Group #34208451 — subgroups

Subgroups — #34208451

Order 20224.

238 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂²×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#23764996	C₂	1	1	yes	maximal, characteristic
3	#23761533	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₇₉	C₂	1	2	yes	maximal, complemented
5	C₂×M₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₄	1	1	yes	characteristic
7	#19098518	C₄	1	1	yes	characteristic
8	C₂×C₇₉⋊C₃₂	C₄	1	4	yes	complemented
9	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
11	#19098338	C₄	1	2	yes	—
12	C₁₆×Dic₇₉	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	M₆₄×C₇₉	C₄	1	4	yes	complemented
15	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
16	C₂×#11929608	C₈	1	1	yes	characteristic
17	#14579120	C₈	1	1	yes	characteristic
18	#14579101	C₈	1	4	yes	—
19	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
20	#14579071	C₈	1	2	yes	—
21	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
22	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	—
23	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
29	C₃₂×C₇₉	C₂×C₄	1	4	yes	—
30	#11929608	C₁₆	1	4	yes	—
31	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
32	C₄⋊Dic₇₉	C₁₆	1	2	yes	—
33	C₂²⋊Dic₇₉	C₁₆	1	2	yes	—
34	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
35	C₇₉⋊C₁₆	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₄²	1	2	yes	—
38	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₇₉	C₄²	1	2	yes	—
42	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₇₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₉	C₂×C₁₆	1	4	yes	—
50	C₈×C₇₉	C₄×C₈	1	2	yes	—
51	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
53	C₇₉⋊C₈	C₂×C₁₆	1	4	yes	—
54	C₂²×C₇₉	—	2	2	no	—
55	C₂²×C₇₉	C₄×C₁₆	1	2	yes	—
56	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₇₉	C₄×C₁₆	1	2	yes	—
58	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
59	C₄×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
60	Dic₇₉	—	2	4	no	—
61	#13549	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	2	2	no	—
63	C₂×C₇₉	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₇₉	#2258	1	2	yes	—
65	C₂×M₆₄	D₇₉	1	1	yes	characteristic
66	C₂²×C₃₂	—	79	1	no	—
67	#2108	—	79	1	no	—
68	#1397	—	79	2	no	—
69	C₄×C₃₂	—	79	2	no	—
70	C₇₉	#13549	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₃₂	—	79	1	no	—
72	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
73	#405	—	79	1	no	—
74	M₆₄	Dic₇₉	1	4	yes	complemented
75	C₂×C₃₂	—	79	1	no	—
76	C₂×C₃₂	—	79	1	no	—
77	C₂×C₃₂	—	79	4	no	—
78	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
79	#346	—	79	2	no	—
80	C₄×C₁₆	—	79	2	no	—
81	C₂×M₁₆	—	79	1	no	—
82	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
83	#117	—	79	1	no	—
84	M₃₂	—	79	4	no	—
85	C₂×C₁₆	—	79	2	no	—
86	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
89	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
90	#97	—	79	2	no	—
91	C₄×C₈	—	79	2	no	—
92	C₃₂	—	79	2	no	—
93	C₃₂	—	79	2	no	—
94	C₃₂	C₂×Dic₇₉	1	4	yes	—
95	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
96	M₁₆	—	79	4	no	—
97	C₂×C₈	—	79	2	no	—
98	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
99	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
100	C₂×C₈	C₈×D₇₉	1	2	yes	—
101	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
102	C₄⋊C₄	—	79	2	no	—
103	C₂²⋊C₄	—	79	2	no	—
104	C₄²	—	79	2	no	—
105	C₁₆	—	79	4	no	—
106	C₁₆	C₄×Dic₇₉	1	2	yes	—
107	C₁₆	C₂×C₄×D₇₉	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₇₉	1	1	yes	characteristic
109	C₂³	C₁₆×D₇₉	1	1	yes	characteristic
110	C₂×C₄	—	79	4	no	—
111	C₂×C₄	C₈×Dic₇₉	1	2	yes	—
112	C₂×C₄	C₁₆×D₇₉	1	2	yes	—
113	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
114	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
115	C₈	—	79	4	no	—
116	C₈	C₈×Dic₇₉	1	2	yes	—
117	C₈	C₂×C₄×Dic₇₉	1	1	yes	characteristic
118	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₇₉	1	2	yes	—
121	C₂²	C₂×C₁₆×D₇₉	1	1	yes	characteristic
122	C₄	—	158	4	no	—
123	C₄	C₁₆×Dic₇₉	1	2	yes	—
124	C₄	C₂×C₈×Dic₇₉	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₇₉	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₇₉	1	1	yes	characteristic
128	C₂	#23765695	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic

#23764996

C₂

yes

maximal, characteristic

#23761533

C₂

yes

maximal, complemented

C₃₂×Dic₇₉

C₂

yes

maximal, complemented

C₂×M₆₄×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579101

C₄

yes

characteristic

#19098518

C₄

yes

characteristic

C₂×C₇₉⋊C₃₂

C₄

yes

complemented

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

#19098338

C₄

yes

—

C₁₆×Dic₇₉

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

M₆₄×C₇₉

C₄

yes

complemented

C₂×C₃₂×C₇₉

C₂²

yes

—

C₂×#11929608

C₈

yes

characteristic

#14579120

C₈

yes

characteristic

#14579101

C₈

yes

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

#14579071

C₈

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₇₉⋊C₃₂

C₂×C₄

yes

—

C₇₉⋊C₃₂

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₃₂×C₇₉

C₂×C₄

yes

—

#11929608

C₁₆

yes

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₇₉

C₁₆

yes

—

C₂²⋊Dic₇₉

C₁₆

yes

—

C₄×Dic₇₉

C₂×C₈

yes

—

C₇₉⋊C₁₆

C₂×C₈

yes

—

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₄²

yes

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₄²

yes

—

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₂³×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

C₄×C₈

yes

—

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×C₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×Dic₇₉

C₂×C₁₆

yes

—

C₈×C₇₉

C₄×C₈

yes

—

C₈×C₇₉

C₂×C₄²

yes

characteristic

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₇₉⋊C₈

C₂×C₁₆

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×C₁₆

yes

—

C₂²×C₇₉

C₂²×C₁₆

yes

characteristic

C₄×C₇₉

C₄×C₁₆

yes

—

C₄×C₇₉

C₂×C₄×C₈

yes

characteristic

C₄×C₇₉

C₂²×C₁₆

yes

characteristic

Dic₇₉

—

#13549

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄×C₁₆

yes

characteristic

C₂×C₇₉

#2258

yes

—

C₂×M₆₄

D₇₉

yes

characteristic

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₇₉

#13549

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#405

—

M₆₄

Dic₇₉

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₉

yes

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

C₂×Dic₇₉

yes

—

C₂²×C₄

C₈×D₇₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₉

yes

—

C₂×C₈

C₈×D₇₉

yes

characteristic

100

C₂×C₈

C₈×D₇₉

yes

—

101

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₇₉

yes

—

107

C₁₆

C₂×C₄×D₇₉

yes

characteristic

108

C₁₆

C₂²×Dic₇₉

yes

characteristic

109

C₂³

C₁₆×D₇₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₈×Dic₇₉

yes

—

112

C₂×C₄

C₁₆×D₇₉

yes

—

113

C₂×C₄

C₁₆×D₇₉

yes

characteristic

114

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₇₉

yes

—

117

C₈

C₂×C₄×Dic₇₉

yes

characteristic

118

C₈

C₂×C₈×D₇₉

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₇₉

yes

—

121

C₂²

C₂×C₁₆×D₇₉

yes

characteristic

122

C₄

—

158

—

123

C₄

C₁₆×Dic₇₉

yes

—

124

C₄

C₂×C₈×Dic₇₉

yes

characteristic

125

C₄

C₂×C₁₆×D₇₉

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₇₉

yes

characteristic

128

C₂

#23765695

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₉
commutator	C₂×C₇₉	C₂×C₄×C₁₆
Fitting	C₂×M₆₄×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₁₆
prime radical	C₂³×C₇₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂×M₆₄	D₇₉
hypocenter	C₇₉	#13549
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₉

commutator

C₂×C₇₉

C₂×C₄×C₁₆

Fitting

C₂×M₆₄×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×C₁₆

prime radical

C₂³×C₇₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

hypercenter

C₂×M₆₄

D₇₉

C₇₉

G ▷ C₂×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #23761533	C₂	semidirect
4. C₃₂×Dic₇₉	C₂	semidirect
8. C₂×C₇₉⋊C₃₂	C₄	semidirect
14. M₆₄×C₇₉	C₄	semidirect
70. C₇₉	#13549	semidirect
74. M₆₄	Dic₇₉	semidirect

H ≅ G/N

type

3. #23761533

C₂

semidirect

4. C₃₂×Dic₇₉

C₂

semidirect

8. C₂×C₇₉⋊C₃₂

C₄

semidirect

14. M₆₄×C₇₉

C₄

semidirect

70. C₇₉

#13549

semidirect

74. M₆₄

Dic₇₉

semidirect

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

subgroup	index
C₂²×C₇₉⋊C₃₂	2
#23764996	2
#23761533	2
C₃₂×Dic₇₉	2
C₂×M₆₄×C₇₉	2
#13549	79

subgroup

index

C₂²×C₇₉⋊C₃₂

#23764996

#23761533

C₃₂×Dic₇₉

C₂×M₆₄×C₇₉

#13549

Subgroups — #34208451

Conjugacy classes of subgroups

Complements

Maximal subgroups