Group #26021806 — subgroups

Subgroups — #26021806

Order 11392.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#19151647	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151625	C₂	1	2	yes	maximal, complemented
3	#19151616	C₂	1	2	yes	maximal
4	#19151566	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	C₂×#13083558	C₂²	1	2	yes	—
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	#15667948	—	2	2	no	—
10	C₄⋊#10623494	—	2	4	no	—
11	#15667940	—	2	4	no	—
12	#15667924	—	2	2	no	—
13	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	#117×C₈₉	C₂²	1	2	yes	—
15	#13083574	—	2	2	no	—
16	#13083574	—	2	2	no	—
17	C₂×Dic₁₇₈	Q₈	1	1	yes	characteristic
18	C₂×D₃₅₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₈₉	—	2	1	no	—
20	C₂×C₄×D₈₉	—	2	1	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	C₂×#10623494	—	2	1	no	—
24	C₂×#10623494	—	2	2	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂²⋊C₄×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	D₄	1	4	yes	—
32	C₂³×C₈₉	#41	1	1	yes	characteristic
33	C₂²×D₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	2	4	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#41	1	2	yes	—
39	C₂²⋊D₈₉	—	4	2	no	—
40	C₂×Dic₈₉	—	2	1	no	—
41	D₃₅₆	—	2	2	no	—
42	C₄×D₈₉	—	4	1	no	—
43	C₄×D₈₉	—	4	1	no	—
44	Dic₁₇₈	—	2	2	no	—
45	C₈×C₈₉	—	2	2	no	—
46	#10623494	—	4	1	no	—
47	#10623494	—	4	2	no	—
48	C₂²×C₈₉	—	4	1	no	—
49	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₉	#123	1	2	yes	—
51	C₂×D₈₉	—	4	2	no	—
52	C₂×D₈₉	—	4	1	no	—
53	C₄×C₈₉	—	4	4	no	—
54	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₈₉	#123	1	1	yes	characteristic
56	C₄×C₈₉	#127	1	1	yes	characteristic
57	Dic₈₉	—	4	1	no	—
58	C₂×C₈₉	—	2	2	no	—
59	C₂×C₈₉	—	2	1	no	—
60	C₂×C₈₉	#399	1	1	yes	characteristic
61	D₈₉	—	8	1	no	—
62	#2098	—	89	1	no	maximal, Sylow
63	C₈₉	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	89	1	no	—
65	#424	—	89	2	no	—
66	#421	—	89	2	no	—
67	#408	—	89	1	no	—
68	H(3,ℤ/4)	D₈₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	89	1	no	—
70	C₂×M₁₆	—	89	2	no	—
71	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
72	C₂²×C₈	—	89	1	no	—
73	#117	C₂×D₈₉	1	2	yes	—
74	#108	—	178	2	no	—
75	C₄⋊C₈	—	178	4	no	—
76	#104	—	178	4	no	—
77	C₂²⋊C₈	—	178	2	no	—
78	#41	—	178	2	no	—
79	#41	—	178	2	no	—
80	C₂×Q₈	—	89	1	no	—
81	C₂×D₄	—	89	1	no	—
82	C₂×D₄	—	178	1	no	—
83	C₂²×C₄	—	178	1	no	—
84	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	178	2	no	—
87	M₁₆	—	178	2	no	—
88	C₂×C₈	—	178	1	no	—
89	C₂×C₈	—	178	2	no	—
90	C₂×C₈	—	178	2	no	—
91	C₂×C₈	D₃₅₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₃₅₆	1	4	yes	—
95	C₂³	—	178	1	no	—
96	C₂³	#13083573	1	1	yes	characteristic
97	Q₈	—	178	2	no	—
98	D₄	—	356	2	no	—
99	D₄	—	178	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	356	1	no	—
102	C₂×C₄	—	356	1	no	—
103	C₂×C₄	—	178	1	no	—
104	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
105	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
107	C₂×C₄	#13083573	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	356	1	no	—
110	C₈	—	356	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	356	2	no	—
113	C₂²	—	356	1	no	—
114	C₂²	#15667992	1	1	yes	characteristic
115	C₂²	#15667996	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	356	1	no	—
118	C₄	#15667992	1	2	yes	—
119	C₄	#15667996	1	1	yes	characteristic
120	C₄	#15668009	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	712	1	no	—
124	C₂	#19151535	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151647

C₂

yes

maximal, characteristic, complemented

#19151625

C₂

yes

maximal, complemented

#19151616

C₂

yes

maximal

#19151566

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

Q₈

yes

characteristic

C₂×D₃₅₆

Q₈

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#399

yes

characteristic

D₈₉

—

#2098

—

maximal, Sylow

C₈₉

#2098

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

#117

C₂×D₈₉

yes

—

#108

—

178

—

C₄⋊C₈

—

178

—

#104

—

178

—

C₂²⋊C₈

—

178

—

#41

—

178

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₃₅₆

yes

—

C₂³

—

178

—

C₂³

#13083573

yes

characteristic

Q₈

—

178

—

D₄

—

356

—

D₄

—

178

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

—

178

—

104

C₂×C₄

C₂×D₃₅₆

yes

—

105

C₂×C₄

C₂×D₃₅₆

yes

characteristic

106

C₂×C₄

Q₈×D₈₉

yes

characteristic

107

C₂×C₄

#13083573

yes

—

108

C₈

—

109

C₈

—

356

—

110

C₈

—

356

—

111

C₂²

—

112

C₂²

—

356

—

113

C₂²

—

356

—

114

C₂²

#15667992

yes

characteristic

115

C₂²

#15667996

yes

—

116

C₄

—

117

C₄

—

356

—

118

C₄

#15667992

yes

—

119

C₄

#15667996

yes

characteristic

120

C₄

#15668009

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

712

—

124

C₂

#19151535

yes

characteristic

	subgroup	G/H
center	C₄	#15668009
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	H(3,ℤ/4)×C₈₉	C₂
socle	C₂×C₈₉	#399
prime radical	C₂×#13083574	C₂²
hypercenter	H(3,ℤ/4)	D₈₉
hypocenter	C₈₉	#2098
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15668009

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

H(3,ℤ/4)×C₈₉

C₂

C₂×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #19151647	C₂	semidirect
2. #19151625	C₂	semidirect
5. H(3,ℤ/4)×C₈₉	C₂	semidirect
63. C₈₉	#2098	semidirect
68. H(3,ℤ/4)	D₈₉	semidirect

H ≅ G/N

type

1. #19151647

C₂

semidirect

2. #19151625

C₂

semidirect

5. H(3,ℤ/4)×C₈₉

C₂

semidirect

63. C₈₉

#2098

semidirect

68. H(3,ℤ/4)

D₈₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19151647	2
#19151625	2
#19151616	2
#19151566	2
H(3,ℤ/4)×C₈₉	2
#2098	89

subgroup

index

H(3,ℤ/4)×C₈₉

Subgroups — #26021806

Conjugacy classes of subgroups

Complements

Maximal subgroups