Group #34200148 — subgroups

Subgroups — #34200148

Order 20224.

214 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 77 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	#23765272	C₂	1	1	yes	maximal, characteristic
2	#23762218	C₂	1	2	yes	maximal
3	#23761369	C₂	1	2	yes	maximal
4	#23761320	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1368	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	C₄	1	1	yes	characteristic
7	#19098687	—	2	2	no	—
8	C₄×Dic₃₁₆	C₂²	1	2	yes	—
9	#19098391	C₄	1	1	yes	characteristic
10	#19098368	C₄	1	2	yes	—
11	#19098339	C₂²	1	1	yes	characteristic
12	#19098293	—	2	2	no	—
13	#19098284	C₂²	1	1	yes	characteristic
14	#346×C₇₉	C₂²	1	1	yes	characteristic
15	#334×C₇₉	C₂²	1	2	yes	—
16	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₅₈	—	2	2	no	—
18	Dic₆₃₂	C₈	1	4	yes	complemented
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	#14579095	C₂×C₄	1	2	yes	—
21	Dic₇₉⋊C₈	—	4	2	no	—
22	#14579086	—	2	2	no	—
23	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
24	C₈×Dic₇₉	—	4	2	no	—
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	2	2	no	—
30	Dic₃₁₆	—	2	2	no	—
31	Dic₃₁₆	C₂×C₈	1	4	yes	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₄⋊Dic₇₉	—	2	2	no	—
34	Dic₇₉⋊C₄	—	4	2	no	—
35	C₄×Dic₇₉	—	4	2	no	—
36	C₇₉⋊C₁₆	—	4	1	no	—
37	C₂×C₈×C₇₉	—	4	2	no	—
38	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₉	—	2	1	no	—
42	C₁₆×C₇₉	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₇₉	—	4	2	no	—
46	C₂×Dic₇₉	—	4	2	no	—
47	Dic₁₅₈	—	2	4	no	—
48	Dic₁₅₈	—	4	2	no	—
49	C₈×C₇₉	—	2	1	no	—
50	C₈×C₇₉	—	8	2	no	—
51	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₇₉	#131	1	1	yes	characteristic
53	C₇₉⋊C₈	—	4	2	no	—
54	C₂²×C₇₉	#442	1	1	yes	characteristic
55	C₄×C₇₉	—	2	2	no	—
56	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
57	C₄×C₇₉	#444	1	1	yes	characteristic
58	Dic₇₉	—	4	4	no	—
59	Dic₇₉	—	8	2	no	—
60	#12066	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	#1580	1	1	yes	characteristic
62	C₂×C₇₉	#2180	1	2	yes	—
63	#2176	—	79	1	no	—
64	#1579	—	79	2	no	—
65	#1368	D₇₉	1	1	yes	characteristic
66	#1334	—	79	2	no	—
67	#1313	—	79	1	no	—
68	C₇₉	#12066	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	79	1	no	—
70	C₈×Q₈	—	158	2	no	—
71	C₄×Q₁₆	—	79	2	no	—
72	#367	—	79	1	no	—
73	#358	—	79	2	no	—
74	#346	—	79	1	no	—
75	#346	C₂×D₇₉	1	1	yes	characteristic
76	#334	C₂×D₇₉	1	2	yes	—
77	Q₈⋊C₈	—	158	2	no	—
78	#322	—	79	1	no	—
79	C₂×Q₁₆	—	79	2	no	—
80	C₄×Q₈	—	158	2	no	—
81	Q₃₂	—	79	4	no	—
82	C₂×C₁₆	—	158	1	no	—
83	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
85	#107	—	79	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	316	2	no	—
88	#103	—	158	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	—	158	2	no	—
93	Q₁₆	—	79	4	no	—
94	Q₁₆	—	158	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	158	2	no	—
97	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
98	C₂×C₈	#11929625	1	1	yes	characteristic
99	C₄⋊C₄	—	316	2	no	—
100	C₄⋊C₄	—	158	2	no	—
101	C₄²	—	316	2	no	—
102	C₄²	D₄×D₇₉	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	316	1	no	—
105	C₁₆	C₈×D₇₉	1	2	yes	—
106	Q₈	—	316	2	no	—
107	Q₈	—	158	4	no	—
108	C₂×C₄	—	316	2	no	—
109	C₂×C₄	—	316	2	no	—
110	C₂×C₄	#14579123	1	1	yes	characteristic
111	C₂×C₄	#14579188	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	8	2	no	—
114	C₈	—	316	2	no	—
115	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
116	C₈	#14579169	1	1	yes	characteristic
117	C₂²	#19098674	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	316	4	no	—
120	C₄	—	632	2	no	—
121	C₄	#19098628	1	1	yes	characteristic
122	C₄	#19098684	1	1	yes	characteristic
123	C₂	#23762223	1	1	yes	characteristic
124	C₂	#23765290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765272

C₂

yes

maximal, characteristic

#23762218

C₂

yes

maximal

#23761369

C₂

yes

maximal

#23761320

C₂

yes

maximal, characteristic

C₇₉×#1368

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₄

yes

characteristic

#19098687

—

C₄×Dic₃₁₆

C₂²

yes

—

#19098391

C₄

yes

characteristic

#19098368

C₄

yes

—

#19098339

C₂²

yes

characteristic

#19098293

—

#19098284

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

characteristic

#334×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

C₂×C₄

yes

—

C₄×Dic₁₅₈

—

Dic₆₃₂

C₈

yes

complemented

C₂×C₇₉⋊C₁₆

—

#14579095

C₂×C₄

yes

—

Dic₇₉⋊C₈

—

#14579086

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₈×Dic₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

C₂×C₈

yes

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#442

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

#444

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2176

—

#1579

—

#1368

D₇₉

yes

characteristic

#1334

—

#1313

—

C₇₉

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

158

—

C₄×Q₁₆

—

—

—

—

C₂×D₇₉

yes

characteristic

#334

C₂×D₇₉

yes

—

Q₈⋊C₈

—

158

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

158

—

Q₃₂

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

316

—

#103

—

158

—

C₄×C₈

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

158

—

Q₁₆

—

Q₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

C₂×C₈

#11929625

yes

characteristic

C₄⋊C₄

—

316

—

100

C₄⋊C₄

—

158

—

101

C₄²

—

316

—

102

C₄²

D₄×D₇₉

yes

characteristic

103

C₁₆

—

104

C₁₆

—

316

—

105

C₁₆

C₈×D₇₉

yes

—

106

Q₈

—

316

—

107

Q₈

—

158

—

108

C₂×C₄

—

316

—

109

C₂×C₄

—

316

—

110

C₂×C₄

#14579123

yes

characteristic

111

C₂×C₄

#14579188

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

316

—

115

C₈

C₂×C₈×D₇₉

yes

characteristic

116

C₈

#14579169

yes

characteristic

117

C₂²

#19098674

yes

characteristic

118

C₄

—

119

C₄

—

316

—

120

C₄

—

632

—

121

C₄

#19098628

yes

characteristic

122

C₄

#19098684

yes

characteristic

123

C₂

#23762223

yes

characteristic

124

C₂

#23765290

yes

—

	subgroup	G/H
center	C₂²	#19098674
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1368	C₂
socle, prime radical	C₂²×C₇₉	#442
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1368	D₇₉
hypocenter	C₇₉	#12066
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#19098674

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1368

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 ◁ #1368

N	H ≅ G/N	type
18. Dic₆₃₂	C₈	semidirect
68. C₇₉	#12066	semidirect

H ≅ G/N

type

18. Dic₆₃₂

C₈

semidirect

68. C₇₉

#12066

semidirect

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

subgroup	index
#23765272	2
#23762218	2
#23761369	2
#23761320	2
C₇₉×#1368	2
#12066	79

subgroup

index

C₇₉×#1368

Subgroups — #34200148

Conjugacy classes of subgroups

Complements

Maximal subgroups