Group #23765757 — subgroups

Subgroups — #23765757

Order 10112.

696 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 389 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	C₂×#14579120	C₂	1	2	yes	maximal
2	C₄⋊C₄×Dic₇₉	C₂	1	8	yes	maximal, complemented
3	C₄×C₂²⋊Dic₇₉	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#14579120	C₄	1	16	yes	complemented
6	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
8	C₂×C₄×Dic₇₉	C₂²	1	8	yes	—
9	C₂×C₄×Dic₇₉	C₂²	1	4	yes	—
10	C₇₉⋊#95	C₂²	1	8	yes	—
11	C₂³×C₄×C₇₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₇₉	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
15	C₂²×Dic₇₉	C₂³	1	8	yes	—
16	C₄⋊Dic₇₉	C₂×C₄	1	16	yes	complemented
17	C₂²⋊Dic₇₉	C₂×C₄	1	16	yes	—
18	C₄×Dic₇₉	—	4	8	no	—
19	C₄×Dic₇₉	C₂×C₄	1	16	yes	complemented
20	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₇₉	—	4	4	no	—
22	C₂²×C₄×C₇₉	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₇₉	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
27	C₄⋊C₄×C₇₉	C₂×C₄	1	16	yes	—
28	C₂³×C₇₉	—	2	2	no	—
29	C₂³×C₇₉	—	2	2	no	—
30	C₂³×C₇₉	C₂²×C₄	1	4	yes	—
31	C₂³×C₇₉	C₂²×C₄	1	2	yes	—
32	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
33	C₂×C₄×C₇₉	—	2	8	no	—
34	C₂×C₄×C₇₉	—	4	8	no	—
35	C₂×C₄×C₇₉	C₄²	1	16	yes	complemented
36	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
37	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
38	C₂×C₄×C₇₉	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
40	C₂×Dic₇₉	—	2	8	no	—
41	C₂×Dic₇₉	—	2	8	no	—
42	C₂×Dic₇₉	C₂²×C₄	1	16	yes	—
43	C₂²×C₇₉	—	2	2	no	—
44	C₂²×C₇₉	—	2	2	no	—
45	C₂²×C₇₉	—	2	8	no	—
46	C₂²×C₇₉	C₂×C₄²	1	4	yes	—
47	C₂²×C₇₉	C₂³×C₄	1	2	yes	—
48	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
50	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
51	C₄×C₇₉	—	2	8	no	—
52	C₄×C₇₉	C₂×C₄²	1	8	yes	—
53	Dic₇₉	—	2	16	no	—
54	C₂×C₇₉	—	2	4	no	—
55	C₂×C₇₉	C₂²×C₄²	1	1	yes	characteristic
56	C₂×C₇₉	#518	1	1	yes	characteristic
57	C₂×C₇₉	#519	1	1	yes	characteristic
58	C₂×C₇₉	#520	1	4	yes	—
59	#2274	—	79	1	no	maximal, Sylow
60	C₇₉	#2274	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	79	2	no	—
62	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
63	C₄×C₄⋊C₄	—	79	8	no	—
64	C₄×C₂²⋊C₄	—	79	4	no	—
65	C₂³×C₄	C₂×D₇₉	1	2	yes	—
66	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
67	#117	—	79	16	no	—
68	C₂×C₄⋊C₄	—	79	4	no	—
69	C₂×C₄⋊C₄	Dic₇₉	1	8	yes	complemented
70	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
71	C₂×C₂²⋊C₄	—	79	4	no	—
72	C₂×C₄²	—	79	8	no	—
73	C₂×C₄²	—	79	4	no	—
74	#95	—	79	8	no	—
75	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
76	C₂²×C₄	—	4	4	no	—
77	C₂²×C₄	—	79	8	no	—
78	C₂²×C₄	C₄×D₇₉	1	8	yes	—
79	C₂²×C₄	C₂×Dic₇₉	1	8	yes	—
80	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
81	C₂²×C₄	C₂²×D₇₉	1	4	yes	—
82	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
83	C₄⋊C₄	—	79	16	no	—
84	C₄⋊C₄	C₂×Dic₇₉	1	16	yes	—
85	C₂²⋊C₄	—	79	16	no	—
86	C₄²	—	316	8	no	—
87	C₄²	—	79	16	no	—
88	C₂³	—	2	2	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₂×C₄×D₇₉	1	4	yes	—
91	C₂³	C₂²×Dic₇₉	1	2	yes	—
92	C₂³	C₂³×D₇₉	1	1	yes	characteristic
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	4	8	no	—
95	C₂×C₄	—	79	16	no	—
96	C₂×C₄	—	158	8	no	—
97	C₂×C₄	—	158	8	no	—
98	C₂×C₄	C₄×Dic₇₉	1	16	yes	complemented
99	C₂×C₄	C₂×C₄×D₇₉	1	4	yes	—
100	C₂×C₄	C₂×C₄×D₇₉	1	4	yes	—
101	C₂×C₄	C₂²×Dic₇₉	1	8	yes	—
102	C₂×C₄	C₂²×Dic₇₉	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	8	no	—
106	C₂²	C₂×C₄×Dic₇₉	1	4	yes	—
107	C₂²	C₂²×C₄×D₇₉	1	2	yes	—
108	C₂²	C₂³×Dic₇₉	1	1	yes	characteristic
109	C₂²	#14579206	1	2	yes	—
110	C₂²	#14579209	1	2	yes	—
111	C₄	—	2	8	no	—
112	C₄	—	158	16	no	—
113	C₄	C₂×C₄×Dic₇₉	1	8	yes	—
114	C₂	—	2	4	no	—
115	C₂	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
116	C₂	#19099038	1	1	yes	characteristic
117	C₂	#19099041	1	1	yes	characteristic
118	C₂	#19099044	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579120

C₂

yes

maximal

C₄⋊C₄×Dic₇₉

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₇₉

C₂

yes

maximal

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

#14579120

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

complemented

C₂²⋊Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

complemented

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂²×C₄

yes

—

C₂³×C₇₉

C₂²×C₄

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄²

yes

complemented

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄²

yes

—

C₂²×C₇₉

C₂³×C₄

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄²

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂²×C₄²

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2274

—

maximal, Sylow

C₇₉

#2274

yes

characteristic, complemented, Sylow

C₂×#117

—

C₂²×C₄⋊C₄

D₇₉

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

C₂³×C₄

C₂×D₇₉

yes

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₇₉

yes

complemented

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

C₂⁴

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₇₉

yes

—

C₂²⋊C₄

—

C₄²

—

316

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×C₄×D₇₉

yes

—

C₂³

C₂²×Dic₇₉

yes

—

C₂³

C₂³×D₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

—

158

—

C₂×C₄

C₄×Dic₇₉

yes

complemented

C₂×C₄

C₂×C₄×D₇₉

yes

—

100

C₂×C₄

C₂×C₄×D₇₉

yes

—

101

C₂×C₄

C₂²×Dic₇₉

yes

—

102

C₂×C₄

C₂²×Dic₇₉

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₄×Dic₇₉

yes

—

107

C₂²

C₂²×C₄×D₇₉

yes

—

108

C₂²

C₂³×Dic₇₉

yes

characteristic

109

C₂²

#14579206

yes

—

110

C₂²

#14579209

yes

—

111

C₄

—

112

C₄

—

158

—

113

C₄

C₂×C₄×Dic₇₉

yes

—

114

C₂

—

115

C₂

C₂²×C₄×Dic₇₉

yes

characteristic

116

C₂

#19099038

yes

characteristic

117

C₂

#19099041

yes

characteristic

118

C₂

#19099044

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator	C₂×C₇₉	C₂²×C₄²
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle, cosocle	C₂³×C₇₉	C₂⁴
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#2274
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator

C₂×C₇₉

C₂²×C₄²

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle, cosocle

C₂³×C₇₉

C₂⁴

prime radical

C₂⁴×C₇₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₇₉

C₇₉

G ▷ C₂×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. C₄⋊C₄×Dic₇₉	C₂	semidirect
5. #14579120	C₄	semidirect
13. C₂×C₄⋊C₄×C₇₉	C₄	semidirect
16. C₄⋊Dic₇₉	C₂×C₄	semidirect
19. C₄×Dic₇₉	C₂×C₄	semidirect
35. C₂×C₄×C₇₉	C₄²	semidirect
60. C₇₉	#2274	semidirect
69. C₂×C₄⋊C₄	Dic₇₉	semidirect
98. C₂×C₄	C₄×Dic₇₉	semidirect

H ≅ G/N

type

2. C₄⋊C₄×Dic₇₉

C₂

semidirect

5. #14579120

C₄

semidirect

13. C₂×C₄⋊C₄×C₇₉

C₄

semidirect

16. C₄⋊Dic₇₉

C₂×C₄

semidirect

19. C₄×Dic₇₉

C₂×C₄

semidirect

35. C₂×C₄×C₇₉

C₄²

semidirect

60. C₇₉

#2274

semidirect

69. C₂×C₄⋊C₄

Dic₇₉

semidirect

98. C₂×C₄

C₄×Dic₇₉

semidirect

Subgroups — #23765757

Conjugacy classes of subgroups

Complements