Group #23762071 — subgroups

Subgroups — #23762071

Order 10112.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#14579161	C₂	1	1	yes	maximal, characteristic
2	#19098587	C₂	1	1	yes	maximal, characteristic
3	#19098308	C₂	1	4	yes	maximal, complemented
4	#431×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	—	2	1	no	—
6	C₂×#11929608	—	2	1	no	—
7	#14579161	—	2	1	no	—
8	#14579161	—	2	1	no	—
9	#14579161	C₄	1	2	yes	—
10	#14579161	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₉	—	2	1	no	—
12	C₂×C₄⋊Dic₇₉	C₄	1	2	yes	—
13	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₇₉	—	2	1	no	—
18	C₄×C₈×C₇₉	—	2	1	no	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	—	4	2	no	—
22	C₂×Dic₁₅₈	—	4	2	no	—
23	C₂²×Dic₇₉	—	2	2	no	—
24	#11929608	—	4	2	no	—
25	C₂×C₇₉⋊C₈	—	2	2	no	—
26	C₄⋊Dic₇₉	—	2	2	no	—
27	C₄⋊Dic₇₉	—	2	2	no	—
28	C₄⋊Dic₇₉	—	2	2	no	—
29	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₇₉	D₄	1	1	yes	characteristic
36	C₄²×C₇₉	D₄	1	1	yes	characteristic
37	C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	D₈	1	2	yes	—
45	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₇₉	—	2	4	no	—
50	C₂×Dic₇₉	—	4	4	no	—
51	Dic₁₅₈	—	4	4	no	—
52	Dic₁₅₈	—	4	2	no	—
53	Dic₁₅₈	—	4	2	no	—
54	C₈×C₇₉	—	4	2	no	—
55	C₇₉⋊C₈	—	4	2	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	#102	1	2	yes	—
59	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	—	2	1	no	—
62	C₄×C₇₉	#101	1	2	yes	—
63	C₄×C₇₉	#102	1	2	yes	—
64	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
65	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₇₉	#137	1	2	yes	—
67	Dic₇₉	—	4	4	no	—
68	C₂×C₇₉	—	2	2	no	—
69	C₂×C₇₉	C₂×#101	1	1	yes	characteristic
70	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₇₉	#419	1	1	yes	characteristic
72	#1546	—	79	1	no	maximal, Sylow
73	C₇₉	#1546	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	79	1	no	—
75	#431	D₇₉	1	1	yes	characteristic
76	#423	—	79	1	no	—
77	#332	—	79	4	no	—
78	C₂²×Q₈	—	158	1	no	—
79	C₂×M₁₆	—	158	1	no	—
80	C₄⋊Q₈	—	79	2	no	—
81	C₄⋊Q₈	—	79	2	no	—
82	C₄⋊Q₈	—	158	1	no	—
83	C₄⋊Q₈	—	158	1	no	—
84	C₂×C₄⋊C₄	—	79	2	no	—
85	C₂×C₄⋊C₄	—	158	1	no	—
86	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
87	C₄⋊C₈	—	79	2	no	—
88	C₄⋊C₈	—	158	1	no	—
89	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	316	2	no	—
93	C₂×Q₈	—	316	2	no	—
94	C₂×Q₈	—	158	2	no	—
95	C₂×Q₈	—	158	2	no	—
96	C₂²×C₄	—	158	2	no	—
97	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
98	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
100	M₁₆	—	316	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	158	2	no	—
103	C₄⋊C₄	—	79	4	no	—
104	C₄⋊C₄	—	158	2	no	—
105	C₄⋊C₄	—	158	2	no	—
106	C₄⋊C₄	—	158	2	no	—
107	C₄²	C₄×D₇₉	1	1	yes	characteristic
108	C₄²	D₃₁₆	1	1	yes	characteristic
109	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₄²	C₂²×D₇₉	1	1	yes	characteristic
111	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
112	Q₈	—	316	4	no	—
113	Q₈	—	316	2	no	—
114	Q₈	—	316	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	316	4	no	—
118	C₂×C₄	—	158	4	no	—
119	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	D₆₃₂	1	2	yes	—
123	C₂×C₄	#11929613	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	316	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#14579083	1	2	yes	—
132	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	316	4	no	—
136	C₄	#14579079	1	2	yes	—
137	C₄	#14579083	1	2	yes	—
138	C₄	C₂×D₆₃₂	1	1	yes	characteristic
139	C₄	C₂×#11929613	1	1	yes	characteristic
140	C₄	#14579191	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#14579079	1	1	yes	characteristic
143	C₂	C₂×#14579083	1	1	yes	characteristic
144	C₂	#19098571	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

#19098587

C₂

yes

maximal, characteristic

#19098308

C₂

yes

maximal, complemented

#431×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

—

—

—

C₄

yes

—

#14579161

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#102

yes

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#101

yes

—

C₄×C₇₉

#102

yes

—

C₄×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

#137

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

D₇₉

yes

characteristic

#423

—

#332

—

C₂²×Q₈

—

158

—

C₂×M₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

100

M₁₆

—

316

—

101

C₂×C₈

—

102

C₂×C₈

—

158

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

158

—

105

C₄⋊C₄

—

158

—

106

C₄⋊C₄

—

158

—

107

C₄²

C₄×D₇₉

yes

characteristic

108

C₄²

D₃₁₆

yes

characteristic

109

C₄²

C₂²⋊D₇₉

yes

characteristic

110

C₄²

C₂²×D₇₉

yes

characteristic

111

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

112

Q₈

—

316

—

113

Q₈

—

316

—

114

Q₈

—

316

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

316

—

118

C₂×C₄

—

158

—

119

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

122

C₂×C₄

D₆₃₂

yes

—

123

C₂×C₄

#11929613

yes

—

124

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

126

C₂×C₄

C₂×D₃₁₆

yes

characteristic

127

C₈

—

128

C₈

—

316

—

129

C₂²

—

130

C₂²

—

131

C₂²

#14579083

yes

—

132

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

316

—

136

C₄

#14579079

yes

—

137

C₄

#14579083

yes

—

138

C₄

C₂×D₆₃₂

yes

characteristic

139

C₄

C₂×#11929613

yes

characteristic

140

C₄

#14579191

yes

—

141

C₂

—

142

C₂

C₂×#14579079

yes

characteristic

143

C₂

C₂×#14579083

yes

characteristic

144

C₂

#19098571

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#431×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#431	D₇₉
hypocenter	C₇₉	#1546
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#431×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₇₉

C₂²⋊C₄

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #19098308	C₂	semidirect
73. C₇₉	#1546	semidirect

H ≅ G/N

type

3. #19098308

C₂

semidirect

73. C₇₉

#1546

semidirect

Subgroups — #23762071

Conjugacy classes of subgroups

Complements