Group #34178704 — subgroups

Subgroups — #34178704

Order 20224.

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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	1	yes	maximal, characteristic
2	#23763146	C₂	1	1	yes	maximal, characteristic
3	C₇₉⋊#1273	C₂	1	4	yes	maximal
4	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
7	#19098318	C₄	1	4	yes	—
8	#19098318	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
10	C₄³×C₇₉	C₂²	1	1	yes	characteristic
11	#336×C₇₉	C₄	1	4	yes	—
12	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
13	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
14	C₄×C₇₉⋊C₈	—	2	4	no	—
15	C₄×C₇₉⋊C₈	—	2	4	no	—
16	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₇₉	—	2	4	no	—
22	C₂×C₇₉⋊C₈	—	2	4	no	—
23	C₂×C₇₉⋊C₈	—	2	4	no	—
24	C₂×C₇₉⋊C₈	—	2	8	no	—
25	C₂×C₇₉⋊C₈	—	2	8	no	—
26	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
27	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	—	2	4	no	—
31	C₂×C₈×C₇₉	—	2	8	no	—
32	C₄²×C₇₉	—	2	4	no	—
33	C₄²×C₇₉	—	2	4	no	—
34	C₄²×C₇₉	—	2	4	no	—
35	C₄²×C₇₉	#41	1	4	yes	—
36	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇₉	—	2	4	no	—
38	C₂×C₄×C₇₉	—	2	4	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	#117	1	2	yes	—
41	C₂×C₄×C₇₉	#117	1	2	yes	—
42	C₂×C₄×C₇₉	#117	1	2	yes	—
43	C₂×C₄×C₇₉	#131	1	4	yes	—
44	C₂×C₄×C₇₉	#131	1	4	yes	—
45	C₂×C₄×C₇₉	#131	1	4	yes	—
46	C₈×C₇₉	—	4	4	no	—
47	C₇₉⋊C₈	—	4	4	no	—
48	C₇₉⋊C₈	—	4	4	no	—
49	C₂²×C₇₉	#376	1	1	yes	characteristic
50	C₂²×C₇₉	#405	1	2	yes	—
51	C₂²×C₇₉	#405	1	2	yes	—
52	C₂²×C₇₉	#405	1	2	yes	—
53	C₄×C₇₉	—	4	4	no	—
54	C₄×C₇₉	#433	1	4	yes	—
55	C₄×C₇₉	#433	1	4	yes	—
56	C₄×C₇₉	#433	1	4	yes	—
57	#8135	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	#1454	1	4	yes	—
59	C₂×C₇₉	#1746	1	1	yes	characteristic
60	C₂×C₇₉	#1746	1	1	yes	characteristic
61	C₂×C₇₉	#1746	1	1	yes	characteristic
62	#1744	—	79	1	no	—
63	#1744	—	79	1	no	—
64	#1744	D₇₉	1	1	yes	characteristic
65	#1273	—	79	4	no	—
66	C₇₉	#8135	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄×C₈	—	79	2	no	—
68	C₂×C₄×C₈	—	79	2	no	—
69	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
70	C₄³	C₂×D₇₉	1	1	yes	characteristic
71	#336	—	79	4	no	—
72	#336	—	79	4	no	—
73	#336	Dic₇₉	1	4	yes	—
74	C₂²×C₈	—	79	4	no	—
75	C₂²×C₈	—	79	4	no	—
76	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
77	C₂×C₄²	C₄×D₇₉	1	2	yes	—
78	C₂×C₄²	C₄×D₇₉	1	2	yes	—
79	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
80	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	158	4	no	—
83	C₄×C₈	—	158	4	no	—
84	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	158	4	no	—
91	C₂×C₈	—	158	4	no	—
92	C₂×C₈	—	158	8	no	—
93	C₂×C₈	—	158	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11929624	1	4	yes	—
98	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#14579118	1	2	yes	—
103	C₂×C₄	#14579118	1	2	yes	—
104	C₂×C₄	#14579120	1	2	yes	—
105	C₂×C₄	#14579167	1	4	yes	—
106	C₂×C₄	#14579168	1	4	yes	—
107	C₂×C₄	#14579169	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	316	4	no	—
110	C₈	—	316	4	no	—
111	C₂²	#19098409	1	1	yes	characteristic
112	C₂²	#19098518	1	2	yes	—
113	C₂²	#19098519	1	2	yes	—
114	C₂²	#19098521	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#19098622	1	4	yes	—
117	C₄	#19098624	1	4	yes	—
118	C₄	#19098625	1	4	yes	—
119	C₂	#23761700	1	4	yes	—
120	C₂	#23763151	1	1	yes	characteristic
121	C₂	#23763152	1	1	yes	characteristic
122	C₂	#23763153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763146

C₂

yes

maximal, characteristic

#23763146

C₂

yes

maximal, characteristic

C₇₉⋊#1273

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098318

C₄

yes

—

#19098318

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

C₄

yes

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄²

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#376

yes

characteristic

C₂²×C₇₉

#405

yes

—

C₂²×C₇₉

#405

yes

—

C₂²×C₇₉

#405

yes

—

C₄×C₇₉

—

C₄×C₇₉

#433

yes

—

C₄×C₇₉

#433

yes

—

C₄×C₇₉

#433

yes

—

#8135

—

maximal, Sylow

C₂×C₇₉

#1454

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1273

—

C₇₉

#8135

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₄³

C₂×D₇₉

yes

—

—

Dic₇₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

158

—

C₄×C₈

—

158

—

C₂²×C₄

C₄×Dic₇₉

yes

—

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11929624

yes

—

C₂³

C₂×C₄×Dic₇₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#14579118

yes

—

103

C₂×C₄

#14579118

yes

—

104

C₂×C₄

#14579120

yes

—

105

C₂×C₄

#14579167

yes

—

106

C₂×C₄

#14579168

yes

—

107

C₂×C₄

#14579169

yes

—

108

C₈

—

109

C₈

—

316

—

110

C₈

—

316

—

111

C₂²

#19098409

yes

characteristic

112

C₂²

#19098518

yes

—

113

C₂²

#19098519

yes

—

114

C₂²

#19098521

yes

—

115

C₄

—

116

C₄

#19098622

yes

—

117

C₄

#19098624

yes

—

118

C₄

#19098625

yes

—

119

C₂

#23761700

yes

—

120

C₂

#23763151

yes

characteristic

121

C₂

#23763152

yes

characteristic

122

C₂

#23763153

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂×C₄²
Fitting	C₇₉×#1744	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#8135
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₇₉

commutator, socle, prime radical

C₂³×C₇₉

C₂×C₄²

Fitting

C₇₉×#1744

C₂

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
66. C₇₉	#8135	semidirect

H ≅ G/N

type

66. C₇₉

#8135

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
C₇₉×#1744	2
#8135	79

subgroup

index

#23763146

C₇₉⋊#1273

C₇₉×#1744

#8135

Subgroups — #34178704

Conjugacy classes of subgroups

Complements

Maximal subgroups