Group #23762721 — subgroups

Subgroups — #23762721

Order 10112.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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	#19099311	C₂	1	1	yes	maximal, characteristic
2	#19098925	C₂	1	2	yes	maximal, complemented
3	#19098305	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098302	C₂	1	2	yes	maximal, complemented
5	#491×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579149	C₂²	1	2	yes	—
7	#14579146	—	2	2	no	—
8	#14579139	—	2	2	no	—
9	#14579136	—	2	2	no	—
10	#14579084	—	2	2	no	—
11	#14579081	—	2	2	no	—
12	#14579072	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
14	#125×C₇₉	C₂²	1	1	yes	characteristic
15	#103×C₇₉	—	2	2	no	—
16	#97×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	D₄	1	2	yes	—
18	C₂×D₃₁₆	—	2	2	no	—
19	C₂×D₃₁₆	D₄	1	2	yes	—
20	C₂×C₄×D₇₉	—	2	2	no	—
21	C₂×C₇₉⋊C₈	—	2	2	no	—
22	Dic₇₉⋊C₄	—	2	2	no	—
23	(C₂×D₇₉)⋊C₄	—	2	2	no	—
24	(C₂×D₇₉)⋊C₄	—	2	2	no	—
25	(C₂×D₇₉)⋊C₄	—	2	2	no	—
26	C₂×Q₈×C₇₉	—	2	1	no	—
27	C₂×C₈×C₇₉	—	2	1	no	—
28	C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
31	C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	C₂²×D₇₉	—	2	2	no	—
33	Q₈×C₇₉	—	4	2	no	—
34	C₂×C₄×C₇₉	—	2	2	no	—
35	C₂×C₄×C₇₉	—	4	1	no	—
36	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₇₉	—	2	2	no	—
39	D₃₁₆	—	4	2	no	—
40	D₃₁₆	—	8	2	no	—
41	C₄×D₇₉	—	8	2	no	—
42	Dic₁₅₈	—	4	2	no	—
43	C₈×C₇₉	—	4	1	no	—
44	C₇₉⋊C₈	—	4	2	no	—
45	C₂²×C₇₉	#120	1	1	yes	characteristic
46	C₂×D₇₉	—	4	2	no	—
47	C₂×D₇₉	—	4	2	no	—
48	C₂×D₇₉	—	4	2	no	—
49	C₄×C₇₉	—	2	1	no	—
50	C₄×C₇₉	—	2	2	no	—
51	C₄×C₇₉	—	4	2	no	—
52	C₄×C₇₉	—	8	1	no	—
53	Dic₇₉	—	4	2	no	—
54	C₂×C₇₉	#454	1	1	yes	characteristic
55	C₂×C₇₉	#455	1	2	yes	—
56	D₇₉	—	8	2	no	—
57	#1662	—	79	1	no	maximal, Sylow
58	C₇₉	#1662	1	1	yes	characteristic, complemented, Sylow
59	#562	—	79	1	no	—
60	#491	D₇₉	1	1	yes	characteristic, complemented
61	#489	—	79	2	no	—
62	#330	—	79	1	no	—
63	#329	—	79	2	no	—
64	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
65	#125	C₂×D₇₉	1	1	yes	characteristic
66	#124	—	79	2	no	—
67	#123	—	158	2	no	—
68	C₂²⋊Q₈	—	158	2	no	—
69	#121	—	158	2	no	—
70	#103	—	2	2	no	—
71	#103	—	158	2	no	—
72	#102	—	158	2	no	—
73	#97	—	79	2	no	—
74	#97	C₂×D₇₉	1	1	yes	characteristic
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	79	2	no	—
77	C₂×D₄	—	79	2	no	—
78	C₂×D₄	—	158	2	no	—
79	C₂²×C₄	—	158	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	158	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	158	2	no	—
85	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
86	C₂²⋊C₄	—	158	2	no	—
87	C₂²⋊C₄	—	158	2	no	—
88	C₂²⋊C₄	—	158	2	no	—
89	C₄²	C₂²×D₇₉	1	1	yes	characteristic
90	C₂³	—	158	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	316	2	no	—
93	D₄	—	316	2	no	—
94	D₄	—	632	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	632	2	no	—
98	C₂×C₄	—	158	2	no	—
99	C₂×C₄	D₄×D₇₉	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	—	316	2	no	—
103	C₂²	—	316	2	no	—
104	C₂²	—	316	2	no	—
105	C₂²	—	316	2	no	—
106	C₂²	#14579129	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	8	1	no	—
111	C₄	—	316	2	no	—
112	C₂	—	632	2	no	—
113	C₂	#19098735	1	1	yes	characteristic
114	C₂	#19098739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099311

C₂

yes

maximal, characteristic

#19098925

C₂

yes

maximal, complemented

#19098305

C₂

yes

maximal, characteristic, complemented

#19098302

C₂

yes

maximal, complemented

#491×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579149

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#125×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

#97×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

—

C₂×C₄×D₇₉

—

C₂×C₇₉⋊C₈

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

#454

yes

characteristic

C₂×C₇₉

#455

yes

—

D₇₉

—

#1662

—

maximal, Sylow

C₇₉

#1662

yes

characteristic, complemented, Sylow

#562

—

#491

D₇₉

yes

characteristic, complemented

#489

—

#330

—

#329

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

#125

C₂×D₇₉

yes

characteristic

#124

—

#123

—

158

—

C₂²⋊Q₈

—

158

—

#121

—

158

—

#103

—

#103

—

158

—

#102

—

158

—

#97

—

#97

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

158

—

C₂²×C₄

—

158

—

C₂×C₈

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂²⋊C₄

—

158

—

C₂²⋊C₄

—

158

—

C₂²⋊C₄

—

158

—

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

—

158

—

Q₈

—

Q₈

—

316

—

D₄

—

316

—

D₄

—

632

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

632

—

C₂×C₄

—

158

—

C₂×C₄

D₄×D₇₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₈

—

102

C₈

—

316

—

103

C₂²

—

316

—

104

C₂²

—

316

—

105

C₂²

—

316

—

106

C₂²

#14579129

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

316

—

112

C₂

—

632

—

113

C₂

#19098735

yes

characteristic

114

C₂

#19098739

yes

—

	subgroup	G/H
center	C₂²	#14579129
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#491×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	#19099311	C₂
hypercenter	#491	D₇₉
hypocenter	C₇₉	#1662
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#14579129

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#491×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
2. #19098925	C₂	semidirect
3. #19098305	C₂	semidirect
4. #19098302	C₂	semidirect
5. #491×C₇₉	C₂	semidirect
58. C₇₉	#1662	semidirect
60. #491	D₇₉	semidirect

H ≅ G/N

type

2. #19098925

C₂

semidirect

3. #19098305

C₂

semidirect

4. #19098302

C₂

semidirect

5. #491×C₇₉

C₂

semidirect

58. C₇₉

#1662

semidirect

60. #491

D₇₉

semidirect

Subgroups — #23762721

Conjugacy classes of subgroups

Complements