Group #23765108 — subgroups

Subgroups — #23765108

Order 10112.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14579183	C₂	1	1	yes	maximal, characteristic
2	#19098601	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098368	C₂	1	2	yes	maximal, complemented
4	#19098365	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929624	—	2	1	no	—
7	#14579183	—	2	2	no	—
8	#14579183	C₄	1	4	yes	complemented
9	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×#11929613	—	2	1	no	—
11	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
12	#14579120	—	2	1	no	—
13	#14579095	C₂²	1	2	yes	—
14	#14579093	—	2	1	no	—
15	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	—	2	2	no	—
17	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
18	#11929624	—	2	4	no	—
19	#11929624	—	4	2	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×C₂²⋊D₇₉	—	4	1	no	—
23	C₂×C₄×D₇₉	—	4	1	no	—
24	Dic₃₁₆	—	2	1	no	—
25	Dic₃₁₆	C₂×C₄	1	2	yes	—
26	#11929613	—	2	4	no	—
27	D₆₃₂	—	2	1	no	—
28	D₆₃₂	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₂²⋊Dic₇₉	—	4	1	no	—
31	C₄×Dic₇₉	—	4	1	no	—
32	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₉	D₄	1	2	yes	—
35	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₇₉	—	2	2	no	—
38	C₂³×C₇₉	D₈	1	1	yes	characteristic
39	C₂²×D₇₉	—	4	1	no	—
40	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊D₇₉	—	4	4	no	—
45	C₂×Dic₇₉	—	4	1	no	—
46	C₂×Dic₇₉	—	4	2	no	—
47	D₃₁₆	—	2	2	no	—
48	D₃₁₆	—	4	1	no	—
49	C₄×D₇₉	—	4	4	no	—
50	Dic₁₅₈	—	2	2	no	—
51	Dic₁₅₈	—	4	1	no	—
52	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
53	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₇₉	—	2	2	no	—
56	C₂²×C₇₉	#102	1	2	yes	—
57	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₉	—	4	2	no	—
59	C₂×D₇₉	—	8	2	no	—
60	C₄×C₇₉	#102	1	2	yes	—
61	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₇₉	—	4	2	no	—
64	Dic₇₉	—	8	2	no	—
65	C₂×C₇₉	—	2	2	no	—
66	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
67	C₂×C₇₉	#508	1	2	yes	—
68	D₇₉	—	8	2	no	—
69	#2138	—	79	1	no	maximal, Sylow
70	C₇₉	#2138	1	1	yes	characteristic, complemented, Sylow
71	C₂×#135	—	79	1	no	—
72	C₂²×C₁₆	D₇₉	1	1	yes	characteristic, complemented
73	#427	—	79	1	no	—
74	#358	—	79	2	no	—
75	#357	—	79	2	no	—
76	C₂×#41	—	158	1	no	—
77	#135	—	79	4	no	—
78	#135	—	158	2	no	—
79	C₂×Q₁₆	—	79	1	no	—
80	C₂×SD₁₆	—	158	1	no	—
81	C₂×D₈	—	79	1	no	—
82	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
83	#117	—	158	1	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×D₇₉	1	2	yes	—
86	#107	—	79	2	no	—
87	#106	—	158	1	no	—
88	#41	—	316	2	no	—
89	#41	—	158	4	no	—
90	C₂×Q₈	—	158	1	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	2	no	—
96	Q₁₆	—	158	1	no	—
97	SD₁₆	—	158	4	no	—
98	D₈	—	79	2	no	—
99	D₈	—	158	1	no	—
100	C₂×C₈	C₄×D₇₉	1	2	yes	—
101	C₂×C₈	D₃₁₆	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
103	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₄⋊C₄	—	158	2	no	—
105	C₂²⋊C₄	—	316	1	no	—
106	C₄²	—	316	1	no	—
107	C₁₆	—	2	2	no	—
108	C₂³	—	316	1	no	—
109	C₂³	D₆₃₂	1	1	yes	characteristic
110	Q₈	—	316	1	no	—
111	Q₈	—	158	2	no	—
112	D₄	—	316	4	no	—
113	D₄	—	316	1	no	—
114	D₄	—	158	2	no	—
115	C₂×C₄	—	316	1	no	—
116	C₂×C₄	—	316	4	no	—
117	C₂×C₄	—	316	2	no	—
118	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
119	C₂×C₄	D₆₃₂	1	1	yes	characteristic
120	C₂×C₄	#11929613	1	2	yes	—
121	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
122	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
123	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	316	2	no	—
127	C₂²	—	632	2	no	—
128	C₂²	#14579083	1	2	yes	—
129	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
130	C₄	—	316	2	no	—
131	C₄	—	632	2	no	—
132	C₄	#14579083	1	2	yes	—
133	C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
134	C₄	C₂×#11929613	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	632	2	no	—
137	C₂	C₂×#14579083	1	1	yes	characteristic
138	C₂	#19098995	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579183

C₂

yes

maximal, characteristic

#19098601

C₂

yes

maximal, characteristic, complemented

#19098368

C₂

yes

maximal, complemented

#19098365

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

—

#14579183

—

#14579183

C₄

yes

complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929613

—

C₂×D₆₃₂

C₂²

yes

characteristic

#14579120

—

#14579095

C₂²

yes

—

#14579093

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂²

yes

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

Dic₃₁₆

C₂×C₄

yes

—

#11929613

—

D₆₃₂

—

D₆₃₂

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#102

yes

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#102

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#102

yes

characteristic

C₂×C₇₉

#508

yes

—

D₇₉

—

#2138

—

maximal, Sylow

C₇₉

#2138

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂²×C₁₆

D₇₉

yes

characteristic, complemented

#427

—

#358

—

#357

—

C₂×#41

—

158

—

#135

—

#135

—

158

—

C₂×Q₁₆

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

#117

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

—

#107

—

#106

—

158

—

#41

—

316

—

#41

—

158

—

C₂×Q₈

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

C₂²×C₄

D₃₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

158

—

SD₁₆

—

158

—

D₈

—

D₈

—

158

—

100

C₂×C₈

C₄×D₇₉

yes

—

101

C₂×C₈

D₃₁₆

yes

characteristic

102

C₂×C₈

C₂²⋊D₇₉

yes

—

103

C₂×C₈

C₂²×D₇₉

yes

characteristic

104

C₄⋊C₄

—

158

—

105

C₂²⋊C₄

—

316

—

106

C₄²

—

316

—

107

C₁₆

—

108

C₂³

—

316

—

109

C₂³

D₆₃₂

yes

characteristic

110

Q₈

—

316

—

111

Q₈

—

158

—

112

D₄

—

316

—

113

D₄

—

316

—

114

D₄

—

158

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

316

—

118

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

119

C₂×C₄

D₆₃₂

yes

characteristic

120

C₂×C₄

#11929613

yes

—

121

C₂×C₄

C₂×D₃₁₆

yes

characteristic

122

C₈

(C₂×D₇₉)⋊C₄

yes

—

123

C₈

C₂×C₄×D₇₉

yes

characteristic

124

C₈

C₂×C₂²⋊D₇₉

yes

characteristic

125

C₂²

—

126

C₂²

—

316

—

127

C₂²

—

632

—

128

C₂²

#14579083

yes

—

129

C₂²

C₂×D₆₃₂

yes

characteristic

130

C₄

—

316

—

131

C₄

—

632

—

132

C₄

#14579083

yes

—

133

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

134

C₄

C₂×#11929613

yes

characteristic

135

C₂

—

136

C₂

—

632

—

137

C₂

C₂×#14579083

yes

characteristic

138

C₂

#19098995

yes

—

	subgroup	G/H
center	C₂×C₄	D₆₃₂
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₂²×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂×#14579183	C₂
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂²×C₁₆	D₇₉
hypocenter	C₇₉	#2138
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

D₆₃₂

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₂²×C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂×#14579183

C₂

cosocle

C₂×C₈×C₇₉

C₂³

hypercenter

C₂²×C₁₆

D₇₉

hypocenter

C₇₉

#2138

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #19098601	C₂	semidirect
3. #19098368	C₂	semidirect
4. #19098365	C₂	semidirect
5. C₂²×C₁₆×C₇₉	C₂	semidirect
8. #14579183	C₄	semidirect
70. C₇₉	#2138	semidirect
72. C₂²×C₁₆	D₇₉	semidirect

H ≅ G/N

type

2. #19098601

C₂

semidirect

3. #19098368

C₂

semidirect

4. #19098365

C₂

semidirect

5. C₂²×C₁₆×C₇₉

C₂

semidirect

8. #14579183

C₄

semidirect

70. C₇₉

#2138

semidirect

72. C₂²×C₁₆

D₇₉

semidirect

Subgroups — #23765108

Conjugacy classes of subgroups

Complements