Group #26019374 — subgroups

Subgroups — #26019374

Order 11392.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152334	C₂	1	1	yes	maximal, characteristic
2	#19151970	C₂	1	1	yes	maximal, characteristic
3	#19151894	C₂	1	2	yes	maximal, complemented
4	#19151328	C₂	1	2	yes	maximal, complemented
5	#331×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083562	—	2	2	no	—
7	#15668012	C₂²	1	1	yes	characteristic
8	#15667999	—	2	2	no	—
9	#15667993	—	2	2	no	—
10	C₄×Dic₁₇₈	C₂²	1	2	yes	—
11	D₄×Dic₈₉	—	2	2	no	—
12	#15667946	—	2	2	no	—
13	#15667937	—	2	2	no	—
14	C₄×#10623494	C₂²	1	1	yes	characteristic
15	#127×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₁₇₈	—	2	1	no	—
18	C₂×Dic₁₇₈	D₄	1	2	yes	—
19	C₂²×Dic₈₉	—	2	2	no	—
20	#13083562	—	4	4	no	—
21	C₂×#10623494	—	2	1	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	D₄	1	2	yes	—
24	#13083553	—	2	2	no	—
25	C₂²⋊Dic₈₉	—	2	4	no	—
26	C₂²⋊Dic₈₉	—	2	2	no	—
27	C₄×Dic₈₉	—	2	2	no	—
28	C₂×D₄×C₈₉	—	2	2	no	—
29	C₂×D₄×C₈₉	D₄	1	2	yes	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂³×C₈₉	—	2	2	no	—
33	D₄×C₈₉	—	4	2	no	—
34	D₄×C₈₉	—	4	2	no	—
35	D₄×C₈₉	—	4	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	C₂×Dic₈₉	—	4	1	no	—
41	C₂×Dic₈₉	—	8	2	no	—
42	Dic₁₇₈	—	2	2	no	—
43	Dic₁₇₈	—	2	2	no	—
44	Dic₁₇₈	—	4	2	no	—
45	C₈×C₈₉	—	4	2	no	—
46	#10623494	—	2	2	no	—
47	C₂²×C₈₉	—	4	4	no	—
48	C₂²×C₈₉	—	4	2	no	—
49	C₂²×C₈₉	#120	1	1	yes	characteristic
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	#135	1	2	yes	—
52	C₄×C₈₉	#137	1	2	yes	—
53	Dic₈₉	—	4	2	no	—
54	Dic₈₉	—	4	2	no	—
55	Dic₈₉	—	8	1	no	—
56	C₂×C₈₉	—	8	2	no	—
57	C₂×C₈₉	#452	1	2	yes	—
58	C₂×C₈₉	#453	1	1	yes	characteristic
59	#1630	—	89	1	no	maximal, Sylow
60	C₈₉	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	89	1	no	—
62	#492	—	89	1	no	—
63	#477	—	89	2	no	—
64	#331	D₈₉	1	1	yes	characteristic
65	Q₈⋊C₈	—	89	2	no	—
66	C₂×SD₁₆	—	178	2	no	—
67	C₄⋊Q₈	—	89	1	no	—
68	#127	C₂×D₈₉	1	1	yes	characteristic
69	#124	—	178	2	no	—
70	#123	—	178	2	no	—
71	C₄×Q₈	—	89	2	no	—
72	C₄×D₄	—	178	2	no	—
73	#107	—	178	2	no	—
74	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
75	#103	—	178	2	no	—
76	C₄×C₈	—	89	1	no	—
77	C₂×Q₈	—	89	2	no	—
78	C₂×Q₈	—	178	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₃₅₆	1	2	yes	—
81	C₂²×C₄	—	178	2	no	—
82	SD₁₆	—	356	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	178	1	no	—
85	C₄⋊C₄	—	89	2	no	—
86	C₄⋊C₄	—	178	2	no	—
87	C₄⋊C₄	—	178	2	no	—
88	C₂²⋊C₄	—	178	4	no	—
89	C₂²⋊C₄	—	178	2	no	—
90	C₄²	—	178	2	no	—
91	C₄²	C₂²×D₈₉	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	356	2	no	—
94	Q₈	—	178	2	no	—
95	Q₈	—	178	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	356	1	no	—
100	C₂×C₄	—	178	2	no	—
101	C₂×C₄	—	178	2	no	—
102	C₂×C₄	—	712	2	no	—
103	C₂×C₄	D₄×D₈₉	1	2	yes	—
104	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	178	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#15667982	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	356	2	no	—
112	C₄	—	356	2	no	—
113	C₄	—	712	1	no	—
114	C₄	#15668031	1	2	yes	—
115	C₄	#15668041	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#19151761	1	2	yes	—
118	C₂	#19151768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152334

C₂

yes

maximal, characteristic

#19151970

C₂

yes

maximal, characteristic

#19151894

C₂

yes

maximal, complemented

#19151328

C₂

yes

maximal, complemented

#331×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083562

—

#15668012

C₂²

yes

characteristic

#15667999

—

#15667993

—

C₄×Dic₁₇₈

C₂²

yes

—

D₄×Dic₈₉

—

#15667946

—

#15667937

—

C₄×#10623494

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

#13083562

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#452

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

Q₈⋊C₈

—

C₂×SD₁₆

—

178

—

C₄⋊Q₈

—

#127

C₂×D₈₉

yes

characteristic

#124

—

178

—

#123

—

178

—

C₄×Q₈

—

C₄×D₄

—

178

—

#107

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

C₂×D₄

D₃₅₆

yes

—

C₂²×C₄

—

178

—

SD₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

Q₈

—

356

—

Q₈

—

178

—

Q₈

—

178

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

356

—

100

C₂×C₄

—

178

—

101

C₂×C₄

—

178

—

102

C₂×C₄

—

712

—

103

C₂×C₄

D₄×D₈₉

yes

—

104

C₂×C₄

C₂×D₃₅₆

yes

characteristic

105

C₈

—

106

C₈

—

178

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15667982

yes

characteristic

110

C₄

—

111

C₄

—

356

—

112

C₄

—

356

—

113

C₄

—

712

—

114

C₄

#15668031

yes

—

115

C₄

#15668041

yes

—

116

C₂

—

117

C₂

#19151761

yes

—

118

C₂

#19151768

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#331×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#127×C₈₉	C₂²
hypercenter	#331	D₈₉
hypocenter	C₈₉	#1630
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#331×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

#127×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
3. #19151894	C₂	semidirect
4. #19151328	C₂	semidirect
60. C₈₉	#1630	semidirect

H ≅ G/N

type

3. #19151894

C₂

semidirect

4. #19151328

C₂

semidirect

60. C₈₉

#1630

semidirect

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

subgroup	index
#19152334	2
#19151970	2
#19151894	2
#19151328	2
#331×C₈₉	2
#1630	89

subgroup

index

#331×C₈₉

Subgroups — #26019374

Conjugacy classes of subgroups

Complements

Maximal subgroups