Group #19099302 — subgroups

Subgroups — #19099302

Order 5056.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#14579157	C₂	1	1	yes	maximal, characteristic, complemented
2	#14579154	C₂	1	2	yes	maximal, complemented
3	#14579143	C₂	1	2	yes	maximal, complemented
4	#14579142	C₂	1	1	yes	maximal, characteristic, complemented
5	#14579134	C₂	1	2	yes	maximal, complemented
6	#14579133	C₂	1	2	yes	maximal, complemented
7	#14579132	C₂	1	2	yes	maximal, complemented
8	#14579130	C₂	1	2	yes	maximal, complemented
9	#120×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₃₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₇₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₇₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₇₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₇₉	C₂²	1	2	yes	complemented
15	C₂²×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₄×D₇₉	C₂²	1	2	yes	complemented
17	C₄⋊Dic₇₉	C₂²	1	2	yes	complemented
18	Dic₇₉⋊C₄	C₂²	1	2	yes	complemented
19	Dic₇₉⋊C₄	C₂²	1	2	yes	complemented
20	C₂²⋊Dic₇₉	C₂²	1	2	yes	—
21	C₂²⋊Dic₇₉	C₂²	1	2	yes	—
22	C₂²⋊Dic₇₉	C₂²	1	2	yes	—
23	C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic, complemented
24	(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	complemented
25	C₄×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
26	C₂⁴×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₇₉	C₂²	1	2	yes	—
28	C₂×D₄×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×C₇₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
31	C₂³×C₇₉	—	4	2	no	—
32	C₂³×C₇₉	—	4	1	no	—
33	C₂³×C₇₉	C₂³	1	2	yes	—
34	C₂³×C₇₉	C₂³	1	1	yes	characteristic
35	C₂³×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²×D₇₉	C₂³	1	2	yes	—
37	D₄×C₇₉	—	4	2	no	—
38	D₄×C₇₉	—	4	1	no	—
39	C₂×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂³	1	2	yes	—
41	C₂²⋊D₇₉	—	4	2	no	—
42	C₂²⋊D₇₉	—	4	2	no	—
43	C₂²⋊D₇₉	—	4	2	no	—
44	C₂²⋊D₇₉	—	4	2	no	—
45	C₂×Dic₇₉	—	4	1	no	—
46	C₂×Dic₇₉	C₂³	1	2	yes	—
47	C₂×Dic₇₉	C₂³	1	2	yes	—
48	C₂×Dic₇₉	C₂³	1	2	yes	—
49	D₃₁₆	—	4	1	no	—
50	C₄×D₇₉	—	4	2	no	—
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	—	2	2	no	—
53	C₂²×C₇₉	—	2	2	no	—
54	C₂²×C₇₉	—	2	2	no	—
55	C₂²×C₇₉	—	2	2	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	—	4	4	no	—
59	C₂²×C₇₉	C₂⁴	1	1	yes	characteristic
60	C₂×D₇₉	—	2	2	no	—
61	C₂×D₇₉	—	2	2	no	—
62	C₂×D₇₉	—	2	2	no	—
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	—	2	1	no	—
65	Dic₇₉	—	2	2	no	—
66	Dic₇₉	—	2	2	no	—
67	Dic₇₉	—	2	2	no	—
68	C₂×C₇₉	—	4	1	no	—
69	C₂×C₇₉	—	4	1	no	—
70	C₂×C₇₉	—	4	2	no	—
71	C₂×C₇₉	H(5,𝔽₂)	1	2	yes	—
72	C₂×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
73	D₇₉	—	4	2	no	—
74	C₇₉	#560	1	1	yes	characteristic, complemented, Sylow
75	#560	—	79	1	no	maximal, Sylow
76	#127	—	79	1	no	—
77	#126	—	79	2	no	—
78	#123	—	79	1	no	—
79	#123	—	79	2	no	—
80	#121	—	79	2	no	—
81	#121	—	79	2	no	—
82	#121	—	79	2	no	—
83	#120	—	79	2	no	—
84	#120	D₇₉	1	1	yes	characteristic, complemented
85	C₂⁴	C₂×D₇₉	1	1	yes	characteristic
86	C₂×D₄	—	79	2	no	—
87	C₂×D₄	—	79	2	no	—
88	C₂×D₄	—	79	2	no	—
89	C₂×D₄	—	79	2	no	—
90	C₂×D₄	—	79	1	no	—
91	C₂×D₄	C₂×D₇₉	1	2	yes	—
92	C₂×D₄	C₂×D₇₉	1	1	yes	characteristic
93	C₂²×C₄	—	79	1	no	—
94	C₂²×C₄	—	79	2	no	—
95	C₄⋊C₄	—	79	2	no	—
96	C₄⋊C₄	—	79	2	no	—
97	C₄⋊C₄	—	79	2	no	—
98	C₂²⋊C₄	—	79	2	no	—
99	C₂²⋊C₄	—	79	2	no	—
100	C₂²⋊C₄	—	79	2	no	—
101	C₂²⋊C₄	—	79	1	no	—
102	C₂²⋊C₄	—	79	2	no	—
103	C₂²⋊C₄	C₂×D₇₉	1	2	yes	—
104	C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
105	C₄²	—	79	1	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	79	2	no	—
109	C₂³	C₂²×D₇₉	1	2	yes	—
110	C₂³	C₂²×D₇₉	1	1	yes	characteristic
111	C₂³	C₂²×D₇₉	1	1	yes	characteristic
112	D₄	—	4	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	316	1	no	—
115	D₄	—	316	2	no	—
116	D₄	—	316	2	no	—
117	D₄	—	316	2	no	—
118	D₄	—	316	2	no	—
119	C₂×C₄	—	316	2	no	—
120	C₂×C₄	—	316	1	no	—
121	C₂×C₄	—	79	2	no	—
122	C₂×C₄	—	79	2	no	—
123	C₂×C₄	—	79	2	no	—
124	C₂×C₄	C₂²×D₇₉	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₇₉	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	158	2	no	—
135	C₂²	—	158	2	no	—
136	C₂²	—	158	2	no	—
137	C₂²	C₂³×D₇₉	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	158	2	no	—
141	C₄	—	158	2	no	—
142	C₄	—	158	2	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	—	316	2	no	—
147	C₂	#14579206	1	2	yes	—
148	C₂	#14579206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14579157

C₂

yes

maximal, characteristic, complemented

#14579154

C₂

yes

maximal, complemented

#14579143

C₂

yes

maximal, complemented

#14579142

C₂

yes

maximal, characteristic, complemented

#14579134

C₂

yes

maximal, complemented

#14579133

C₂

yes

maximal, complemented

#14579132

C₂

yes

maximal, complemented

#14579130

C₂

yes

maximal, complemented

#120×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₃₁₆

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₇₉

C₂²

yes

—

C₂×C₂²⋊D₇₉

C₂²

yes

—

C₂×C₂²⋊D₇₉

C₂²

yes

—

C₂×C₂²⋊D₇₉

C₂²

yes

complemented

C₂²×Dic₇₉

C₂²

yes

characteristic, complemented

C₂×C₄×D₇₉

C₂²

yes

complemented

C₄⋊Dic₇₉

C₂²

yes

complemented

Dic₇₉⋊C₄

C₂²

yes

complemented

Dic₇₉⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₇₉

C₂²

yes

—

C₂²⋊Dic₇₉

C₂²

yes

—

C₂²⋊Dic₇₉

C₂²

yes

—

C₂²⋊Dic₇₉

C₂²

yes

characteristic, complemented

(C₂×D₇₉)⋊C₄

C₂²

yes

complemented

C₄×Dic₇₉

C₂²

yes

characteristic, complemented

C₂⁴×C₇₉

C₂²

yes

characteristic

C₂×D₄×C₇₉

C₂²

yes

—

C₂×D₄×C₇₉

C₂²

yes

characteristic

C₂²⋊C₄×C₇₉

C₂²

yes

—

C₂²⋊C₄×C₇₉

C₂²

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

C₂³

yes

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄×C₇₉

C₂³

yes

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂³

yes

—

C₂×Dic₇₉

C₂³

yes

—

C₂×Dic₇₉

C₂³

yes

—

D₃₁₆

—

C₄×D₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂⁴

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

H(5,𝔽₂)

yes

—

C₂×C₇₉

H(5,𝔽₂)

yes

characteristic

D₇₉

—

C₇₉

#560

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₂⁴

C₂×D₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇₉

yes

—

C₂×D₄

C₂×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₇₉

yes

—

104

C₂²⋊C₄

C₂×D₇₉

yes

characteristic

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₇₉

yes

—

110

C₂³

C₂²×D₇₉

yes

characteristic

111

C₂³

C₂²×D₇₉

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

316

—

115

D₄

—

316

—

116

D₄

—

316

—

117

D₄

—

316

—

118

D₄

—

316

—

119

C₂×C₄

—

316

—

120

C₂×C₄

—

316

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×D₇₉

yes

characteristic

125

C₂×C₄

C₂²×D₇₉

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

158

—

135

C₂²

—

158

—

136

C₂²

—

158

—

137

C₂²

C₂³×D₇₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

158

—

141

C₄

—

158

—

142

C₄

—

158

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

316

—

147

C₂

#14579206

yes

—

148

C₂

#14579206

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂⁴
Fitting	#120×C₇₉	C₂
hypercenter	#120	D₇₉
hypocenter	C₇₉	#560
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂⁴

Fitting

#120×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. #14579157	C₂	semidirect
2. #14579154	C₂	semidirect
3. #14579143	C₂	semidirect
4. #14579142	C₂	semidirect
5. #14579134	C₂	semidirect
6. #14579133	C₂	semidirect
7. #14579132	C₂	semidirect
8. #14579130	C₂	semidirect
9. #120×C₇₉	C₂	semidirect
10. C₂×D₃₁₆	C₂²	semidirect
14. C₂×C₂²⋊D₇₉	C₂²	semidirect
15. C₂²×Dic₇₉	C₂²	semidirect
16. C₂×C₄×D₇₉	C₂²	semidirect
17. C₄⋊Dic₇₉	C₂²	semidirect
18. Dic₇₉⋊C₄	C₂²	semidirect
19. Dic₇₉⋊C₄	C₂²	semidirect
23. C₂²⋊Dic₇₉	C₂²	semidirect
24. (C₂×D₇₉)⋊C₄	C₂²	semidirect
25. C₄×Dic₇₉	C₂²	semidirect
74. C₇₉	#560	semidirect
84. #120	D₇₉	semidirect

H ≅ G/N

type

1. #14579157

C₂

semidirect

2. #14579154

C₂

semidirect

3. #14579143

C₂

semidirect

4. #14579142

C₂

semidirect

5. #14579134

C₂

semidirect

6. #14579133

C₂

semidirect

7. #14579132

C₂

semidirect

8. #14579130

C₂

semidirect

9. #120×C₇₉

C₂

semidirect

10. C₂×D₃₁₆

C₂²

semidirect

14. C₂×C₂²⋊D₇₉

C₂²

semidirect

15. C₂²×Dic₇₉

C₂²

semidirect

16. C₂×C₄×D₇₉

C₂²

semidirect

17. C₄⋊Dic₇₉

C₂²

semidirect

18. Dic₇₉⋊C₄

C₂²

semidirect

19. Dic₇₉⋊C₄

C₂²

semidirect

23. C₂²⋊Dic₇₉

C₂²

semidirect

24. (C₂×D₇₉)⋊C₄

C₂²

semidirect

25. C₄×Dic₇₉

C₂²

semidirect

74. C₇₉

#560

semidirect

84. #120

D₇₉

semidirect

Subgroups — #19099302

Conjugacy classes of subgroups

Complements