Group #23765319 — subgroups

Subgroups — #23765319

Order 10112.

226 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579106	C₂	1	2	yes	maximal, complemented
3	#19098799	C₂	1	1	yes	maximal, characteristic
4	#19098368	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₅₈	—	2	1	no	—
7	C₂×Dic₃₁₆	—	2	1	no	—
8	C₂×Dic₃₁₆	—	2	4	no	—
9	C₂×Dic₃₁₆	C₂²	1	2	yes	—
10	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
11	#14579135	—	2	1	no	—
12	#14579106	—	2	4	no	—
13	#14579095	C₂²	1	1	yes	characteristic
14	#14579083	—	2	1	no	—
15	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
17	C₂×Dic₁₅₈	—	2	4	no	—
18	C₂×Dic₁₅₈	—	2	2	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂×Dic₁₅₈	—	4	1	no	—
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×C₂²⋊D₇₉	—	4	1	no	—
23	C₂²×Dic₇₉	—	4	1	no	—
24	Dic₃₁₆	—	2	4	no	—
25	Dic₃₁₆	—	2	2	no	—
26	Dic₃₁₆	D₄	1	4	yes	complemented
27	D₆₃₂	—	2	2	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	(C₂×D₇₉)⋊C₄	—	4	1	no	—
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₁₆×C₇₉	—	2	2	no	—
35	C₂³×C₇₉	D₈	1	1	yes	characteristic
36	C₂²×D₇₉	—	4	1	no	—
37	C₂×C₄×C₇₉	—	2	1	no	—
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₇₉	—	8	2	no	—
42	C₂×Dic₇₉	—	4	4	no	—
43	C₂×Dic₇₉	—	4	2	no	—
44	C₂×Dic₇₉	—	4	1	no	—
45	D₃₁₆	—	4	1	no	—
46	D₃₁₆	—	4	1	no	—
47	Dic₁₅₈	—	2	4	no	—
48	Dic₁₅₈	—	4	4	no	—
49	Dic₁₅₈	—	4	2	no	—
50	C₈×C₇₉	—	2	1	no	—
51	C₈×C₇₉	C₂×D₄	1	2	yes	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	SD₃₂	1	2	yes	complemented
55	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
56	C₂×D₇₉	—	8	1	no	—
57	C₂×D₇₉	—	8	1	no	—
58	C₂×D₇₉	—	8	1	no	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	#120	1	1	yes	characteristic
61	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₉	—	4	4	no	—
63	Dic₇₉	—	8	1	no	—
64	C₂×C₇₉	—	2	2	no	—
65	C₂×C₇₉	#447	1	1	yes	characteristic
66	C₂×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
67	C₂×C₇₉	#510	1	1	yes	characteristic
68	D₇₉	—	16	1	no	—
69	#2185	—	79	1	no	maximal, Sylow
70	C₇₉	#2185	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₁₆	—	79	1	no	—
72	C₂×SD₃₂	—	79	2	no	—
73	#466	—	79	1	no	—
74	#358	—	79	2	no	—
75	C₂²⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
76	C₂²×Q₈	—	158	1	no	—
77	C₂×Q₁₆	—	79	2	no	—
78	C₂×Q₁₆	—	158	4	no	—
79	C₂×Q₁₆	—	158	1	no	—
80	C₂×D₈	—	79	1	no	—
81	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
82	#121	—	158	1	no	—
83	SD₃₂	—	158	4	no	—
84	C₂×C₁₆	C₂×D₇₉	1	2	yes	—
85	#107	—	79	1	no	—
86	#102	—	158	1	no	—
87	C₂×Q₈	—	316	2	no	—
88	C₂×Q₈	—	316	1	no	—
89	C₂×Q₈	—	158	4	no	—
90	C₂×Q₈	—	158	2	no	—
91	C₂×D₄	—	316	1	no	—
92	C₂×D₄	—	158	1	no	—
93	C₂²×C₄	—	316	1	no	—
94	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
95	Q₁₆	—	79	4	no	—
96	Q₁₆	—	158	4	no	—
97	Q₁₆	—	158	2	no	—
98	D₈	—	158	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	D₃₁₆	1	1	yes	characteristic
101	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
102	C₄⋊C₄	—	158	1	no	—
103	C₂²⋊C₄	—	316	1	no	—
104	C₁₆	—	2	2	no	—
105	C₂³	—	316	1	no	—
106	C₂³	D₆₃₂	1	1	yes	characteristic
107	Q₈	—	316	4	no	—
108	Q₈	—	316	2	no	—
109	Q₈	—	158	4	no	—
110	D₄	—	316	1	no	—
111	D₄	—	316	1	no	—
112	D₄	—	632	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	316	4	no	—
116	C₂×C₄	—	316	2	no	—
117	C₂×C₄	—	316	1	no	—
118	C₂×C₄	D₆₃₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	D₄×D₇₉	1	2	yes	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	632	1	no	—
125	C₂²	—	632	1	no	—
126	C₂²	—	632	1	no	—
127	C₂²	#14579106	1	2	yes	complemented
128	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	316	4	no	—
131	C₄	—	632	1	no	—
132	C₄	#14579131	1	1	yes	characteristic
133	C₄	#14579189	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	—	1264	1	no	—
136	C₂	#19098699	1	1	yes	characteristic
137	C₂	C₂×#14579106	1	1	yes	characteristic
138	C₂	#19099006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#14579106

C₂

yes

maximal, complemented

#19098799

C₂

yes

maximal, characteristic

#19098368

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

—

—

C₂²

yes

characteristic

#14579083

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

D₄

yes

complemented

D₆₃₂

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

SD₃₂

yes

complemented

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#447

yes

characteristic

C₂×C₇₉

C₂×SD₃₂

yes

characteristic

C₂×C₇₉

#510

yes

characteristic

D₇₉

—

#2185

—

maximal, Sylow

C₇₉

#2185

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×SD₃₂

—

#466

—

#358

—

C₂²⋊C₁₆

D₇₉

yes

characteristic, complemented

C₂²×Q₈

—

158

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂×D₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

#121

—

158

—

SD₃₂

—

158

—

C₂×C₁₆

C₂×D₇₉

yes

—

#107

—

#102

—

158

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

C₂²×C₄

D₃₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

158

—

Q₁₆

—

158

—

D₈

—

158

—

C₂×C₈

—

100

C₂×C₈

D₃₁₆

yes

characteristic

101

C₂×C₈

C₂²×D₇₉

yes

characteristic

102

C₄⋊C₄

—

158

—

103

C₂²⋊C₄

—

316

—

104

C₁₆

—

105

C₂³

—

316

—

106

C₂³

D₆₃₂

yes

characteristic

107

Q₈

—

316

—

108

Q₈

—

316

—

109

Q₈

—

158

—

110

D₄

—

316

—

111

D₄

—

316

—

112

D₄

—

632

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

316

—

118

C₂×C₄

D₆₃₂

yes

characteristic

119

C₂×C₄

C₂×D₃₁₆

yes

characteristic

120

C₈

—

121

C₈

D₄×D₇₉

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

632

—

125

C₂²

—

632

—

126

C₂²

—

632

—

127

C₂²

#14579106

yes

complemented

128

C₂²

C₂×D₆₃₂

yes

characteristic

129

C₄

—

130

C₄

—

316

—

131

C₄

—

632

—

132

C₄

#14579131

yes

characteristic

133

C₄

#14579189

yes

characteristic

134

C₂

—

135

C₂

—

1264

—

136

C₂

#19098699

yes

characteristic

137

C₂

C₂×#14579106

yes

characteristic

138

C₂

#19099006

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₂²⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	#19098799	C₂
hypercenter	C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#2185
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₆₃₂

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₂²⋊C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

#19098799

C₂

hypercenter

C₂²⋊C₁₆

D₇₉

hypocenter

C₇₉

#2185

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
1. C₂²×Dic₃₁₆	C₂	semidirect
2. C₂×#14579106	C₂	semidirect
4. #19098368	C₂	semidirect
5. C₂²⋊C₁₆×C₇₉	C₂	semidirect
26. Dic₃₁₆	D₄	semidirect
54. C₂²×C₇₉	SD₃₂	semidirect
70. C₇₉	#2185	semidirect
75. C₂²⋊C₁₆	D₇₉	semidirect
127. C₂²	#14579106	semidirect

H ≅ G/N

type

1. C₂²×Dic₃₁₆

C₂

semidirect

2. C₂×#14579106

C₂

semidirect

4. #19098368

C₂

semidirect

5. C₂²⋊C₁₆×C₇₉

C₂

semidirect

26. Dic₃₁₆

D₄

semidirect

54. C₂²×C₇₉

SD₃₂

semidirect

70. C₇₉

#2185

semidirect

75. C₂²⋊C₁₆

D₇₉

semidirect

127. C₂²

#14579106

semidirect

Subgroups — #23765319

Conjugacy classes of subgroups

Complements