Group #19098749 — subgroups

Subgroups — #19098749

Order 5056.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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	#14579206	C₂	1	1	yes	maximal, characteristic, complemented
2	#14579130	C₂	1	2	yes	maximal, complemented
3	#14579076	C₂	1	1	yes	maximal, characteristic, complemented
4	#14579075	C₂	1	2	yes	maximal, complemented
5	#120×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#11929624	—	2	1	no	—
7	#11929623	—	2	2	no	—
8	C₂×C₂²⋊D₇₉	—	2	1	no	—
9	C₂×C₂²⋊D₇₉	—	2	2	no	—
10	C₂×C₂²⋊D₇₉	C₂²	1	2	yes	—
11	D₄×D₇₉	—	2	2	no	—
12	C₂²⋊Dic₇₉	—	2	2	no	—
13	C₂²⋊Dic₇₉	C₂²	1	2	yes	complemented
14	C₂⁴×C₇₉	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₇₉	—	2	1	no	—
16	C₂×D₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₇₉	—	2	1	no	—
18	C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic, complemented
19	C₂³×C₇₉	—	2	2	no	—
20	C₂³×C₇₉	—	2	1	no	—
21	C₂³×C₇₉	—	4	2	no	—
22	C₂³×C₇₉	D₄	1	1	yes	characteristic, complemented
23	C₂³×C₇₉	C₂³	1	1	yes	characteristic
24	C₂²×D₇₉	—	2	1	no	—
25	C₂²×D₇₉	D₄	1	2	yes	complemented
26	D₄×C₇₉	—	2	2	no	—
27	D₄×C₇₉	—	4	2	no	—
28	C₂×C₄×C₇₉	—	2	1	no	—
29	C₂×C₄×C₇₉	D₄	1	1	yes	characteristic, complemented
30	C₂²⋊D₇₉	—	2	2	no	—
31	C₂²⋊D₇₉	—	2	2	no	—
32	C₂²⋊D₇₉	—	2	2	no	—
33	C₂²⋊D₇₉	—	4	4	no	—
34	C₂×Dic₇₉	—	2	1	no	—
35	C₂×Dic₇₉	—	2	2	no	—
36	C₂×Dic₇₉	D₄	1	2	yes	complemented
37	D₃₁₆	—	2	1	no	—
38	C₄×D₇₉	—	2	2	no	—
39	Dic₁₅₈	—	2	1	no	—
40	C₂²×C₇₉	—	2	4	no	—
41	C₂²×C₇₉	—	2	2	no	—
42	C₂²×C₇₉	—	2	1	no	—
43	C₂²×C₇₉	—	4	1	no	—
44	C₂²×C₇₉	—	4	1	no	—
45	C₂²×C₇₉	—	4	2	no	—
46	C₂²×C₇₉	—	4	1	no	—
47	C₂²×C₇₉	—	4	1	no	—
48	C₂²×C₇₉	C₂×D₄	1	2	yes	—
49	C₂²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×D₇₉	—	2	2	no	—
51	C₂×D₇₉	—	4	2	no	—
52	C₂×D₇₉	—	8	1	no	—
53	C₄×C₇₉	—	2	1	no	—
54	C₄×C₇₉	—	4	1	no	—
55	Dic₇₉	—	2	2	no	—
56	Dic₇₉	—	4	2	no	—
57	C₂×C₇₉	—	2	2	no	—
58	C₂×C₇₉	—	2	1	no	—
59	C₂×C₇₉	—	4	1	no	—
60	C₂×C₇₉	—	4	2	no	—
61	C₂×C₇₉	#120	1	1	yes	characteristic
62	D₇₉	—	4	2	no	—
63	C₇₉	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
64	B(4,𝔽₂)	—	79	1	no	maximal, Sylow
65	H(5,𝔽₂)	—	79	1	no	—
66	#120	—	79	2	no	—
67	#120	D₇₉	1	1	yes	characteristic, complemented
68	#99	—	79	1	no	—
69	#99	—	79	2	no	—
70	C₂⁴	C₂×D₇₉	1	1	yes	characteristic, complemented
71	#41	—	158	2	no	—
72	#41	—	158	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	79	2	no	—
75	C₂×D₄	—	158	2	no	—
76	C₂×D₄	—	158	2	no	—
77	C₂×D₄	—	158	1	no	—
78	C₂×D₄	C₂×D₇₉	1	1	yes	characteristic
79	C₂²⋊C₄	—	2	1	no	—
80	C₂²⋊C₄	—	79	2	no	—
81	C₂²⋊C₄	—	158	2	no	—
82	C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic, complemented
83	C₂³	—	2	2	no	—
84	C₂³	—	2	1	no	—
85	C₂³	—	4	2	no	—
86	C₂³	—	79	2	no	—
87	C₂³	—	158	1	no	—
88	C₂³	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
89	C₂³	C₂²×D₇₉	1	1	yes	characteristic
90	Q₈	—	158	1	no	—
91	D₄	—	2	2	no	—
92	D₄	—	4	2	no	—
93	D₄	—	316	4	no	—
94	D₄	—	158	1	no	—
95	D₄	—	158	2	no	—
96	D₄	—	158	2	no	—
97	D₄	—	158	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	79	2	no	—
100	C₂×C₄	—	158	2	no	—
101	C₂×C₄	—	158	2	no	—
102	C₂×C₄	—	158	1	no	—
103	C₂×C₄	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	316	2	no	—
113	C₂²	—	632	1	no	—
114	C₂²	—	158	2	no	—
115	C₂²	D₄×D₇₉	1	2	yes	—
116	C₂²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	4	1	no	—
119	C₄	—	316	2	no	—
120	C₄	—	158	2	no	—
121	C₂	—	2	1	no	—
122	C₂	—	2	2	no	—
123	C₂	—	4	2	no	—
124	C₂	—	4	1	no	—
125	C₂	—	316	2	no	—
126	C₂	#14579129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14579206

C₂

yes

maximal, characteristic, complemented

#14579130

C₂

yes

maximal, complemented

#14579076

C₂

yes

maximal, characteristic, complemented

#14579075

C₂

yes

maximal, complemented

#120×C₇₉

C₂

yes

maximal, characteristic, complemented

#11929624

—

#11929623

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

C₂²

yes

—

D₄×D₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

C₂²

yes

complemented

C₂⁴×C₇₉

C₂²

yes

characteristic, complemented

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂²

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

C₂²

yes

characteristic, complemented

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

D₄

yes

characteristic, complemented

C₂³×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

D₄

yes

complemented

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₄

yes

characteristic, complemented

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₄

yes

complemented

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

C₂×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#120

yes

characteristic

D₇₉

—

C₇₉

B(4,𝔽₂)

yes

characteristic, complemented, Sylow

—

—

—

D₇₉

yes

characteristic, complemented

#99

—

#99

—

C₂⁴

C₂×D₇₉

yes

characteristic, complemented

#41

—

158

—

#41

—

158

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

C₂×D₇₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

158

—

C₂²⋊C₄

C₂×D₇₉

yes

characteristic, complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

158

—

C₂³

C₂²⋊D₇₉

yes

characteristic, complemented

C₂³

C₂²×D₇₉

yes

characteristic

Q₈

—

158

—

D₄

—

D₄

—

D₄

—

316

—

D₄

—

158

—

D₄

—

158

—

D₄

—

158

—

D₄

—

158

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

158

—

101

C₂×C₄

—

158

—

102

C₂×C₄

—

158

—

103

C₂×C₄

C₂²⋊D₇₉

yes

characteristic, complemented

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

316

—

113

C₂²

—

632

—

114

C₂²

—

158

—

115

C₂²

D₄×D₇₉

yes

—

116

C₂²

C₂×C₂²⋊D₇₉

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

316

—

120

C₄

—

158

—

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

316

—

126

C₂

#14579129

yes

characteristic

	subgroup	G/H
center	C₂	#14579129
commutator	C₂³×C₇₉	C₂³
Frattini	C₂³	C₂²×D₇₉
Fitting	#120×C₇₉	C₂
socle	C₂×C₇₉	#120
hypercenter	#120	D₇₉
hypocenter	C₇₉	B(4,𝔽₂)
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

subgroup

G/H

center

C₂

#14579129

commutator

C₂³×C₇₉

C₂³

Frattini

C₂³

C₂²×D₇₉

Fitting

#120×C₇₉

C₂

C₂×C₇₉

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ #120

N	H ≅ G/N	type
1. #14579206	C₂	semidirect
2. #14579130	C₂	semidirect
3. #14579076	C₂	semidirect
4. #14579075	C₂	semidirect
5. #120×C₇₉	C₂	semidirect
13. C₂²⋊Dic₇₉	C₂²	semidirect
14. C₂⁴×C₇₉	C₂²	semidirect
18. C₂²⋊C₄×C₇₉	C₂²	semidirect
22. C₂³×C₇₉	D₄	semidirect
25. C₂²×D₇₉	D₄	semidirect
29. C₂×C₄×C₇₉	D₄	semidirect
36. C₂×Dic₇₉	D₄	semidirect
63. C₇₉	B(4,𝔽₂)	semidirect
67. #120	D₇₉	semidirect
70. C₂⁴	C₂×D₇₉	semidirect
82. C₂²⋊C₄	C₂×D₇₉	semidirect
88. C₂³	C₂²⋊D₇₉	semidirect
103. C₂×C₄	C₂²⋊D₇₉	semidirect

H ≅ G/N

type

1. #14579206

C₂

semidirect

2. #14579130

C₂

semidirect

3. #14579076

C₂

semidirect

4. #14579075

C₂

semidirect

5. #120×C₇₉

C₂

semidirect

13. C₂²⋊Dic₇₉

C₂²

semidirect

14. C₂⁴×C₇₉

C₂²

semidirect

18. C₂²⋊C₄×C₇₉

C₂²

semidirect

22. C₂³×C₇₉

D₄

semidirect

25. C₂²×D₇₉

D₄

semidirect

29. C₂×C₄×C₇₉

D₄

semidirect

36. C₂×Dic₇₉

D₄

semidirect

63. C₇₉

B(4,𝔽₂)

semidirect

67. #120

D₇₉

semidirect

70. C₂⁴

C₂×D₇₉

semidirect

82. C₂²⋊C₄

C₂×D₇₉

semidirect

88. C₂³

C₂²⋊D₇₉

semidirect

103. C₂×C₄

C₂²⋊D₇₉

semidirect

Subgroups — #19098749

Conjugacy classes of subgroups

Complements