Group #19152161 — subgroups

Subgroups — #19152161

Order 5696.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#13083573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#15668008	C₂	1	1	yes	maximal, characteristic, complemented
4	#15668000	C₂	1	1	yes	maximal, characteristic, complemented
5	#15667999	C₂	1	1	yes	maximal, characteristic, complemented
6	#15667984	C₂	1	4	yes	maximal, complemented
7	#15667980	C₂	1	4	yes	maximal, complemented
8	#15667969	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₈₉	C₂²	1	2	yes	—
11	#13083573	—	2	4	no	—
12	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₈₉	C₂²	1	4	yes	complemented
17	D₄×D₈₉	—	2	4	no	—
18	C₂²×Dic₈₉	C₂²	1	2	yes	complemented
19	C₂×C₄×D₈₉	C₂²	1	1	yes	characteristic
20	#13083553	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₈₉	C₂²	1	4	yes	complemented
22	#13083551	C₂²	1	2	yes	—
23	#13083551	C₂²	1	4	yes	complemented
24	C₄×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×C₈₉	C₂²	1	4	yes	complemented
28	C₄²×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₈₉	C₂³	1	2	yes	—
30	C₂³×C₈₉	C₂³	1	2	yes	—
31	C₂²×D₈₉	—	2	4	no	—
32	C₂²×D₈₉	—	4	4	no	—
33	C₂²×D₈₉	C₂³	1	2	yes	—
34	C₂²×D₈₉	C₂³	1	1	yes	characteristic
35	D₄×C₈₉	—	2	2	no	—
36	D₄×C₈₉	—	2	2	no	—
37	D₄×C₈₉	—	4	4	no	—
38	C₂×C₄×C₈₉	C₂³	1	2	yes	—
39	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²⋊D₈₉	—	2	4	no	—
41	C₂²⋊D₈₉	—	2	4	no	—
42	C₂²⋊D₈₉	—	4	4	no	—
43	C₂×Dic₈₉	—	2	4	no	—
44	C₂×Dic₈₉	C₂³	1	2	yes	—
45	C₂×Dic₈₉	C₂³	1	2	yes	—
46	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
47	D₃₅₆	—	2	2	no	—
48	C₄×D₈₉	D₄	1	4	yes	complemented
49	Dic₁₇₈	—	2	2	no	—
50	C₂²×C₈₉	—	2	2	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	—	2	4	no	—
53	C₂²×C₈₉	—	2	4	no	—
54	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
55	C₂×D₈₉	—	2	2	no	—
56	C₂×D₈₉	—	2	2	no	—
57	C₂×D₈₉	—	2	4	no	—
58	C₂×D₈₉	—	4	8	no	—
59	C₂×D₈₉	C₂×D₄	1	2	yes	—
60	C₄×C₈₉	—	2	2	no	—
61	C₄×C₈₉	C₂×D₄	1	2	yes	—
62	Dic₈₉	—	2	2	no	—
63	Dic₈₉	—	2	2	no	—
64	Dic₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₈₉	—	4	2	no	—
66	C₂×C₈₉	—	4	2	no	—
67	C₂×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₈₉	H(5,𝔽₂)	1	2	yes	—
69	D₈₉	—	2	2	no	—
70	D₈₉	—	4	2	no	—
71	C₈₉	#535	1	1	yes	characteristic, complemented, Sylow
72	#535	—	89	1	no	maximal, Sylow
73	C₂×#41	—	89	1	no	—
74	C₂²×D₄	—	89	1	no	—
75	#127	—	89	1	no	—
76	#127	D₈₉	1	1	yes	characteristic, complemented
77	#124	—	89	1	no	—
78	#124	—	89	1	no	—
79	#121	—	89	4	no	—
80	#120	—	89	4	no	—
81	#117	—	89	1	no	—
82	C₂⁴	—	89	2	no	—
83	#41	—	178	4	no	—
84	C₂×Q₈	—	89	1	no	—
85	C₂×D₄	—	89	4	no	—
86	C₂×D₄	—	89	2	no	—
87	C₂×D₄	—	89	2	no	—
88	C₂×D₄	—	89	1	no	—
89	C₂×D₄	—	178	4	no	—
90	C₂×D₄	C₂×D₈₉	1	1	yes	characteristic
91	C₂×D₄	C₂×D₈₉	1	1	yes	characteristic, complemented
92	C₂×D₄	C₂×D₈₉	1	4	yes	complemented
93	C₂²×C₄	—	89	1	no	—
94	C₂²×C₄	—	89	2	no	—
95	C₄⋊C₄	—	89	2	no	—
96	C₂²⋊C₄	—	89	4	no	—
97	C₂²⋊C₄	—	89	2	no	—
98	C₂²⋊C₄	—	89	4	no	—
99	C₄²	—	89	1	no	—
100	C₄²	C₂×D₈₉	1	1	yes	characteristic, complemented
101	C₂³	—	89	1	no	—
102	C₂³	—	89	2	no	—
103	C₂³	—	356	4	no	—
104	C₂³	—	178	4	no	—
105	C₂³	C₂²×D₈₉	1	2	yes	—
106	C₂³	C₂²×D₈₉	1	2	yes	—
107	Q₈	—	178	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	4	4	no	—
111	D₄	—	356	4	no	—
112	D₄	—	178	4	no	—
113	D₄	—	178	4	no	—
114	D₄	—	178	2	no	—
115	C₂×C₄	—	89	1	no	—
116	C₂×C₄	—	89	4	no	—
117	C₂×C₄	—	89	2	no	—
118	C₂×C₄	—	89	2	no	—
119	C₂×C₄	—	178	4	no	—
120	C₂×C₄	C₂²×D₈₉	1	2	yes	—
121	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	89	2	no	—
127	C₂²	—	356	8	no	—
128	C₂²	—	178	2	no	—
129	C₂²	—	178	4	no	—
130	C₂²	—	178	2	no	—
131	C₂²	C₂³×D₈₉	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	89	2	no	—
134	C₄	—	178	2	no	—
135	C₄	—	178	2	no	—
136	C₄	D₄×D₈₉	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	4	2	no	—
139	C₂	—	356	2	no	—
140	C₂	—	178	2	no	—
141	C₂	C₂×D₄×D₈₉	1	1	yes	characteristic
142	C₂	#15668057	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13083573

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₈₉

C₂

yes

maximal, characteristic, complemented

#15668008

C₂

yes

maximal, characteristic, complemented

#15668000

C₂

yes

maximal, characteristic, complemented

#15667999

C₂

yes

maximal, characteristic, complemented

#15667984

C₂

yes

maximal, complemented

#15667980

C₂

yes

maximal, complemented

#15667969

C₂

yes

maximal, characteristic, complemented

#127×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₈₉

C₂²

yes

—

#13083573

—

C₂×Dic₁₇₈

C₂²

yes

characteristic, complemented

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

complemented

D₄×D₈₉

—

C₂²×Dic₈₉

C₂²

yes

complemented

C₂×C₄×D₈₉

C₂²

yes

characteristic

#13083553

C₂²

yes

complemented

C₂²⋊Dic₈₉

C₂²

yes

complemented

#13083551

C₂²

yes

—

#13083551

C₂²

yes

complemented

C₄×Dic₈₉

C₂²

yes

characteristic, complemented

C₂×D₄×C₈₉

C₂²

yes

characteristic

C₂×D₄×C₈₉

C₂²

yes

characteristic, complemented

C₂×D₄×C₈₉

C₂²

yes

complemented

C₄²×C₈₉

C₂²

yes

characteristic, complemented

C₂³×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂³

yes

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂³

yes

—

C₂²×D₈₉

C₂³

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

characteristic

D₃₅₆

—

C₄×D₈₉

D₄

yes

complemented

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×D₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₄

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×D₄

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

—

D₈₉

—

D₈₉

—

C₈₉

#535

yes

characteristic, complemented, Sylow

#535

—

maximal, Sylow

C₂×#41

—

C₂²×D₄

—

#127

—

#127

D₈₉

yes

characteristic, complemented

—

—

—

—

—

C₂⁴

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

C₂×D₈₉

yes

characteristic

C₂×D₄

C₂×D₈₉

yes

characteristic, complemented

C₂×D₄

C₂×D₈₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂×D₈₉

yes

characteristic, complemented

101

C₂³

—

102

C₂³

—

103

C₂³

—

356

—

104

C₂³

—

178

—

105

C₂³

C₂²×D₈₉

yes

—

106

C₂³

C₂²×D₈₉

yes

—

107

Q₈

—

178

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

356

—

112

D₄

—

178

—

113

D₄

—

178

—

114

D₄

—

178

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

178

—

120

C₂×C₄

C₂²×D₈₉

yes

—

121

C₂×C₄

C₂²×D₈₉

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

356

—

128

C₂²

—

178

—

129

C₂²

—

178

—

130

C₂²

—

178

—

131

C₂²

C₂³×D₈₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

178

—

135

C₄

—

178

—

136

C₄

D₄×D₈₉

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

356

—

140

C₂

—

178

—

141

C₂

C₂×D₄×D₈₉

yes

characteristic

142

C₂

#15668057

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#127×C₈₉	C₂
hypercenter	#127	D₈₉
hypocenter	C₈₉	#535
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#127×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂×#13083573	C₂	semidirect
2. C₂×D₄×D₈₉	C₂	semidirect
3. #15668008	C₂	semidirect
4. #15668000	C₂	semidirect
5. #15667999	C₂	semidirect
6. #15667984	C₂	semidirect
7. #15667980	C₂	semidirect
8. #15667969	C₂	semidirect
9. #127×C₈₉	C₂	semidirect
12. C₂×Dic₁₇₈	C₂²	semidirect
16. C₂×C₂²⋊D₈₉	C₂²	semidirect
18. C₂²×Dic₈₉	C₂²	semidirect
20. #13083553	C₂²	semidirect
21. C₂²⋊Dic₈₉	C₂²	semidirect
23. #13083551	C₂²	semidirect
24. C₄×Dic₈₉	C₂²	semidirect
26. C₂×D₄×C₈₉	C₂²	semidirect
27. C₂×D₄×C₈₉	C₂²	semidirect
28. C₄²×C₈₉	C₂²	semidirect
48. C₄×D₈₉	D₄	semidirect
71. C₈₉	#535	semidirect
76. #127	D₈₉	semidirect
91. C₂×D₄	C₂×D₈₉	semidirect
92. C₂×D₄	C₂×D₈₉	semidirect
100. C₄²	C₂×D₈₉	semidirect

H ≅ G/N

type

1. C₂×#13083573

C₂

semidirect

2. C₂×D₄×D₈₉

C₂

semidirect

3. #15668008

C₂

semidirect

4. #15668000

C₂

semidirect

5. #15667999

C₂

semidirect

6. #15667984

C₂

semidirect

7. #15667980

C₂

semidirect

8. #15667969

C₂

semidirect

9. #127×C₈₉

C₂

semidirect

12. C₂×Dic₁₇₈

C₂²

semidirect

16. C₂×C₂²⋊D₈₉

C₂²

semidirect

18. C₂²×Dic₈₉

C₂²

semidirect

20. #13083553

C₂²

semidirect

21. C₂²⋊Dic₈₉

C₂²

semidirect

23. #13083551

C₂²

semidirect

24. C₄×Dic₈₉

C₂²

semidirect

26. C₂×D₄×C₈₉

C₂²

semidirect

27. C₂×D₄×C₈₉

C₂²

semidirect

28. C₄²×C₈₉

C₂²

semidirect

48. C₄×D₈₉

D₄

semidirect

71. C₈₉

#535

semidirect

76. #127

D₈₉

semidirect

91. C₂×D₄

C₂×D₈₉

semidirect

92. C₂×D₄

C₂×D₈₉

semidirect

100. C₄²

C₂×D₈₉

semidirect

Subgroups — #19152161

Conjugacy classes of subgroups

Complements