Group #34175671 — subgroups

Subgroups — #34175671

Order 20224.

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

normal subgroups only

This group has 126 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	#23765155	C₂	1	1	yes	maximal, characteristic
2	#23761540	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1400	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	C₄	1	1	yes	characteristic
5	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
6	#19098397	—	2	1	no	—
7	Dic₇₉⋊C₁₆	—	2	1	no	—
8	Dic₇₉⋊C₁₆	—	2	1	no	—
9	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
10	M₆₄×C₇₉	—	2	1	no	—
11	C₂²×C₇₉⋊C₈	C₈	1	1	yes	characteristic
12	C₂×C₄×Dic₇₉	C₈	1	1	yes	characteristic
13	#14579101	—	4	1	no	—
14	C₂×C₇₉⋊C₁₆	—	2	1	no	—
15	C₈×Dic₇₉	—	2	1	no	—
16	C₈×Dic₇₉	—	2	1	no	—
17	C₇₉⋊C₃₂	—	4	1	no	—
18	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₇₉	—	4	1	no	—
20	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
22	C₃₂×C₇₉	—	4	1	no	—
23	C₂²×Dic₇₉	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	4	1	no	—
26	C₄×Dic₇₉	—	2	1	no	—
27	C₄×Dic₇₉	C₁₆	1	1	yes	characteristic
28	C₄×Dic₇₉	C₁₆	1	1	yes	characteristic
29	C₇₉⋊C₁₆	—	4	1	no	—
30	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₇₉	—	2	1	no	—
32	C₂×C₈×C₇₉	—	2	1	no	—
33	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
35	C₁₆×C₇₉	—	2	1	no	—
36	C₁₆×C₇₉	—	2	1	no	—
37	C₂³×C₇₉	M₃₂	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×Dic₇₉	—	2	1	no	—
43	C₂×Dic₇₉	—	2	1	no	—
44	C₂×Dic₇₉	—	4	2	no	—
45	C₈×C₇₉	—	2	1	no	—
46	C₈×C₇₉	#99	1	1	yes	characteristic
47	C₈×C₇₉	#101	1	1	yes	characteristic
48	C₇₉⋊C₈	—	4	1	no	—
49	C₂²×C₇₉	—	2	1	no	—
50	C₂²×C₇₉	—	4	1	no	—
51	C₂²×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
52	C₄×C₇₉	—	2	1	no	—
53	C₄×C₇₉	#324	1	1	yes	characteristic
54	C₄×C₇₉	#349	1	1	yes	characteristic
55	Dic₇₉	—	2	2	no	—
56	Dic₇₉	—	4	1	no	—
57	#7364	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	—	2	1	no	—
59	C₂×C₇₉	—	4	1	no	—
60	C₂×C₇₉	#1322	1	1	yes	characteristic
61	#2150	—	79	1	no	—
62	#1400	—	79	1	no	—
63	#1400	D₇₉	1	1	yes	characteristic
64	C₇₉	#7364	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₃₂	—	79	1	no	—
66	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
67	C₂×C₄×C₈	—	79	1	no	—
68	M₆₄	—	2	1	no	—
69	M₆₄	—	158	1	no	—
70	C₄⋊C₁₆	—	158	1	no	—
71	C₄⋊C₁₆	—	158	1	no	—
72	C₂²×C₈	—	79	1	no	—
73	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
74	C₂×C₄²	—	79	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	—	316	1	no	—
77	C₂×C₁₆	—	158	1	no	—
78	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
79	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
80	C₄×C₈	—	158	1	no	—
81	C₄×C₈	—	158	1	no	—
82	C₃₂	—	4	1	no	—
83	C₃₂	—	316	1	no	—
84	C₂²×C₄	—	158	1	no	—
85	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	316	1	no	—
89	C₂×C₈	—	158	1	no	—
90	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
91	C₂×C₈	#11929607	1	1	yes	characteristic
92	C₄²	—	79	1	no	—
93	C₄²	—	79	1	no	—
94	C₄²	—	158	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	316	1	no	—
98	C₂³	#14579100	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	316	2	no	—
102	C₂×C₄	—	158	1	no	—
103	C₂×C₄	—	158	1	no	—
104	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
105	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	316	1	no	—
108	C₈	#14579075	1	1	yes	characteristic
109	C₈	#14579079	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	(C₂×D₇₉)⋊C₁₆	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	316	1	no	—
115	C₄	—	158	2	no	—
116	C₄	#19098288	1	1	yes	characteristic
117	C₄	#19098344	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	4	1	no	—
120	C₂	#23761339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765155

C₂

yes

maximal, characteristic

#23761540

C₂

yes

maximal, characteristic

C₇₉×#1400

C₂

yes

maximal, characteristic

C₂×#14579101

C₄

yes

characteristic

C₂×C₈×Dic₇₉

C₄

yes

characteristic

#19098397

—

Dic₇₉⋊C₁₆

—

Dic₇₉⋊C₁₆

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

M₆₄×C₇₉

—

C₂²×C₇₉⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₇₉

C₈

yes

characteristic

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₇₉⋊C₃₂

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₁₆

yes

characteristic

C₄×Dic₇₉

C₁₆

yes

characteristic

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

M₃₂

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#7364

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7364

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

158

—

C₄⋊C₁₆

—

158

—

C₄⋊C₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

316

—

C₂×C₁₆

—

158

—

C₂×C₁₆

D₃₁₆

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

—

158

—

C₄×C₈

—

158

—

C₃₂

—

C₃₂

—

316

—

C₂²×C₄

—

158

—

C₂²×C₄

C₈×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₈

#11929607

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

158

—

C₁₆

—

C₁₆

—

C₁₆

—

316

—

C₂³

#14579100

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

158

—

103

C₂×C₄

—

158

—

104

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

105

C₂×C₄

C₁₆×D₇₉

yes

characteristic

106

C₈

—

107

C₈

—

316

—

108

C₈

#14579075

yes

characteristic

109

C₈

#14579079

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

(C₂×D₇₉)⋊C₁₆

yes

characteristic

113

C₄

—

114

C₄

—

316

—

115

C₄

—

158

—

116

C₄

#19098288

yes

characteristic

117

C₄

#19098344

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#23761339

yes

characteristic

	subgroup	G/H
center	C₈	#14579075
commutator	C₂×C₄×C₇₉	C₂×C₁₆
Frattini	C₂×M₃₂	C₂×D₇₉
Fitting	C₇₉×#1400	C₂
socle	C₂×C₇₉	#1322
prime radical	C₂³×C₇₉	M₃₂
cosocle	C₂×M₃₂×C₇₉	C₂²
hypercenter	#1400	D₇₉
hypocenter	C₇₉	#7364
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

subgroup

G/H

center

C₈

#14579075

commutator

C₂×C₄×C₇₉

C₂×C₁₆

Frattini

C₂×M₃₂

C₂×D₇₉

Fitting

C₇₉×#1400

C₂

socle

C₂×C₇₉

#1322

prime radical

C₂³×C₇₉

M₃₂

cosocle

C₂×M₃₂×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

N	H ≅ G/N	type
64. C₇₉	#7364	semidirect

H ≅ G/N

type

64. C₇₉

#7364

semidirect

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

subgroup	index
#23765155	2
#23761540	2
C₇₉×#1400	2
#7364	79

subgroup

index

#23765155

#23761540

C₇₉×#1400

#7364

Subgroups — #34175671

Conjugacy classes of subgroups

Complements

Maximal subgroups