Group #19152304 — subgroups

Subgroups — #19152304

Order 5696.

228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 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	#15668011	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668003	C₂	1	1	yes	maximal, characteristic, complemented
3	#15667992	C₂	1	2	yes	maximal, complemented
4	#15667990	C₂	1	2	yes	maximal, complemented
5	#15667990	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
8	#15667974	C₂	1	1	yes	maximal, characteristic
9	#15667974	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₈₉	C₂	1	2	yes	maximal, complemented
11	#125×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×C₄×D₈₉	C₂²	1	2	yes	—
16	C₂×C₄×D₈₉	C₂²	1	2	yes	—
17	C₂×C₄×D₈₉	C₂²	1	2	yes	—
18	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₈₉	C₂²	1	2	yes	—
21	#13083553	C₂²	1	1	yes	characteristic
22	#13083553	C₂²	1	1	yes	characteristic
23	#13083553	C₂²	1	2	yes	—
24	#13083553	C₂²	1	2	yes	—
25	#13083551	C₂²	1	2	yes	—
26	#13083551	C₂²	1	2	yes	—
27	#13083551	C₂²	1	2	yes	—
28	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×D₈₉	C₂³	1	2	yes	—
36	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
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×Dic₈₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₈₉	C₂³	1	2	yes	—
45	C₂×Dic₈₉	C₂³	1	1	yes	characteristic
46	D₃₅₆	Q₈	1	4	yes	complemented
47	C₄×D₈₉	—	2	4	no	—
48	C₄×D₈₉	—	4	2	no	—
49	C₄×D₈₉	—	4	2	no	—
50	Dic₁₇₈	—	2	2	no	—
51	Dic₁₇₈	—	4	2	no	—
52	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×D₈₉	—	2	2	no	—
54	C₂×D₈₉	—	2	2	no	—
55	C₂×D₈₉	C₂×Q₈	1	4	yes	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	C₂×Q₈	1	2	yes	—
61	Dic₈₉	—	2	1	no	—
62	Dic₈₉	—	2	1	no	—
63	Dic₈₉	—	2	2	no	—
64	Dic₈₉	—	2	1	no	—
65	Dic₈₉	#41	1	2	yes	—
66	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₈₉	C₂×#41	1	1	yes	characteristic
68	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
69	D₈₉	—	2	4	no	—
70	C₈₉	#554	1	1	yes	characteristic, complemented, Sylow
71	#554	—	89	1	no	maximal, Sylow
72	C₄⋊Q₈	—	89	1	no	—
73	#125	—	89	1	no	—
74	#125	D₈₉	1	1	yes	characteristic, complemented
75	C₂²⋊Q₈	—	89	2	no	—
76	C₂²⋊Q₈	—	89	2	no	—
77	C₂²⋊Q₈	—	89	2	no	—
78	C₄×Q₈	—	89	1	no	—
79	C₄×D₄	—	89	1	no	—
80	C₄×D₄	—	89	1	no	—
81	C₄×D₄	—	89	1	no	—
82	C₂×C₄⋊C₄	—	89	2	no	—
83	C₂×Q₈	—	89	2	no	—
84	C₂×Q₈	—	89	1	no	—
85	C₂×D₄	—	89	1	no	—
86	C₂²×C₄	—	89	2	no	—
87	C₂²×C₄	—	89	2	no	—
88	C₂²×C₄	—	89	2	no	—
89	C₄⋊C₄	—	89	1	no	—
90	C₄⋊C₄	—	89	2	no	—
91	C₄⋊C₄	—	89	2	no	—
92	C₄⋊C₄	—	89	1	no	—
93	C₄⋊C₄	—	89	1	no	—
94	C₄⋊C₄	—	89	1	no	—
95	C₄⋊C₄	—	89	2	no	—
96	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
97	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
100	C₂²⋊C₄	—	89	2	no	—
101	C₂²⋊C₄	—	89	2	no	—
102	C₂²⋊C₄	—	89	2	no	—
103	C₄²	—	89	1	no	—
104	C₄²	—	89	1	no	—
105	C₄²	C₂×D₈₉	1	1	yes	characteristic
106	C₂³	—	89	2	no	—
107	Q₈	—	356	2	no	—
108	Q₈	—	178	2	no	—
109	D₄	—	89	4	no	—
110	C₂×C₄	—	89	1	no	—
111	C₂×C₄	—	89	1	no	—
112	C₂×C₄	—	89	1	no	—
113	C₂×C₄	—	89	1	no	—
114	C₂×C₄	—	89	2	no	—
115	C₂×C₄	—	356	2	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₈₉	1	2	yes	—
121	C₂×C₄	C₂²×D₈₉	1	2	yes	—
122	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
123	C₂²	—	89	4	no	—
124	C₂²	—	178	2	no	—
125	C₂²	—	178	2	no	—
126	C₂²	C₂³×D₈₉	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	89	2	no	—
132	C₄	—	178	1	no	—
133	C₄	—	178	1	no	—
134	C₄	—	178	2	no	—
135	C₄	—	178	1	no	—
136	C₄	Q₈×D₈₉	1	2	yes	—
137	C₂	—	178	4	no	—
138	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
139	C₂	#41×D₈₉	1	1	yes	characteristic
140	C₂	#15668058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668011

C₂

yes

maximal, characteristic, complemented

#15668003

C₂

yes

maximal, characteristic, complemented

#15667992

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

C₄×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

C₄×D₃₅₆

C₂

yes

maximal, characteristic

#15667974

C₂

yes

maximal, characteristic

#15667974

C₂

yes

maximal, characteristic

C₄⋊C₄×D₈₉

C₂

yes

maximal, complemented

#125×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂²

yes

—

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₂×C₄×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

characteristic

#13083553

C₂²

yes

characteristic

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

characteristic

C₄×Dic₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄²×C₈₉

C₂²

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

characteristic

D₃₅₆

Q₈

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#41

yes

—

C₂×C₈₉

C₂²×Q₈

yes

characteristic

C₂×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

D₈₉

—

C₈₉

#554

yes

characteristic, complemented, Sylow

#554

—

maximal, Sylow

C₄⋊Q₈

—

#125

—

#125

D₈₉

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₈₉

yes

characteristic

106

C₂³

—

107

Q₈

—

356

—

108

Q₈

—

178

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

356

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

178

—

118

C₂×C₄

C₂²×D₈₉

yes

characteristic

119

C₂×C₄

C₂²×D₈₉

yes

characteristic

120

C₂×C₄

C₂²×D₈₉

yes

—

121

C₂×C₄

C₂²×D₈₉

yes

—

122

C₂×C₄

C₂²×D₈₉

yes

characteristic

123

C₂²

—

124

C₂²

—

178

—

125

C₂²

—

178

—

126

C₂²

C₂³×D₈₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

178

—

133

C₄

—

178

—

134

C₄

—

178

—

135

C₄

—

178

—

136

C₄

Q₈×D₈₉

yes

—

137

C₂

—

178

—

138

C₂

C₂×Q₈×D₈₉

yes

characteristic

139

C₂

#41×D₈₉

yes

characteristic

140

C₂

#15668058

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#125×C₈₉	C₂
hypercenter	#125	D₈₉
hypocenter	C₈₉	#554
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

#125×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #15668011	C₂	semidirect
2. #15668003	C₂	semidirect
3. #15667992	C₂	semidirect
4. #15667990	C₂	semidirect
5. #15667990	C₂	semidirect
6. C₄×Dic₁₇₈	C₂	semidirect
10. C₄⋊C₄×D₈₉	C₂	semidirect
11. #125×C₈₉	C₂	semidirect
46. D₃₅₆	Q₈	semidirect
70. C₈₉	#554	semidirect
74. #125	D₈₉	semidirect

H ≅ G/N

type

1. #15668011

C₂

semidirect

2. #15668003

C₂

semidirect

3. #15667992

C₂

semidirect

4. #15667990

C₂

semidirect

5. #15667990

C₂

semidirect

6. C₄×Dic₁₇₈

C₂

semidirect

10. C₄⋊C₄×D₈₉

C₂

semidirect

11. #125×C₈₉

C₂

semidirect

46. D₃₅₆

Q₈

semidirect

70. C₈₉

#554

semidirect

74. #125

D₈₉

semidirect

Subgroups — #19152304

Conjugacy classes of subgroups

Complements