Group #26019331 — subgroups

Subgroups — #26019331

Order 11392.

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

normal subgroups only

This group has 228 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	#19152328	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151972	C₂	1	1	yes	maximal, characteristic
3	#19151799	C₂	1	2	yes	maximal
4	#19151328	C₂	1	2	yes	maximal, complemented
5	#333×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083563	—	2	2	no	—
7	C₂×#13083561	—	2	2	no	—
8	#15668011	—	2	2	no	—
9	#15668010	—	2	2	no	—
10	#15668009	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₇₈	C₂²	1	2	yes	—
12	#15667977	—	2	2	no	—
13	#15667934	—	2	2	no	—
14	C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	D₄	1	2	yes	—
19	C₂×D₃₅₆	—	2	1	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	#13083563	—	4	4	no	—
22	#13083561	—	4	4	no	—
23	C₂×#10623494	—	2	1	no	—
24	C₄⋊Dic₈₉	D₄	1	2	yes	—
25	#13083553	—	2	4	no	—
26	#13083553	—	2	2	no	—
27	C₄×Dic₈₉	—	2	2	no	—
28	C₄×Dic₈₉	—	2	2	no	—
29	C₂×Q₈×C₈₉	D₄	1	2	yes	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×D₈₉	—	4	1	no	—
34	Q₈×C₈₉	—	4	2	no	—
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₈₉	—	2	2	no	—
39	C₂×Dic₈₉	—	2	2	no	—
40	D₃₅₆	—	4	2	no	—
41	D₃₅₆	—	4	2	no	—
42	D₃₅₆	—	4	2	no	—
43	Dic₁₇₈	—	2	2	no	—
44	Dic₁₇₈	—	8	2	no	—
45	Dic₁₇₈	SD₁₆	1	4	yes	complemented
46	C₈×C₈₉	—	4	2	no	—
47	#10623494	—	2	2	no	—
48	C₂²×C₈₉	#120	1	1	yes	characteristic
49	C₂×D₈₉	—	8	2	no	—
50	C₂×D₈₉	—	8	1	no	—
51	C₄×C₈₉	—	2	1	no	—
52	C₄×C₈₉	—	4	2	no	—
53	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₈₉	#137	1	2	yes	—
55	Dic₈₉	—	2	4	no	—
56	Dic₈₉	—	4	2	no	—
57	C₂×C₈₉	#448	1	2	yes	—
58	C₂×C₈₉	#453	1	1	yes	characteristic
59	D₈₉	—	16	1	no	—
60	#1623	—	89	1	no	maximal, Sylow
61	C₈₉	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	89	1	no	—
63	#492	—	89	1	no	—
64	#460	—	89	2	no	—
65	#333	D₈₉	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	89	2	no	—
67	C₂×SD₁₆	—	178	2	no	—
68	C₂×SD₁₆	—	178	2	no	—
69	C₄⋊Q₈	—	178	2	no	—
70	C₄⋊Q₈	—	178	2	no	—
71	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
72	#127	—	89	1	no	—
73	C₄×Q₈	—	89	2	no	—
74	C₄×Q₈	—	178	2	no	—
75	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
76	#102	—	178	2	no	—
77	C₄×C₈	—	89	1	no	—
78	C₂×Q₈	—	89	2	no	—
79	C₂×Q₈	—	178	2	no	—
80	C₂×Q₈	D₃₅₆	1	2	yes	—
81	C₂×D₄	—	178	2	no	—
82	C₂×D₄	—	178	1	no	—
83	SD₁₆	—	356	4	no	—
84	SD₁₆	—	356	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	178	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	89	2	no	—
89	C₄⋊C₄	—	178	4	no	—
90	C₄⋊C₄	—	178	2	no	—
91	C₄²	—	178	2	no	—
92	C₄²	—	178	2	no	—
93	C₄²	C₂²×D₈₉	1	1	yes	characteristic
94	C₂³	—	356	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	89	4	no	—
97	Q₈	—	178	2	no	—
98	Q₈	—	712	2	no	—
99	D₄	—	356	2	no	—
100	D₄	—	356	2	no	—
101	D₄	—	356	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	178	2	no	—
104	C₂×C₄	—	178	2	no	—
105	C₂×C₄	D₄×D₈₉	1	2	yes	—
106	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	178	2	no	—
109	C₂²	—	712	2	no	—
110	C₂²	—	712	1	no	—
111	C₂²	#15667982	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	356	2	no	—
115	C₄	—	178	4	no	—
116	C₄	SD₁₆×D₈₉	1	2	yes	—
117	C₄	#15668044	1	2	yes	—
118	C₂	—	1424	1	no	—
119	C₂	#19151738	1	2	yes	—
120	C₂	#19151768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152328

C₂

yes

maximal, characteristic, complemented

#19151972

C₂

yes

maximal, characteristic

#19151799

C₂

yes

maximal

#19151328

C₂

yes

maximal, complemented

#333×C₈₉

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₇₈

C₂²

yes

—

#15667977

—

#15667934

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

#13083561

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

SD₁₆

yes

complemented

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#448

yes

—

C₂×C₈₉

#453

yes

characteristic

D₈₉

—

#1623

—

maximal, Sylow

C₈₉

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

#492

—

#460

—

#333

D₈₉

yes

characteristic, complemented

Q₈⋊C₈

—

C₂×SD₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

D₃₅₆

yes

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

SD₁₆

—

356

—

SD₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

356

—

Q₈

—

Q₈

—

Q₈

—

178

—

Q₈

—

712

—

D₄

—

356

—

100

D₄

—

356

—

101

D₄

—

356

—

102

C₂×C₄

—

103

C₂×C₄

—

178

—

104

C₂×C₄

—

178

—

105

C₂×C₄

D₄×D₈₉

yes

—

106

C₂×C₄

C₂×D₃₅₆

yes

characteristic

107

C₈

—

108

C₈

—

178

—

109

C₂²

—

712

—

110

C₂²

—

712

—

111

C₂²

#15667982

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

356

—

115

C₄

—

178

—

116

C₄

SD₁₆×D₈₉

yes

—

117

C₄

#15668044

yes

—

118

C₂

—

1424

—

119

C₂

#19151738

yes

—

120

C₂

#19151768

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#333×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#15668009	C₂²
hypercenter	#333	D₈₉
hypocenter	C₈₉	#1623
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#333×C₈₉

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #19152328	C₂	semidirect
4. #19151328	C₂	semidirect
5. #333×C₈₉	C₂	semidirect
45. Dic₁₇₈	SD₁₆	semidirect
61. C₈₉	#1623	semidirect
65. #333	D₈₉	semidirect

H ≅ G/N

type

1. #19152328

C₂

semidirect

4. #19151328

C₂

semidirect

5. #333×C₈₉

C₂

semidirect

45. Dic₁₇₈

SD₁₆

semidirect

61. C₈₉

#1623

semidirect

65. #333

D₈₉

semidirect

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

subgroup	index
#19152328	2
#19151972	2
#19151799	2
#19151328	2
#333×C₈₉	2
#1623	89

subgroup

index

#333×C₈₉

Subgroups — #26019331

Conjugacy classes of subgroups

Complements

Maximal subgroups