Group #23761251 — subgroups

Subgroups — #23761251

Order 10112.

160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19098616	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
9	C₄×C₇₉⋊C₈	—	2	1	no	—
10	C₂×M₁₆×C₇₉	C₄	1	2	yes	—
11	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	#97×C₇₉	—	2	1	no	—
13	C₄×C₈×C₇₉	—	2	1	no	—
14	C₂×C₇₉⋊C₈	—	2	2	no	—
15	C₂×C₇₉⋊C₈	—	2	2	no	—
16	C₂×C₇₉⋊C₈	—	4	2	no	—
17	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
20	M₁₆×C₇₉	—	2	2	no	—
21	M₁₆×C₇₉	C₈	1	4	yes	complemented
22	C₂×C₈×C₇₉	—	2	2	no	—
23	C₄²×C₇₉	D₄	1	1	yes	characteristic
24	C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₄²×C₇₉	D₄	1	1	yes	characteristic
26	C₄²×C₇₉	Q₈	1	1	yes	characteristic
27	C₂³×C₇₉	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₇₉	—	2	1	no	—
29	C₂×C₄×C₇₉	—	2	1	no	—
30	C₂×C₄×C₇₉	—	4	1	no	—
31	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₇₉	M₁₆	1	2	yes	—
39	C₈×C₇₉	—	2	4	no	—
40	C₇₉⋊C₈	—	4	2	no	—
41	C₇₉⋊C₈	—	4	2	no	—
42	C₂²×C₇₉	—	2	1	no	—
43	C₂²×C₇₉	—	2	1	no	—
44	C₂²×C₇₉	#95	1	1	yes	characteristic
45	C₂²×C₇₉	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₇₉	#97	1	1	yes	characteristic
47	C₄×C₇₉	—	2	1	no	—
48	C₄×C₇₉	—	2	1	no	—
49	C₄×C₇₉	—	2	1	no	—
50	C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
51	C₄×C₇₉	#100	1	1	yes	characteristic
52	C₄×C₇₉	#101	1	1	yes	characteristic
53	C₄×C₇₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₇₉	—	2	1	no	—
55	C₂×C₇₉	—	2	1	no	—
56	C₂×C₇₉	#336	1	1	yes	characteristic
57	C₂×C₇₉	#343	1	1	yes	characteristic
58	C₂×C₇₉	#344	1	1	yes	characteristic
59	#1283	—	79	1	no	maximal, Sylow
60	C₇₉	#1283	1	1	yes	characteristic, complemented, Sylow
61	#431	—	79	1	no	—
62	C₂×C₄⋊C₈	—	79	1	no	—
63	C₄×M₁₆	D₇₉	1	1	yes	characteristic
64	C₂×M₁₆	Dic₇₉	1	2	yes	—
65	C₂²×C₈	—	79	2	no	—
66	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
67	C₄⋊C₈	—	158	1	no	—
68	C₄⋊C₈	—	158	1	no	—
69	C₄⋊C₈	—	158	1	no	—
70	C₂²⋊C₈	—	79	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	158	1	no	—
74	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	C₇₉⋊C₈	1	4	yes	complemented
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	316	2	no	—
81	C₂×C₈	—	158	2	no	—
82	C₂×C₈	—	158	2	no	—
83	C₄²	Dic₁₅₈	1	1	yes	characteristic
84	C₄²	D₃₁₆	1	1	yes	characteristic
85	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
86	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
87	C₂³	C₄×Dic₇₉	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
94	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
98	C₂×C₄	#11929608	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	—	316	2	no	—
101	C₈	—	316	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₇₉⋊#95	1	1	yes	characteristic
105	C₂²	C₄×C₇₉⋊C₈	1	1	yes	characteristic
106	C₂²	#14579072	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
111	C₄	#14579077	1	1	yes	characteristic
112	C₄	#14579079	1	1	yes	characteristic
113	C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#19098318	1	1	yes	characteristic
117	C₂	#19098331	1	1	yes	characteristic
118	C₂	#19098335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098616

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

C₄×M₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

C₄

yes

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

C₄

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

C₈

yes

complemented

C₂×C₈×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

Q₈

yes

characteristic

C₂³×C₇₉

C₄²

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₈

yes

—

C₂×C₄×C₇₉

M₁₆

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#95

yes

characteristic

C₂²×C₇₉

C₄×C₈

yes

characteristic

C₂²×C₇₉

#97

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊C₈

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₇₉

yes

characteristic

C₂×M₁₆

Dic₇₉

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

158

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂×Dic₇₉

yes

characteristic

M₁₆

—

M₁₆

C₇₉⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄²

Dic₁₅₈

yes

characteristic

C₄²

D₃₁₆

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₂³

C₄×Dic₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

C₂×C₇₉⋊C₈

yes

—

C₂×C₄

#11929608

yes

—

C₈

—

100

C₈

—

316

—

101

C₈

—

316

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₇₉⋊#95

yes

characteristic

105

C₂²

C₄×C₇₉⋊C₈

yes

characteristic

106

C₂²

#14579072

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂²⋊(C₇₉⋊C₈)

yes

—

111

C₄

#14579077

yes

characteristic

112

C₄

#14579079

yes

characteristic

113

C₄

C₄⋊(C₇₉⋊C₈)

yes

—

114

C₂

—

115

C₂

—

116

C₂

#19098318

yes

characteristic

117

C₂

#19098331

yes

characteristic

118

C₂

#19098335

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂²×C₇₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₇₉
Fitting	C₄×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#95
prime radical	C₂³×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂²
hypercenter	C₄×M₁₆	D₇₉
hypocenter	C₇₉	#1283
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₇₉)⋊C₄

commutator

C₂²×C₇₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₇₉

Fitting

C₄×M₁₆×C₇₉

C₂

socle

C₂²×C₇₉

#95

prime radical

C₂³×C₇₉

C₄²

cosocle

C₂×C₄²×C₇₉

C₂²

hypercenter

C₄×M₁₆

D₇₉

hypocenter

C₇₉

#1283

LCS

G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
21. M₁₆×C₇₉	C₈	semidirect
60. C₇₉	#1283	semidirect
78. M₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

21. M₁₆×C₇₉

C₈

semidirect

60. C₇₉

#1283

semidirect

78. M₁₆

C₇₉⋊C₈

semidirect

Subgroups — #23761251

Conjugacy classes of subgroups

Complements