Group #26020413 — subgroups

Subgroups — #26020413

Order 11392.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#15667945	C₂	1	2	yes	maximal
2	C₂×#15667944	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151463	C₂	1	2	yes	maximal
5	C₂×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667945	—	2	4	no	—
10	#15667944	—	2	2	no	—
11	C₈×Dic₈₉	C₄	1	4	yes	complemented
12	#15667919	—	2	4	no	—
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	#106×C₈₉	—	2	2	no	—
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂²×Dic₈₉	D₄	1	2	yes	—
18	C₂×#10623494	C₂×C₄	1	2	yes	—
19	C₂×#10623494	D₄	1	2	yes	—
20	C₂×#10623494	Q₈	1	2	yes	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₈₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₈₉	D₄	1	2	yes	—
29	C₂×C₈×C₈₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	4	4	no	—
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	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₂×Dic₈₉	—	4	4	no	—
44	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
45	C₈×C₈₉	C₄⋊C₄	1	4	yes	—
46	#10623494	C₄⋊C₄	1	4	yes	—
47	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₈₉	#127	1	1	yes	characteristic
49	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₉	#135	1	4	yes	—
51	C₄×C₈₉	—	8	2	no	—
52	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
55	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₈₉	—	2	4	no	—
57	Dic₈₉	—	8	2	no	—
58	C₂×C₈₉	#384	1	1	yes	characteristic
59	C₂×C₈₉	#427	1	2	yes	—
60	C₂×C₈₉	#495	1	2	yes	—
61	C₂×C₈₉	#499	1	2	yes	—
62	#1846	—	89	1	no	maximal, Sylow
63	C₈₉	#1846	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	89	2	no	—
65	C₂×#106	—	89	1	no	—
66	C₂×#106	D₈₉	1	1	yes	characteristic
67	C₂×C₄×C₈	—	89	1	no	—
68	#382	—	89	2	no	—
69	C₂²×C₈	—	89	1	no	—
70	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	89	2	no	—
72	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
73	C₂×C₄²	—	89	1	no	—
74	#107	—	178	4	no	—
75	#106	—	2	2	no	—
76	#106	—	178	2	no	—
77	C₄×C₈	—	89	4	no	—
78	#95	—	178	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	89	2	no	—
81	C₂²×C₄	—	178	2	no	—
82	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
83	C₂×C₈	—	89	2	no	—
84	C₂×C₈	—	89	2	no	—
85	C₂×C₈	—	89	2	no	—
86	C₂×C₈	Dic₁₇₈	1	2	yes	—
87	C₂×C₈	C₄×D₈₉	1	2	yes	—
88	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	178	2	no	—
92	C₄⋊C₄	—	178	2	no	—
93	C₄²	—	89	2	no	—
94	C₄²	—	89	2	no	—
95	C₂³	D₄×D₈₉	1	1	yes	characteristic
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	89	4	no	—
99	C₂×C₄	—	356	2	no	—
100	C₂×C₄	—	356	4	no	—
101	C₂×C₄	—	178	4	no	—
102	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
103	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
108	C₈	—	89	4	no	—
109	C₈	#13083553	1	4	yes	—
110	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
111	C₂²	#15668008	1	1	yes	characteristic
112	C₂²	#15668011	1	1	yes	characteristic
113	C₂²	#15668033	1	4	yes	—
114	C₄	—	8	2	no	—
115	C₄	—	178	4	no	—
116	C₄	—	712	2	no	—
117	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
118	C₄	C₂×#13083553	1	1	yes	characteristic
119	C₄	#15668010	1	1	yes	characteristic
120	C₄	#15668011	1	1	yes	characteristic
121	C₂	#19151467	1	1	yes	characteristic
122	C₂	#19151633	1	2	yes	—
123	C₂	#19151979	1	2	yes	—
124	C₂	#19151998	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667945

C₂

yes

maximal

C₂×#15667944

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151463

C₂

yes

maximal

C₂×#106×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667945

—

#15667944

—

C₈×Dic₈₉

C₄

yes

complemented

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

#106×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

Q₈

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

Q₈

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₈×C₈₉

C₄⋊C₄

yes

—

#10623494

C₄⋊C₄

yes

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#127

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

#135

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#384

yes

characteristic

C₂×C₈₉

#427

yes

—

C₂×C₈₉

#495

yes

—

C₂×C₈₉

#499

yes

—

#1846

—

maximal, Sylow

C₈₉

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×#106

D₈₉

yes

characteristic

C₂×C₄×C₈

—

#382

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

#107

—

178

—

#106

—

#106

—

178

—

C₄×C₈

—

#95

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₇₈

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

C₂³

D₄×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

100

C₂×C₄

—

356

—

101

C₂×C₄

—

178

—

102

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

103

C₂×C₄

D₄×D₈₉

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

105

C₂×C₄

Q₈×D₈₉

yes

characteristic

106

C₂×C₄

Q₈×D₈₉

yes

characteristic

107

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

108

C₈

—

109

C₈

#13083553

yes

—

110

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

111

C₂²

#15668008

yes

characteristic

112

C₂²

#15668011

yes

characteristic

113

C₂²

#15668033

yes

—

114

C₄

—

115

C₄

—

178

—

116

C₄

—

712

—

117

C₄

C₄⋊C₄×D₈₉

yes

characteristic

118

C₄

C₂×#13083553

yes

characteristic

119

C₄

#15668010

yes

characteristic

120

C₄

#15668011

yes

characteristic

121

C₂

#19151467

yes

characteristic

122

C₂

#19151633

yes

—

123

C₂

#19151979

yes

—

124

C₂

#19151998

yes

—

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#106×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#106	D₈₉
hypocenter	C₈₉	#1846
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#106×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
11. C₈×Dic₈₉	C₄	semidirect
63. C₈₉	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₈₉

C₄

semidirect

63. C₈₉

#1846

semidirect

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

subgroup	index
C₂×#15667945	2
C₂×#15667944	2
C₂×C₈×Dic₈₉	2
#19151463	2
C₂×#106×C₈₉	2
#1846	89

subgroup

index

C₂×C₈×Dic₈₉

C₂×#106×C₈₉

Subgroups — #26020413

Conjugacy classes of subgroups

Complements

Maximal subgroups