Group #34178698 — subgroups

Subgroups — #34178698

Order 20224.

446 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 243 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	#23763146	C₂	1	2	yes	maximal
2	C₇₉⋊#1273	C₂	1	4	yes	maximal
3	C₄²×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
5	#19098318	C₄	1	8	yes	—
6	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
9	C₄³×C₇₉	C₂²	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
11	C₄×C₇₉⋊C₈	—	2	8	no	—
12	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₇₉	—	2	4	no	—
18	C₄×C₈×C₇₉	—	2	4	no	—
19	C₄×C₈×C₇₉	C₈	1	8	yes	complemented
20	C₂×C₇₉⋊C₈	—	2	8	no	—
21	C₂×C₇₉⋊C₈	—	2	16	no	—
22	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
23	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
24	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
26	C₂×C₈×C₇₉	—	2	4	no	—
27	C₂×C₈×C₇₉	—	2	4	no	—
28	C₂×C₈×C₇₉	C₂×C₈	1	8	yes	—
29	C₄²×C₇₉	—	2	8	no	—
30	C₄²×C₇₉	C₂×C₈	1	4	yes	—
31	C₄²×C₇₉	M₁₆	1	4	yes	—
32	C₄²×C₇₉	#41	1	4	yes	—
33	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
34	C₂×C₄×C₇₉	—	2	4	no	—
35	C₂×C₄×C₇₉	—	2	4	no	—
36	C₂×C₄×C₇₉	C₄×C₈	1	4	yes	—
37	C₂×C₄×C₇₉	#97	1	4	yes	—
38	C₂×C₄×C₇₉	#117	1	4	yes	—
39	C₂×C₄×C₇₉	#117	1	2	yes	—
40	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
41	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
42	C₂×C₄×C₇₉	#131	1	8	yes	—
43	C₈×C₇₉	—	2	8	no	—
44	C₇₉⋊C₈	—	4	8	no	—
45	C₂²×C₇₉	#376	1	1	yes	characteristic
46	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
47	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
48	C₂²×C₇₉	#405	1	4	yes	—
49	C₄×C₇₉	—	2	4	no	—
50	C₄×C₇₉	—	2	4	no	—
51	C₄×C₇₉	#431	1	4	yes	—
52	C₄×C₇₉	#433	1	8	yes	—
53	#8133	—	79	1	no	maximal, Sylow
54	C₂×C₇₉	#1450	1	2	yes	—
55	C₂×C₇₉	#1452	1	2	yes	—
56	C₂×C₇₉	#1744	1	1	yes	characteristic
57	C₂×C₇₉	#1746	1	2	yes	—
58	#1744	—	79	2	no	—
59	C₄²×C₈	D₇₉	1	1	yes	characteristic
60	#1273	—	79	4	no	—
61	C₇₉	#8133	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	79	4	no	—
63	C₂×C₄×C₈	Dic₇₉	1	2	yes	—
64	C₂×C₄×C₈	Dic₇₉	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
66	C₄³	C₂×D₇₉	1	1	yes	characteristic
67	#336	—	79	8	no	—
68	C₂²×C₈	—	79	8	no	—
69	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
70	C₂×C₄²	C₄×D₇₉	1	4	yes	—
71	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
72	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
73	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	158	8	no	—
77	C₄×C₈	C₇₉⋊C₈	1	8	yes	complemented
78	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
79	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
80	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
81	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	158	8	no	—
85	C₂×C₈	—	158	16	no	—
86	C₂×C₈	C₂×C₇₉⋊C₈	1	8	yes	—
87	C₄²	—	2	8	no	—
88	C₄²	C₂×C₇₉⋊C₈	1	4	yes	—
89	C₄²	#11929608	1	4	yes	—
90	C₄²	#11929624	1	4	yes	—
91	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₇₉⋊C₈	1	4	yes	—
95	C₂×C₄	#14579072	1	4	yes	—
96	C₂×C₄	#14579118	1	4	yes	—
97	C₂×C₄	#14579120	1	2	yes	—
98	C₂×C₄	C₂²×C₇₉⋊C₈	1	2	yes	—
99	C₂×C₄	C₂×#11929608	1	2	yes	—
100	C₂×C₄	#14579167	1	8	yes	—
101	C₈	—	2	8	no	—
102	C₈	—	316	8	no	—
103	C₂²	#19098409	1	1	yes	characteristic
104	C₂²	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
105	C₂²	C₂×#14579072	1	1	yes	characteristic
106	C₂²	#19098518	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	2	4	no	—
109	C₄	#19098616	1	4	yes	—
110	C₄	#19098622	1	8	yes	—
111	C₂	#23761683	1	2	yes	—
112	C₂	#23761692	1	2	yes	—
113	C₂	#23763146	1	1	yes	characteristic
114	C₂	#23763151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763146

C₂

yes

maximal

C₇₉⋊#1273

C₂

yes

maximal

C₄²×C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098318

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄²

yes

—

C₂²×C₄×C₇₉

C₄²

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

—

C₄²×C₇₉

M₁₆

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×C₈

yes

—

C₂×C₄×C₇₉

#97

yes

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#376

yes

characteristic

C₂²×C₇₉

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₉

C₂×#97

yes

characteristic

C₂²×C₇₉

#405

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#431

yes

—

C₄×C₇₉

#433

yes

—

#8133

—

maximal, Sylow

C₂×C₇₉

#1450

yes

—

C₂×C₇₉

#1452

yes

—

C₂×C₇₉

#1744

yes

characteristic

C₂×C₇₉

#1746

yes

—

#1744

—

C₄²×C₈

D₇₉

yes

characteristic

#1273

—

C₇₉

#8133

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₇₉

yes

—

C₂×C₄×C₈

Dic₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

158

—

C₄×C₈

C₇₉⋊C₈

yes

complemented

C₂²×C₄

C₄×Dic₇₉

yes

—

C₂²×C₄

C₄×Dic₇₉

yes

—

C₂²×C₄

C₂×C₄×D₇₉

yes

—

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

C₂×C₇₉⋊C₈

yes

—

C₄²

—

C₄²

C₂×C₇₉⋊C₈

yes

—

C₄²

#11929608

yes

—

C₄²

#11929624

yes

—

C₂³

C₂×C₄×Dic₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₇₉⋊C₈

yes

—

C₂×C₄

#14579072

yes

—

C₂×C₄

#14579118

yes

—

C₂×C₄

#14579120

yes

—

C₂×C₄

C₂²×C₇₉⋊C₈

yes

—

C₂×C₄

C₂×#11929608

yes

—

100

C₂×C₄

#14579167

yes

—

101

C₈

—

102

C₈

—

316

—

103

C₂²

#19098409

yes

characteristic

104

C₂²

C₂×C₄×C₇₉⋊C₈

yes

characteristic

105

C₂²

C₂×#14579072

yes

characteristic

106

C₂²

#19098518

yes

—

107

C₄

—

108

C₄

—

109

C₄

#19098616

yes

—

110

C₄

#19098622

yes

—

111

C₂

#23761683

yes

—

112

C₂

#23761692

yes

—

113

C₂

#23763146

yes

characteristic

114

C₂

#23763151

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₄²×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄²×C₈	D₇₉
hypocenter	C₇₉	#8133
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₇₉

commutator

C₂²×C₇₉

C₂×C₄×C₈

Fitting

C₄²×C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₄²

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₄²×C₈

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄² ◁ C₄²×C₈

N	H ≅ G/N	type
19. C₄×C₈×C₇₉	C₈	semidirect
61. C₇₉	#8133	semidirect
77. C₄×C₈	C₇₉⋊C₈	semidirect

H ≅ G/N

type

19. C₄×C₈×C₇₉

C₈

semidirect

61. C₇₉

#8133

semidirect

77. C₄×C₈

C₇₉⋊C₈

semidirect

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

subgroup	index
#23763146	2
C₇₉⋊#1273	2
C₄²×C₈×C₇₉	2
#8133	79

subgroup

index

#23763146

C₇₉⋊#1273

C₄²×C₈×C₇₉

#8133

Subgroups — #34178698

Conjugacy classes of subgroups

Complements

Maximal subgroups