Group #34180837 — subgroups

Subgroups — #34180837

Order 20224.

656 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 139 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	#23764378	C₂	1	3	yes	maximal
2	#23762138	C₂	1	3	yes	maximal
3	C₇₉×#3556	C₂	1	1	yes	maximal, characteristic
4	#19098587	C₂²	1	3	yes	—
5	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
6	#19098322	—	2	6	no	—
7	#19098309	—	2	6	no	—
8	#564×C₇₉	C₄	1	4	yes	complemented
9	#557×C₇₉	—	2	6	no	—
10	#541×C₇₉	—	2	6	no	—
11	#533×C₇₉	C₂²	1	3	yes	—
12	C₂×#11929608	—	2	3	no	—
13	C₂×C₄×Dic₇₉	—	2	3	no	—
14	C₄⋊(C₇₉⋊C₈)	—	2	6	no	—
15	C₄⋊Q₈×C₇₉	—	2	6	no	—
16	C₄⋊Q₈×C₇₉	D₄	1	6	yes	—
17	#126×C₇₉	—	2	8	no	—
18	#125×C₇₉	—	2	6	no	—
19	#125×C₇₉	—	2	24	no	—
20	#125×C₇₉	C₂×C₄	1	6	yes	—
21	C₂²⋊Q₈×C₇₉	—	2	6	no	—
22	C₄×Q₈×C₇₉	—	2	12	no	—
23	C₄×Q₈×C₇₉	—	2	6	no	—
24	#117×C₇₉	D₄	1	6	yes	—
25	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₇₉	—	4	1	no	—
27	#11929608	—	4	6	no	—
28	C₂×C₇₉⋊C₈	—	4	3	no	—
29	C₄×Dic₇₉	—	2	6	no	—
30	C₄×Dic₇₉	—	4	3	no	—
31	C₂×Q₈×C₇₉	—	2	6	no	—
32	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
33	C₄⋊C₄×C₇₉	—	2	24	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	C₂²⋊C₄×C₇₉	—	2	6	no	—
38	C₄²×C₇₉	—	2	8	no	—
39	C₄²×C₇₉	—	2	6	no	—
40	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇₉	C₂×D₄	1	3	yes	—
42	C₂³×C₇₉	#120	1	1	yes	characteristic
43	Q₈×C₇₉	—	4	6	no	—
44	Q₈×C₇₉	—	4	6	no	—
45	C₂×C₄×C₇₉	—	2	3	no	—
46	C₂×C₄×C₇₉	—	2	3	no	—
47	C₂×C₄×C₇₉	—	2	12	no	—
48	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
49	C₂×C₄×C₇₉	#120	1	3	yes	—
50	C₂×Dic₇₉	—	4	6	no	—
51	C₇₉⋊C₈	—	8	3	no	—
52	C₂²×C₇₉	—	2	3	no	—
53	C₂²×C₇₉	#379	1	1	yes	characteristic
54	C₄×C₇₉	—	2	3	no	—
55	C₄×C₇₉	—	4	12	no	—
56	C₄×C₇₉	#454	1	3	yes	—
57	C₄×C₇₉	#456	1	3	yes	—
58	Dic₇₉	—	4	4	no	—
59	#8533	—	79	1	no	maximal, Sylow
60	C₂×C₇₉	—	4	1	no	—
61	C₂×C₇₉	#1793	1	3	yes	—
62	#3556	D₇₉	1	1	yes	characteristic
63	#1994	—	79	3	no	—
64	#1561	—	79	3	no	—
65	C₇₉	#8533	1	1	yes	characteristic, complemented, Sylow
66	#564	Dic₇₉	1	4	yes	complemented
67	#557	—	2	6	no	—
68	#541	—	2	6	no	—
69	#533	C₂×D₇₉	1	3	yes	—
70	#423	—	79	3	no	—
71	C₄³	—	79	1	no	—
72	#339	—	158	6	no	—
73	#332	—	158	6	no	—
74	C₂×M₁₆	—	158	3	no	—
75	C₄⋊Q₈	—	2	6	no	—
76	C₄⋊Q₈	C₂²⋊D₇₉	1	6	yes	—
77	#126	—	2	8	no	—
78	#125	—	2	6	no	—
79	#125	—	2	24	no	—
80	#125	C₂×Dic₇₉	1	6	yes	—
81	C₂²⋊Q₈	—	2	6	no	—
82	C₄×Q₈	—	2	12	no	—
83	C₄×Q₈	—	2	6	no	—
84	#117	C₂²⋊D₇₉	1	6	yes	—
85	C₂×C₄²	—	158	3	no	—
86	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
87	C₄⋊C₈	—	158	6	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂²×C₄	—	316	1	no	—
90	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
91	M₁₆	—	316	6	no	—
92	C₂×C₈	—	316	3	no	—
93	C₄⋊C₄	—	2	24	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	C₂²⋊Dic₇₉	1	12	yes	—
97	C₂²⋊C₄	—	2	6	no	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	6	no	—
100	C₄²	—	316	3	no	—
101	C₄²	—	158	6	no	—
102	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
103	C₄²	C₂×C₂²⋊D₇₉	1	3	yes	—
104	C₂³	#14579130	1	1	yes	characteristic
105	Q₈	—	4	6	no	—
106	Q₈	—	4	6	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	316	6	no	—
111	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	3	yes	—
112	C₂×C₄	#14579130	1	3	yes	—
113	C₈	—	632	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	#19098419	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	4	12	no	—
118	C₄	—	316	4	no	—
119	C₄	#19098736	1	3	yes	—
120	C₄	#19098746	1	3	yes	—
121	C₂	—	4	1	no	—
122	C₂	#23763359	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764378

C₂

yes

maximal

#23762138

C₂

yes

maximal

C₇₉×#3556

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

C₄²×Dic₇₉

C₂²

yes

characteristic

#19098322

—

#19098309

—

#564×C₇₉

C₄

yes

complemented

#557×C₇₉

—

#541×C₇₉

—

#533×C₇₉

C₂²

yes

—

C₂×#11929608

—

C₂×C₄×Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#126×C₇₉

—

#125×C₇₉

—

#125×C₇₉

—

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#120

yes

characteristic

Q₈×C₇₉

—

Q₈×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

—

C₂²×C₇₉

#379

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#454

yes

—

C₄×C₇₉

#456

yes

—

Dic₇₉

—

#8533

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1793

yes

—

#3556

D₇₉

yes

characteristic

#1994

—

#1561

—

C₇₉

#8533

yes

characteristic, complemented, Sylow

#564

Dic₇₉

yes

—

—

C₂×D₇₉

yes

—

#423

—

C₄³

—

#339

—

158

—

#332

—

158

—

C₂×M₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

—

—

—

C₂×Dic₇₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂²⋊D₇₉

yes

—

C₂×C₄²

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

158

—

C₂×Q₈

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

M₁₆

—

316

—

C₂×C₈

—

316

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

316

—

101

C₄²

—

158

—

102

C₄²

C₂²×Dic₇₉

yes

characteristic

103

C₄²

C₂×C₂²⋊D₇₉

yes

—

104

C₂³

#14579130

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

316

—

111

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

—

112

C₂×C₄

#14579130

yes

—

113

C₈

—

632

—

114

C₂²

—

115

C₂²

#19098419

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

316

—

119

C₄

#19098736

yes

—

120

C₄

#19098746

yes

—

121

C₂

—

122

C₂

#23763359

yes

—

	subgroup	G/H
center	C₂²	#19098419
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3556	C₂
socle	C₂²×C₇₉	#379
prime radical	C₂³×C₇₉	#120
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3556	D₇₉
hypocenter	C₇₉	#8533
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

#19098419

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3556

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
8. #564×C₇₉	C₄	semidirect
65. C₇₉	#8533	semidirect
66. #564	Dic₇₉	semidirect

H ≅ G/N

type

8. #564×C₇₉

C₄

semidirect

65. C₇₉

#8533

semidirect

66. #564

Dic₇₉

semidirect

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

subgroup	index
#23764378	2
#23762138	2
C₇₉×#3556	2
#8533	79

subgroup

index

#23764378

#23762138

C₇₉×#3556

#8533

Subgroups — #34180837

Conjugacy classes of subgroups

Complements

Maximal subgroups