Group #26019397 — subgroups

Subgroups — #26019397

Order 11392.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151977	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151888	C₂	1	2	yes	maximal, complemented
3	#19151347	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#10623494	C₂	1	2	yes	maximal, complemented
5	#555×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083564	—	2	2	no	—
7	#15668012	C₂²	1	1	yes	characteristic
8	#15667943	—	2	2	no	—
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	#15667933	—	2	2	no	—
11	C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	#124×C₈₉	—	2	2	no	—
14	#121×C₈₉	—	2	2	no	—
15	C₄×D₄×C₈₉	C₂²	1	2	yes	—
16	#117×C₈₉	—	2	2	no	—
17	C₂×Dic₁₇₈	—	2	1	no	—
18	#13083564	—	4	4	no	—
19	C₂×#10623494	—	2	1	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×Q₈×C₈₉	D₄	1	2	yes	—
23	C₂×D₄×C₈₉	—	2	2	no	—
24	C₂×D₄×C₈₉	D₄	1	2	yes	—
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
28	C₂²⋊C₄×C₈₉	—	2	4	no	—
29	C₂²⋊C₄×C₈₉	—	2	2	no	—
30	C₄²×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂³×C₈₉	—	2	2	no	—
33	Q₈×C₈₉	—	4	2	no	—
34	D₄×C₈₉	—	2	2	no	—
35	D₄×C₈₉	—	2	2	no	—
36	D₄×C₈₉	—	8	2	no	—
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
43	C₂×Dic₈₉	—	4	1	no	—
44	Dic₁₇₈	—	4	2	no	—
45	#10623494	—	2	2	no	—
46	#10623494	—	4	2	no	—
47	C₂²×C₈₉	—	4	2	no	—
48	C₂²×C₈₉	—	4	2	no	—
49	C₂²×C₈₉	—	4	2	no	—
50	C₂²×C₈₉	#120	1	1	yes	characteristic
51	C₄×C₈₉	—	2	1	no	—
52	C₄×C₈₉	—	4	2	no	—
53	C₄×C₈₉	—	4	2	no	—
54	C₄×C₈₉	#135	1	2	yes	—
55	C₄×C₈₉	#137	1	2	yes	—
56	Dic₈₉	—	8	1	no	—
57	C₂×C₈₉	—	8	2	no	—
58	C₂×C₈₉	#452	1	2	yes	—
59	C₂×C₈₉	#453	1	1	yes	characteristic
60	#1634	—	89	1	no	maximal, Sylow
61	C₈₉	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	D₈₉	1	1	yes	characteristic
63	#494	—	89	1	no	—
64	#476	—	89	2	no	—
65	#333	—	89	1	no	—
66	D₄⋊C₈	—	89	2	no	—
67	C₂×Q₁₆	—	178	2	no	—
68	C₄⋊Q₈	—	89	1	no	—
69	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₈₉	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	178	2	no	—
75	C₄⋊C₈	—	89	2	no	—
76	#102	—	178	2	no	—
77	C₄×C₈	—	89	1	no	—
78	C₂×Q₈	—	178	1	no	—
79	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	356	4	no	—
84	C₂×C₈	—	178	2	no	—
85	C₂×C₈	—	178	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	178	2	no	—
88	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₈₉	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	356	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	356	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
106	C₈	—	356	2	no	—
107	C₈	—	178	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#15667981	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	712	1	no	—
116	C₄	#15668032	1	2	yes	—
117	C₄	#15668042	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#19151760	1	2	yes	—
120	C₂	#19151766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151977

C₂

yes

maximal, characteristic, complemented

#19151888

C₂

yes

maximal, complemented

#19151347

C₂

yes

maximal, characteristic, complemented

D₄⋊#10623494

C₂

yes

maximal, complemented

#555×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083564

—

#15668012

C₂²

yes

characteristic

#15667943

—

C₄⋊#10623494

C₂²

yes

—

#15667933

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

#121×C₈₉

—

C₄×D₄×C₈₉

C₂²

yes

—

#117×C₈₉

—

C₂×Dic₁₇₈

—

#13083564

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#452

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#555

D₈₉

yes

—

—

—

D₄⋊C₈

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₈₉

yes

—

#117

—

#106

—

178

—

C₄⋊C₈

—

#102

—

178

—

C₄×C₈

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

—

Q₁₆

—

356

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

356

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

356

—

104

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

106

C₈

—

356

—

107

C₈

—

178

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#15667981

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

712

—

116

C₄

#15668032

yes

—

117

C₄

#15668042

yes

—

118

C₂

—

119

C₂

#19151760

yes

—

120

C₂

#19151766

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting, prime radical	#555×C₈₉	C₂
socle	C₂²×C₈₉	#120
hypercenter	#555	D₈₉
hypocenter	C₈₉	#1634
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting, prime radical

#555×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #19151977	C₂	semidirect
2. #19151888	C₂	semidirect
3. #19151347	C₂	semidirect
4. D₄⋊#10623494	C₂	semidirect
61. C₈₉	#1634	semidirect

H ≅ G/N

type

1. #19151977

C₂

semidirect

2. #19151888

C₂

semidirect

3. #19151347

C₂

semidirect

4. D₄⋊#10623494

C₂

semidirect

61. C₈₉

#1634

semidirect

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

subgroup	index
#19151977	2
#19151888	2
#19151347	2
D₄⋊#10623494	2
#555×C₈₉	2
#1634	89

subgroup

index

#555×C₈₉

Subgroups — #26019397

Conjugacy classes of subgroups

Complements

Maximal subgroups