Group #23762704 — subgroups

Subgroups — #23762704

Order 10112.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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	#19099305	C₂	1	1	yes	maximal, characteristic
2	#19098925	C₂	1	2	yes	maximal, complemented
3	#19098303	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098302	C₂	1	2	yes	maximal, complemented
5	#497×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579150	—	2	1	no	—
7	#14579149	C₂²	1	2	yes	—
8	#14579147	—	2	2	no	—
9	#14579131	—	2	2	no	—
10	#14579129	—	2	1	no	—
11	#14579084	—	2	2	no	—
12	#14579081	—	2	2	no	—
13	#14579072	C₂²	1	2	yes	complemented
14	C₂×Q₁₆×C₇₉	—	2	1	no	—
15	C₂×SD₁₆×C₇₉	—	2	1	no	—
16	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
17	#124×C₇₉	C₂²	1	1	yes	characteristic
18	#97×C₇₉	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₇₉	—	2	1	no	—
20	C₂×Dic₁₅₈	D₄	1	2	yes	—
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×D₃₁₆	D₄	1	2	yes	—
23	C₂×C₂²⋊D₇₉	—	2	2	no	—
24	C₂×C₇₉⋊C₈	—	2	2	no	—
25	C₂²⋊Dic₇₉	—	2	1	no	—
26	(C₂×D₇₉)⋊C₄	—	2	2	no	—
27	(C₂×D₇₉)⋊C₄	—	2	2	no	—
28	(C₂×D₇₉)⋊C₄	—	2	2	no	—
29	C₄×Dic₇₉	—	2	1	no	—
30	C₂×Q₈×C₇₉	—	2	1	no	—
31	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
33	Q₁₆×C₇₉	—	4	2	no	—
34	SD₁₆×C₇₉	—	4	2	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	—	2	2	no	—
37	C₂²⋊C₄×C₇₉	—	2	2	no	—
38	C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₂³×C₇₉	—	2	1	no	—
40	C₂²×D₇₉	—	2	2	no	—
41	C₂²×D₇₉	—	8	1	no	—
42	C₂²×D₇₉	—	8	2	no	—
43	Q₈×C₇₉	—	4	1	no	—
44	Q₈×C₇₉	—	4	1	no	—
45	Q₈×C₇₉	—	4	1	no	—
46	D₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊D₇₉	—	8	2	no	—
52	C₂×Dic₇₉	—	2	2	no	—
53	D₃₁₆	—	4	2	no	—
54	D₃₁₆	—	8	1	no	—
55	Dic₁₅₈	—	4	2	no	—
56	C₈×C₇₉	—	2	2	no	—
57	C₇₉⋊C₈	—	4	2	no	—
58	C₂²×C₇₉	—	4	1	no	—
59	C₂²×C₇₉	—	4	2	no	—
60	C₂²×C₇₉	#120	1	1	yes	characteristic
61	C₂×D₇₉	—	4	2	no	—
62	C₂×D₇₉	—	4	2	no	—
63	C₂×D₇₉	—	4	2	no	—
64	C₂×D₇₉	—	8	4	no	—
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	2	no	—
67	C₄×C₇₉	—	4	1	no	—
68	C₄×C₇₉	—	8	1	no	—
69	Dic₇₉	—	4	2	no	—
70	C₂×C₇₉	—	8	1	no	—
71	C₂×C₇₉	#453	1	1	yes	characteristic
72	C₂×C₇₉	#455	1	2	yes	—
73	D₇₉	—	8	2	no	—
74	#1659	—	79	1	no	maximal, Sylow
75	C₇₉	#1659	1	1	yes	characteristic, complemented, Sylow
76	H(3,𝔽₄)	—	79	1	no	—
77	#497	D₇₉	1	1	yes	characteristic, complemented
78	#489	—	79	2	no	—
79	#329	—	79	1	no	—
80	#329	—	79	2	no	—
81	C₂×Q₁₆	—	2	1	no	—
82	C₂×SD₁₆	—	2	1	no	—
83	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic, complemented
84	#124	—	79	2	no	—
85	#124	—	158	2	no	—
86	#124	—	158	1	no	—
87	#124	C₂×D₇₉	1	1	yes	characteristic
88	#120	—	158	2	no	—
89	#120	—	158	1	no	—
90	#103	—	158	2	no	—
91	#102	—	158	2	no	—
92	#97	—	79	2	no	—
93	#97	C₂×D₇₉	1	1	yes	characteristic, complemented
94	C₂⁴	—	158	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	79	2	no	—
97	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
98	C₂×D₄	—	79	2	no	—
99	C₂×D₄	—	158	1	no	—
100	C₂×D₄	—	158	2	no	—
101	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
102	Q₁₆	—	4	2	no	—
103	SD₁₆	—	4	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	158	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	158	2	no	—
109	C₂²⋊C₄	—	158	1	no	—
110	C₂²⋊C₄	—	158	2	no	—
111	C₂²⋊C₄	—	158	2	no	—
112	C₄²	—	158	1	no	—
113	C₄²	C₂²×D₇₉	1	1	yes	characteristic
114	C₂³	—	2	1	no	—
115	C₂³	—	632	1	no	—
116	C₂³	—	632	2	no	—
117	C₂³	—	158	2	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	316	2	no	—
122	D₄	—	4	1	no	—
123	D₄	—	316	2	no	—
124	D₄	—	632	1	no	—
125	D₄	—	632	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	158	2	no	—
129	C₂×C₄	D₄×D₇₉	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	316	2	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	2	no	—
135	C₂²	—	316	2	no	—
136	C₂²	—	316	2	no	—
137	C₂²	—	316	2	no	—
138	C₂²	—	632	4	no	—
139	C₂²	#14579129	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	316	2	no	—
145	C₂	—	8	1	no	—
146	C₂	—	632	2	no	—
147	C₂	#19098730	1	1	yes	characteristic
148	C₂	#19098739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099305

C₂

yes

maximal, characteristic

#19098925

C₂

yes

maximal, complemented

#19098303

C₂

yes

maximal, characteristic, complemented

#19098302

C₂

yes

maximal, complemented

#497×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579150

—

#14579149

C₂²

yes

—

—

—

—

—

—

C₂²

yes

complemented

C₂×Q₁₆×C₇₉

—

C₂×SD₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic, complemented

#124×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

characteristic, complemented

C₂³×D₇₉

—

C₂×Dic₁₅₈

D₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₇₉⋊C₈

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

D₄

yes

characteristic

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#453

yes

characteristic

C₂×C₇₉

#455

yes

—

D₇₉

—

#1659

—

maximal, Sylow

C₇₉

#1659

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

#497

D₇₉

yes

characteristic, complemented

#489

—

#329

—

#329

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic, complemented

#124

—

#124

—

158

—

#124

—

158

—

#124

C₂×D₇₉

yes

characteristic

#120

—

158

—

#120

—

158

—

#103

—

158

—

#102

—

158

—

#97

—

#97

C₂×D₇₉

yes

characteristic, complemented

C₂⁴

—

158

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

158

—

100

C₂×D₄

—

158

—

101

C₂×D₄

C₂²⋊D₇₉

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

158

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

158

—

109

C₂²⋊C₄

—

158

—

110

C₂²⋊C₄

—

158

—

111

C₂²⋊C₄

—

158

—

112

C₄²

—

158

—

113

C₄²

C₂²×D₇₉

yes

characteristic

114

C₂³

—

115

C₂³

—

632

—

116

C₂³

—

632

—

117

C₂³

—

158

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

316

—

122

D₄

—

123

D₄

—

316

—

124

D₄

—

632

—

125

D₄

—

632

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

158

—

129

C₂×C₄

D₄×D₇₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

131

C₈

—

132

C₈

—

316

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

316

—

136

C₂²

—

316

—

137

C₂²

—

316

—

138

C₂²

—

632

—

139

C₂²

#14579129

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

316

—

145

C₂

—

146

C₂

—

632

—

147

C₂

#19098730

yes

characteristic

148

C₂

#19098739

yes

—

	subgroup	G/H
center	C₂²	#14579129
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#497×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	#19099305	C₂
hypercenter	#497	D₇₉
hypocenter	C₇₉	#1659
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

subgroup

G/H

center

C₂²

#14579129

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#497×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #497

N	H ≅ G/N	type
2. #19098925	C₂	semidirect
3. #19098303	C₂	semidirect
4. #19098302	C₂	semidirect
5. #497×C₇₉	C₂	semidirect
13. #14579072	C₂²	semidirect
16. C₄⋊Q₈×C₇₉	C₂²	semidirect
18. #97×C₇₉	C₂²	semidirect
75. C₇₉	#1659	semidirect
77. #497	D₇₉	semidirect
83. C₄⋊Q₈	C₂×D₇₉	semidirect
93. #97	C₂×D₇₉	semidirect

H ≅ G/N

type

2. #19098925

C₂

semidirect

3. #19098303

C₂

semidirect

4. #19098302

C₂

semidirect

5. #497×C₇₉

C₂

semidirect

13. #14579072

C₂²

semidirect

16. C₄⋊Q₈×C₇₉

C₂²

semidirect

18. #97×C₇₉

C₂²

semidirect

75. C₇₉

#1659

semidirect

77. #497

D₇₉

semidirect

83. C₄⋊Q₈

C₂×D₇₉

semidirect

93. #97

C₂×D₇₉

semidirect

Subgroups — #23762704

Conjugacy classes of subgroups

Complements