Group #26021054 — subgroups

Subgroups — #26021054

Order 11392.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667937	C₂	1	4	yes	maximal
3	#19151468	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
6	#15668012	—	2	2	no	—
7	#15668012	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₈₉	—	2	2	no	—
9	C₂×C₄⋊Dic₈₉	—	2	2	no	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#15667937	—	2	8	no	—
12	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₉	—	2	2	no	—
15	C₂×Dic₁₇₈	—	2	4	no	—
16	C₂×Dic₁₇₈	—	2	4	no	—
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₈₉	—	2	2	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	8	no	—
22	C₄⋊Dic₈₉	—	2	4	no	—
23	C₄⋊Dic₈₉	—	4	4	no	—
24	C₂²×C₄×C₈₉	D₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	D₄	1	4	yes	—
29	C₄²×C₈₉	C₂×C₄	1	2	yes	—
30	C₄²×C₈₉	D₄	1	2	yes	—
31	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₈₉	SD₁₆	1	4	yes	—
34	C₂×C₄×C₈₉	Q₁₆	1	4	yes	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₉	#41	1	2	yes	—
39	C₂×Dic₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	4	2	no	—
41	C₂×Dic₈₉	—	4	2	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₂×Dic₈₉	—	4	4	no	—
44	Dic₁₇₈	—	2	8	no	—
45	Dic₁₇₈	—	4	4	no	—
46	C₈×C₈₉	—	2	4	no	—
47	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈₉	#124	1	1	yes	characteristic
49	C₂²×C₈₉	#127	1	1	yes	characteristic
50	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₈₉	#103	1	8	yes	complemented
53	C₄×C₈₉	C₄×D₄	1	2	yes	—
54	C₄×C₈₉	#121	1	2	yes	—
55	Dic₈₉	—	4	4	no	—
56	Dic₈₉	—	8	2	no	—
57	C₂×C₈₉	#390	1	1	yes	characteristic
58	C₂×C₈₉	C₂×#103	1	2	yes	—
59	C₂×C₈₉	#487	1	2	yes	—
60	C₂×C₈₉	#492	1	1	yes	characteristic
61	C₂×C₈₉	#494	1	1	yes	characteristic
62	#1956	—	89	1	no	maximal, Sylow
63	C₈₉	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	89	1	no	—
65	C₂×#103	—	89	4	no	—
66	C₂×C₄×C₈	D₈₉	1	1	yes	characteristic
67	#384	—	89	1	no	—
68	C₂²×Q₈	—	89	2	no	—
69	C₂²×C₈	C₂×D₈₉	1	2	yes	—
70	C₄⋊Q₈	—	89	4	no	—
71	C₄⋊Q₈	—	178	2	no	—
72	C₂×C₄⋊C₄	—	89	2	no	—
73	C₂×C₄⋊C₄	—	178	2	no	—
74	C₂×C₄⋊C₄	—	178	2	no	—
75	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
76	#103	—	178	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	89	4	no	—
79	C₂×Q₈	—	178	4	no	—
80	C₂×Q₈	—	178	4	no	—
81	C₂×Q₈	—	178	2	no	—
82	C₂²×C₄	—	178	2	no	—
83	C₂²×C₄	—	178	2	no	—
84	C₂²×C₄	D₃₅₆	1	2	yes	—
85	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₃₅₆	1	4	yes	—
89	C₄⋊C₄	—	356	4	no	—
90	C₄⋊C₄	—	178	4	no	—
91	C₄⋊C₄	—	178	8	no	—
92	C₄²	C₄×D₈₉	1	2	yes	—
93	C₄²	C₂²⋊D₈₉	1	2	yes	—
94	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
95	Q₈	—	356	4	no	—
96	Q₈	—	178	8	no	—
97	C₂×C₄	—	356	2	no	—
98	C₂×C₄	—	356	2	no	—
99	C₂×C₄	—	356	2	no	—
100	C₂×C₄	—	356	4	no	—
101	C₂×C₄	—	178	4	no	—
102	C₂×C₄	#13083551	1	4	yes	—
103	C₂×C₄	#13083563	1	4	yes	—
104	C₂×C₄	Dic₃₅₆	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
108	C₂×C₄	#13083574	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×#13083551	1	1	yes	characteristic
111	C₂²	#15668000	1	1	yes	characteristic
112	C₂²	#15668009	1	1	yes	characteristic
113	C₂²	C₂×#13083563	1	2	yes	—
114	C₂²	C₂×Dic₃₅₆	1	2	yes	—
115	C₄	—	356	4	no	—
116	C₄	—	712	2	no	—
117	C₄	#15667937	1	8	yes	complemented
118	C₄	C₄×D₃₅₆	1	2	yes	—
119	C₄	#15667986	1	2	yes	—
120	C₂	#19151500	1	1	yes	characteristic
121	C₂	C₂×#15667937	1	2	yes	—
122	C₂	#19151949	1	2	yes	—
123	C₂	#19151971	1	1	yes	characteristic
124	C₂	#19151978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

#19151468

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

#15668012

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667937

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

#103

yes

complemented

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

#121

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

C₂×#103

yes

—

C₂×C₈₉

#487

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₈₉

yes

characteristic

#384

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

C₄×D₈₉

yes

—

C₄²

C₂²⋊D₈₉

yes

—

C₂³

C₂×D₃₅₆

yes

characteristic

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

100

C₂×C₄

—

356

—

101

C₂×C₄

—

178

—

102

C₂×C₄

#13083551

yes

—

103

C₂×C₄

#13083563

yes

—

104

C₂×C₄

Dic₃₅₆

yes

—

105

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₂×C₄

C₂×D₃₅₆

yes

—

108

C₂×C₄

#13083574

yes

—

109

C₈

—

110

C₂²

C₂×#13083551

yes

characteristic

111

C₂²

#15668000

yes

characteristic

112

C₂²

#15668009

yes

characteristic

113

C₂²

C₂×#13083563

yes

—

114

C₂²

C₂×Dic₃₅₆

yes

—

115

C₄

—

356

—

116

C₄

—

712

—

117

C₄

#15667937

yes

complemented

118

C₄

C₄×D₃₅₆

yes

—

119

C₄

#15667986

yes

—

120

C₂

#19151500

yes

characteristic

121

C₂

C₂×#15667937

yes

—

122

C₂

#19151949

yes

—

123

C₂

#19151971

yes

characteristic

124

C₂

#19151978

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₄×C₈	D₈₉
hypocenter	C₈₉	#1956
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄×C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₄×C₈

D₈₉

hypocenter

C₈₉

#1956

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #15668012	C₄	semidirect
52. C₄×C₈₉	#103	semidirect
63. C₈₉	#1956	semidirect
117. C₄	#15667937	semidirect

H ≅ G/N

type

7. #15668012

C₄

semidirect

52. C₄×C₈₉

#103

semidirect

63. C₈₉

#1956

semidirect

117. C₄

#15667937

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×#15667937	2
#19151468	2
C₂×C₄×C₈×C₈₉	2
#1956	89

subgroup

index

C₂×C₄×C₈×C₈₉

Subgroups — #26021054

Conjugacy classes of subgroups

Complements

Maximal subgroups