Group #23762459 — subgroups

Subgroups — #23762459

Order 10112.

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

normal subgroups only

This group has 240 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	#19098939	C₂	1	1	yes	maximal, characteristic
2	#19098763	C₂	1	2	yes	maximal, complemented
3	#19098307	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929611	—	2	2	no	—
7	C₂×#11929609	—	2	2	no	—
8	#14579158	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	#14579081	—	2	2	no	—
11	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₇₉	C₂²	1	1	yes	characteristic
13	#124×C₇₉	—	2	2	no	—
14	#123×C₇₉	—	2	2	no	—
15	C₄×Q₈×C₇₉	C₂²	1	2	yes	—
16	C₄×D₄×C₇₉	—	2	2	no	—
17	C₂×D₃₁₆	—	2	2	no	—
18	C₂×D₃₁₆	—	2	1	no	—
19	#11929611	—	4	4	no	—
20	#11929609	—	4	4	no	—
21	C₂×C₇₉⋊C₈	—	2	1	no	—
22	C₂×C₇₉⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₇₉	D₄	1	2	yes	—
24	C₂×D₄×C₇₉	—	2	2	no	—
25	C₂×D₄×C₇₉	D₄	1	2	yes	—
26	C₂²×C₄×C₇₉	—	2	2	no	—
27	C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
29	C₂²⋊C₄×C₇₉	—	2	4	no	—
30	C₂²⋊C₄×C₇₉	—	2	2	no	—
31	C₄²×C₇₉	—	2	2	no	—
32	C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₂³×C₇₉	—	2	2	no	—
34	C₂²×D₇₉	—	4	1	no	—
35	Q₈×C₇₉	—	2	2	no	—
36	Q₈×C₇₉	—	2	2	no	—
37	D₄×C₇₉	—	4	2	no	—
38	D₄×C₇₉	—	4	2	no	—
39	D₄×C₇₉	—	4	2	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	8	2	no	—
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
45	D₃₁₆	—	4	2	no	—
46	D₃₁₆	—	4	2	no	—
47	D₃₁₆	—	4	2	no	—
48	C₇₉⋊C₈	—	2	2	no	—
49	C₇₉⋊C₈	—	4	2	no	—
50	C₂²×C₇₉	—	4	4	no	—
51	C₂²×C₇₉	—	4	2	no	—
52	C₂²×C₇₉	#120	1	1	yes	characteristic
53	C₂×D₇₉	—	8	2	no	—
54	C₂×D₇₉	—	8	1	no	—
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	—	4	2	no	—
57	C₄×C₇₉	—	4	2	no	—
58	C₄×C₇₉	#135	1	2	yes	—
59	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₇₉	—	8	2	no	—
61	C₂×C₇₉	#449	1	2	yes	—
62	C₂×C₇₉	#453	1	1	yes	characteristic
63	D₇₉	—	16	1	no	—
64	#1620	—	79	1	no	maximal, Sylow
65	C₇₉	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	D₇₉	1	1	yes	characteristic, complemented
67	#493	—	79	1	no	—
68	#460	—	79	2	no	—
69	#331	—	79	1	no	—
70	Q₈⋊C₈	—	79	2	no	—
71	C₂×SD₁₆	—	158	2	no	—
72	C₂×D₈	—	158	2	no	—
73	#127	—	79	1	no	—
74	#127	C₂×D₇₉	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₇₉	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	79	2	no	—
80	#102	—	158	2	no	—
81	C₄×C₈	—	79	1	no	—
82	C₂×Q₈	C₂²⋊D₇₉	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	158	2	no	—
85	C₂×D₄	—	158	1	no	—
86	C₂×D₄	C₂²⋊D₇₉	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	316	4	no	—
89	D₈	—	316	4	no	—
90	C₂×C₈	—	158	2	no	—
91	C₂×C₈	—	158	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₇₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	316	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	316	2	no	—
106	D₄	—	316	2	no	—
107	D₄	—	316	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
113	C₈	—	316	2	no	—
114	C₈	—	158	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	632	2	no	—
118	C₂²	—	632	1	no	—
119	C₂²	#14579130	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#14579181	1	2	yes	—
124	C₄	#14579186	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	1264	1	no	—
127	C₂	#19098708	1	2	yes	—
128	C₂	#19098731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098939

C₂

yes

maximal, characteristic

#19098763

C₂

yes

maximal, complemented

#19098307

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₉⋊C₈)

C₂

yes

maximal, complemented

#559×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

—

C₂×#11929609

—

#14579158

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579081

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

#124×C₇₉

—

#123×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

—

C₄×D₄×C₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929611

—

#11929609

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

D₄

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₇₉

—

C₂×C₇₉

#449

yes

—

C₂×C₇₉

#453

yes

characteristic

D₇₉

—

#1620

—

maximal, Sylow

C₇₉

#1620

yes

characteristic, complemented, Sylow

#559

D₇₉

yes

characteristic, complemented

#493

—

#460

—

#331

—

Q₈⋊C₈

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

158

—

#127

—

#127

C₂×D₇₉

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₇₉

yes

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

158

—

C₄×C₈

—

C₂×Q₈

C₂²⋊D₇₉

yes

—

C₂×D₄

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

—

SD₁₆

—

316

—

D₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

—

C₂³

—

316

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

316

—

106

D₄

—

316

—

107

D₄

—

316

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

113

C₈

—

316

—

114

C₈

—

158

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

632

—

118

C₂²

—

632

—

119

C₂²

#14579130

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#14579181

yes

—

124

C₄

#14579186

yes

—

125

C₂

—

126

C₂

—

1264

—

127

C₂

#19098708

yes

—

128

C₂

#19098731

yes

characteristic

	subgroup	G/H
center	C₂²	#14579130
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#559×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	#19098939	C₂
hypercenter	#559	D₇₉
hypocenter	C₇₉	#1620
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#14579130

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#559×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #19098763	C₂	semidirect
3. #19098307	C₂	semidirect
4. Q₈⋊(C₇₉⋊C₈)	C₂	semidirect
5. #559×C₇₉	C₂	semidirect
65. C₇₉	#1620	semidirect
66. #559	D₇₉	semidirect

H ≅ G/N

type

2. #19098763

C₂

semidirect

3. #19098307

C₂

semidirect

4. Q₈⋊(C₇₉⋊C₈)

C₂

semidirect

5. #559×C₇₉

C₂

semidirect

65. C₇₉

#1620

semidirect

66. #559

D₇₉

semidirect

Subgroups — #23762459

Conjugacy classes of subgroups

Complements