Group #23776293 — subgroups

Subgroups — #23776293

Order 10112.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19099288	C₂	1	2	yes	maximal
2	#19098964	C₂	1	2	yes	maximal
3	#19098869	C₂	1	4	yes	maximal
4	#19098845	C₂	1	2	yes	maximal
5	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
7	#19098657	C₂	1	2	yes	maximal
8	#500×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	#14579159	—	2	2	no	—
11	#14579152	C₂²	1	4	yes	—
12	#14579151	—	2	4	no	—
13	Q₈×Dic₇₉	—	2	2	no	—
14	C₄×Dic₁₅₈	C₂²	1	2	yes	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	#14579126	C₂²	1	4	yes	—
17	#14579095	C₂²	1	1	yes	characteristic
18	#14579092	C₂²	1	4	yes	—
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₉⋊C₈	C₂²	1	2	yes	—
21	#14579086	C₂²	1	2	yes	—
22	#14579084	C₂²	1	4	yes	—
23	C₈×Dic₇₉	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
25	#107×C₇₉	C₂²	1	2	yes	—
26	#107×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₅₈	C₂³	1	2	yes	—
30	Dic₃₁₆	Q₈	1	4	yes	complemented
31	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₇₉	—	2	4	no	—
33	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₇₉	C₂³	1	2	yes	—
35	Dic₇₉⋊C₄	—	2	4	no	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	Dic₇₉⋊C₄	—	2	2	no	—
38	Dic₇₉⋊C₄	D₄	1	2	yes	—
39	C₄×Dic₇₉	—	2	4	no	—
40	C₄×Dic₇₉	—	2	2	no	—
41	C₄×Dic₇₉	C₂³	1	2	yes	—
42	C₂×Q₈×C₇₉	—	2	2	no	—
43	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	Q₈×C₇₉	—	4	4	no	—
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	2	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
57	Dic₁₅₈	—	2	2	no	—
58	Dic₁₅₈	—	2	2	no	—
59	Dic₁₅₈	C₂×Q₈	1	4	yes	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	C₂×Q₈	1	2	yes	—
62	C₇₉⋊C₈	—	2	2	no	—
63	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	4	4	no	—
66	C₄×C₇₉	#135	1	2	yes	—
67	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₇₉	—	2	4	no	—
70	Dic₇₉	—	2	2	no	—
71	Dic₇₉	—	2	2	no	—
72	Dic₇₉	—	4	2	no	—
73	C₂×C₇₉	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
75	C₂×C₇₉	#578	1	1	yes	characteristic
76	#3383	—	79	1	no	maximal, Sylow
77	C₇₉	#3383	1	1	yes	characteristic, complemented, Sylow
78	#557	—	79	2	no	—
79	#500	D₇₉	1	1	yes	characteristic
80	#499	—	79	2	no	—
81	#479	—	79	4	no	—
82	#475	—	79	2	no	—
83	C₈×Q₈	—	79	1	no	—
84	C₄×Q₁₆	—	79	2	no	—
85	C₄×Q₁₆	—	79	1	no	—
86	C₂×Q₁₆	—	79	1	no	—
87	C₄⋊Q₈	—	158	2	no	—
88	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
89	#125	—	79	4	no	—
90	#125	—	158	4	no	—
91	C₄×Q₈	—	79	1	no	—
92	C₄×Q₈	—	79	4	no	—
93	C₄×Q₈	—	79	2	no	—
94	C₄×Q₈	—	158	2	no	—
95	#107	—	79	1	no	—
96	#107	C₂×D₇₉	1	2	yes	—
97	#107	C₂×D₇₉	1	2	yes	—
98	#106	—	79	4	no	—
99	C₄⋊C₈	—	79	2	no	—
100	C₄⋊C₈	—	79	1	no	—
101	#103	—	79	2	no	—
102	#103	—	79	4	no	—
103	C₄×C₈	—	79	2	no	—
104	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	79	1	no	—
107	C₂×Q₈	—	79	2	no	—
108	Q₁₆	—	79	4	no	—
109	C₂×C₈	—	79	2	no	—
110	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	79	2	no	—
114	C₄⋊C₄	—	79	1	no	—
115	C₄⋊C₄	—	79	2	no	—
116	C₄⋊C₄	—	158	4	no	—
117	C₄⋊C₄	—	158	4	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	—	158	2	no	—
120	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
121	C₄²	—	79	2	no	—
122	C₄²	—	158	4	no	—
123	C₄²	—	158	2	no	—
124	C₄²	C₂²×D₇₉	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	Q₈	—	79	4	no	—
127	Q₈	—	158	2	no	—
128	Q₈	—	158	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	79	2	no	—
131	C₂×C₄	—	79	2	no	—
132	C₂×C₄	—	158	2	no	—
133	C₂×C₄	—	158	2	no	—
134	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	158	2	no	—
139	C₈	Q₈×D₇₉	1	2	yes	—
140	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	4	no	—
143	C₄	—	316	2	no	—
144	C₄	—	158	4	no	—
145	C₄	—	158	2	no	—
146	C₄	—	158	2	no	—
147	C₄	#14579180	1	2	yes	—
148	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
149	C₄	#14579209	1	1	yes	characteristic
150	C₂	#19099228	1	1	yes	characteristic
151	C₂	C₂×#14579180	1	1	yes	characteristic
152	C₂	#19099376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099288

C₂

yes

maximal

#19098964

C₂

yes

maximal

#19098869

C₂

yes

maximal

#19098845

C₂

yes

maximal

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic

#19098657

C₂

yes

maximal

#500×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

#14579159

—

#14579152

C₂²

yes

—

#14579151

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579126

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579086

C₂²

yes

—

#14579084

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

Dic₃₁₆

Q₈

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

D₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₂×Q₈

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

D₄×Q₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#557

—

#500

D₇₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

#125

—

#125

—

158

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

158

—

#107

—

#107

C₂×D₇₉

yes

—

#107

C₂×D₇₉

yes

—

#106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₇₉

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₇₉

yes

characteristic

111

C₂×C₈

C₂²×D₇₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

—

158

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₄⋊C₄

C₂²×D₇₉

yes

—

121

C₄²

—

122

C₄²

—

158

—

123

C₄²

—

158

—

124

C₄²

C₂²×D₇₉

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

158

—

128

Q₈

—

158

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

158

—

133

C₂×C₄

—

158

—

134

C₂×C₄

D₄×D₇₉

yes

characteristic

135

C₂×C₄

D₄×D₇₉

yes

characteristic

136

C₂×C₄

C₂³×D₇₉

yes

characteristic

137

C₈

—

138

C₈

—

158

—

139

C₈

Q₈×D₇₉

yes

—

140

C₂²

C₂×D₄×D₇₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

316

—

144

C₄

—

158

—

145

C₄

—

158

—

146

C₄

—

158

—

147

C₄

#14579180

yes

—

148

C₄

C₂×Q₈×D₇₉

yes

characteristic

149

C₄

#14579209

yes

characteristic

150

C₂

#19099228

yes

characteristic

151

C₂

C₂×#14579180

yes

characteristic

152

C₂

#19099376

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#500×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂²×D₄
hypercenter	#500	D₇₉
hypocenter	C₇₉	#3383
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#500×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
30. Dic₃₁₆	Q₈	semidirect
77. C₇₉	#3383	semidirect

H ≅ G/N

type

30. Dic₃₁₆

Q₈

semidirect

77. C₇₉

#3383

semidirect

Subgroups — #23776293

Conjugacy classes of subgroups

Complements