Group #34175370 — subgroups

Subgroups — #34175370

Order 20224.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23762163	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098620	—	2	2	no	—
4	#19098316	—	2	2	no	—
5	#19098314	C₄	1	4	yes	—
6	C₂²×C₁₆×C₇₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₇₉	—	2	2	no	—
9	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
11	#14579072	—	4	2	no	—
12	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₉	—	2	2	no	—
16	C₂×C₁₆×C₇₉	—	4	2	no	—
17	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₉	D₄	1	2	yes	—
19	C₂×C₇₉⋊C₈	—	4	4	no	—
20	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₇₉	—	2	1	no	—
24	C₂×C₈×C₇₉	—	2	1	no	—
25	C₂×C₈×C₇₉	—	4	1	no	—
26	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
29	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₇₉	SD₁₆	1	2	yes	—
31	C₁₆×C₇₉	—	4	2	no	—
32	C₂³×C₇₉	#107	1	1	yes	characteristic
33	C₂×C₄×C₇₉	—	2	2	no	—
34	C₂×C₄×C₇₉	—	2	1	no	—
35	C₂×C₄×C₇₉	—	2	1	no	—
36	C₂×C₄×C₇₉	—	4	1	no	—
37	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
38	C₂×C₄×C₇₉	#102	1	2	yes	—
39	C₂×C₄×C₇₉	#103	1	2	yes	—
40	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
41	C₈×C₇₉	—	2	2	no	—
42	C₈×C₇₉	—	2	1	no	—
43	C₈×C₇₉	#100	1	1	yes	characteristic
44	C₈×C₇₉	#101	1	1	yes	characteristic
45	C₇₉⋊C₈	—	8	4	no	—
46	C₂²×C₇₉	—	2	1	no	—
47	C₂²×C₇₉	—	2	1	no	—
48	C₂²×C₇₉	#340	1	1	yes	characteristic
49	C₂²×C₇₉	#368	1	2	yes	—
50	C₄×C₇₉	—	2	1	no	—
51	C₄×C₇₉	—	2	2	no	—
52	C₄×C₇₉	—	2	2	no	—
53	C₄×C₇₉	#338	1	1	yes	characteristic
54	C₄×C₇₉	#343	1	1	yes	characteristic
55	#7242	—	79	1	no	maximal, Sylow
56	C₂×C₇₉	—	2	2	no	—
57	C₂×C₇₉	#1292	1	1	yes	characteristic
58	C₂×C₇₉	#1381	1	1	yes	characteristic
59	C₂×C₇₉	#1384	1	1	yes	characteristic
60	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
61	#1568	—	79	2	no	—
62	C₇₉	#7242	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₁₆	Dic₇₉	1	2	yes	—
64	#432	—	158	2	no	—
65	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
66	C₄⋊C₁₆	—	2	2	no	—
67	#335	—	158	2	no	—
68	#334	—	79	4	no	—
69	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
71	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	158	4	no	—
75	C₂²⋊C₈	—	316	2	no	—
76	#97	—	316	2	no	—
77	C₄×C₈	C₄×D₇₉	1	2	yes	—
78	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
79	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
80	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
81	C₂²×C₄	Dic₃₁₆	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	316	4	no	—
86	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
88	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
89	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
90	C₄²	#11929613	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#14579095	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
98	C₂×C₄	#14579083	1	2	yes	—
99	C₂×C₄	#14579086	1	2	yes	—
100	C₂×C₄	#14579093	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	632	4	no	—
104	C₈	#14579078	1	1	yes	characteristic
105	C₈	#14579080	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#19098326	1	1	yes	characteristic
109	C₂²	#19098393	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	C₇₉⋊#338	1	1	yes	characteristic
114	C₄	#19098332	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#23761275	1	1	yes	characteristic
117	C₂	#23761494	1	1	yes	characteristic
118	C₂	#23761502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762163

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic

#19098620

—

#19098316

—

#19098314

C₄

yes

—

C₂²×C₁₆×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#107

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

#106

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₂²×C₇₉

#368

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₄⋊C₁₆

D₇₉

yes

characteristic

#1568

—

C₇₉

#7242

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₇₉

yes

—

#432

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

158

—

#334

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

316

—

#97

—

316

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

D₆₃₂

yes

characteristic

C₂²×C₄

Dic₃₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄×Dic₇₉

yes

characteristic

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

Dic₇₉⋊C₄

yes

—

C₄²

(C₂×D₇₉)⋊C₄

yes

—

C₄²

#11929613

yes

—

C₁₆

—

C₂³

#14579095

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇₉⋊#95

yes

characteristic

C₂×C₄

#14579083

yes

—

C₂×C₄

#14579086

yes

—

100

C₂×C₄

#14579093

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

632

—

104

C₈

#14579078

yes

characteristic

105

C₈

#14579080

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#19098326

yes

characteristic

109

C₂²

#19098393

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₇₉⋊#338

yes

characteristic

114

C₄

#19098332

yes

characteristic

115

C₂

—

116

C₂

#23761275

yes

characteristic

117

C₂

#23761494

yes

characteristic

118

C₂

#23761502

yes

characteristic

	subgroup	G/H
center	C₂²	#19098326
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#107
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#7242
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#19098326

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×C₄⋊C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₇₉

hypocenter

C₇₉

#7242

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
62. C₇₉	#7242	semidirect

H ≅ G/N

type

62. C₇₉

#7242

semidirect

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

subgroup	index
#23762163	2
C₂×C₄⋊C₁₆×C₇₉	2
#7242	79

subgroup

index

#23762163

C₂×C₄⋊C₁₆×C₇₉

#7242

Subgroups — #34175370

Conjugacy classes of subgroups

Complements

Maximal subgroups