Group #23764189 — subgroups

Subgroups — #23764189

Order 10112.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#19098612	C₂	1	1	yes	maximal, characteristic
2	#19098611	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579096	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579096	C₂	1	1	yes	maximal, characteristic
5	#19098519	C₂	1	1	yes	maximal, characteristic
6	#19098333	C₂	1	1	yes	maximal, characteristic
7	#344×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929608	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	C₂×#11929608	C₂²	1	1	yes	characteristic
11	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	#14579097	—	2	2	no	—
13	#14579096	—	2	2	no	—
14	#14579096	—	2	1	no	—
15	#14579096	—	2	1	no	—
16	#14579096	C₄	1	4	yes	complemented
17	#14579072	—	2	1	no	—
18	C₄×C₇₉⋊C₈	—	2	1	no	—
19	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	#117×C₇₉	C₂²	1	1	yes	characteristic
22	#11929608	—	2	1	no	—
23	#11929608	—	2	1	no	—
24	#11929608	—	2	1	no	—
25	#11929608	—	2	1	no	—
26	#11929608	C₂×C₄	1	2	yes	—
27	#11929608	D₄	1	2	yes	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₉	—	2	1	no	—
37	M₁₆×C₇₉	—	2	1	no	—
38	M₁₆×C₇₉	—	2	1	no	—
39	M₁₆×C₇₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₂²⋊C₄×C₇₉	Q₈	1	2	yes	—
44	C₄²×C₇₉	—	2	1	no	—
45	C₂³×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	—	4	1	no	—
55	C₇₉⋊C₈	—	2	2	no	—
56	C₇₉⋊C₈	—	2	2	no	—
57	C₇₉⋊C₈	—	4	1	no	—
58	C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₇₉	—	4	1	no	—
61	C₂²×C₇₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₇₉	—	4	2	no	—
65	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
67	C₄×C₇₉	#121	1	1	yes	characteristic
68	C₄×C₇₉	#125	1	1	yes	characteristic
69	C₂×C₇₉	—	2	1	no	—
70	C₂×C₇₉	—	2	1	no	—
71	C₂×C₇₉	—	2	1	no	—
72	C₂×C₇₉	#389	1	1	yes	characteristic
73	#1953	—	79	1	no	maximal, Sylow
74	C₇₉	#1953	1	1	yes	characteristic, complemented, Sylow
75	#430	—	79	1	no	—
76	#430	—	79	1	no	—
77	C₂×#108	—	79	1	no	—
78	C₂×#108	—	79	1	no	—
79	#405	—	79	1	no	—
80	#344	—	79	1	no	—
81	#344	D₇₉	1	1	yes	characteristic
82	C₂×M₁₆	—	79	1	no	—
83	C₂×M₁₆	—	79	1	no	—
84	C₂×M₁₆	—	79	1	no	—
85	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
87	C₂²×C₈	—	79	1	no	—
88	#117	C₂×D₇₉	1	1	yes	characteristic
89	#108	—	79	4	no	—
90	#108	—	158	2	no	—
91	#108	—	158	1	no	—
92	#108	—	158	1	no	—
93	#108	—	158	2	no	—
94	#97	—	158	1	no	—
95	C₄×C₈	—	158	1	no	—
96	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	79	2	no	—
101	M₁₆	—	79	2	no	—
102	M₁₆	—	158	1	no	—
103	M₁₆	—	158	1	no	—
104	M₁₆	—	158	1	no	—
105	M₁₆	—	158	1	no	—
106	M₁₆	C₄×D₇₉	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	79	2	no	—
110	C₂×C₈	—	79	1	no	—
111	C₂×C₈	—	79	1	no	—
112	C₂×C₈	—	79	2	no	—
113	C₂×C₈	—	158	1	no	—
114	C₂×C₈	—	158	1	no	—
115	C₂×C₈	—	158	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₁₅₈	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₁₅₈	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
122	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
123	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
125	C₂×C₄	#11929623	1	1	yes	characteristic
126	C₂×C₄	#11929624	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	316	1	no	—
130	C₈	—	79	2	no	—
131	C₈	—	79	2	no	—
132	C₈	—	158	2	no	—
133	C₈	—	158	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₁₅₈	1	1	yes	characteristic
136	C₂²	#14579138	1	1	yes	characteristic
137	C₂²	#14579160	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
140	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
141	C₄	#14579134	1	1	yes	characteristic
142	C₄	#14579151	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#19098460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098612

C₂

yes

maximal, characteristic

#19098611

C₂

yes

maximal, characteristic

C₂×#14579096

C₂

yes

maximal, characteristic

C₂×#14579096

C₂

yes

maximal, characteristic

#19098519

C₂

yes

maximal, characteristic

#19098333

C₂

yes

maximal, characteristic

#344×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

—

—

—

C₄

yes

complemented

#14579072

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

—

—

—

—

C₂×C₄

yes

—

#11929608

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

Q₈

yes

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₄⋊C₄

yes

—

C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×Q₈

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

#117

C₂×D₇₉

yes

characteristic

#108

—

#108

—

158

—

#108

—

158

—

#108

—

158

—

#108

—

158

—

#97

—

158

—

C₄×C₈

—

158

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

158

—

103

M₁₆

—

158

—

104

M₁₆

—

158

—

105

M₁₆

—

158

—

106

M₁₆

C₄×D₇₉

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

158

—

114

C₂×C₈

—

158

—

115

C₂×C₈

—

158

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₁₅₈

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₁₅₈

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

122

C₂×C₄

D₄×D₇₉

yes

characteristic

123

C₂×C₄

D₄×D₇₉

yes

characteristic

124

C₂×C₄

Q₈×D₇₉

yes

characteristic

125

C₂×C₄

#11929623

yes

characteristic

126

C₂×C₄

#11929624

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

316

—

130

C₈

—

131

C₈

—

132

C₈

—

158

—

133

C₈

—

158

—

134

C₂²

—

135

C₂²

C₄×Dic₁₅₈

yes

characteristic

136

C₂²

#14579138

yes

characteristic

137

C₂²

#14579160

yes

characteristic

138

C₄

—

139

C₄

C₄⋊C₄×D₇₉

yes

characteristic

140

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

141

C₄

#14579134

yes

characteristic

142

C₄

#14579151

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#19098460

yes

characteristic

	subgroup	G/H
center	C₄	#14579134
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#344×C₇₉	C₂
socle	C₂×C₇₉	#389
prime radical	C₂³×C₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#344	D₇₉
hypocenter	C₇₉	#1953
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14579134

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#344×C₇₉

C₂

socle

C₂×C₇₉

#389

prime radical

C₂³×C₇₉

C₂×Q₈

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #14579096	C₄	semidirect
74. C₇₉	#1953	semidirect

H ≅ G/N

type

16. #14579096

C₄

semidirect

74. C₇₉

#1953

semidirect

Subgroups — #23764189

Conjugacy classes of subgroups

Complements