Group #34208898 — subgroups

Subgroups — #34208898

Order 20224.

166 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	Dic₇₉⋊C₃₂	C₂	1	2	yes	maximal
3	C₄×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₇₉	C₂	1	2	yes	maximal
5	C₇₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₅₈	C₄	1	1	yes	characteristic
7	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₉⋊C₁₆)	C₄	1	1	yes	characteristic
10	Dic₇₉⋊C₁₆	C₄	1	2	yes	—
11	C₁₆×Dic₇₉	C₂²	1	2	yes	—
12	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₅₈	C₈	1	1	yes	characteristic
15	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
16	C₄⋊(C₇₉⋊C₈)	C₈	1	1	yes	characteristic
17	Dic₇₉⋊C₈	C₈	1	2	yes	—
18	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
19	C₇₉⋊C₃₂	—	2	1	no	—
20	C₇₉⋊C₃₂	Q₈	1	2	yes	complemented
21	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₇₉	—	2	2	no	—
25	C₂×Dic₁₅₈	C₁₆	1	1	yes	characteristic
26	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
27	C₄⋊Dic₇₉	C₁₆	1	1	yes	characteristic
28	Dic₇₉⋊C₄	C₁₆	1	2	yes	—
29	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
30	C₇₉⋊C₁₆	—	2	2	no	—
31	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₉	—	2	1	no	—
35	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₇₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₇₉	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₇₉	C₂×C₁₆	1	2	yes	—
42	Dic₁₅₈	C₃₂	1	4	yes	complemented
43	C₈×C₇₉	—	2	1	no	—
44	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₇₉	#131	1	1	yes	characteristic
46	C₇₉⋊C₈	—	2	2	no	—
47	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₇₉	—	2	1	no	—
49	C₄×C₇₉	C₂×C₃₂	1	2	yes	—
50	C₄×C₇₉	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₇₉	#504	1	1	yes	characteristic
52	Dic₇₉	—	2	2	no	—
53	Dic₇₉	C₂×C₃₂	1	4	yes	—
54	Q₈×C₃₂	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₇₉	C₂²×C₃₂	1	1	yes	characteristic
57	C₂×C₇₉	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	—	79	1	no	—
59	C₄⋊C₃₂	—	79	2	no	—
60	C₄⋊C₃₂	D₇₉	1	1	yes	characteristic
61	C₄×C₃₂	—	79	2	no	—
62	C₄×C₃₂	—	79	1	no	—
63	C₇₉	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	79	1	no	—
65	C₂×C₃₂	—	79	1	no	—
66	C₂×C₃₂	—	79	1	no	—
67	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
68	C₄⋊C₁₆	—	79	2	no	—
69	C₄⋊C₁₆	—	79	1	no	—
70	C₄×C₁₆	—	79	2	no	—
71	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
72	C₄×Q₈	—	79	1	no	—
73	C₂×C₁₆	—	79	2	no	—
74	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
76	C₄⋊C₈	—	79	2	no	—
77	C₄⋊C₈	—	79	1	no	—
78	C₄×C₈	—	79	2	no	—
79	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
80	C₃₂	—	2	2	no	—
81	C₃₂	—	79	2	no	—
82	C₃₂	—	158	1	no	—
83	C₂×Q₈	—	79	1	no	—
84	C₂×C₈	—	79	2	no	—
85	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
87	C₄⋊C₄	—	79	2	no	—
88	C₄⋊C₄	—	79	1	no	—
89	C₄²	—	79	2	no	—
90	C₄²	C₈×D₇₉	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	158	2	no	—
93	C₁₆	Q₈×D₇₉	1	1	yes	characteristic
94	C₁₆	#11929623	1	1	yes	characteristic
95	Q₈	—	79	4	no	—
96	C₂×C₄	—	79	2	no	—
97	C₂×C₄	—	79	2	no	—
98	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	158	2	no	—
103	C₈	#14579126	1	1	yes	characteristic
104	C₈	#14579169	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₇₉	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	79	4	no	—
108	C₄	—	158	2	no	—
109	C₄	C₃₂×D₇₉	1	2	yes	—
110	C₄	#19098687	1	1	yes	characteristic
111	C₄	#19098981	1	1	yes	characteristic
112	C₂	#23765302	1	1	yes	characteristic
113	C₂	C₂×C₃₂×D₇₉	1	1	yes	characteristic
114	C₂	#23765695	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₅₈

C₂

yes

maximal, characteristic

Dic₇₉⋊C₃₂

C₂

yes

maximal

C₄×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic

C₃₂×Dic₇₉

C₂

yes

maximal

C₇₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₅₈

C₄

yes

characteristic

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

C₄

yes

characteristic

Dic₇₉⋊C₁₆

C₄

yes

—

C₁₆×Dic₇₉

C₂²

yes

—

C₂×C₃₂×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×Dic₁₅₈

C₈

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₈

yes

characteristic

Dic₇₉⋊C₈

C₈

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₇₉⋊C₃₂

—

C₇₉⋊C₃₂

Q₈

yes

complemented

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₃₂×C₇₉

—

C₂×Dic₁₅₈

C₁₆

yes

characteristic

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₇₉

C₁₆

yes

characteristic

Dic₇₉⋊C₄

C₁₆

yes

—

C₄×Dic₇₉

C₂×C₈

yes

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₁₆×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×Dic₇₉

C₂×C₁₆

yes

—

C₂×Dic₇₉

C₂×C₁₆

yes

—

Dic₁₅₈

C₃₂

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

C₄×Q₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×C₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×C₃₂

yes

—

C₄×C₇₉

C₈×Q₈

yes

characteristic

C₄×C₇₉

#504

yes

characteristic

Dic₇₉

—

Dic₇₉

C₂×C₃₂

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂×C₇₉

Q₈×C₁₆

yes

characteristic

C₂×C₇₉

C₂²×C₃₂

yes

characteristic

C₂×C₇₉

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₇₉

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₇₉

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₉

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

158

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₉

yes

characteristic

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₇₉

yes

characteristic

C₁₆

—

C₁₆

—

158

—

C₁₆

Q₈×D₇₉

yes

characteristic

C₁₆

#11929623

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₇₉

yes

characteristic

C₂×C₄

C₁₆×D₇₉

yes

characteristic

100

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

101

C₈

—

102

C₈

—

158

—

103

C₈

#14579126

yes

characteristic

104

C₈

#14579169

yes

characteristic

105

C₂²

C₂×C₁₆×D₇₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

158

—

109

C₄

C₃₂×D₇₉

yes

—

110

C₄

#19098687

yes

characteristic

111

C₄

#19098981

yes

characteristic

112

C₂

#23765302

yes

characteristic

113

C₂

C₂×C₃₂×D₇₉

yes

characteristic

114

C₂

#23765695

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₉
commutator	C₂×C₇₉	C₂²×C₃₂
Fitting	C₇₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₉	C₂²×C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₄⋊C₃₂	D₇₉
hypocenter	C₇₉	Q₈×C₃₂
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₉

commutator

C₂×C₇₉

C₂²×C₃₂

Fitting

C₇₉×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₉

C₂²×C₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

hypercenter

C₄⋊C₃₂

D₇₉

hypocenter

C₇₉

Q₈×C₃₂

LCS

G ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
20. C₇₉⋊C₃₂	Q₈	semidirect
42. Dic₁₅₈	C₃₂	semidirect
63. C₇₉	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. C₇₉⋊C₃₂

Q₈

semidirect

42. Dic₁₅₈

C₃₂

semidirect

63. C₇₉

Q₈×C₃₂

semidirect

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

subgroup	index
C₁₆×Dic₁₅₈	2
Dic₇₉⋊C₃₂	2
C₄×C₇₉⋊C₃₂	2
C₃₂×Dic₇₉	2
C₇₉×C₄⋊C₃₂	2
Q₈×C₃₂	79

subgroup

index

C₁₆×Dic₁₅₈

Dic₇₉⋊C₃₂

C₄×C₇₉⋊C₃₂

C₃₂×Dic₇₉

C₇₉×C₄⋊C₃₂

Q₈×C₃₂

Subgroups — #34208898

Conjugacy classes of subgroups

Complements

Maximal subgroups