Group #23762082 — subgroups

Subgroups — #23762082

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	#19098616	C₂	1	1	yes	maximal, characteristic
3	#19098311	C₂	1	4	yes	maximal, complemented
4	#423×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	C₄	1	2	yes	—
6	#14579161	—	2	1	no	—
7	#14579161	—	2	1	no	—
8	#14579161	C₄	1	2	yes	—
9	#14579161	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₉	—	2	1	no	—
11	C₂×C₄⋊Dic₇₉	—	2	1	no	—
12	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
14	C₄×C₇₉⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₇₉	—	2	1	no	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	4	4	no	—
21	C₂×Dic₁₅₈	C₂×C₄	1	4	yes	—
22	C₂²×Dic₇₉	—	2	2	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₄⋊Dic₇₉	—	2	2	no	—
26	C₄⋊Dic₇₉	—	2	2	no	—
27	C₄⋊Dic₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	M₁₆×C₇₉	—	4	2	no	—
32	C₂×C₈×C₇₉	—	2	2	no	—
33	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₇₉	D₄	1	1	yes	characteristic
35	C₄²×C₇₉	D₄	1	1	yes	characteristic
36	C₄²×C₇₉	C₂³	1	1	yes	characteristic
37	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	2	no	—
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
45	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	4	no	—
49	C₂×Dic₇₉	—	4	4	no	—
50	Dic₁₅₈	—	4	4	no	—
51	Dic₁₅₈	—	4	4	no	—
52	C₈×C₇₉	—	4	2	no	—
53	C₇₉⋊C₈	—	4	2	no	—
54	C₂²×C₇₉	—	2	1	no	—
55	C₂²×C₇₉	—	2	1	no	—
56	C₂²×C₇₉	#103	1	2	yes	—
57	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₇₉	—	2	1	no	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	#100	1	2	yes	—
61	C₄×C₇₉	#103	1	2	yes	—
62	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₇₉	#137	1	2	yes	—
65	Dic₇₉	—	4	4	no	—
66	C₂×C₇₉	—	2	2	no	—
67	C₂×C₇₉	C₂×#100	1	1	yes	characteristic
68	C₂×C₇₉	C₂×#103	1	1	yes	characteristic
69	C₂×C₇₉	#419	1	1	yes	characteristic
70	#1548	—	79	1	no	maximal, Sylow
71	C₇₉	#1548	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊Q₈	—	79	1	no	—
73	#431	—	79	1	no	—
74	#423	D₇₉	1	1	yes	characteristic
75	#333	—	79	4	no	—
76	C₂²×Q₈	—	79	2	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	79	2	no	—
79	C₄⋊Q₈	—	79	2	no	—
80	C₄⋊Q₈	—	158	1	no	—
81	C₄⋊Q₈	—	158	1	no	—
82	C₂×C₄⋊C₄	—	158	1	no	—
83	C₂×C₄⋊C₄	—	158	1	no	—
84	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	79	2	no	—
87	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
88	C₂²⋊C₈	—	158	1	no	—
89	C₄×C₈	—	158	1	no	—
90	C₂×Q₈	—	316	4	no	—
91	C₂×Q₈	—	79	4	no	—
92	C₂×Q₈	—	158	2	no	—
93	C₂²×C₄	—	158	2	no	—
94	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
95	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
97	M₁₆	—	4	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	158	2	no	—
100	C₄⋊C₄	—	158	2	no	—
101	C₄⋊C₄	—	158	2	no	—
102	C₄⋊C₄	—	158	2	no	—
103	C₄⋊C₄	—	158	2	no	—
104	C₄²	C₄×D₇₉	1	1	yes	characteristic
105	C₄²	D₃₁₆	1	1	yes	characteristic
106	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₄²	C₂²×D₇₉	1	1	yes	characteristic
108	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
109	Q₈	—	316	4	no	—
110	Q₈	—	316	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	316	4	no	—
114	C₂×C₄	—	158	4	no	—
115	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
116	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	#11929612	1	2	yes	—
119	C₂×C₄	#11929614	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	316	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#14579084	1	2	yes	—
128	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	316	4	no	—
132	C₄	#14579077	1	2	yes	—
133	C₄	#14579084	1	2	yes	—
134	C₄	C₂×#11929612	1	1	yes	characteristic
135	C₄	C₂×#11929614	1	1	yes	characteristic
136	C₄	#14579194	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#14579077	1	1	yes	characteristic
139	C₂	C₂×#14579084	1	1	yes	characteristic
140	C₂	#19098573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

#19098616

C₂

yes

maximal, characteristic

#19098311

C₂

yes

maximal, complemented

#423×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₄

yes

—

#14579161

—

#14579161

—

#14579161

C₄

yes

—

#14579161

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

M₁₆×C₇₉

—

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

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

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

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#103

yes

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#100

yes

—

C₄×C₇₉

#103

yes

—

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

C₂×Q₁₆

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

—

#423

D₇₉

yes

characteristic

#333

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

158

—

C₄×C₈

—

158

—

C₂×Q₈

—

316

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

158

—

100

C₄⋊C₄

—

158

—

101

C₄⋊C₄

—

158

—

102

C₄⋊C₄

—

158

—

103

C₄⋊C₄

—

158

—

104

C₄²

C₄×D₇₉

yes

characteristic

105

C₄²

D₃₁₆

yes

characteristic

106

C₄²

C₂²⋊D₇₉

yes

characteristic

107

C₄²

C₂²×D₇₉

yes

characteristic

108

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

109

Q₈

—

316

—

110

Q₈

—

316

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

316

—

114

C₂×C₄

—

158

—

115

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

116

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

117

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

118

C₂×C₄

#11929612

yes

—

119

C₂×C₄

#11929614

yes

—

120

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

122

C₂×C₄

C₂×D₃₁₆

yes

characteristic

123

C₈

—

124

C₈

—

316

—

125

C₂²

—

126

C₂²

—

127

C₂²

#14579084

yes

—

128

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

316

—

132

C₄

#14579077

yes

—

133

C₄

#14579084

yes

—

134

C₄

C₂×#11929612

yes

characteristic

135

C₄

C₂×#11929614

yes

characteristic

136

C₄

#14579194

yes

—

137

C₂

—

138

C₂

C₂×#14579077

yes

characteristic

139

C₂

C₂×#14579084

yes

characteristic

140

C₂

#19098573

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1548
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#423×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 ◁ #423

N	H ≅ G/N	type
3. #19098311	C₂	semidirect
71. C₇₉	#1548	semidirect

H ≅ G/N

type

3. #19098311

C₂

semidirect

71. C₇₉

#1548

semidirect

Subgroups — #23762082

Conjugacy classes of subgroups

Complements