Group #34180725 — subgroups

Subgroups — #34180725

Order 20224.

656 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 147 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	#23764370	C₂	1	3	yes	maximal
2	C₂×#19098305	C₂	1	3	yes	maximal
3	C₂×#564×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579072	C₂²	1	3	yes	—
5	#19098433	C₂²	1	1	yes	characteristic
6	#19098325	—	2	6	no	—
7	#19098305	—	2	6	no	—
8	#564×C₇₉	—	2	6	no	—
9	#564×C₇₉	C₄	1	4	yes	complemented
10	C₂×#125×C₇₉	—	2	6	no	—
11	C₂×#125×C₇₉	C₂²	1	3	yes	—
12	C₂²×C₇₉⋊C₈	—	2	3	no	—
13	C₂×C₄⋊Dic₇₉	—	2	3	no	—
14	#14579072	—	2	6	no	—
15	#125×C₇₉	—	2	24	no	—
16	#125×C₇₉	—	2	24	no	—
17	#125×C₇₉	—	2	6	no	—
18	#125×C₇₉	C₂×C₄	1	6	yes	—
19	#125×C₇₉	D₄	1	6	yes	—
20	C₂×C₄⋊C₄×C₇₉	—	2	6	no	—
21	C₂×C₄⋊C₄×C₇₉	—	2	6	no	—
22	C₂×C₄⋊C₄×C₇₉	D₄	1	6	yes	—
23	C₂×C₄²×C₇₉	—	2	2	no	—
24	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₇₉	—	4	1	no	—
26	C₂×C₇₉⋊C₈	—	2	6	no	—
27	C₂×C₇₉⋊C₈	—	4	3	no	—
28	C₂×C₇₉⋊C₈	—	4	3	no	—
29	C₄⋊Dic₇₉	—	4	6	no	—
30	C₂²×C₄×C₇₉	—	2	6	no	—
31	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
32	C₄⋊C₄×C₇₉	—	2	24	no	—
33	C₄⋊C₄×C₇₉	—	2	12	no	—
34	C₄⋊C₄×C₇₉	—	2	12	no	—
35	C₄⋊C₄×C₇₉	—	2	6	no	—
36	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	12	yes	—
37	C₄²×C₇₉	—	2	8	no	—
38	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₇₉	C₂×D₄	1	3	yes	—
40	C₂³×C₇₉	#120	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	3	no	—
42	C₂×C₄×C₇₉	—	2	3	no	—
43	C₂×C₄×C₇₉	—	2	12	no	—
44	C₂×C₄×C₇₉	—	4	6	no	—
45	C₂×C₄×C₇₉	—	4	6	no	—
46	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
47	C₂×C₄×C₇₉	#120	1	3	yes	—
48	C₂×Dic₇₉	—	8	3	no	—
49	C₇₉⋊C₈	—	4	6	no	—
50	C₂²×C₇₉	#379	1	1	yes	characteristic
51	C₂²×C₇₉	#454	1	3	yes	—
52	C₂²×C₇₉	#456	1	3	yes	—
53	C₄×C₇₉	—	2	3	no	—
54	C₄×C₇₉	—	2	3	no	—
55	C₄×C₇₉	—	4	12	no	—
56	Dic₇₉	—	16	1	no	—
57	#8509	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	#1658	1	1	yes	characteristic
59	C₂×C₇₉	#1663	1	3	yes	—
60	C₂×C₇₉	#1793	1	3	yes	—
61	C₂×#564	D₇₉	1	1	yes	characteristic
62	#1992	—	79	3	no	—
63	C₂×#330	—	79	3	no	—
64	C₇₉	#8509	1	1	yes	characteristic, complemented, Sylow
65	#564	—	2	6	no	—
66	#564	Dic₇₉	1	4	yes	complemented
67	C₂×#125	—	2	6	no	—
68	C₂×#125	C₂×D₇₉	1	3	yes	—
69	C₂×#97	—	79	3	no	—
70	#384	—	79	1	no	—
71	#340	—	158	6	no	—
72	#330	—	158	6	no	—
73	C₂²×C₈	—	158	3	no	—
74	#125	—	2	24	no	—
75	#125	—	2	24	no	—
76	#125	—	2	6	no	—
77	#125	C₂×Dic₇₉	1	6	yes	—
78	#125	C₂²⋊D₇₉	1	6	yes	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	158	3	no	—
82	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	6	yes	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
85	#97	—	158	6	no	—
86	C₂²×C₄	—	2	6	no	—
87	C₂²×C₄	—	316	1	no	—
88	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
89	C₂×C₈	—	316	3	no	—
90	C₂×C₈	—	316	3	no	—
91	C₂×C₈	—	158	6	no	—
92	C₄⋊C₄	—	2	24	no	—
93	C₄⋊C₄	—	2	12	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	—	316	6	no	—
97	C₄⋊C₄	C₂²⋊Dic₇₉	1	12	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
100	C₄²	C₂×C₂²⋊D₇₉	1	3	yes	—
101	C₂³	#14579130	1	1	yes	characteristic
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	—	632	3	no	—
108	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	3	yes	—
109	C₂×C₄	#14579130	1	3	yes	—
110	C₈	—	316	6	no	—
111	C₂²	#19098419	1	1	yes	characteristic
112	C₂²	#19098736	1	3	yes	—
113	C₂²	#19098746	1	3	yes	—
114	C₄	—	2	3	no	—
115	C₄	—	2	3	no	—
116	C₄	—	4	12	no	—
117	C₄	—	1264	1	no	—
118	C₂	#23762702	1	1	yes	characteristic
119	C₂	#23762728	1	3	yes	—
120	C₂	#23763359	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764370

C₂

yes

maximal

C₂×#19098305

C₂

yes

maximal

C₂×#564×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579072

C₂²

yes

—

#19098433

C₂²

yes

characteristic

#19098325

—

#19098305

—

#564×C₇₉

—

#564×C₇₉

C₄

yes

complemented

C₂×#125×C₇₉

—

C₂×#125×C₇₉

C₂²

yes

—

C₂²×C₇₉⋊C₈

—

C₂×C₄⋊Dic₇₉

—

#14579072

—

#125×C₇₉

—

#125×C₇₉

—

#125×C₇₉

—

#125×C₇₉

C₂×C₄

yes

—

#125×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

#120

yes

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#379

yes

characteristic

C₂²×C₇₉

#454

yes

—

C₂²×C₇₉

#456

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1793

yes

—

C₂×#564

D₇₉

yes

characteristic

#1992

—

C₂×#330

—

C₇₉

#8509

yes

characteristic, complemented, Sylow

#564

—

#564

Dic₇₉

yes

complemented

C₂×#125

—

C₂×#125

C₂×D₇₉

yes

—

C₂×#97

—

#384

—

#340

—

158

—

#330

—

158

—

C₂²×C₈

—

158

—

—

—

—

C₂×Dic₇₉

yes

—

#125

C₂²⋊D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#97

—

158

—

C₂²×C₄

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

C₂×C₈

—

316

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

C₄²

—

C₄²

C₂²×Dic₇₉

yes

characteristic

100

C₄²

C₂×C₂²⋊D₇₉

yes

—

101

C₂³

#14579130

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

632

—

108

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

—

109

C₂×C₄

#14579130

yes

—

110

C₈

—

316

—

111

C₂²

#19098419

yes

characteristic

112

C₂²

#19098736

yes

—

113

C₂²

#19098746

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

1264

—

118

C₂

#23762702

yes

characteristic

119

C₂

#23762728

yes

—

120

C₂

#23763359

yes

—

	subgroup	G/H
center	C₂³	#14579130
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#564×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#120
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#564	D₇₉
hypocenter	C₇₉	#8509
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

subgroup

G/H

center

C₂³

#14579130

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#564×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#120

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

N	H ≅ G/N	type
9. #564×C₇₉	C₄	semidirect
64. C₇₉	#8509	semidirect
66. #564	Dic₇₉	semidirect

H ≅ G/N

type

9. #564×C₇₉

C₄

semidirect

64. C₇₉

#8509

semidirect

66. #564

Dic₇₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23764370	2
C₂×#19098305	2
C₂×#564×C₇₉	2
#8509	79

subgroup

index

#23764370

C₂×#19098305

C₂×#564×C₇₉

#8509

Subgroups — #34180725

Conjugacy classes of subgroups

Complements

Maximal subgroups