Group #23774745 — subgroups

Subgroups — #23774745

Order 10112.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#19099120	C₂	1	2	yes	maximal
2	#19098859	C₂	1	4	yes	maximal, complemented
3	#19098804	C₂	1	4	yes	maximal
4	#19098774	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₁₆	C₂²	1	4	yes	—
7	#14579161	C₂²	1	2	yes	—
8	#14579161	C₂²	1	2	yes	—
9	#14579153	—	2	2	no	—
10	#14579140	—	2	2	no	—
11	#14579140	C₂²	1	4	yes	—
12	C₄×Dic₁₅₈	—	2	2	no	—
13	C₄×Dic₁₅₈	C₂²	1	4	yes	—
14	#14579120	—	2	2	no	—
15	#14579095	C₂²	1	4	yes	—
16	#14579086	C₂²	1	8	yes	—
17	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	C₂³	1	4	yes	—
22	Dic₃₁₆	—	4	4	no	—
23	C₄⋊Dic₇₉	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₄⋊Dic₇₉	—	2	2	no	—
26	C₄⋊Dic₇₉	C₂³	1	4	yes	—
27	Dic₇₉⋊C₄	—	2	4	no	—
28	Dic₇₉⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₇₉	—	2	4	no	—
30	C₄×Dic₇₉	—	2	4	no	—
31	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₇₉	—	4	2	no	—
35	C₂×C₈×C₇₉	C₂³	1	4	yes	—
36	C₄²×C₇₉	D₄	1	2	yes	—
37	C₄²×C₇₉	C₂³	1	2	yes	—
38	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	4	1	no	—
42	C₂×C₄×C₇₉	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₇₉	—	2	4	no	—
48	C₂×Dic₇₉	—	2	4	no	—
49	Dic₁₅₈	—	2	4	no	—
50	Dic₁₅₈	—	2	4	no	—
51	Dic₁₅₈	—	4	4	no	—
52	C₈×C₇₉	—	2	4	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	—	2	1	no	—
55	C₂²×C₇₉	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₇₉	—	2	1	no	—
58	C₄×C₇₉	—	2	2	no	—
59	C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
60	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₇₉	—	4	4	no	—
63	Dic₇₉	—	4	4	no	—
64	C₂×C₇₉	—	2	2	no	—
65	C₂×C₇₉	#537	1	1	yes	characteristic
66	C₂×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₇₉	#575	1	1	yes	characteristic
68	#3249	—	79	1	no	maximal, Sylow
69	C₇₉	#3249	1	1	yes	characteristic, complemented, Sylow
70	#533	—	79	2	no	—
71	#477	—	79	4	no	—
72	#467	—	79	4	no	—
73	#462	—	79	4	no	—
74	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
75	C₂×Q₁₆	—	79	4	no	—
76	C₂²×C₈	C₂×D₇₉	1	2	yes	—
77	C₄⋊Q₈	—	79	2	no	—
78	C₄⋊Q₈	—	79	2	no	—
79	#125	—	158	2	no	—
80	C₂²⋊Q₈	—	79	4	no	—
81	C₂²⋊Q₈	—	158	2	no	—
82	C₄×Q₈	—	79	4	no	—
83	C₄×Q₈	—	158	2	no	—
84	#117	—	158	2	no	—
85	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
86	#107	—	79	4	no	—
87	C₄⋊C₈	C₂×D₇₉	1	4	yes	—
88	#103	—	79	8	no	—
89	C₂×Q₈	—	79	4	no	—
90	C₂×Q₈	—	158	2	no	—
91	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
92	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
94	Q₁₆	—	316	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₇₉	1	4	yes	—
97	C₄⋊C₄	—	79	4	no	—
98	C₄⋊C₄	—	158	4	no	—
99	C₄⋊C₄	—	158	4	no	—
100	C₄⋊C₄	—	158	2	no	—
101	C₄⋊C₄	—	158	2	no	—
102	C₄⋊C₄	—	158	2	no	—
103	C₂²⋊C₄	—	158	4	no	—
104	C₄²	—	158	4	no	—
105	C₄²	D₃₁₆	1	2	yes	—
106	C₄²	C₂²×D₇₉	1	2	yes	—
107	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
108	Q₈	—	316	4	no	—
109	Q₈	—	158	4	no	—
110	Q₈	—	158	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	158	4	no	—
115	C₂×C₄	—	158	4	no	—
116	C₂×C₄	Dic₃₁₆	1	4	yes	complemented
117	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
118	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
120	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₃₁₆	1	2	yes	—
125	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	—	316	4	no	—
129	C₄	—	316	4	no	—
130	C₄	C₂×Dic₃₁₆	1	4	yes	—
131	C₄	#14579206	1	1	yes	characteristic
132	C₄	#14579209	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#19099144	1	1	yes	characteristic
135	C₂	C₂²×Dic₃₁₆	1	1	yes	characteristic
136	C₂	#19099363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal

#19098859

C₂

yes

maximal, complemented

#19098804

C₂

yes

maximal

#19098774

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

—

#14579161

C₂²

yes

—

#14579161

C₂²

yes

—

#14579153

—

#14579140

—

#14579140

C₂²

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₂²

yes

—

#14579120

—

#14579095

C₂²

yes

—

#14579086

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

—

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#537

yes

characteristic

C₂×C₇₉

C₂²×Q₁₆

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄⋊C₈

D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₇₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

#117

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₇₉

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

D₃₁₆

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

characteristic

Q₁₆

—

316

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

100

C₄⋊C₄

—

158

—

101

C₄⋊C₄

—

158

—

102

C₄⋊C₄

—

158

—

103

C₂²⋊C₄

—

158

—

104

C₄²

—

158

—

105

C₄²

D₃₁₆

yes

—

106

C₄²

C₂²×D₇₉

yes

—

107

C₂³

C₂×D₃₁₆

yes

characteristic

108

Q₈

—

316

—

109

Q₈

—

158

—

110

Q₈

—

158

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

158

—

115

C₂×C₄

—

158

—

116

C₂×C₄

Dic₃₁₆

yes

complemented

117

C₂×C₄

C₂×D₃₁₆

yes

—

118

C₂×C₄

C₂×D₃₁₆

yes

characteristic

119

C₂×C₄

C₂×D₃₁₆

yes

—

120

C₂×C₄

C₂³×D₇₉

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₃₁₆

yes

—

125

C₂²

C₂²×D₃₁₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

316

—

129

C₄

—

316

—

130

C₄

C₂×Dic₃₁₆

yes

—

131

C₄

#14579206

yes

characteristic

132

C₄

#14579209

yes

characteristic

133

C₂

—

134

C₂

#19099144

yes

characteristic

135

C₂

C₂²×Dic₃₁₆

yes

characteristic

136

C₂

#19099363

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#3249
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#3249

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #19098859	C₂	semidirect
4. #19098774	C₂	semidirect
42. C₂×C₄×C₇₉	Q₁₆	semidirect
69. C₇₉	#3249	semidirect
116. C₂×C₄	Dic₃₁₆	semidirect

H ≅ G/N

type

2. #19098859

C₂

semidirect

4. #19098774

C₂

semidirect

42. C₂×C₄×C₇₉

Q₁₆

semidirect

69. C₇₉

#3249

semidirect

116. C₂×C₄

Dic₃₁₆

semidirect

Subgroups — #23774745

Conjugacy classes of subgroups

Complements