Group #23763000 — subgroups

Subgroups — #23763000

Order 10112.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19098935	C₂	1	1	yes	maximal, characteristic
3	#19098914	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098913	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098909	C₂	1	1	yes	maximal, characteristic
6	C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929612	—	2	2	no	—
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579092	—	2	2	no	—
12	#14579086	—	2	2	no	—
13	#14579085	—	2	2	no	—
14	#14579082	—	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	#127×C₇₉	C₂²	1	1	yes	characteristic
19	#102×C₇₉	—	2	2	no	—
20	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×Dic₁₅₈	—	2	1	no	—
23	#11929612	—	4	4	no	—
24	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	—	2	1	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₂×Q₈×C₇₉	—	2	1	no	—
31	C₂×D₄×C₇₉	—	2	1	no	—
32	C₂×D₄×C₇₉	—	2	1	no	—
33	C₂×D₄×C₇₉	—	2	1	no	—
34	C₂×C₈×C₇₉	D₄	1	2	yes	—
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₂³×C₇₉	—	4	1	no	—
39	Q₈×C₇₉	—	4	2	no	—
40	D₄×C₇₉	—	4	2	no	—
41	D₄×C₇₉	—	4	2	no	—
42	D₄×C₇₉	—	4	2	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	C₂×Dic₇₉	—	4	1	no	—
49	Dic₁₅₈	—	4	2	no	—
50	Dic₁₅₈	—	4	2	no	—
51	C₈×C₇₉	—	2	2	no	—
52	C₇₉⋊C₈	—	2	2	no	—
53	C₇₉⋊C₈	SD₁₆	1	4	yes	—
54	C₂²×C₇₉	—	8	1	no	—
55	C₂²×C₇₉	—	8	1	no	—
56	C₂²×C₇₉	—	8	1	no	—
57	C₂²×C₇₉	#127	1	1	yes	characteristic
58	C₄×C₇₉	—	8	1	no	—
59	C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
60	C₄×C₇₉	#135	1	2	yes	—
61	C₄×C₇₉	#135	1	2	yes	—
62	Dic₇₉	—	8	1	no	—
63	Dic₇₉	—	8	1	no	—
64	C₂×C₇₉	—	16	1	no	—
65	C₂×C₇₉	#492	1	1	yes	characteristic
66	C₂×C₇₉	#495	1	1	yes	characteristic
67	C₂×C₇₉	#495	1	1	yes	characteristic
68	#1708	—	79	1	no	maximal, Sylow
69	C₇₉	#1708	1	1	yes	characteristic, complemented, Sylow
70	#498	—	79	1	no	—
71	#492	—	79	1	no	—
72	#487	—	79	1	no	—
73	#487	—	79	1	no	—
74	#486	—	79	1	no	—
75	#486	D₇₉	1	1	yes	characteristic
76	C₈²	—	79	1	no	—
77	C₂×SD₁₆	—	158	2	no	—
78	C₄⋊Q₈	—	79	1	no	—
79	C₄⋊Q₈	—	79	1	no	—
80	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
81	#127	C₂×D₇₉	1	1	yes	characteristic
82	#106	—	158	2	no	—
83	#103	—	158	2	no	—
84	#103	—	158	2	no	—
85	#102	—	2	2	no	—
86	#102	—	158	2	no	—
87	C₄×C₈	—	79	1	no	—
88	C₄×C₈	—	79	1	no	—
89	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	158	1	no	—
92	C₂×Q₈	—	158	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	SD₁₆	—	316	4	no	—
97	C₂×C₈	—	79	2	no	—
98	C₂×C₈	—	79	2	no	—
99	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	158	1	no	—
103	C₄⋊C₄	—	158	1	no	—
104	C₄⋊C₄	—	158	1	no	—
105	C₄⋊C₄	—	158	1	no	—
106	C₄²	C₂²×D₇₉	1	1	yes	characteristic
107	C₂³	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	316	2	no	—
110	Q₈	—	316	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	316	1	no	—
116	C₂×C₄	—	316	1	no	—
117	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
118	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	79	4	no	—
122	C₈	—	158	2	no	—
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	#14579157	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	632	1	no	—
129	C₄	—	632	1	no	—
130	C₄	SD₁₆×D₇₉	1	2	yes	—
131	C₄	#14579180	1	2	yes	—
132	C₄	#14579181	1	2	yes	—
133	C₂	—	16	1	no	—
134	C₂	#19098934	1	1	yes	characteristic
135	C₂	#19098945	1	1	yes	characteristic
136	C₂	#19098946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098959

C₂

yes

maximal, characteristic, complemented

#19098935

C₂

yes

maximal, characteristic

#19098914

C₂

yes

maximal, characteristic, complemented

#19098913

C₂

yes

maximal, characteristic, complemented

#19098909

C₂

yes

maximal, characteristic

C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#486×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929612

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

—

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

#102×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

#11929612

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×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

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

SD₁₆

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#127

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

#135

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₉

yes

characteristic

C₈²

—

C₂×SD₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

#127

C₂×D₇₉

yes

characteristic

#106

—

158

—

#103

—

158

—

#103

—

158

—

#102

—

#102

—

158

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₉

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

158

—

103

C₄⋊C₄

—

158

—

104

C₄⋊C₄

—

158

—

105

C₄⋊C₄

—

158

—

106

C₄²

C₂²×D₇₉

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

316

—

110

Q₈

—

316

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

D₄×D₇₉

yes

characteristic

118

C₂×C₄

D₄×D₇₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

158

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#14579157

yes

characteristic

127

C₄

—

128

C₄

—

632

—

129

C₄

—

632

—

130

C₄

SD₁₆×D₇₉

yes

—

131

C₄

#14579180

yes

—

132

C₄

#14579181

yes

—

133

C₂

—

134

C₂

#19098934

yes

characteristic

135

C₂

#19098945

yes

characteristic

136

C₂

#19098946

yes

characteristic

	subgroup	G/H
center	C₂²	#14579157
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#486×C₇₉	C₂
socle	C₂²×C₇₉	#127
prime radical	#127×C₇₉	C₂²
hypercenter	#486	D₇₉
hypocenter	C₇₉	#1708
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14579157

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#486×C₇₉

C₂

socle

C₂²×C₇₉

#127

prime radical

#127×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #19098959	C₂	semidirect
3. #19098914	C₂	semidirect
4. #19098913	C₂	semidirect
6. C₈×C₇₉⋊C₈	C₂	semidirect
69. C₇₉	#1708	semidirect

H ≅ G/N

type

1. #19098959

C₂

semidirect

3. #19098914

C₂

semidirect

4. #19098913

C₂

semidirect

6. C₈×C₇₉⋊C₈

C₂

semidirect

69. C₇₉

#1708

semidirect

Subgroups — #23763000

Conjugacy classes of subgroups

Complements