Group #23763023 — subgroups

Subgroups — #23763023

Order 10112.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19098959	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098937	C₂	1	1	yes	maximal, characteristic
3	#19098936	C₂	1	1	yes	maximal, characteristic
4	#19098913	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098911	C₂	1	1	yes	maximal, characteristic
6	#19098284	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	—	2	2	no	—
9	C₂×#11929611	—	2	2	no	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579158	C₂²	1	1	yes	characteristic
12	#14579092	—	2	2	no	—
13	#14579085	—	2	2	no	—
14	#14579081	—	2	2	no	—
15	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
19	#107×C₇₉	—	2	2	no	—
20	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×D₃₁₆	—	2	1	no	—
23	C₂×D₃₁₆	—	2	1	no	—
24	C₂×D₃₁₆	—	2	1	no	—
25	#11929613	—	4	4	no	—
26	#11929611	—	4	4	no	—
27	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
28	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₂×Q₈×C₇₉	—	2	1	no	—
32	C₂×Q₈×C₇₉	—	2	1	no	—
33	C₂×C₈×C₇₉	D₄	1	2	yes	—
34	C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₂²×D₇₉	—	4	1	no	—
40	Q₈×C₇₉	—	4	2	no	—
41	Q₈×C₇₉	—	4	2	no	—
42	C₂×C₄×C₇₉	—	4	1	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₇₉	—	4	1	no	—
48	D₃₁₆	—	4	2	no	—
49	D₃₁₆	—	4	2	no	—
50	D₃₁₆	—	4	2	no	—
51	Dic₁₅₈	—	4	2	no	—
52	C₈×C₇₉	SD₁₆	1	4	yes	complemented
53	C₇₉⋊C₈	—	2	2	no	—
54	C₇₉⋊C₈	—	2	2	no	—
55	C₂²×C₇₉	#127	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	—	8	1	no	—
60	C₄×C₇₉	—	8	1	no	—
61	C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
62	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₇₉	#137	1	2	yes	—
64	Dic₇₉	—	8	1	no	—
65	C₂×C₇₉	#492	1	1	yes	characteristic
66	C₂×C₇₉	#496	1	1	yes	characteristic
67	C₂×C₇₉	#497	1	1	yes	characteristic
68	D₇₉	—	16	1	no	—
69	#1714	—	79	1	no	maximal, Sylow
70	C₇₉	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	D₇₉	1	1	yes	characteristic, complemented
72	#498	—	79	1	no	—
73	#492	—	79	1	no	—
74	#492	—	79	1	no	—
75	#487	—	79	1	no	—
76	#486	—	79	1	no	—
77	#322	—	79	1	no	—
78	C₂×SD₁₆	—	158	2	no	—
79	C₂×SD₁₆	—	158	2	no	—
80	C₄⋊Q₈	—	79	1	no	—
81	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
83	#127	—	79	1	no	—
84	#107	—	2	2	no	—
85	#106	—	158	2	no	—
86	#103	—	158	2	no	—
87	#102	—	158	2	no	—
88	C₄×C₈	—	79	1	no	—
89	C₄×C₈	—	79	1	no	—
90	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	158	1	no	—
94	C₂×D₄	—	158	1	no	—
95	C₂×D₄	—	158	1	no	—
96	C₂×D₄	—	158	1	no	—
97	SD₁₆	—	316	4	no	—
98	SD₁₆	—	316	4	no	—
99	C₂×C₈	—	79	2	no	—
100	C₂×C₈	—	79	2	no	—
101	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	158	1	no	—
107	C₄⋊C₄	—	158	1	no	—
108	C₄²	C₂²×D₇₉	1	1	yes	characteristic
109	C₂³	—	316	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	316	2	no	—
113	D₄	—	316	2	no	—
114	D₄	—	316	2	no	—
115	D₄	—	316	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	316	1	no	—
119	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
122	C₈	—	158	2	no	—
123	C₈	—	158	2	no	—
124	C₈	#11929611	1	4	yes	complemented
125	C₂²	—	632	1	no	—
126	C₂²	—	632	1	no	—
127	C₂²	—	632	1	no	—
128	C₂²	#14579157	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	632	1	no	—
132	C₄	C₂×#11929611	1	2	yes	—
133	C₄	#14579185	1	2	yes	—
134	C₄	#14579193	1	2	yes	—
135	C₂	—	1264	1	no	—
136	C₂	#19098937	1	1	yes	characteristic
137	C₂	#19098951	1	1	yes	characteristic
138	C₂	#19098956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098959

C₂

yes

maximal, characteristic, complemented

#19098937

C₂

yes

maximal, characteristic

#19098936

C₂

yes

maximal, characteristic

#19098913

C₂

yes

maximal, characteristic, complemented

#19098911

C₂

yes

maximal, characteristic

#19098284

C₂

yes

maximal, characteristic, complemented

#500×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929613

—

C₂×#11929611

—

#14579161

C₂²

yes

characteristic

#14579158

C₂²

yes

—

—

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#107×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929613

—

#11929611

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

SD₁₆

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#127

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

#137

yes

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1714

—

maximal, Sylow

C₇₉

#1714

yes

characteristic, complemented, Sylow

#500

D₇₉

yes

characteristic, complemented

—

—

—

—

—

—

C₂×SD₁₆

—

158

—

C₂×SD₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

C₂×D₇₉

yes

—

—

—

158

—

#103

—

158

—

#102

—

158

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

SD₁₆

—

316

—

SD₁₆

—

316

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₇₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

158

—

107

C₄⋊C₄

—

158

—

108

C₄²

C₂²×D₇₉

yes

characteristic

109

C₂³

—

316

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

316

—

113

D₄

—

316

—

114

D₄

—

316

—

115

D₄

—

316

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

316

—

119

C₂×C₄

D₄×D₇₉

yes

characteristic

120

C₂×C₄

D₄×D₇₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

122

C₈

—

158

—

123

C₈

—

158

—

124

C₈

#11929611

yes

complemented

125

C₂²

—

632

—

126

C₂²

—

632

—

127

C₂²

—

632

—

128

C₂²

#14579157

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

632

—

132

C₄

C₂×#11929611

yes

—

133

C₄

#14579185

yes

—

134

C₄

#14579193

yes

—

135

C₂

—

1264

—

136

C₂

#19098937

yes

characteristic

137

C₂

#19098951

yes

characteristic

138

C₂

#19098956

yes

characteristic

	subgroup	G/H
center	C₂²	#14579157
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#500×C₇₉	C₂
socle	C₂²×C₇₉	#127
prime radical	#14579158	C₂²
hypercenter	#500	D₇₉
hypocenter	C₇₉	#1714
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14579157

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#500×C₇₉

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #19098959	C₂	semidirect
4. #19098913	C₂	semidirect
6. #19098284	C₂	semidirect
7. #500×C₇₉	C₂	semidirect
52. C₈×C₇₉	SD₁₆	semidirect
70. C₇₉	#1714	semidirect
71. #500	D₇₉	semidirect
124. C₈	#11929611	semidirect

H ≅ G/N

type

1. #19098959

C₂

semidirect

4. #19098913

C₂

semidirect

6. #19098284

C₂

semidirect

7. #500×C₇₉

C₂

semidirect

52. C₈×C₇₉

SD₁₆

semidirect

70. C₇₉

#1714

semidirect

71. #500

D₇₉

semidirect

124. C₈

#11929611

semidirect

Subgroups — #23763023

Conjugacy classes of subgroups

Complements