Group #34200149 — subgroups

Subgroups — #34200149

Order 20224.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23762219	C₂	1	2	yes	maximal
2	#23761466	C₂	1	1	yes	maximal, characteristic
3	#23761370	C₂	1	2	yes	maximal
4	#23761320	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2176	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₉⋊C₈	—	2	2	no	—
7	#19098339	C₂²	1	1	yes	characteristic
8	#19098314	C₂²	1	2	yes	—
9	Q₈⋊(C₇₉⋊C₈)	—	2	2	no	—
10	#19098284	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₉	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
13	#367×C₇₉	C₄	1	1	yes	characteristic
14	#358×C₇₉	C₄	1	2	yes	—
15	#346×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₇₉⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
19	C₄×C₇₉⋊C₈	—	4	2	no	—
20	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
21	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₇₉	—	2	2	no	—
23	Q₃₂×C₇₉	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	#107×C₇₉	C₂×C₄	1	2	yes	—
27	#103×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₇₉⋊C₈	—	2	2	no	—
30	C₂×C₇₉⋊C₈	—	4	2	no	—
31	C₇₉⋊C₁₆	—	4	1	no	—
32	C₂×Q₈×C₇₉	—	2	2	no	—
33	Q₁₆×C₇₉	—	2	2	no	—
34	Q₁₆×C₇₉	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₄⋊C₄×C₇₉	—	4	2	no	—
39	C₄²×C₇₉	—	4	2	no	—
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₉	—	2	1	no	—
42	C₁₆×C₇₉	C₂×C₈	1	2	yes	—
43	Q₈×C₇₉	—	2	4	no	—
44	Q₈×C₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	—	4	2	no	—
46	C₂×C₄×C₇₉	—	4	2	no	—
47	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₇₉	—	2	1	no	—
50	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₇₉	#131	1	1	yes	characteristic
52	C₇₉⋊C₈	—	4	2	no	—
53	C₇₉⋊C₈	—	8	2	no	—
54	C₂²×C₇₉	#442	1	1	yes	characteristic
55	C₄×C₇₉	—	2	2	no	—
56	C₄×C₇₉	—	4	4	no	—
57	C₄×C₇₉	—	8	2	no	—
58	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
59	C₄×C₇₉	#444	1	1	yes	characteristic
60	#12066	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	#1580	1	1	yes	characteristic
62	C₂×C₇₉	#2180	1	2	yes	—
63	#2176	D₇₉	1	1	yes	characteristic
64	#1579	—	79	2	no	—
65	#1368	—	79	1	no	—
66	#1334	—	79	2	no	—
67	#1313	—	79	1	no	—
68	C₇₉	#12066	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	Dic₇₉	1	1	yes	characteristic
70	C₈×Q₈	—	158	2	no	—
71	C₄×Q₁₆	C₂×D₇₉	1	2	yes	—
72	#367	Dic₇₉	1	1	yes	characteristic
73	#358	Dic₇₉	1	2	yes	—
74	#346	—	79	1	no	—
75	#346	C₂×D₇₉	1	1	yes	characteristic
76	#334	—	79	2	no	—
77	Q₈⋊C₈	—	158	2	no	—
78	#322	—	79	1	no	—
79	C₂×Q₁₆	C₂×Dic₇₉	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	C₇₉⋊C₈	1	4	yes	complemented
82	C₂×C₁₆	—	158	1	no	—
83	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
85	#107	C₂×Dic₇₉	1	2	yes	—
86	C₄⋊C₈	—	316	2	no	—
87	C₄⋊C₈	—	158	2	no	—
88	#103	—	2	2	no	—
89	C₄×C₈	—	316	2	no	—
90	C₄×C₈	—	79	2	no	—
91	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
95	C₂×C₈	—	316	2	no	—
96	C₂×C₈	—	158	2	no	—
97	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
98	C₂×C₈	#11929623	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₇₉	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	316	1	no	—
105	C₁₆	C₂×C₇₉⋊C₈	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
111	C₂×C₄	#14579185	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	316	2	no	—
114	C₈	—	632	2	no	—
115	C₈	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
116	C₈	#14579168	1	1	yes	characteristic
117	C₂²	#19098675	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
122	C₄	#19098685	1	1	yes	characteristic
123	C₂	#23762224	1	1	yes	characteristic
124	C₂	#23765292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762219

C₂

yes

maximal

#23761466

C₂

yes

maximal, characteristic

#23761370

C₂

yes

maximal

#23761320

C₂

yes

maximal, characteristic

C₇₉×#2176

C₂

yes

maximal, characteristic

Q₈×C₇₉⋊C₈

—

#19098339

C₂²

yes

characteristic

#19098314

C₂²

yes

—

Q₈⋊(C₇₉⋊C₈)

—

#19098284

C₂²

yes

characteristic

C₂×Q₃₂×C₇₉

C₄

yes

characteristic

C₄×Q₁₆×C₇₉

C₂²

yes

—

#367×C₇₉

C₄

yes

characteristic

#358×C₇₉

C₄

yes

—

#346×C₇₉

C₂²

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×Q₁₆×C₇₉

C₂×C₄

yes

—

C₄×Q₈×C₇₉

—

Q₃₂×C₇₉

C₈

yes

complemented

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

C₂×C₄

yes

—

#103×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#442

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

C₄×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2176

D₇₉

yes

—

—

—

—

C₇₉

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₉

yes

characteristic

C₈×Q₈

—

158

—

C₄×Q₁₆

C₂×D₇₉

yes

—

#367

Dic₇₉

yes

characteristic

#358

Dic₇₉

yes

—

#346

—

#346

C₂×D₇₉

yes

characteristic

#334

—

Q₈⋊C₈

—

158

—

#322

—

C₂×Q₁₆

C₂×Dic₇₉

yes

—

C₄×Q₈

—

Q₃₂

C₇₉⋊C₈

yes

complemented

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

#107

C₂×Dic₇₉

yes

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

#103

—

C₄×C₈

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×C₇₉⋊C₈

yes

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

#11929623

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₇₉

yes

characteristic

103

C₁₆

—

104

C₁₆

—

316

—

105

C₁₆

C₂×C₇₉⋊C₈

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₇₉

yes

characteristic

111

C₂×C₄

#14579185

yes

—

112

C₈

—

113

C₈

—

316

—

114

C₈

—

632

—

115

C₈

C₂²×C₇₉⋊C₈

yes

characteristic

116

C₈

#14579168

yes

characteristic

117

C₂²

#19098675

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×C₇₉⋊C₈

yes

characteristic

122

C₄

#19098685

yes

characteristic

123

C₂

#23762224

yes

characteristic

124

C₂

#23765292

yes

—

	subgroup	G/H
center	C₂²	#19098675
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2176	C₂
socle, prime radical	C₂²×C₇₉	#442
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#2176	D₇₉
hypocenter	C₇₉	#12066
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#19098675

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2176

C₂

socle, prime radical

C₂²×C₇₉

#442

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
23. Q₃₂×C₇₉	C₈	semidirect
68. C₇₉	#12066	semidirect
81. Q₃₂	C₇₉⋊C₈	semidirect

H ≅ G/N

type

23. Q₃₂×C₇₉

C₈

semidirect

68. C₇₉

#12066

semidirect

81. Q₃₂

C₇₉⋊C₈

semidirect

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

subgroup	index
#23762219	2
#23761466	2
#23761370	2
#23761320	2
C₇₉×#2176	2
#12066	79

subgroup

index

C₇₉×#2176

Subgroups — #34200149

Conjugacy classes of subgroups

Complements

Maximal subgroups