Group #23771393 — subgroups

Subgroups — #23771393

Order 10112.

432 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 157 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	#19098484	C₂	1	4	yes	maximal
2	#19098483	C₂	1	2	yes	maximal
3	#19098470	C₂	1	2	yes	maximal
4	#19098469	C₂	1	2	yes	maximal
5	#19098467	C₂	1	4	yes	maximal
6	#391×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
8	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
13	C₇₉⋊#95	C₂²	1	2	yes	—
14	C₇₉⋊#95	C₂²	1	4	yes	—
15	C₇₉⋊#95	C₂²	1	4	yes	—
16	C₇₉⋊#95	C₂²	1	4	yes	—
17	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	#95×C₇₉	C₂²	1	4	yes	—
21	C₂×Dic₁₅₈	—	4	2	no	—
22	C₂×Dic₁₅₈	—	4	2	no	—
23	C₂×Dic₁₅₈	—	4	4	no	—
24	C₂×Dic₁₅₈	—	4	2	no	—
25	C₂×Dic₁₅₈	Q₈	1	8	yes	complemented
26	C₂²×Dic₇₉	C₂³	1	4	yes	—
27	C₂²×Dic₇₉	C₂³	1	4	yes	—
28	C₄⋊Dic₇₉	—	4	2	no	—
29	C₄⋊Dic₇₉	—	4	2	no	—
30	Dic₇₉⋊C₄	—	4	4	no	—
31	C₄×Dic₇₉	—	4	2	no	—
32	C₂×Q₈×C₇₉	—	4	1	no	—
33	C₂×Q₈×C₇₉	—	4	1	no	—
34	C₂×Q₈×C₇₉	Q₈	1	4	yes	complemented
35	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	4	1	no	—
40	C₄²×C₇₉	—	4	1	no	—
41	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
42	Q₈×C₇₉	—	4	4	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂×Dic₇₉	—	2	4	no	—
55	C₂×Dic₇₉	—	2	4	no	—
56	C₂×Dic₇₉	—	2	4	no	—
57	C₂×Dic₇₉	—	2	4	no	—
58	C₂×Dic₇₉	—	2	4	no	—
59	C₂×Dic₇₉	C₂×Q₈	1	8	yes	—
60	Dic₁₅₈	—	4	8	no	—
61	Dic₁₅₈	—	4	8	no	—
62	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
63	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
64	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
67	C₄×C₇₉	—	2	4	no	—
68	C₄×C₇₉	—	2	2	no	—
69	C₄×C₇₉	—	2	4	no	—
70	C₄×C₇₉	—	4	2	no	—
71	Dic₇₉	—	2	8	no	—
72	Dic₇₉	—	4	4	no	—
73	C₂×C₇₉	#543	1	2	yes	—
74	C₂×C₇₉	#543	1	1	yes	characteristic
75	C₂×C₇₉	Q₈²	1	1	yes	characteristic
76	C₂×C₇₉	Q₈²	1	2	yes	—
77	C₂×C₇₉	H(3,𝔽₄)	1	1	yes	characteristic
78	#2811	—	79	1	no	maximal, Sylow
79	C₇₉	#2811	1	1	yes	characteristic, complemented, Sylow
80	#395	—	79	2	no	—
81	#395	—	79	4	no	—
82	#391	—	79	2	no	—
83	#391	—	79	2	no	—
84	#391	—	79	4	no	—
85	#391	D₇₉	1	1	yes	characteristic
86	C₂²×Q₈	—	79	2	no	—
87	C₂²×Q₈	—	79	2	no	—
88	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	79	4	no	—
90	C₂×C₄⋊C₄	—	79	2	no	—
91	C₂×C₄⋊C₄	—	79	2	no	—
92	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
93	C₂×C₄²	—	79	2	no	—
94	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
95	#95	—	79	2	no	—
96	#95	—	79	4	no	—
97	#95	—	79	4	no	—
98	#95	—	79	4	no	—
99	#95	C₂×D₇₉	1	4	yes	—
100	C₂×Q₈	—	4	1	no	—
101	C₂×Q₈	—	4	1	no	—
102	C₂×Q₈	—	316	2	no	—
103	C₂×Q₈	—	316	2	no	—
104	C₂×Q₈	—	316	2	no	—
105	C₂×Q₈	—	316	4	no	—
106	C₂×Q₈	—	79	8	no	—
107	C₂×Q₈	Dic₁₅₈	1	4	yes	complemented
108	C₂²×C₄	—	79	4	no	—
109	C₂²×C₄	—	79	4	no	—
110	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	316	4	no	—
116	C₄⋊C₄	—	316	2	no	—
117	C₄⋊C₄	—	316	2	no	—
118	C₄²	—	4	1	no	—
119	C₄²	—	316	2	no	—
120	C₂³	C₂³×D₇₉	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	Q₈	—	316	8	no	—
123	Q₈	—	316	8	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	79	8	no	—
133	C₂×C₄	—	158	4	no	—
134	C₂×C₄	—	158	4	no	—
135	C₂×C₄	—	158	4	no	—
136	C₂×C₄	—	158	4	no	—
137	C₂×C₄	—	158	4	no	—
138	C₂×C₄	Q₈×D₇₉	1	4	yes	—
139	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
141	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
142	C₂²	C₂²×Dic₁₅₈	1	1	yes	characteristic
143	C₂²	#14579206	1	1	yes	characteristic
144	C₂²	#14579206	1	1	yes	characteristic
145	C₂²	#14579207	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	316	4	no	—
151	C₄	—	158	8	no	—
152	C₂	#19099188	1	2	yes	—
153	C₂	#19099189	1	1	yes	characteristic
154	C₂	#19099293	1	1	yes	characteristic
155	C₂	Q₈×Dic₁₅₈	1	2	yes	—
156	C₂	#19099305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098484

C₂

yes

maximal

#19098483

C₂

yes

maximal

#19098470

C₂

yes

maximal

#19098469

C₂

yes

maximal

#19098467

C₂

yes

maximal

#391×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂²×Dic₁₅₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

Q₈

yes

complemented

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

Q₈

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂⁴

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×Q₈

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂²×Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#543

yes

—

C₂×C₇₉

#543

yes

characteristic

C₂×C₇₉

Q₈²

yes

characteristic

C₂×C₇₉

Q₈²

yes

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₉

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

—

—

—

—

C₂×D₇₉

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

316

—

103

C₂×Q₈

—

316

—

104

C₂×Q₈

—

316

—

105

C₂×Q₈

—

316

—

106

C₂×Q₈

—

107

C₂×Q₈

Dic₁₅₈

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₇₉

yes

—

111

C₂²×C₄

C₂²×D₇₉

yes

—

112

C₂²×C₄

C₂²×D₇₉

yes

characteristic

113

C₂²×C₄

C₂²×D₇₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

316

—

116

C₄⋊C₄

—

316

—

117

C₄⋊C₄

—

316

—

118

C₄²

—

119

C₄²

—

316

—

120

C₂³

C₂³×D₇₉

yes

characteristic

121

Q₈

—

122

Q₈

—

316

—

123

Q₈

—

316

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

—

158

—

138

C₂×C₄

Q₈×D₇₉

yes

—

139

C₂×C₄

C₂×Dic₁₅₈

yes

—

140

C₂×C₄

C₂×Dic₁₅₈

yes

—

141

C₂²

C₂×Q₈×D₇₉

yes

—

142

C₂²

C₂²×Dic₁₅₈

yes

characteristic

143

C₂²

#14579206

yes

characteristic

144

C₂²

#14579206

yes

characteristic

145

C₂²

#14579207

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

316

—

151

C₄

—

158

—

152

C₂

#19099188

yes

—

153

C₂

#19099189

yes

characteristic

154

C₂

#19099293

yes

characteristic

155

C₂

Q₈×Dic₁₅₈

yes

—

156

C₂

#19099305

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#391×C₇₉	C₂
hypercenter	#391	D₇₉
hypocenter	C₇₉	#2811
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, prime radical, cosocle

C₂³×C₇₉

C₂⁴

Fitting

#391×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
25. C₂×Dic₁₅₈	Q₈	semidirect
34. C₂×Q₈×C₇₉	Q₈	semidirect
79. C₇₉	#2811	semidirect
107. C₂×Q₈	Dic₁₅₈	semidirect

H ≅ G/N

type

25. C₂×Dic₁₅₈

Q₈

semidirect

34. C₂×Q₈×C₇₉

Q₈

semidirect

79. C₇₉

#2811

semidirect

107. C₂×Q₈

Dic₁₅₈

semidirect

Subgroups — #23771393

Conjugacy classes of subgroups

Complements