Group #34175678 — subgroups

Subgroups — #34175678

Order 20224.

162 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 47 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	#23761346	C₂	1	2	yes	maximal
2	C₇₉×#1836	C₂	1	1	yes	maximal, characteristic
3	#19098289	—	2	2	no	—
4	C₂×#97×C₇₉	C₄	1	1	yes	characteristic
5	#384×C₇₉	C₂²	1	1	yes	characteristic
6	#382×C₇₉	C₄	1	1	yes	characteristic
7	#340×C₇₉	—	2	2	no	—
8	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
9	C₂²×C₈×C₇₉	—	2	1	no	—
10	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
12	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
13	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
14	#97×C₇₉	—	2	1	no	—
15	#97×C₇₉	C₈	1	2	yes	complemented
16	#95×C₇₉	—	4	1	no	—
17	C₂×C₇₉⋊C₈	—	8	2	no	—
18	C₂²×C₄×C₇₉	—	2	1	no	—
19	C₂²×C₄×C₇₉	—	2	2	no	—
20	C₂²×C₄×C₇₉	—	4	1	no	—
21	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₉	—	2	2	no	—
23	C₂×C₈×C₇₉	—	4	1	no	—
24	C₂×C₈×C₇₉	—	4	1	no	—
25	C₄⋊C₄×C₇₉	—	4	2	no	—
26	C₄⋊C₄×C₇₉	—	4	2	no	—
27	C₄⋊C₄×C₇₉	—	8	2	no	—
28	C₄²×C₇₉	—	2	1	no	—
29	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
31	C₂³×C₇₉	#99	1	1	yes	characteristic
32	C₂×C₄×C₇₉	—	2	1	no	—
33	C₂×C₄×C₇₉	—	2	1	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	—	4	1	no	—
38	C₂×C₄×C₇₉	—	4	2	no	—
39	C₂×C₄×C₇₉	—	8	2	no	—
40	C₂×C₄×C₇₉	—	8	1	no	—
41	C₂×C₄×C₇₉	—	8	1	no	—
42	C₂×C₄×C₇₉	—	8	1	no	—
43	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
45	C₈×C₇₉	—	4	2	no	—
46	C₇₉⋊C₈	—	16	2	no	—
47	C₂²×C₇₉	—	2	1	no	—
48	C₂²×C₇₉	—	2	1	no	—
49	C₂²×C₇₉	#324	1	1	yes	characteristic
50	C₂²×C₇₉	#354	1	1	yes	characteristic
51	C₂²×C₇₉	#355	1	1	yes	characteristic
52	C₄×C₇₉	—	2	1	no	—
53	C₄×C₇₉	—	2	1	no	—
54	C₄×C₇₉	—	4	1	no	—
55	C₄×C₇₉	—	4	1	no	—
56	C₄×C₇₉	—	8	2	no	—
57	C₄×C₇₉	—	16	1	no	—
58	#7367	—	79	1	no	maximal, Sylow
59	C₂×C₇₉	—	2	1	no	—
60	C₂×C₇₉	—	2	1	no	—
61	C₂×C₇₉	#1324	1	1	yes	characteristic
62	C₂×C₇₉	#1412	1	1	yes	characteristic
63	C₂×C₇₉	#1413	1	1	yes	characteristic
64	#1836	D₇₉	1	1	yes	characteristic
65	#1325	—	79	2	no	—
66	C₇₉	#7367	1	1	yes	characteristic, complemented, Sylow
67	C₂×#97	Dic₇₉	1	1	yes	characteristic
68	#384	C₂×D₇₉	1	1	yes	characteristic
69	#382	Dic₇₉	1	1	yes	characteristic
70	#340	—	2	2	no	—
71	#324	—	158	2	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
76	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
77	C₂²⋊C₈	—	316	2	no	—
78	#97	—	2	1	no	—
79	#97	C₇₉⋊C₈	1	2	yes	complemented
80	#95	—	4	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	632	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
94	C₄²	#11929608	1	1	yes	characteristic
95	C₂³	#14579076	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	—	8	1	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
108	C₂×C₄	#14579080	1	1	yes	characteristic
109	C₈	—	4	2	no	—
110	C₈	—	1264	2	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#19098289	1	1	yes	characteristic
114	C₂²	#19098358	1	1	yes	characteristic
115	C₂²	#19098360	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	8	2	no	—
121	C₄	—	16	1	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	#23761344	1	1	yes	characteristic
125	C₂	#23761571	1	1	yes	characteristic
126	C₂	#23761573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761346

C₂

yes

maximal

C₇₉×#1836

C₂

yes

maximal, characteristic

#19098289

—

C₂×#97×C₇₉

C₄

yes

characteristic

#384×C₇₉

C₂²

yes

characteristic

#382×C₇₉

C₄

yes

characteristic

#340×C₇₉

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

#97×C₇₉

—

#97×C₇₉

C₈

yes

complemented

#95×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

#99

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

#101

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#7367

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1325

—

C₇₉

#7367

yes

characteristic, complemented, Sylow

C₂×#97

Dic₇₉

yes

characteristic

#384

C₂×D₇₉

yes

characteristic

#382

Dic₇₉

yes

characteristic

#340

—

#324

—

158

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂²⋊C₈

—

316

—

#97

—

#97

C₇₉⋊C₈

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

632

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₇₉⋊C₈

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

108

C₂×C₄

#14579080

yes

characteristic

109

C₈

—

110

C₈

—

1264

—

111

C₂²

—

112

C₂²

—

113

C₂²

#19098289

yes

characteristic

114

C₂²

#19098358

yes

characteristic

115

C₂²

#19098360

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₂

—

123

C₂

—

124

C₂

#23761344

yes

characteristic

125

C₂

#23761571

yes

characteristic

126

C₂

#23761573

yes

characteristic

	subgroup	G/H
center	C₂²	#19098358
commutator	C₄²×C₇₉	C₂×C₈
Frattini	#384	C₂×D₇₉
Fitting	C₇₉×#1836	C₂
socle	C₂²×C₇₉	#354
prime radical	C₂³×C₇₉	#99
cosocle	#384×C₇₉	C₂²
hypercenter	#1836	D₇₉
hypocenter	C₇₉	#7367
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1836

subgroup

G/H

center

C₂²

#19098358

commutator

C₄²×C₇₉

C₂×C₈

Frattini

#384

C₂×D₇₉

Fitting

C₇₉×#1836

C₂

C₂²×C₇₉

C₂³×C₇₉

#384×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1836

N	H ≅ G/N	type
15. #97×C₇₉	C₈	semidirect
66. C₇₉	#7367	semidirect
79. #97	C₇₉⋊C₈	semidirect

H ≅ G/N

type

15. #97×C₇₉

C₈

semidirect

66. C₇₉

#7367

semidirect

79. #97

C₇₉⋊C₈

semidirect

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

subgroup	index
#23761346	2
C₇₉×#1836	2
#7367	79

subgroup

index

#23761346

C₇₉×#1836

#7367

Subgroups — #34175678

Conjugacy classes of subgroups

Complements

Maximal subgroups