Group #34175676 — subgroups

Subgroups — #34175676

Order 20224.

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

normal subgroups only

This group has 158 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	#23761344	C₂	1	2	yes	maximal
2	C₇₉×#1835	C₂	1	1	yes	maximal, characteristic
3	#19098289	—	2	2	no	—
4	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
5	#382×C₇₉	C₄	1	1	yes	characteristic
6	#382×C₇₉	C₂²	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	D₄	1	2	yes	—
12	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₇₉	—	2	1	no	—
14	C₄×C₈×C₇₉	C₈	1	2	yes	complemented
15	#95×C₇₉	—	4	1	no	—
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	C₄²×C₇₉	—	2	1	no	—
28	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
29	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
30	C₂³×C₇₉	#99	1	1	yes	characteristic
31	C₂×C₄×C₇₉	—	2	1	no	—
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	—	4	2	no	—
36	C₂×C₄×C₇₉	—	4	2	no	—
37	C₂×C₄×C₇₉	—	8	2	no	—
38	C₂×C₄×C₇₉	—	8	2	no	—
39	C₂×C₄×C₇₉	—	8	1	no	—
40	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
42	C₈×C₇₉	—	4	2	no	—
43	C₇₉⋊C₈	—	16	2	no	—
44	C₂²×C₇₉	—	2	2	no	—
45	C₂²×C₇₉	#324	1	1	yes	characteristic
46	C₂²×C₇₉	#354	1	1	yes	characteristic
47	C₂²×C₇₉	#355	1	1	yes	characteristic
48	C₄×C₇₉	—	2	1	no	—
49	C₄×C₇₉	—	2	1	no	—
50	C₄×C₇₉	—	4	1	no	—
51	C₄×C₇₉	—	4	1	no	—
52	C₄×C₇₉	—	8	2	no	—
53	C₄×C₇₉	—	16	1	no	—
54	#7366	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	—	2	1	no	—
56	C₂×C₇₉	—	2	1	no	—
57	C₂×C₇₉	#1324	1	1	yes	characteristic
58	C₂×C₇₉	#1414	1	2	yes	—
59	#1835	D₇₉	1	1	yes	characteristic
60	#1324	—	79	2	no	—
61	C₇₉	#7366	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
63	#382	Dic₇₉	1	1	yes	characteristic
64	#382	C₂×D₇₉	1	1	yes	characteristic
65	#340	—	2	2	no	—
66	#324	—	158	2	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
70	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
71	C₂²⋊C₈	—	316	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₇₉⋊C₈	1	2	yes	complemented
74	#95	—	4	1	no	—
75	#95	—	4	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	632	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	—	4	2	no	—
86	C₄²	—	2	1	no	—
87	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
88	C₄²	#11929608	1	1	yes	characteristic
89	C₂³	#14579076	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	1	no	—
99	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
100	C₂×C₄	#14579080	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₈	—	1264	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	#19098289	1	1	yes	characteristic
105	C₂²	#19098358	1	1	yes	characteristic
106	C₂²	#19098360	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	1	no	—
110	C₄	—	4	1	no	—
111	C₄	—	8	2	no	—
112	C₄	—	16	1	no	—
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#23761344	1	1	yes	characteristic
116	C₂	#23761575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23761344

C₂

yes

maximal

C₇₉×#1835

C₂

yes

maximal, characteristic

#19098289

—

C₂×C₄×C₈×C₇₉

C₄

yes

characteristic

#382×C₇₉

C₄

yes

characteristic

#382×C₇₉

C₂²

yes

characteristic

#340×C₇₉

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₈

yes

complemented

#95×C₇₉

—

#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

—

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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

#101

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#7366

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1324

yes

characteristic

C₂×C₇₉

#1414

yes

—

#1835

D₇₉

yes

characteristic

#1324

—

C₇₉

#7366

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

Dic₇₉

yes

characteristic

#382

Dic₇₉

yes

characteristic

#382

C₂×D₇₉

yes

characteristic

#340

—

#324

—

158

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂²⋊C₈

—

316

—

C₄×C₈

—

C₄×C₈

C₇₉⋊C₈

yes

complemented

#95

—

#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

—

C₄²

—

C₄²

C₂×C₇₉⋊C₈

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

100

C₂×C₄

#14579080

yes

characteristic

101

C₈

—

102

C₈

—

1264

—

103

C₂²

—

104

C₂²

#19098289

yes

characteristic

105

C₂²

#19098358

yes

characteristic

106

C₂²

#19098360

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₂

—

114

C₂

—

115

C₂

#23761344

yes

characteristic

116

C₂

#23761575

yes

—

	subgroup	G/H
center	C₂²	#19098358
commutator	C₄²×C₇₉	C₂×C₈
Frattini	#382	C₂×D₇₉
Fitting	C₇₉×#1835	C₂
socle	C₂²×C₇₉	#354
prime radical	C₂³×C₇₉	#99
cosocle	#382×C₇₉	C₂²
hypercenter	#1835	D₇₉
hypocenter	C₇₉	#7366
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1835

subgroup

G/H

center

C₂²

#19098358

commutator

C₄²×C₇₉

C₂×C₈

Frattini

#382

C₂×D₇₉

Fitting

C₇₉×#1835

C₂

C₂²×C₇₉

C₂³×C₇₉

#382×C₇₉

C₂²

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1835

N	H ≅ G/N	type
14. C₄×C₈×C₇₉	C₈	semidirect
61. C₇₉	#7366	semidirect
73. C₄×C₈	C₇₉⋊C₈	semidirect

H ≅ G/N

type

14. C₄×C₈×C₇₉

C₈

semidirect

61. C₇₉

#7366

semidirect

73. C₄×C₈

C₇₉⋊C₈

semidirect

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

subgroup	index
#23761344	2
C₇₉×#1835	2
#7366	79

subgroup

index

#23761344

C₇₉×#1835

#7366

Subgroups — #34175676

Conjugacy classes of subgroups

Complements

Maximal subgroups