Group #19152175 — subgroups

Subgroups — #19152175

Order 5696.

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

normal subgroups only

This group has 294 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	2	yes	maximal, complemented
2	#15667990	C₂	1	2	yes	maximal, complemented
3	#15667989	C₂	1	2	yes	maximal, complemented
4	#15667986	C₂	1	1	yes	maximal, characteristic, complemented
5	#15667985	C₂	1	2	yes	maximal, complemented
6	#15667984	C₂	1	2	yes	maximal, complemented
7	#15667984	C₂	1	2	yes	maximal, complemented
8	C₂×#13083553	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#13083573	—	2	8	no	—
11	C₂×Dic₁₇₈	C₂²	1	2	yes	—
12	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
16	C₂²×Dic₈₉	C₂²	1	2	yes	—
17	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₈₉	C₂²	1	2	yes	—
20	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
21	#13083553	C₂²	1	2	yes	—
22	#13083553	C₂²	1	2	yes	—
23	#13083553	C₂²	1	2	yes	—
24	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
25	#13083551	C₂²	1	2	yes	—
26	#13083551	C₂²	1	2	yes	—
27	C₂×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×D₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
32	C₂³×C₈₉	C₂³	1	2	yes	—
33	C₂³×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²×D₈₉	C₂³	1	2	yes	—
35	D₄×C₈₉	—	2	4	no	—
36	D₄×C₈₉	—	4	1	no	—
37	C₂×C₄×C₈₉	—	4	1	no	—
38	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂³	1	2	yes	—
40	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²⋊D₈₉	—	2	4	no	—
42	C₂²⋊D₈₉	—	2	4	no	—
43	C₂²⋊D₈₉	—	4	2	no	—
44	C₂×Dic₈₉	—	2	4	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	C₂×Dic₈₉	D₄	1	4	yes	complemented
47	C₂×Dic₈₉	C₂³	1	2	yes	—
48	C₂×Dic₈₉	C₂³	1	2	yes	—
49	C₂×Dic₈₉	C₂³	1	2	yes	—
50	D₃₅₆	—	4	1	no	—
51	C₄×D₈₉	—	2	4	no	—
52	Dic₁₇₈	—	2	4	no	—
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	—	2	1	no	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	2	2	no	—
58	C₂²×C₈₉	C₂×D₄	1	2	yes	—
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	C₄×C₈₉	—	2	1	no	—
66	Dic₈₉	—	2	2	no	—
67	Dic₈₉	—	2	2	no	—
68	Dic₈₉	C₂×D₄	1	4	yes	—
69	C₂×C₈₉	—	2	2	no	—
70	C₂×C₈₉	—	4	2	no	—
71	C₂×C₈₉	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
74	D₈₉	—	4	2	no	—
75	C₈₉	#537	1	1	yes	characteristic, complemented, Sylow
76	#537	—	89	1	no	maximal, Sylow
77	C₂×#41	—	89	2	no	—
78	C₂²⋊Q₈	—	89	2	no	—
79	C₂²⋊Q₈	—	89	2	no	—
80	#121	—	89	2	no	—
81	#121	—	89	2	no	—
82	#121	—	89	1	no	—
83	#121	—	89	2	no	—
84	#121	D₈₉	1	1	yes	characteristic, complemented
85	C₂×C₄⋊C₄	—	89	1	no	—
86	#41	—	178	8	no	—
87	C₂×Q₈	—	89	2	no	—
88	C₂×D₄	—	89	2	no	—
89	C₂×D₄	—	89	2	no	—
90	C₂×D₄	—	89	2	no	—
91	C₂×D₄	—	89	1	no	—
92	C₂×D₄	C₂×D₈₉	1	2	yes	—
93	C₂×D₄	C₂×D₈₉	1	1	yes	characteristic
94	C₂²×C₄	—	89	1	no	—
95	C₂²×C₄	—	89	1	no	—
96	C₂²×C₄	—	89	2	no	—
97	C₂²×C₄	—	89	2	no	—
98	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₄	—	89	2	no	—
100	C₄⋊C₄	—	89	2	no	—
101	C₄⋊C₄	—	89	2	no	—
102	C₄⋊C₄	—	89	1	no	—
103	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
104	C₂²⋊C₄	—	89	2	no	—
105	C₂²⋊C₄	—	89	2	no	—
106	C₂²⋊C₄	—	89	2	no	—
107	C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
108	C₂³	—	89	2	no	—
109	C₂³	C₂²×D₈₉	1	2	yes	—
110	C₂³	C₂²×D₈₉	1	1	yes	characteristic
111	Q₈	—	178	4	no	—
112	D₄	—	2	4	no	—
113	D₄	—	4	1	no	—
114	D₄	—	356	1	no	—
115	D₄	—	356	2	no	—
116	D₄	—	178	4	no	—
117	D₄	—	178	4	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	89	4	no	—
120	C₂×C₄	—	89	2	no	—
121	C₂×C₄	—	89	2	no	—
122	C₂×C₄	—	89	2	no	—
123	C₂×C₄	—	356	1	no	—
124	C₂×C₄	—	178	4	no	—
125	C₂×C₄	—	178	4	no	—
126	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₈₉	1	2	yes	—
128	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	178	2	no	—
135	C₂²	—	178	2	no	—
136	C₂²	—	178	2	no	—
137	C₂²	D₄×D₈₉	1	2	yes	—
138	C₂²	C₂³×D₈₉	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	89	4	no	—
143	C₄	—	178	2	no	—
144	C₄	—	178	2	no	—
145	C₂	—	2	2	no	—
146	C₂	—	4	2	no	—
147	C₂	—	356	2	no	—
148	C₂	C₂×D₄×D₈₉	1	1	yes	characteristic
149	C₂	#15668057	1	1	yes	characteristic
150	C₂	#15668059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13083573

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

#15667989

C₂

yes

maximal, complemented

#15667986

C₂

yes

maximal, characteristic, complemented

#15667985

C₂

yes

maximal, complemented

#15667984

C₂

yes

maximal, complemented

#15667984

C₂

yes

maximal, complemented

C₂×#13083553

C₂

yes

maximal, characteristic, complemented

#121×C₈₉

C₂

yes

maximal, characteristic, complemented

#13083573

—

C₂×Dic₁₇₈

C₂²

yes

—

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

C₂²⋊Dic₈₉

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₂×D₄×C₈₉

C₂²

yes

—

C₂×D₄×C₈₉

C₂²

yes

characteristic

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂³×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₄

yes

complemented

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₄

yes

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×D₄

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

characteristic

D₈₉

—

C₈₉

#537

yes

characteristic, complemented, Sylow

#537

—

maximal, Sylow

C₂×#41

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×C₄⋊C₄

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₈₉

yes

—

C₂×D₄

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₈₉

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

C₂×D₈₉

yes

—

108

C₂³

—

109

C₂³

C₂²×D₈₉

yes

—

110

C₂³

C₂²×D₈₉

yes

characteristic

111

Q₈

—

178

—

112

D₄

—

113

D₄

—

114

D₄

—

356

—

115

D₄

—

356

—

116

D₄

—

178

—

117

D₄

—

178

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

178

—

125

C₂×C₄

—

178

—

126

C₂×C₄

C₂²×D₈₉

yes

characteristic

127

C₂×C₄

C₂²×D₈₉

yes

—

128

C₂×C₄

C₂²×D₈₉

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

178

—

135

C₂²

—

178

—

136

C₂²

—

178

—

137

C₂²

D₄×D₈₉

yes

—

138

C₂²

C₂³×D₈₉

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

178

—

144

C₄

—

178

—

145

C₂

—

146

C₂

—

147

C₂

—

356

—

148

C₂

C₂×D₄×D₈₉

yes

characteristic

149

C₂

#15668057

yes

characteristic

150

C₂

#15668059

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#121×C₈₉	C₂
hypercenter	#121	D₈₉
hypocenter	C₈₉	#537
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#121×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂×#13083573	C₂	semidirect
2. #15667990	C₂	semidirect
3. #15667989	C₂	semidirect
4. #15667986	C₂	semidirect
5. #15667985	C₂	semidirect
6. #15667984	C₂	semidirect
7. #15667984	C₂	semidirect
8. C₂×#13083553	C₂	semidirect
9. #121×C₈₉	C₂	semidirect
46. C₂×Dic₈₉	D₄	semidirect
75. C₈₉	#537	semidirect
84. #121	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#13083573

C₂

semidirect

2. #15667990

C₂

semidirect

3. #15667989

C₂

semidirect

4. #15667986

C₂

semidirect

5. #15667985

C₂

semidirect

6. #15667984

C₂

semidirect

7. #15667984

C₂

semidirect

8. C₂×#13083553

C₂

semidirect

9. #121×C₈₉

C₂

semidirect

46. C₂×Dic₈₉

D₄

semidirect

75. C₈₉

#537

semidirect

84. #121

D₈₉

semidirect

Subgroups — #19152175

Conjugacy classes of subgroups

Complements