Group #23773923 — subgroups

Subgroups — #23773923

Order 10112.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19099116	C₂	1	2	yes	maximal
2	#19098971	C₂	1	4	yes	maximal, complemented
3	#19098970	C₂	1	4	yes	maximal, complemented
4	#19098601	C₂	1	1	yes	maximal, characteristic
5	#19098599	C₂	1	2	yes	maximal
6	C₂×#14579071	C₂	1	1	yes	maximal, characteristic
7	#427×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14579160	C₂²	1	4	yes	—
10	#14579152	C₂²	1	4	yes	—
11	#14579138	—	2	4	no	—
12	C₄×Dic₁₅₈	—	2	4	no	—
13	#14579120	C₂²	1	2	yes	—
14	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
15	#14579095	C₂²	1	2	yes	—
16	#14579094	C₂²	1	4	yes	—
17	#14579093	C₂²	1	2	yes	—
18	#14579092	C₂²	1	4	yes	—
19	#14579071	C₂²	1	2	yes	—
20	#14579071	C₂²	1	2	yes	—
21	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	#117×C₇₉	C₂²	1	2	yes	—
23	#107×C₇₉	C₂²	1	2	yes	—
24	#106×C₇₉	C₂²	1	2	yes	—
25	C₂×Dic₁₅₈	—	2	4	no	—
26	C₂²×Dic₇₉	D₄	1	2	yes	—
27	C₂×C₇₉⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₇₉	—	2	4	no	—
30	C₄⋊Dic₇₉	C₂³	1	4	yes	—
31	Dic₇₉⋊C₄	—	2	4	no	—
32	Dic₇₉⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₇₉	—	2	2	no	—
34	C₄×Dic₇₉	—	2	2	no	—
35	C₄×Dic₇₉	D₄	1	2	yes	—
36	C₄×Dic₇₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
39	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₇₉	—	2	2	no	—
43	C₄²×C₇₉	—	2	2	no	—
44	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	—	2	4	no	—
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₇₉	—	2	4	no	—
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	C₂×D₄	1	4	yes	—
53	Dic₁₅₈	—	4	8	no	—
54	C₈×C₇₉	C₂×Q₈	1	2	yes	—
55	C₈×C₇₉	C₂×Q₈	1	2	yes	—
56	C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
57	C₂²×C₇₉	—	2	2	no	—
58	C₂²×C₇₉	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₇₉	—	4	4	no	—
61	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
62	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
64	Dic₇₉	—	2	4	no	—
65	Dic₇₉	—	4	4	no	—
66	C₂×C₇₉	—	2	2	no	—
67	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₇₉	#575	1	2	yes	—
69	#3162	—	79	1	no	maximal, Sylow
70	C₇₉	#3162	1	1	yes	characteristic, complemented, Sylow
71	#533	—	79	2	no	—
72	#501	—	79	4	no	—
73	#501	—	79	4	no	—
74	#427	—	79	1	no	—
75	#427	—	79	2	no	—
76	#427	D₇₉	1	1	yes	characteristic
77	C₂×#97	—	79	1	no	—
78	C₂²×C₈	—	79	1	no	—
79	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
80	C₄⋊Q₈	—	79	4	no	—
81	#125	—	79	4	no	—
82	C₂²⋊Q₈	—	158	4	no	—
83	C₄×Q₈	—	158	4	no	—
84	#117	—	79	2	no	—
85	#117	C₂×D₇₉	1	2	yes	—
86	C₂×C₄²	—	79	1	no	—
87	#107	—	79	2	no	—
88	#107	—	79	4	no	—
89	#107	C₂×D₇₉	1	2	yes	—
90	#106	—	79	2	no	—
91	#106	—	79	4	no	—
92	#106	C₂×D₇₉	1	2	yes	—
93	#97	—	79	2	no	—
94	#97	—	79	2	no	—
95	C₂×Q₈	—	158	4	no	—
96	C₂²×C₄	—	79	2	no	—
97	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×C₈	—	79	4	no	—
99	C₂×C₈	—	79	2	no	—
100	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
101	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	C₄⋊C₄	—	79	4	no	—
104	C₄⋊C₄	—	158	4	no	—
105	C₄⋊C₄	—	158	4	no	—
106	C₄⋊C₄	—	158	4	no	—
107	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	158	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	79	2	no	—
112	C₄²	—	79	2	no	—
113	C₄²	—	158	2	no	—
114	C₂³	D₄×D₇₉	1	1	yes	characteristic
115	Q₈	—	316	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	316	2	no	—
118	C₂×C₄	—	79	4	no	—
119	C₂×C₄	—	158	4	no	—
120	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₇₉	1	2	yes	—
122	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
123	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
124	C₈	—	79	4	no	—
125	C₈	C₂×Dic₁₅₈	1	2	yes	—
126	C₈	C₂×Dic₁₅₈	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#14579160	1	2	yes	—
129	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	316	4	no	—
132	C₄	—	158	4	no	—
133	C₄	#14579160	1	2	yes	—
134	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
135	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#14579160	1	1	yes	characteristic
138	C₂	#19099360	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099116

C₂

yes

maximal

#19098971

C₂

yes

maximal, complemented

#19098970

C₂

yes

maximal, complemented

#19098601

C₂

yes

maximal, characteristic

#19098599

C₂

yes

maximal

C₂×#14579071

C₂

yes

maximal, characteristic

#427×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579160

C₂²

yes

—

#14579152

C₂²

yes

—

#14579138

—

C₄×Dic₁₅₈

—

#14579120

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

C₂²

yes

—

#14579094

C₂²

yes

—

#14579093

C₂²

yes

—

#14579092

C₂²

yes

—

#14579071

C₂²

yes

—

#14579071

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

#106×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

Q₈

yes

complemented

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

C₈×C₇₉

C₂×Q₈

yes

—

C₈×C₇₉

C₂×Q₈

yes

—

C₇₉⋊C₈

C₂×Q₈

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄⋊Q₈

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄⋊Q₈

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂²×Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₉

#575

yes

—

#3162

—

maximal, Sylow

C₇₉

#3162

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₉

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

#117

C₂×D₇₉

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₇₉

yes

—

#106

—

#106

—

#106

C₂×D₇₉

yes

—

#97

—

#97

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₁₅₈

yes

complemented

101

C₂×C₈

C₂²×D₇₉

yes

characteristic

102

C₂×C₈

C₂²×D₇₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

158

—

105

C₄⋊C₄

—

158

—

106

C₄⋊C₄

—

158

—

107

C₄⋊C₄

C₂²×D₇₉

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

158

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

158

—

114

C₂³

D₄×D₇₉

yes

characteristic

115

Q₈

—

316

—

116

C₂×C₄

—

117

C₂×C₄

—

316

—

118

C₂×C₄

—

119

C₂×C₄

—

158

—

120

C₂×C₄

D₄×D₇₉

yes

characteristic

121

C₂×C₄

Q₈×D₇₉

yes

—

122

C₂×C₄

C₂×Dic₁₅₈

yes

—

123

C₂×C₄

C₂³×D₇₉

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₁₅₈

yes

—

126

C₈

C₂×Dic₁₅₈

yes

—

127

C₂²

—

128

C₂²

#14579160

yes

—

129

C₂²

C₂×D₄×D₇₉

yes

characteristic

130

C₄

—

131

C₄

—

316

—

132

C₄

—

158

—

133

C₄

#14579160

yes

—

134

C₄

C₂×Q₈×D₇₉

yes

characteristic

135

C₄

C₂²×Dic₁₅₈

yes

characteristic

136

C₂

—

137

C₂

C₂×#14579160

yes

characteristic

138

C₂

#19099360

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#427×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#427	D₇₉
hypocenter	C₇₉	#3162
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#427×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #19098971	C₂	semidirect
3. #19098970	C₂	semidirect
27. C₂×C₇₉⋊C₈	Q₈	semidirect
38. C₂×C₈×C₇₉	Q₈	semidirect
70. C₇₉	#3162	semidirect
100. C₂×C₈	Dic₁₅₈	semidirect

H ≅ G/N

type

2. #19098971

C₂

semidirect

3. #19098970

C₂

semidirect

27. C₂×C₇₉⋊C₈

Q₈

semidirect

38. C₂×C₈×C₇₉

Q₈

semidirect

70. C₇₉

#3162

semidirect

100. C₂×C₈

Dic₁₅₈

semidirect

Subgroups — #23773923

Conjugacy classes of subgroups

Complements