Group #19152197 — subgroups

Subgroups — #19152197

Order 5696.

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

normal subgroups only

This group has 218 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	#15668007	C₂	1	2	yes	maximal, complemented
2	#15668003	C₂	1	2	yes	maximal, complemented
3	#15667997	C₂	1	2	yes	maximal, complemented
4	#15667996	C₂	1	2	yes	maximal, complemented
5	#15667988	C₂	1	2	yes	maximal, complemented
6	#15667986	C₂	1	1	yes	maximal, characteristic
7	#15667972	C₂	1	2	yes	maximal, complemented
8	C₂×#13083553	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₈₉	C₂²	1	2	yes	—
12	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₈₉	C₂²	1	2	yes	—
15	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₈₉	C₂²	1	2	yes	—
17	C₄⋊Dic₈₉	C₂²	1	2	yes	—
18	#13083553	C₂²	1	2	yes	—
19	#13083553	C₂²	1	2	yes	—
20	#13083553	C₂²	1	2	yes	—
21	#13083551	C₂²	1	2	yes	—
22	#13083551	C₂²	1	2	yes	—
23	#13083551	C₂²	1	2	yes	—
24	#13083551	C₂²	1	2	yes	—
25	C₄×Dic₈₉	C₂²	1	2	yes	—
26	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₂³×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×D₈₉	C₂³	1	2	yes	—
33	Q₈×C₈₉	—	4	1	no	—
34	C₂×C₄×C₈₉	—	4	1	no	—
35	C₂×C₄×C₈₉	C₂³	1	2	yes	—
36	C₂×C₄×C₈₉	C₂³	1	2	yes	—
37	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂²⋊D₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	2	2	no	—
41	C₂×Dic₈₉	—	4	1	no	—
42	C₂×Dic₈₉	C₂³	1	2	yes	—
43	C₂×Dic₈₉	C₂³	1	2	yes	—
44	C₂×Dic₈₉	C₂³	1	2	yes	—
45	D₃₅₆	—	4	1	no	—
46	C₄×D₈₉	—	4	2	no	—
47	C₂²×C₈₉	—	2	1	no	—
48	C₂²×C₈₉	—	2	1	no	—
49	C₂²×C₈₉	#41	1	2	yes	—
50	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
51	C₂×D₈₉	—	2	2	no	—
52	C₂×D₈₉	—	2	2	no	—
53	C₂×D₈₉	—	2	2	no	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	2	2	no	—
58	Dic₈₉	—	2	2	no	—
59	Dic₈₉	—	2	2	no	—
60	Dic₈₉	—	2	2	no	—
61	C₂×C₈₉	—	2	2	no	—
62	C₂×C₈₉	C₂×#41	1	1	yes	characteristic
63	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
65	D₈₉	—	4	2	no	—
66	C₈₉	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	89	1	no	maximal, Sylow
68	#126	—	89	2	no	—
69	#125	—	89	2	no	—
70	#123	—	89	2	no	—
71	#123	—	89	2	no	—
72	C₂²⋊Q₈	—	89	2	no	—
73	C₂²⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
74	#121	—	89	1	no	—
75	C₄×D₄	—	89	2	no	—
76	C₂×C₄⋊C₄	—	89	1	no	—
77	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
78	C₂×D₄	—	89	2	no	—
79	C₂×D₄	—	89	1	no	—
80	C₂²×C₄	—	89	1	no	—
81	C₂²×C₄	—	89	1	no	—
82	C₂²×C₄	—	89	2	no	—
83	C₂²×C₄	C₂×D₈₉	1	1	yes	characteristic
84	C₄⋊C₄	—	89	2	no	—
85	C₄⋊C₄	—	89	2	no	—
86	C₄⋊C₄	—	89	2	no	—
87	C₄⋊C₄	—	89	1	no	—
88	C₄⋊C₄	—	89	2	no	—
89	C₄⋊C₄	—	89	2	no	—
90	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
91	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
92	C₂²⋊C₄	—	89	2	no	—
93	C₂²⋊C₄	—	89	2	no	—
94	C₂²⋊C₄	—	89	2	no	—
95	C₂²⋊C₄	—	89	2	no	—
96	C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
97	C₄²	—	89	2	no	—
98	C₂³	—	89	2	no	—
99	C₂³	C₂²×D₈₉	1	1	yes	characteristic
100	Q₈	—	4	1	no	—
101	D₄	—	356	1	no	—
102	D₄	—	178	4	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	89	2	no	—
105	C₂×C₄	—	89	2	no	—
106	C₂×C₄	—	89	2	no	—
107	C₂×C₄	—	356	2	no	—
108	C₂×C₄	—	356	1	no	—
109	C₂×C₄	—	178	2	no	—
110	C₂×C₄	C₂²×D₈₉	1	2	yes	—
111	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₈₉	1	2	yes	—
113	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	178	2	no	—
117	C₂²	—	178	2	no	—
118	C₂²	—	178	2	no	—
119	C₂²	#13083575	1	2	yes	—
120	C₂²	C₂³×D₈₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	178	2	no	—
126	C₄	—	178	2	no	—
127	C₄	—	178	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	356	2	no	—
130	C₂	C₂×#13083575	1	1	yes	characteristic
131	C₂	#15668057	1	1	yes	characteristic
132	C₂	#15668059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668007

C₂

yes

maximal, complemented

#15668003

C₂

yes

maximal, complemented

#15667997

C₂

yes

maximal, complemented

#15667996

C₂

yes

maximal, complemented

#15667988

C₂

yes

maximal, complemented

#15667986

C₂

yes

maximal, characteristic

#15667972

C₂

yes

maximal, complemented

C₂×#13083553

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

—

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₂²×C₄×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂³×C₈₉

C₂³

yes

characteristic

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

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#41

yes

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

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

#539

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₈₉

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₈₉

yes

—

C₄²

—

C₂³

—

C₂³

C₂²×D₈₉

yes

characteristic

100

Q₈

—

101

D₄

—

356

—

102

D₄

—

178

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂²×D₈₉

yes

—

111

C₂×C₄

C₂²×D₈₉

yes

characteristic

112

C₂×C₄

C₂²×D₈₉

yes

—

113

C₂×C₄

C₂²×D₈₉

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

178

—

117

C₂²

—

178

—

118

C₂²

—

178

—

119

C₂²

#13083575

yes

—

120

C₂²

C₂³×D₈₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

178

—

126

C₄

—

178

—

127

C₄

—

178

—

128

C₂

—

129

C₂

—

356

—

130

C₂

C₂×#13083575

yes

characteristic

131

C₂

#15668057

yes

characteristic

132

C₂

#15668059

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	#539
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. #15668007	C₂	semidirect
2. #15668003	C₂	semidirect
3. #15667997	C₂	semidirect
4. #15667996	C₂	semidirect
5. #15667988	C₂	semidirect
7. #15667972	C₂	semidirect
8. C₂×#13083553	C₂	semidirect
9. C₂²⋊Q₈×C₈₉	C₂	semidirect
66. C₈₉	#539	semidirect
73. C₂²⋊Q₈	D₈₉	semidirect

H ≅ G/N

type

1. #15668007

C₂

semidirect

2. #15668003

C₂

semidirect

3. #15667997

C₂

semidirect

4. #15667996

C₂

semidirect

5. #15667988

C₂

semidirect

7. #15667972

C₂

semidirect

8. C₂×#13083553

C₂

semidirect

9. C₂²⋊Q₈×C₈₉

C₂

semidirect

66. C₈₉

#539

semidirect

73. C₂²⋊Q₈

D₈₉

semidirect

Subgroups — #19152197

Conjugacy classes of subgroups

Complements