Group #19152337 — subgroups

Subgroups — #19152337

Order 5696.

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	#15668008	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668005	C₂	1	2	yes	maximal, complemented
3	#15667994	C₂	1	2	yes	maximal, complemented
4	#15667993	C₂	1	1	yes	maximal, characteristic, complemented
5	#15667985	C₂	1	2	yes	maximal, complemented
6	#15667984	C₂	1	2	yes	maximal, complemented
7	#15667983	C₂	1	2	yes	maximal, complemented
8	#15667981	C₂	1	2	yes	maximal, complemented
9	#120×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	complemented
15	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₄×D₈₉	C₂²	1	2	yes	complemented
17	C₄⋊Dic₈₉	C₂²	1	2	yes	complemented
18	#13083553	C₂²	1	2	yes	complemented
19	#13083553	C₂²	1	2	yes	complemented
20	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
21	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
22	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
23	C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic, complemented
24	#13083551	C₂²	1	2	yes	complemented
25	C₄×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₂⁴×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×D₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂³×C₈₉	—	4	2	no	—
32	C₂³×C₈₉	—	4	1	no	—
33	C₂³×C₈₉	C₂³	1	2	yes	—
34	C₂³×C₈₉	C₂³	1	1	yes	characteristic
35	C₂³×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×D₈₉	C₂³	1	2	yes	—
37	D₄×C₈₉	—	4	2	no	—
38	D₄×C₈₉	—	4	1	no	—
39	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂³	1	2	yes	—
41	C₂²⋊D₈₉	—	4	2	no	—
42	C₂²⋊D₈₉	—	4	2	no	—
43	C₂²⋊D₈₉	—	4	2	no	—
44	C₂²⋊D₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	C₂×Dic₈₉	C₂³	1	2	yes	—
47	C₂×Dic₈₉	C₂³	1	2	yes	—
48	C₂×Dic₈₉	C₂³	1	2	yes	—
49	D₃₅₆	—	4	1	no	—
50	C₄×D₈₉	—	4	2	no	—
51	C₂²×C₈₉	—	2	1	no	—
52	C₂²×C₈₉	—	2	2	no	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	4	4	no	—
59	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂×D₈₉	—	2	2	no	—
61	C₂×D₈₉	—	2	2	no	—
62	C₂×D₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	2	1	no	—
65	Dic₈₉	—	2	2	no	—
66	Dic₈₉	—	2	2	no	—
67	Dic₈₉	—	2	2	no	—
68	C₂×C₈₉	—	4	1	no	—
69	C₂×C₈₉	—	4	2	no	—
70	C₂×C₈₉	—	4	1	no	—
71	C₂×C₈₉	H(5,𝔽₂)	1	2	yes	—
72	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
73	D₈₉	—	4	2	no	—
74	C₈₉	#560	1	1	yes	characteristic, complemented, Sylow
75	#560	—	89	1	no	maximal, Sylow
76	#127	—	89	1	no	—
77	#126	—	89	2	no	—
78	#123	—	89	1	no	—
79	#123	—	89	2	no	—
80	#121	—	89	2	no	—
81	#121	—	89	2	no	—
82	#121	—	89	2	no	—
83	#120	—	89	2	no	—
84	#120	D₈₉	1	1	yes	characteristic, complemented
85	C₂⁴	C₂×D₈₉	1	1	yes	characteristic
86	C₂×D₄	—	89	2	no	—
87	C₂×D₄	—	89	2	no	—
88	C₂×D₄	—	89	2	no	—
89	C₂×D₄	—	89	2	no	—
90	C₂×D₄	—	89	1	no	—
91	C₂×D₄	C₂×D₈₉	1	2	yes	—
92	C₂×D₄	C₂×D₈₉	1	1	yes	characteristic
93	C₂²×C₄	—	89	1	no	—
94	C₂²×C₄	—	89	2	no	—
95	C₄⋊C₄	—	89	2	no	—
96	C₄⋊C₄	—	89	2	no	—
97	C₄⋊C₄	—	89	2	no	—
98	C₂²⋊C₄	—	89	2	no	—
99	C₂²⋊C₄	—	89	2	no	—
100	C₂²⋊C₄	—	89	2	no	—
101	C₂²⋊C₄	—	89	1	no	—
102	C₂²⋊C₄	—	89	2	no	—
103	C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
104	C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
105	C₄²	—	89	1	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	89	2	no	—
109	C₂³	C₂²×D₈₉	1	2	yes	—
110	C₂³	C₂²×D₈₉	1	1	yes	characteristic
111	C₂³	C₂²×D₈₉	1	1	yes	characteristic
112	D₄	—	4	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	356	1	no	—
115	D₄	—	356	2	no	—
116	D₄	—	356	2	no	—
117	D₄	—	356	2	no	—
118	D₄	—	356	2	no	—
119	C₂×C₄	—	89	2	no	—
120	C₂×C₄	—	89	2	no	—
121	C₂×C₄	—	89	2	no	—
122	C₂×C₄	—	356	2	no	—
123	C₂×C₄	—	356	1	no	—
124	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₈₉	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	—	178	2	no	—
135	C₂²	—	178	2	no	—
136	C₂²	—	178	2	no	—
137	C₂²	C₂³×D₈₉	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	178	2	no	—
141	C₄	—	178	2	no	—
142	C₄	—	178	2	no	—
143	C₂	—	4	1	no	—
144	C₂	—	4	2	no	—
145	C₂	—	4	1	no	—
146	C₂	—	356	2	no	—
147	C₂	#15668057	1	2	yes	—
148	C₂	#15668057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668008

C₂

yes

maximal, characteristic, complemented

#15668005

C₂

yes

maximal, complemented

#15667994

C₂

yes

maximal, complemented

#15667993

C₂

yes

maximal, characteristic, complemented

#15667985

C₂

yes

maximal, complemented

#15667984

C₂

yes

maximal, complemented

#15667983

C₂

yes

maximal, complemented

#15667981

C₂

yes

maximal, complemented

#120×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₃₅₆

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

complemented

C₂²×Dic₈₉

C₂²

yes

characteristic, complemented

C₂×C₄×D₈₉

C₂²

yes

complemented

C₄⋊Dic₈₉

C₂²

yes

complemented

#13083553

C₂²

yes

complemented

#13083553

C₂²

yes

complemented

C₂²⋊Dic₈₉

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

characteristic, complemented

#13083551

C₂²

yes

complemented

C₄×Dic₈₉

C₂²

yes

characteristic, complemented

C₂⁴×C₈₉

C₂²

yes

characteristic

C₂×D₄×C₈₉

C₂²

yes

—

C₂×D₄×C₈₉

C₂²

yes

characteristic

C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂²⋊C₄×C₈₉

C₂²

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

H(5,𝔽₂)

yes

—

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

D₈₉

—

C₈₉

#560

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂⁴

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₈₉

yes

—

C₂×D₄

C₂×D₈₉

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×D₈₉

yes

—

104

C₂²⋊C₄

C₂×D₈₉

yes

characteristic

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₈₉

yes

—

110

C₂³

C₂²×D₈₉

yes

characteristic

111

C₂³

C₂²×D₈₉

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

356

—

115

D₄

—

356

—

116

D₄

—

356

—

117

D₄

—

356

—

118

D₄

—

356

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

C₂²×D₈₉

yes

characteristic

125

C₂×C₄

C₂²×D₈₉

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

178

—

135

C₂²

—

178

—

136

C₂²

—

178

—

137

C₂²

C₂³×D₈₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

178

—

141

C₄

—

178

—

142

C₄

—

178

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

356

—

147

C₂

#15668057

yes

—

148

C₂

#15668057

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#120×C₈₉	C₂
hypercenter	#120	D₈₉
hypocenter	C₈₉	#560
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#120×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. #15668008	C₂	semidirect
2. #15668005	C₂	semidirect
3. #15667994	C₂	semidirect
4. #15667993	C₂	semidirect
5. #15667985	C₂	semidirect
6. #15667984	C₂	semidirect
7. #15667983	C₂	semidirect
8. #15667981	C₂	semidirect
9. #120×C₈₉	C₂	semidirect
10. C₂×D₃₅₆	C₂²	semidirect
14. C₂×C₂²⋊D₈₉	C₂²	semidirect
15. C₂²×Dic₈₉	C₂²	semidirect
16. C₂×C₄×D₈₉	C₂²	semidirect
17. C₄⋊Dic₈₉	C₂²	semidirect
18. #13083553	C₂²	semidirect
19. #13083553	C₂²	semidirect
23. C₂²⋊Dic₈₉	C₂²	semidirect
24. #13083551	C₂²	semidirect
25. C₄×Dic₈₉	C₂²	semidirect
74. C₈₉	#560	semidirect
84. #120	D₈₉	semidirect

H ≅ G/N

type

1. #15668008

C₂

semidirect

2. #15668005

C₂

semidirect

3. #15667994

C₂

semidirect

4. #15667993

C₂

semidirect

5. #15667985

C₂

semidirect

6. #15667984

C₂

semidirect

7. #15667983

C₂

semidirect

8. #15667981

C₂

semidirect

9. #120×C₈₉

C₂

semidirect

10. C₂×D₃₅₆

C₂²

semidirect

14. C₂×C₂²⋊D₈₉

C₂²

semidirect

15. C₂²×Dic₈₉

C₂²

semidirect

16. C₂×C₄×D₈₉

C₂²

semidirect

17. C₄⋊Dic₈₉

C₂²

semidirect

18. #13083553

C₂²

semidirect

19. #13083553

C₂²

semidirect

23. C₂²⋊Dic₈₉

C₂²

semidirect

24. #13083551

C₂²

semidirect

25. C₄×Dic₈₉

C₂²

semidirect

74. C₈₉

#560

semidirect

84. #120

D₈₉

semidirect

Subgroups — #19152337

Conjugacy classes of subgroups

Complements