Group #19152219 — subgroups

Subgroups — #19152219

Order 5696.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#15668012	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668007	C₂	1	2	yes	maximal, complemented
3	#15668000	C₂	1	1	yes	maximal, characteristic, complemented
4	#15667998	C₂	1	2	yes	maximal, complemented
5	#15667994	C₂	1	2	yes	maximal, complemented
6	#15667991	C₂	1	1	yes	maximal, characteristic, complemented
7	#15667990	C₂	1	2	yes	maximal, complemented
8	#15667986	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₇₈	C₂²	1	2	yes	—
14	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
16	C₂×C₄×D₈₉	C₂²	1	2	yes	—
17	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₈₉	C₂²	1	2	yes	—
20	#13083553	C₂²	1	2	yes	—
21	#13083553	C₂²	1	2	yes	—
22	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
23	#13083551	C₂²	1	2	yes	—
24	#13083551	C₂²	1	2	yes	—
25	#13083551	C₂²	1	2	yes	—
26	#13083551	C₂²	1	2	yes	—
27	C₄×Dic₈₉	C₂²	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	C₂³×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×D₈₉	C₂³	1	2	yes	—
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²⋊D₈₉	—	4	2	no	—
45	C₂×Dic₈₉	C₂³	1	2	yes	—
46	C₂×Dic₈₉	C₂³	1	2	yes	—
47	C₂×Dic₈₉	C₂³	1	2	yes	—
48	D₃₅₆	—	2	2	no	—
49	C₄×D₈₉	—	4	2	no	—
50	Dic₁₇₈	—	2	2	no	—
51	Dic₁₇₈	—	4	2	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	—	2	1	no	—
55	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂×D₈₉	—	2	2	no	—
57	C₂×D₈₉	—	2	2	no	—
58	C₂×D₈₉	—	2	2	no	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	—	2	1	no	—
64	C₄×C₈₉	#41	1	2	yes	—
65	Dic₈₉	—	2	2	no	—
66	Dic₈₉	—	2	2	no	—
67	Dic₈₉	—	2	2	no	—
68	C₂×C₈₉	—	4	1	no	—
69	C₂×C₈₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
72	D₈₉	—	4	2	no	—
73	C₈₉	#542	1	1	yes	characteristic, complemented, Sylow
74	#542	—	89	1	no	maximal, Sylow
75	C₄⋊Q₈	—	89	1	no	—
76	#126	—	89	2	no	—
77	#124	—	89	2	no	—
78	#124	—	89	1	no	—
79	#123	—	89	2	no	—
80	C₂²⋊Q₈	—	89	1	no	—
81	C₂²⋊Q₈	—	89	2	no	—
82	#121	—	89	1	no	—
83	C₄×Q₈	—	89	1	no	—
84	C₄×D₄	—	89	1	no	—
85	#117	D₈₉	1	1	yes	characteristic, complemented
86	C₂×Q₈	—	89	2	no	—
87	C₂×Q₈	—	89	1	no	—
88	C₂×D₄	—	89	2	no	—
89	C₂×D₄	—	89	1	no	—
90	C₂²×C₄	—	89	2	no	—
91	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
92	C₄⋊C₄	—	89	1	no	—
93	C₄⋊C₄	—	89	1	no	—
94	C₄⋊C₄	—	89	2	no	—
95	C₄⋊C₄	—	89	2	no	—
96	C₄⋊C₄	—	89	2	no	—
97	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
99	C₂²⋊C₄	—	89	2	no	—
100	C₂²⋊C₄	—	89	2	no	—
101	C₂²⋊C₄	—	89	2	no	—
102	C₂²⋊C₄	—	89	2	no	—
103	C₂²⋊C₄	—	89	2	no	—
104	C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
105	C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
106	C₄²	—	89	2	no	—
107	C₄²	C₂×D₈₉	1	1	yes	characteristic
108	C₄²	C₂×D₈₉	1	1	yes	characteristic
109	C₂³	—	89	2	no	—
110	C₂³	C₂²×D₈₉	1	1	yes	characteristic
111	Q₈	—	356	2	no	—
112	Q₈	—	178	2	no	—
113	D₄	—	356	2	no	—
114	D₄	—	178	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	89	2	no	—
117	C₂×C₄	—	89	2	no	—
118	C₂×C₄	—	89	2	no	—
119	C₂×C₄	—	356	2	no	—
120	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	178	2	no	—
130	C₂²	—	178	2	no	—
131	C₂²	—	178	2	no	—
132	C₂²	C₂³×D₈₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	178	2	no	—
139	C₄	—	178	2	no	—
140	C₄	—	178	2	no	—
141	C₄	#13083574	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	356	2	no	—
144	C₂	C₂×#13083574	1	1	yes	characteristic
145	C₂	#15668058	1	1	yes	characteristic
146	C₂	#15668059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668012

C₂

yes

maximal, characteristic, complemented

#15668007

C₂

yes

maximal, complemented

#15668000

C₂

yes

maximal, characteristic, complemented

#15667998

C₂

yes

maximal, complemented

#15667994

C₂

yes

maximal, complemented

#15667991

C₂

yes

maximal, characteristic, complemented

#15667990

C₂

yes

maximal, complemented

#15667986

C₂

yes

maximal, characteristic

C₄×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

C₄×D₃₅₆

C₂

yes

maximal, characteristic, complemented

#117×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂²

yes

—

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂²⋊C₄×C₈₉

C₂²

yes

characteristic

C₂²⋊C₄×C₈₉

C₂²

yes

characteristic

C₄²×C₈₉

C₂²

yes

characteristic

C₄²×C₈₉

C₂²

yes

characteristic

C₂³×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#41

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

characteristic

D₈₉

—

C₈₉

#542

yes

characteristic, complemented, Sylow

#542

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

D₈₉

yes

characteristic, complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₈₉

yes

characteristic

105

C₂²⋊C₄

C₂×D₈₉

yes

characteristic

106

C₄²

—

107

C₄²

C₂×D₈₉

yes

characteristic

108

C₄²

C₂×D₈₉

yes

characteristic

109

C₂³

—

110

C₂³

C₂²×D₈₉

yes

characteristic

111

Q₈

—

356

—

112

Q₈

—

178

—

113

D₄

—

356

—

114

D₄

—

178

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

356

—

120

C₂×C₄

C₂²×D₈₉

yes

characteristic

121

C₂×C₄

C₂²×D₈₉

yes

characteristic

122

C₂×C₄

C₂²×D₈₉

yes

characteristic

123

C₂×C₄

C₂²×D₈₉

yes

characteristic

124

C₂×C₄

C₂²×D₈₉

yes

characteristic

125

C₂×C₄

C₂²×D₈₉

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

178

—

130

C₂²

—

178

—

131

C₂²

—

178

—

132

C₂²

C₂³×D₈₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

178

—

139

C₄

—

178

—

140

C₄

—

178

—

141

C₄

#13083574

yes

—

142

C₂

—

143

C₂

—

356

—

144

C₂

C₂×#13083574

yes

characteristic

145

C₂

#15668058

yes

characteristic

146

C₂

#15668059

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#117×C₈₉	C₂
hypercenter	#117	D₈₉
hypocenter	C₈₉	#542
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#117×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #15668012	C₂	semidirect
2. #15668007	C₂	semidirect
3. #15668000	C₂	semidirect
4. #15667998	C₂	semidirect
5. #15667994	C₂	semidirect
6. #15667991	C₂	semidirect
7. #15667990	C₂	semidirect
9. C₄×Dic₁₇₈	C₂	semidirect
10. C₄×D₃₅₆	C₂	semidirect
11. #117×C₈₉	C₂	semidirect
73. C₈₉	#542	semidirect
85. #117	D₈₉	semidirect

H ≅ G/N

type

1. #15668012

C₂

semidirect

2. #15668007

C₂

semidirect

3. #15668000

C₂

semidirect

4. #15667998

C₂

semidirect

5. #15667994

C₂

semidirect

6. #15667991

C₂

semidirect

7. #15667990

C₂

semidirect

9. C₄×Dic₁₇₈

C₂

semidirect

10. C₄×D₃₅₆

C₂

semidirect

11. #117×C₈₉

C₂

semidirect

73. C₈₉

#542

semidirect

85. #117

D₈₉

semidirect

Subgroups — #19152219

Conjugacy classes of subgroups

Complements