Group #23771474 — subgroups

Subgroups — #23771474

Order 10112.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14579155	C₂	1	2	yes	maximal, complemented
2	#19098493	C₂	1	3	yes	maximal, complemented
3	#19098485	C₂	1	6	yes	maximal, complemented
4	#19098480	C₂	1	3	yes	maximal
5	#384×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
7	#14579155	—	2	8	no	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	3	yes	—
9	C₂×Dic₇₉⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
11	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
12	C₇₉⋊#95	C₂²	1	3	yes	—
13	C₇₉⋊#95	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂³×D₇₉	C₂³	1	2	yes	—
18	C₂×D₃₁₆	—	4	3	no	—
19	C₂²×Dic₇₉	C₂³	1	6	yes	—
20	C₄⋊Dic₇₉	—	4	3	no	—
21	Dic₇₉⋊C₄	—	2	12	no	—
22	(C₂×D₇₉)⋊C₄	—	2	12	no	—
23	(C₂×D₇₉)⋊C₄	—	4	6	no	—
24	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
26	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
27	C₄⋊C₄×C₇₉	—	4	3	no	—
28	C₄⋊C₄×C₇₉	—	4	3	no	—
29	C₄²×C₇₉	D₄	1	4	yes	complemented
30	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
31	C₂²×D₇₉	—	2	2	no	—
32	C₂²×D₇₉	—	2	6	no	—
33	C₂²×D₇₉	—	2	6	no	—
34	C₂×C₄×C₇₉	—	2	3	no	—
35	C₂×C₄×C₇₉	—	2	3	no	—
36	C₂×C₄×C₇₉	—	2	6	no	—
37	C₂×C₄×C₇₉	—	2	3	no	—
38	C₂×C₄×C₇₉	—	2	3	no	—
39	C₂×C₄×C₇₉	C₂×D₄	1	6	yes	—
40	C₂×Dic₇₉	—	2	6	no	—
41	C₂×Dic₇₉	—	2	6	no	—
42	C₂×Dic₇₉	—	2	6	no	—
43	D₃₁₆	—	4	4	no	—
44	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₇₉	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₇₉	2₋¹⁺⁴	1	3	yes	—
47	C₂×D₇₉	—	4	6	no	—
48	C₂×D₇₉	—	4	2	no	—
49	C₂×D₇₉	—	4	6	no	—
50	C₄×C₇₉	—	2	6	no	—
51	C₄×C₇₉	—	4	3	no	—
52	C₄×C₇₉	—	4	1	no	—
53	Dic₇₉	—	4	6	no	—
54	C₂×C₇₉	#537	1	3	yes	—
55	C₂×C₇₉	#560	1	1	yes	characteristic
56	C₂×C₇₉	#563	1	3	yes	—
57	D₇₉	—	8	2	no	—
58	#2823	—	79	1	no	maximal, Sylow
59	C₇₉	#2823	1	1	yes	characteristic, complemented, Sylow
60	C₂×#126	—	79	2	no	—
61	#398	—	79	3	no	—
62	#396	—	79	6	no	—
63	#394	—	79	3	no	—
64	#384	D₇₉	1	1	yes	characteristic, complemented
65	C₂²×D₄	—	79	1	no	—
66	#126	—	158	8	no	—
67	C₂×C₄⋊C₄	—	79	3	no	—
68	C₂×C₄⋊C₄	—	79	6	no	—
69	C₂×C₄⋊C₄	C₂×D₇₉	1	3	yes	—
70	C₂×C₄⋊C₄	C₂×D₇₉	1	3	yes	—
71	C₂×C₂²⋊C₄	—	79	6	no	—
72	C₂×C₂²⋊C₄	—	79	6	no	—
73	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
74	#95	—	79	3	no	—
75	#95	—	79	3	no	—
76	C₂⁴	—	79	2	no	—
77	C₂×D₄	—	316	3	no	—
78	C₂²×C₄	—	79	6	no	—
79	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
81	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	4	3	no	—
84	C₄⋊C₄	—	316	3	no	—
85	C₄⋊C₄	—	158	12	no	—
86	C₂²⋊C₄	—	316	6	no	—
87	C₂²⋊C₄	—	158	12	no	—
88	C₄²	D₃₁₆	1	4	yes	complemented
89	C₂³	—	158	6	no	—
90	C₂³	—	158	6	no	—
91	C₂³	—	158	2	no	—
92	C₂³	C₂³×D₇₉	1	1	yes	characteristic
93	D₄	—	316	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	158	6	no	—
100	C₂×C₄	—	158	6	no	—
101	C₂×C₄	—	158	6	no	—
102	C₂×C₄	C₂×D₃₁₆	1	6	yes	—
103	C₂²	—	316	6	no	—
104	C₂²	—	316	2	no	—
105	C₂²	—	316	6	no	—
106	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
107	C₂²	#14579206	1	3	yes	—
108	C₂²	#14579209	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	3	no	—
112	C₄	—	316	6	no	—
113	C₂	—	632	2	no	—
114	C₂	#19099144	1	3	yes	—
115	C₂	#19099300	1	1	yes	characteristic
116	C₂	#19099312	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579155

C₂

yes

maximal, complemented

#19098493

C₂

yes

maximal, complemented

#19098485

C₂

yes

maximal, complemented

#19098480

C₂

yes

maximal

#384×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₂²

yes

characteristic

#14579155

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

C₂³

yes

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

complemented

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

#537

yes

—

C₂×C₇₉

#560

yes

characteristic

C₂×C₇₉

#563

yes

—

D₇₉

—

#2823

—

maximal, Sylow

C₇₉

#2823

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₉

yes

characteristic, complemented

C₂²×D₄

—

#126

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#95

—

#95

—

C₂⁴

—

C₂×D₄

—

316

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₂²⋊C₄

—

158

—

C₄²

D₃₁₆

yes

complemented

C₂³

—

158

—

C₂³

—

158

—

C₂³

—

158

—

C₂³

C₂³×D₇₉

yes

characteristic

D₄

—

316

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

100

C₂×C₄

—

158

—

101

C₂×C₄

—

158

—

102

C₂×C₄

C₂×D₃₁₆

yes

—

103

C₂²

—

316

—

104

C₂²

—

316

—

105

C₂²

—

316

—

106

C₂²

C₂²×D₃₁₆

yes

characteristic

107

C₂²

#14579206

yes

—

108

C₂²

#14579209

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

316

—

113

C₂

—

632

—

114

C₂

#19099144

yes

—

115

C₂

#19099300

yes

characteristic

116

C₂

#19099312

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#384×C₇₉	C₂
prime radical	C₂²×D₃₁₆	C₂²
hypercenter	#384	D₇₉
hypocenter	C₇₉	#2823
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, cosocle

C₂³×C₇₉

C₂⁴

Fitting

#384×C₇₉

C₂

prime radical

C₂²×D₃₁₆

C₂²

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#14579155	C₂	semidirect
2. #19098493	C₂	semidirect
3. #19098485	C₂	semidirect
5. #384×C₇₉	C₂	semidirect
29. C₄²×C₇₉	D₄	semidirect
59. C₇₉	#2823	semidirect
64. #384	D₇₉	semidirect
88. C₄²	D₃₁₆	semidirect

H ≅ G/N

type

1. C₂×#14579155

C₂

semidirect

2. #19098493

C₂

semidirect

3. #19098485

C₂

semidirect

5. #384×C₇₉

C₂

semidirect

29. C₄²×C₇₉

D₄

semidirect

59. C₇₉

#2823

semidirect

64. #384

D₇₉

semidirect

88. C₄²

D₃₁₆

semidirect

Subgroups — #23771474

Conjugacy classes of subgroups

Complements