Group #23765300 — subgroups

Subgroups — #23765300

Order 10112.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19099001	C₂	1	1	yes	maximal, characteristic
2	#19098686	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098386	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098377	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098374	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14579183	C₂²	1	1	yes	characteristic
9	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
10	#14579167	—	2	1	no	—
11	C₄×D₃₁₆	—	2	1	no	—
12	#14579106	C₄	1	2	yes	complemented
13	D₁₂₆₄	C₄	1	2	yes	complemented
14	#14579097	C₂²	1	1	yes	characteristic
15	#14579093	C₂²	1	1	yes	characteristic
16	#14579089	—	2	1	no	—
17	#14579083	—	2	1	no	—
18	M₃₂×C₇₉	C₂²	1	1	yes	characteristic
19	M₃₂×C₇₉	C₂²	1	1	yes	characteristic
20	#97×C₇₉	C₂²	1	1	yes	characteristic
21	#11929624	—	2	1	no	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×C₄×D₇₉	—	4	1	no	—
24	Dic₃₁₆	C₂×C₄	1	1	yes	characteristic, complemented
25	#11929613	—	2	1	no	—
26	D₆₃₂	—	2	1	no	—
27	D₆₃₂	C₂×C₄	1	2	yes	—
28	D₆₃₂	C₂×C₄	1	1	yes	characteristic, complemented
29	#11929608	—	2	1	no	—
30	#11929607	—	4	1	no	—
31	C₈×D₇₉	—	4	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	(C₂×D₇₉)⋊C₄	—	4	1	no	—
34	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
36	C₄²×C₇₉	D₄	1	1	yes	characteristic
37	C₁₆×C₇₉	—	2	1	no	—
38	C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
39	C₂²×D₇₉	—	4	1	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₇₉	—	4	1	no	—
43	C₂×Dic₇₉	—	4	1	no	—
44	D₃₁₆	—	2	2	no	—
45	D₃₁₆	—	2	1	no	—
46	D₃₁₆	—	4	1	no	—
47	C₄×D₇₉	—	4	1	no	—
48	C₄×D₇₉	—	8	2	no	—
49	Dic₁₅₈	—	2	1	no	—
50	C₈×C₇₉	—	2	1	no	—
51	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₇₉	#41	1	1	yes	characteristic
53	C₇₉⋊C₈	—	4	1	no	—
54	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₇₉	—	4	1	no	—
56	C₂×D₇₉	—	4	2	no	—
57	C₂×D₇₉	—	8	1	no	—
58	C₂×D₇₉	—	8	1	no	—
59	C₄×C₇₉	—	4	1	no	—
60	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
61	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₉	—	4	1	no	—
63	Dic₇₉	—	8	1	no	—
64	C₂×C₇₉	—	2	1	no	—
65	C₂×C₇₉	#442	1	1	yes	characteristic
66	D₇₉	—	8	1	no	—
67	D₇₉	—	8	2	no	—
68	AGL(1,ℤ/16)	—	79	1	no	maximal, Sylow
69	C₇₉	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	—	79	1	no	—
71	#444	—	79	1	no	—
72	#442	—	79	1	no	—
73	#365	—	79	1	no	—
74	#361	—	79	1	no	—
75	#360	—	79	1	no	—
76	#347	D₇₉	1	1	yes	characteristic, complemented
77	#135	—	79	1	no	—
78	C₂×D₈	—	79	1	no	—
79	#131	—	158	1	no	—
80	C₄×D₄	—	158	1	no	—
81	SD₃₂	—	79	2	no	—
82	D₁₆	—	79	2	no	—
83	M₃₂	C₂×D₇₉	1	1	yes	characteristic
84	M₃₂	C₂×D₇₉	1	1	yes	characteristic
85	#108	—	79	1	no	—
86	#106	—	79	1	no	—
87	#104	—	158	1	no	—
88	#102	—	158	1	no	—
89	#97	C₂×D₇₉	1	1	yes	characteristic
90	#41	—	158	1	no	—
91	C₂×D₄	—	158	1	no	—
92	C₂²×C₄	—	316	1	no	—
93	Q₁₆	—	79	1	no	—
94	SD₁₆	—	158	1	no	—
95	D₈	—	79	2	no	—
96	D₈	—	79	1	no	—
97	D₈	—	158	1	no	—
98	M₁₆	—	316	1	no	—
99	M₁₆	—	158	1	no	—
100	C₂×C₈	—	316	1	no	—
101	C₂×C₈	D₃₁₆	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	C₄⋊C₄	—	158	1	no	—
104	C₂²⋊C₄	—	316	1	no	—
105	C₄²	D₃₁₆	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₇₉	1	2	yes	complemented
108	C₂³	—	316	1	no	—
109	Q₈	—	158	1	no	—
110	D₄	—	316	1	no	—
111	D₄	—	316	1	no	—
112	D₄	—	158	2	no	—
113	D₄	—	158	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	316	1	no	—
116	C₂×C₄	—	316	1	no	—
117	C₂×C₄	—	632	2	no	—
118	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	316	1	no	—
121	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
122	C₈	#11929624	1	1	yes	characteristic
123	C₂²	—	316	1	no	—
124	C₂²	—	316	2	no	—
125	C₂²	—	632	1	no	—
126	C₂²	—	632	1	no	—
127	C₂²	#14579189	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	316	1	no	—
130	C₄	—	632	1	no	—
131	C₄	C₄×D₃₁₆	1	1	yes	characteristic
132	C₄	#14579189	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	632	2	no	—
135	C₂	—	632	1	no	—
136	C₂	#19098676	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099001

C₂

yes

maximal, characteristic

#19098686

C₂

yes

maximal, characteristic, complemented

#19098676

C₂

yes

maximal, characteristic, complemented

#19098386

C₂

yes

maximal, characteristic, complemented

#19098377

C₂

yes

maximal, characteristic, complemented

#19098374

C₂

yes

maximal, characteristic, complemented

#347×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579183

C₂²

yes

characteristic

C₂×D₆₃₂

C₂²

yes

characteristic

#14579167

—

C₄×D₃₁₆

—

#14579106

C₄

yes

complemented

D₁₂₆₄

C₄

yes

complemented

#14579097

C₂²

yes

characteristic

#14579093

C₂²

yes

characteristic

#14579089

—

#14579083

—

M₃₂×C₇₉

C₂²

yes

characteristic

M₃₂×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

characteristic

#11929624

—

C₂×D₃₁₆

—

C₂×C₄×D₇₉

—

Dic₃₁₆

C₂×C₄

yes

characteristic, complemented

#11929613

—

D₆₃₂

—

D₆₃₂

C₂×C₄

yes

—

D₆₃₂

C₂×C₄

yes

characteristic, complemented

#11929608

—

#11929607

—

C₈×D₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₄

yes

complemented

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#442

yes

characteristic

D₇₉

—

D₇₉

—

AGL(1,ℤ/16)

—

maximal, Sylow

C₇₉

AGL(1,ℤ/16)

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic, complemented

#135

—

C₂×D₈

—

#131

—

158

—

C₄×D₄

—

158

—

SD₃₂

—

D₁₆

—

M₃₂

C₂×D₇₉

yes

characteristic

M₃₂

C₂×D₇₉

yes

—

—

—

158

—

#102

—

158

—

#97

C₂×D₇₉

yes

characteristic

#41

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

Q₁₆

—

SD₁₆

—

158

—

D₈

—

D₈

—

D₈

—

158

—

M₁₆

—

316

—

M₁₆

—

158

—

100

C₂×C₈

—

316

—

101

C₂×C₈

D₃₁₆

yes

characteristic

102

C₂×C₈

C₂²×D₇₉

yes

characteristic

103

C₄⋊C₄

—

158

—

104

C₂²⋊C₄

—

316

—

105

C₄²

D₃₁₆

yes

characteristic

106

C₁₆

—

107

C₁₆

C₄×D₇₉

yes

complemented

108

C₂³

—

316

—

109

Q₈

—

158

—

110

D₄

—

316

—

111

D₄

—

316

—

112

D₄

—

158

—

113

D₄

—

158

—

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

632

—

118

C₂×C₄

C₂×D₃₁₆

yes

characteristic

119

C₈

—

120

C₈

—

316

—

121

C₈

C₂×C₄×D₇₉

yes

characteristic

122

C₈

#11929624

yes

characteristic

123

C₂²

—

316

—

124

C₂²

—

316

—

125

C₂²

—

632

—

126

C₂²

—

632

—

127

C₂²

#14579189

yes

characteristic

128

C₄

—

129

C₄

—

316

—

130

C₄

—

632

—

131

C₄

C₄×D₃₁₆

yes

characteristic

132

C₄

#14579189

yes

characteristic

133

C₂

—

134

C₂

—

632

—

135

C₂

—

632

—

136

C₂

#19098676

yes

characteristic

	subgroup	G/H
center	C₂	#19098676
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#347×C₇₉	C₂
socle	C₂×C₇₉	#442
prime radical	#19099001	C₂
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	#347	D₇₉
hypocenter	C₇₉	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

subgroup

G/H

center

C₂

#19098676

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#347×C₇₉

C₂

C₂×C₇₉

C₂

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

N	H ≅ G/N	type
2. #19098686	C₂	semidirect
3. #19098676	C₂	semidirect
4. #19098386	C₂	semidirect
5. #19098377	C₂	semidirect
6. #19098374	C₂	semidirect
7. #347×C₇₉	C₂	semidirect
12. #14579106	C₄	semidirect
13. D₁₂₆₄	C₄	semidirect
24. Dic₃₁₆	C₂×C₄	semidirect
28. D₆₃₂	C₂×C₄	semidirect
38. C₁₆×C₇₉	C₂×C₄	semidirect
69. C₇₉	AGL(1,ℤ/16)	semidirect
76. #347	D₇₉	semidirect
107. C₁₆	C₄×D₇₉	semidirect

H ≅ G/N

type

2. #19098686

C₂

semidirect

3. #19098676

C₂

semidirect

4. #19098386

C₂

semidirect

5. #19098377

C₂

semidirect

6. #19098374

C₂

semidirect

7. #347×C₇₉

C₂

semidirect

12. #14579106

C₄

semidirect

13. D₁₂₆₄

C₄

semidirect

24. Dic₃₁₆

C₂×C₄

semidirect

28. D₆₃₂

C₂×C₄

semidirect

38. C₁₆×C₇₉

C₂×C₄

semidirect

69. C₇₉

AGL(1,ℤ/16)

semidirect

76. #347

D₇₉

semidirect

107. C₁₆

C₄×D₇₉

semidirect

Subgroups — #23765300

Conjugacy classes of subgroups

Complements