Group #19152223 — subgroups

Subgroups — #19152223

Order 5696.

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

normal subgroups only

This group has 242 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	#15667992	C₂	1	2	yes	maximal, complemented
2	#15667991	C₂	1	1	yes	maximal, characteristic, complemented
3	#15667990	C₂	1	2	yes	maximal, complemented
4	#15667990	C₂	1	2	yes	maximal, complemented
5	#15667989	C₂	1	2	yes	maximal, complemented
6	#15667988	C₂	1	2	yes	maximal, complemented
7	C₂×#13083551	C₂	1	1	yes	maximal, characteristic
8	C₂²⋊C₄×D₈₉	C₂	1	2	yes	maximal
9	C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₈₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₇₈	C₂²	1	2	yes	complemented
12	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
13	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₈₉	C₂²	1	2	yes	—
15	C₂×C₄×D₈₉	C₂²	1	2	yes	—
16	C₄⋊Dic₈₉	C₂²	1	2	yes	complemented
17	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic, complemented
18	#13083553	C₂²	1	2	yes	complemented
19	#13083553	C₂²	1	2	yes	complemented
20	#13083553	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
22	#13083551	C₂²	1	2	yes	—
23	#13083551	C₂²	1	2	yes	—
24	#13083551	C₂²	1	2	yes	—
25	#13083551	C₂²	1	2	yes	—
26	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
30	C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₂³×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×D₈₉	—	4	1	no	—
33	C₂²×D₈₉	—	4	1	no	—
34	C₂²×D₈₉	Q₈	1	4	yes	complemented
35	C₂²×D₈₉	C₂³	1	2	yes	—
36	Q₈×C₈₉	—	4	1	no	—
37	C₂×C₄×C₈₉	—	4	1	no	—
38	C₂×C₄×C₈₉	C₂³	1	2	yes	—
39	C₂×C₄×C₈₉	C₂³	1	2	yes	—
40	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₈₉	—	4	1	no	—
43	C₂×Dic₈₉	C₂³	1	2	yes	—
44	C₂×Dic₈₉	C₂³	1	2	yes	—
45	C₂×Dic₈₉	C₂³	1	2	yes	—
46	C₄×D₈₉	—	4	2	no	—
47	C₄×D₈₉	—	4	2	no	—
48	Dic₁₇₈	—	4	1	no	—
49	Dic₁₇₈	—	4	2	no	—
50	C₂²×C₈₉	—	2	1	no	—
51	C₂²×C₈₉	—	2	1	no	—
52	C₂²×C₈₉	C₂×Q₈	1	2	yes	—
53	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂×D₈₉	—	2	2	no	—
55	C₂×D₈₉	—	2	2	no	—
56	C₂×D₈₉	—	4	4	no	—
57	C₂×D₈₉	C₂×Q₈	1	4	yes	—
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	—	2	2	no	—
61	C₄×C₈₉	—	2	1	no	—
62	Dic₈₉	—	2	2	no	—
63	Dic₈₉	—	2	2	no	—
64	Dic₈₉	—	2	2	no	—
65	C₂×C₈₉	—	2	2	no	—
66	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
68	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
69	D₈₉	—	2	4	no	—
70	C₈₉	#543	1	1	yes	characteristic, complemented, Sylow
71	#543	—	89	1	no	maximal, Sylow
72	C₂²⋊Q₈	—	89	2	no	—
73	C₂²⋊Q₈	—	89	1	no	—
74	C₂²⋊Q₈	—	89	2	no	—
75	C₂²⋊Q₈	—	89	2	no	—
76	C₂²⋊Q₈	—	89	2	no	—
77	C₂²⋊Q₈	—	89	2	no	—
78	C₂²⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
79	C₂×C₂²⋊C₄	—	89	1	no	—
80	C₂×C₂²⋊C₄	—	89	2	no	—
81	C₂⁴	—	89	1	no	—
82	C₂×Q₈	—	89	1	no	—
83	C₂×Q₈	—	89	2	no	—
84	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
85	C₂²×C₄	—	89	1	no	—
86	C₂²×C₄	—	89	2	no	—
87	C₂²×C₄	—	89	2	no	—
88	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₄⋊C₄	—	89	2	no	—
90	C₄⋊C₄	—	89	2	no	—
91	C₄⋊C₄	—	89	2	no	—
92	C₄⋊C₄	—	89	1	no	—
93	C₄⋊C₄	—	89	2	no	—
94	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic, complemented
95	C₄⋊C₄	C₂×D₈₉	1	2	yes	complemented
96	C₂²⋊C₄	—	89	2	no	—
97	C₂²⋊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	C₂×D₈₉	1	2	yes	—
102	C₂³	—	89	4	no	—
103	C₂³	—	89	2	no	—
104	C₂³	—	356	1	no	—
105	C₂³	—	356	1	no	—
106	C₂³	C₂²×D₈₉	1	1	yes	characteristic
107	Q₈	—	4	1	no	—
108	Q₈	—	356	1	no	—
109	Q₈	—	356	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	89	2	no	—
112	C₂×C₄	—	89	2	no	—
113	C₂×C₄	—	89	2	no	—
114	C₂×C₄	—	356	2	no	—
115	C₂×C₄	—	356	2	no	—
116	C₂×C₄	—	356	1	no	—
117	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₈₉	1	2	yes	—
120	C₂×C₄	C₂²×D₈₉	1	2	yes	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	89	4	no	—
124	C₂²	—	356	4	no	—
125	C₂²	—	178	2	no	—
126	C₂²	—	178	2	no	—
127	C₂²	Q₈×D₈₉	1	2	yes	—
128	C₂²	C₂³×D₈₉	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	178	2	no	—
134	C₄	—	178	2	no	—
135	C₄	—	178	2	no	—
136	C₂	—	2	2	no	—
137	C₂	—	178	4	no	—
138	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
139	C₂	#15668057	1	1	yes	characteristic
140	C₂	#15668058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15667992

C₂

yes

maximal, complemented

#15667991

C₂

yes

maximal, characteristic, complemented

#15667990

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

#15667989

C₂

yes

maximal, complemented

#15667988

C₂

yes

maximal, complemented

C₂×#13083551

C₂

yes

maximal, characteristic

C₂²⋊C₄×D₈₉

C₂

yes

maximal

C₂²⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂²

yes

complemented

C₂×Dic₁₇₈

C₂²

yes

characteristic, complemented

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

complemented

C₄⋊Dic₈₉

C₂²

yes

characteristic, complemented

#13083553

C₂²

yes

complemented

#13083553

C₂²

yes

complemented

#13083553

C₂²

yes

complemented

C₂²⋊Dic₈₉

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₂×Q₈×C₈₉

C₂²

yes

characteristic, complemented

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₈₉

C₂²

yes

complemented

C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂³×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

Q₈

yes

complemented

C₂²×D₈₉

C₂³

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×Q₈

yes

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

C₈₉

#543

yes

characteristic, complemented, Sylow

#543

—

maximal, Sylow

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₈₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₈₉

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic, complemented

C₄⋊C₄

C₂×D₈₉

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

C₂×D₈₉

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

356

—

105

C₂³

—

356

—

106

C₂³

C₂²×D₈₉

yes

characteristic

107

Q₈

—

108

Q₈

—

356

—

109

Q₈

—

356

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

356

—

115

C₂×C₄

—

356

—

116

C₂×C₄

—

356

—

117

C₂×C₄

C₂²×D₈₉

yes

characteristic

118

C₂×C₄

C₂²×D₈₉

yes

characteristic

119

C₂×C₄

C₂²×D₈₉

yes

—

120

C₂×C₄

C₂²×D₈₉

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

356

—

125

C₂²

—

178

—

126

C₂²

—

178

—

127

C₂²

Q₈×D₈₉

yes

—

128

C₂²

C₂³×D₈₉

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

178

—

134

C₄

—

178

—

135

C₄

—

178

—

136

C₂

—

137

C₂

—

178

—

138

C₂

C₂×Q₈×D₈₉

yes

characteristic

139

C₂

#15668057

yes

characteristic

140

C₂

#15668058

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₂²⋊Q₈×C₈₉	C₂
hypercenter	C₂²⋊Q₈	D₈₉
hypocenter	C₈₉	#543
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

C₂²⋊Q₈×C₈₉

C₂

hypercenter

C₂²⋊Q₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15667992	C₂	semidirect
2. #15667991	C₂	semidirect
3. #15667990	C₂	semidirect
4. #15667990	C₂	semidirect
5. #15667989	C₂	semidirect
6. #15667988	C₂	semidirect
9. C₂²⋊Q₈×C₈₉	C₂	semidirect
11. C₂×Dic₁₇₈	C₂²	semidirect
12. C₂×Dic₁₇₈	C₂²	semidirect
16. C₄⋊Dic₈₉	C₂²	semidirect
17. C₄⋊Dic₈₉	C₂²	semidirect
18. #13083553	C₂²	semidirect
19. #13083553	C₂²	semidirect
20. #13083553	C₂²	semidirect
26. C₂×Q₈×C₈₉	C₂²	semidirect
28. C₄⋊C₄×C₈₉	C₂²	semidirect
29. C₄⋊C₄×C₈₉	C₂²	semidirect
34. C₂²×D₈₉	Q₈	semidirect
70. C₈₉	#543	semidirect
78. C₂²⋊Q₈	D₈₉	semidirect
84. C₂×Q₈	C₂×D₈₉	semidirect
94. C₄⋊C₄	C₂×D₈₉	semidirect
95. C₄⋊C₄	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #15667992

C₂

semidirect

2. #15667991

C₂

semidirect

3. #15667990

C₂

semidirect

4. #15667990

C₂

semidirect

5. #15667989

C₂

semidirect

6. #15667988

C₂

semidirect

9. C₂²⋊Q₈×C₈₉

C₂

semidirect

11. C₂×Dic₁₇₈

C₂²

semidirect

12. C₂×Dic₁₇₈

C₂²

semidirect

16. C₄⋊Dic₈₉

C₂²

semidirect

17. C₄⋊Dic₈₉

C₂²

semidirect

18. #13083553

C₂²

semidirect

19. #13083553

C₂²

semidirect

20. #13083553

C₂²

semidirect

26. C₂×Q₈×C₈₉

C₂²

semidirect

28. C₄⋊C₄×C₈₉

C₂²

semidirect

29. C₄⋊C₄×C₈₉

C₂²

semidirect

34. C₂²×D₈₉

Q₈

semidirect

70. C₈₉

#543

semidirect

78. C₂²⋊Q₈

D₈₉

semidirect

84. C₂×Q₈

C₂×D₈₉

semidirect

94. C₄⋊C₄

C₂×D₈₉

semidirect

95. C₄⋊C₄

C₂×D₈₉

semidirect

Subgroups — #19152223

Conjugacy classes of subgroups

Complements