no	—
106	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₇₉	1	3	yes	—
108	C₄⋊C₄	—	79	1	no	—
109	C₄⋊C₄	—	79	3	no	—
110	C₄⋊C₄	—	158	3	no	—
111	C₄⋊C₄	—	158	3	no	—
112	C₄⋊C₄	—	158	3	no	—
113	C₄⋊C₄	—	158	3	no	—
114	C₄⋊C₄	D₃₁₆	1	3	yes	—
115	C₂²⋊C₄	—	158	3	no	—
116	C₄²	—	158	3	no	—
117	C₄²	C₂²×D₇₉	1	3	yes	—
118	C₂³	—	158	1	no	—
119	Q₈	—	316	6	no	—
120	Q₈	—	158	3	no	—
121	Q₈	—	158	3	no	—
122	Q₈	#11929613	1	4	yes	complemented
123	D₄	—	79	2	no	—
124	D₄	—	158	1	no	—
125	C₂×C₄	—	316	6	no	—
126	C₂×C₄	—	158	1	no	—
127	C₂×C₄	—	158	3	no	—
128	C₂×C₄	—	158	3	no	—
129	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
130	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
131	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
132	C₈	—	2	3	no	—
133	C₈	—	2	1	no	—
134	C₂²	—	316	1	no	—
135	C₂²	—	316	1	no	—
136	C₂²	—	158	2	no	—
137	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	316	3	no	—
140	C₄	—	316	3	no	—
141	C₄	—	316	1	no	—
142	C₄	C₂×#11929613	1	6	yes	—
143	C₄	#41×D₇₉	1	1	yes	characteristic
144	C₄	#14579209	1	1	yes	characteristic
145	C₂	—	316	2	no	—
146	C₂	#19099225	1	1	yes	characteristic
147	C₂	C₂²×#11929613	1	1	yes	characteristic
148	C₂	#19099380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099292

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₁₆

C₂

yes

maximal, characteristic

#19098914

C₂

yes

maximal, complemented

#19098849

C₂

yes

maximal, complemented

#19098769

C₂

yes

maximal

C₄×#11929613

C₂

yes

maximal

C₈×Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₉

—

C₂×#11929613

C₂²

yes

—

#14579161

C₂²

yes

—

#14579161

C₂²

yes

—

#14579152

—

#14579141

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

#14579093

C₂²

yes

—

#14579086

C₂²

yes

—

#14579086

C₂²

yes

—

#14579083

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

Q₈×D₇₉

—

C₂×D₃₁₆

C₂³

yes

characteristic

C₂×C₄×D₇₉

—

#11929613

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂²×D₇₉

—

Q₈×C₇₉

SD₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

#41

yes

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#548

yes

characteristic

C₂×C₇₉

C₂²×SD₁₆

yes

characteristic

C₂×C₇₉

#578

yes

characteristic

D₇₉

—

#3359

—

maximal, Sylow

C₇₉

#3359

yes

characteristic, complemented, Sylow

#557

—

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

D₇₉

yes

characteristic, complemented

C₄×SD₁₆

—

C₂²×Q₈

—

158

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

C₂×D₇₉

yes

—

#103

—

#103

—

#102

—

C₄×C₈

C₂×D₇₉

yes

—

C₂×Q₈

—

316

—

100

C₂×Q₈

—

101

C₂×Q₈

—

158

—

102

C₂×Q₈

D₃₁₆

yes

characteristic

103

C₂×D₄

—

104

C₂²×C₄

—

158

—

105

SD₁₆

—

158

—

106

C₂×C₈

C₂²×D₇₉

yes

characteristic

107

C₂×C₈

C₂²×D₇₉

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

158

—

111

C₄⋊C₄

—

158

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

D₃₁₆

yes

—

115

C₂²⋊C₄

—

158

—

116

C₄²

—

158

—

117

C₄²

C₂²×D₇₉

yes

—

118

C₂³

—

158

—

119

Q₈

—

316

—

120

Q₈

—

158

—

121

Q₈

—

158

—

122

Q₈

#11929613

yes

complemented

123

D₄

—

124

D₄

—

158

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

158

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

C₂×D₃₁₆

yes

—

130

C₂×C₄

C₂×D₃₁₆

yes

—

131

C₂×C₄

C₂³×D₇₉

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

316

—

135

C₂²

—

316

—

136

C₂²

—

158

—

137

C₂²

C₂²×D₃₁₆

yes

characteristic

138

C₄

—

139

C₄

—

316

—

140

C₄

—

316

—

141

C₄

—

316

—

142

C₄

C₂×#11929613

yes

—

143

C₄

#41×D₇₉

yes

characteristic

144

C₄

#14579209

yes

characteristic

145

C₂

—

316

—

146

C₂

#19099225

yes

characteristic

147

C₂

C₂²×#11929613

yes

characteristic

148

C₂

#19099380

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₈×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₃₁₆	C₂³
hypercenter	C₈×Q₈	D₇₉
hypocenter	C₇₉	#3359
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₈×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₃₁₆

C₂³

hypercenter

C₈×Q₈

D₇₉

hypocenter

C₇₉

#3359

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #19099292	C₂	semidirect
3. #19098914	C₂	semidirect
4. #19098849	C₂	semidirect
7. C₈×Q₈×C₇₉	C₂	semidirect
44. Q₈×C₇₉	SD₁₆	semidirect
75. C₇₉	#3359	semidirect
81. C₈×Q₈	D₇₉	semidirect
122. Q₈	#11929613	semidirect

H ≅ G/N

type

1. #19099292

C₂

semidirect

3. #19098914

C₂

semidirect

4. #19098849

C₂

semidirect

7. C₈×Q₈×C₇₉

C₂

semidirect

44. Q₈×C₇₉

SD₁₆

semidirect

75. C₇₉

#3359

semidirect

81. C₈×Q₈

D₇₉

semidirect

122. Q₈

#11929613

semidirect

Subgroups — #23776043

Conjugacy classes of subgroups

Complements