Group #34208755 — subgroups

Subgroups — #34208755

Order 20224.

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

normal subgroups only

This group has 206 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	C₂×#19098397	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₃₂)	C₂	1	4	yes	maximal, complemented
4	C₇₉×#2108	C₂	1	1	yes	maximal, characteristic
5	#19098397	—	2	2	no	—
6	C₂×C₇₉⋊C₃₂	—	2	2	no	—
7	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
8	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
9	C₂×M₃₂×C₇₉	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	#405×C₇₉	C₄	1	1	yes	characteristic
12	#346×C₇₉	C₄	1	2	yes	—
13	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₇₉⋊C₃₂	—	2	2	no	—
15	C₇₉⋊C₃₂	—	2	2	no	—
16	C₂×M₁₆×C₇₉	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
18	#117×C₇₉	C₈	1	1	yes	characteristic
19	M₃₂×C₇₉	—	2	2	no	—
20	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
23	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
24	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
25	#97×C₇₉	C₈	1	2	yes	—
26	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₇₉	—	2	2	no	—
29	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
33	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₉	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₇₉	C₁₆	1	2	yes	—
36	C₄²×C₇₉	C₂×C₈	1	2	yes	—
37	C₁₆×C₇₉	—	2	2	no	—
38	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₇₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₇₉	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₉	M₃₂	1	2	yes	—
46	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₇₉	—	2	2	no	—
48	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
49	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₇₉	—	2	4	no	—
55	C₄×C₇₉	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
58	#13628	—	79	1	no	maximal, Sylow
59	C₂×C₇₉	—	2	2	no	—
60	C₂×C₇₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₇₉	#2258	1	2	yes	—
62	C₂×M₆₄	—	79	1	no	—
63	C₂²×C₃₂	—	79	1	no	—
64	#2108	D₇₉	1	1	yes	characteristic
65	C₄⋊C₃₂	—	79	4	no	—
66	C₇₉	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	Dic₇₉	1	1	yes	characteristic
68	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
69	#405	Dic₇₉	1	1	yes	characteristic
70	M₆₄	—	158	2	no	—
71	C₂×C₃₂	—	79	2	no	—
72	C₂×C₃₂	—	79	2	no	—
73	C₂×C₃₂	—	158	2	no	—
74	#346	Dic₇₉	1	2	yes	—
75	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
76	C₂×M₁₆	C₇₉⋊C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
78	#117	C₇₉⋊C₈	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
81	C₂×C₁₆	D₃₁₆	1	2	yes	—
82	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
85	#97	C₇₉⋊C₈	1	2	yes	—
86	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
87	C₃₂	—	158	2	no	—
88	C₃₂	—	158	2	no	—
89	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
92	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
93	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
94	C₂×C₈	#11929608	1	2	yes	—
95	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
96	C₄⋊C₄	C₇₉⋊C₁₆	1	2	yes	—
97	C₂²⋊C₄	C₇₉⋊C₁₆	1	2	yes	—
98	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊Dic₇₉	1	2	yes	—
101	C₁₆	C₂×D₃₁₆	1	1	yes	characteristic
102	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
103	C₂³	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
104	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
105	C₂×C₄	C₂×C₇₉⋊C₁₆	1	4	yes	—
106	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
107	C₂×C₄	#14579101	1	2	yes	—
108	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
111	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
112	C₈	C₂×#11929608	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊(C₇₉⋊C₁₆)	1	2	yes	—
115	C₂²	C₂²×C₇₉⋊C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊(C₇₉⋊C₁₆)	1	2	yes	—
118	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
119	C₄	C₂×#14579101	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
122	C₂	#23765694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098397

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₃₂)

C₂

yes

maximal, complemented

C₇₉×#2108

C₂

yes

maximal, characteristic

#19098397

—

C₂×C₇₉⋊C₃₂

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

C₂×M₃₂×C₇₉

C₄

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#405×C₇₉

C₄

yes

characteristic

#346×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

—

C₇₉⋊C₃₂

—

C₇₉⋊C₃₂

—

C₂×M₁₆×C₇₉

C₈

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

#117×C₇₉

C₈

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#97×C₇₉

C₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₉

C₁₆

yes

—

C₂²⋊C₄×C₇₉

C₁₆

yes

—

C₄²×C₇₉

C₂×C₈

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₁₆

yes

—

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

M₃₂

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₄⋊C₈

yes

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

C₄⋊C₁₆

yes

—

C₂²×C₇₉

C₂²×C₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄⋊C₁₆

yes

—

C₄×C₇₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₉

C₂×M₃₂

yes

characteristic

#13628

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₉

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

D₇₉

yes

characteristic

C₄⋊C₃₂

—

C₇₉

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₉

yes

characteristic

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#405

Dic₇₉

yes

characteristic

M₆₄

—

158

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

158

—

#346

Dic₇₉

yes

—

C₄×C₁₆

C₂×D₇₉

yes

—

C₂×M₁₆

C₇₉⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

#117

C₇₉⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₁₅₈

yes

—

C₂×C₁₆

D₃₁₆

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#97

C₇₉⋊C₈

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₃₂

—

158

—

C₃₂

—

158

—

C₂²×C₄

C₂×C₇₉⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₂×C₇₉⋊C₈

yes

—

C₂×C₈

C₂×C₇₉⋊C₈

yes

characteristic

C₂×C₈

#11929608

yes

—

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₄⋊C₄

C₇₉⋊C₁₆

yes

—

C₂²⋊C₄

C₇₉⋊C₁₆

yes

—

C₄²

C₂×C₇₉⋊C₈

yes

—

C₁₆

—

100

C₁₆

C₄⋊Dic₇₉

yes

—

101

C₁₆

C₂×D₃₁₆

yes

characteristic

102

C₁₆

C₂×Dic₁₅₈

yes

characteristic

103

C₂³

C₂×C₇₉⋊C₁₆

yes

characteristic

104

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

—

105

C₂×C₄

C₂×C₇₉⋊C₁₆

yes

—

106

C₂×C₄

C₂×C₇₉⋊C₁₆

yes

characteristic

107

C₂×C₄

#14579101

yes

—

108

C₂×C₄

C₂²×C₇₉⋊C₈

yes

characteristic

109

C₈

—

110

C₈

C₄⋊(C₇₉⋊C₈)

yes

—

111

C₈

C₂×C₄⋊Dic₇₉

yes

characteristic

112

C₈

C₂×#11929608

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊(C₇₉⋊C₁₆)

yes

—

115

C₂²

C₂²×C₇₉⋊C₁₆

yes

characteristic

116

C₄

—

117

C₄

C₄⋊(C₇₉⋊C₁₆)

yes

—

118

C₄

C₂×C₄⋊(C₇₉⋊C₈)

yes

characteristic

119

C₄

C₂×#14579101

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊(C₇₉⋊C₁₆)

yes

characteristic

122

C₂

#23765694

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₁₆
Fitting	C₇₉×#2108	C₂
prime radical	C₂³×C₇₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	#2108	D₇₉
hypocenter	C₇₉	#13628
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₉

commutator, socle

C₂²×C₇₉

C₂²×C₁₆

Fitting

C₇₉×#2108

C₂

prime radical

C₂³×C₇₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₁₆ ◁ #2108

N	H ≅ G/N	type
3. C₄⋊(C₇₉⋊C₃₂)	C₂	semidirect
66. C₇₉	#13628	semidirect

H ≅ G/N

type

3. C₄⋊(C₇₉⋊C₃₂)

C₂

semidirect

66. C₇₉

#13628

semidirect

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

subgroup	index
C₂×#19098397	2
C₂²×C₇₉⋊C₃₂	2
C₄⋊(C₇₉⋊C₃₂)	2
C₇₉×#2108	2
#13628	79

subgroup

index

C₂×#19098397

C₂²×C₇₉⋊C₃₂

C₄⋊(C₇₉⋊C₃₂)

C₇₉×#2108

#13628

Subgroups — #34208755

Conjugacy classes of subgroups

Complements

Maximal subgroups