Group #26018976 — subgroups

Subgroups — #26018976

Order 11392.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#19151654	C₂	1	2	yes	maximal
3	#19151347	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151346	C₂	1	2	yes	maximal, complemented
5	#333×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₈₉	C₄	1	2	yes	—
7	#15668012	C₂²	1	1	yes	characteristic
8	#15668011	—	2	2	no	—
9	#15668010	C₄	1	2	yes	—
10	C₄⋊C₄×D₈₉	—	2	2	no	—
11	C₄²×D₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊#10623494	C₂²	1	2	yes	—
13	#15667924	—	2	2	no	—
14	#15667922	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
19	Q₈×D₈₉	—	4	4	no	—
20	C₂×C₄×D₈₉	—	2	2	no	—
21	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₈₉	D₄	1	2	yes	—
23	C₂×#10623494	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	#13083553	—	2	4	no	—
26	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₈₉	D₄	1	2	yes	—
28	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×Dic₈₉	—	2	2	no	—
38	C₂×Dic₈₉	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₈₉	—	2	2	no	—
41	C₄×D₈₉	—	2	2	no	—
42	C₄×D₈₉	—	2	2	no	—
43	C₄×D₈₉	—	4	4	no	—
44	Dic₁₇₈	—	4	4	no	—
45	Dic₁₇₈	—	4	2	no	—
46	C₈×C₈₉	—	4	2	no	—
47	#10623494	—	4	2	no	—
48	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₈₉	—	2	2	no	—
50	C₂×D₈₉	#100	1	2	yes	—
51	C₄×C₈₉	—	2	1	no	—
52	C₄×C₈₉	—	4	2	no	—
53	C₄×C₈₉	#137	1	2	yes	—
54	C₄×C₈₉	#137	1	2	yes	—
55	Dic₈₉	—	2	1	no	—
56	Dic₈₉	—	2	2	no	—
57	Dic₈₉	—	4	2	no	—
58	C₂×C₈₉	C₂×#100	1	1	yes	characteristic
59	C₂×C₈₉	#419	1	2	yes	—
60	D₈₉	—	2	2	no	—
61	#1558	—	89	1	no	maximal, Sylow
62	C₈₉	#1558	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	89	1	no	—
64	#432	—	89	2	no	—
65	#333	—	89	1	no	—
66	#333	—	89	2	no	—
67	#333	D₈₉	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	89	2	no	—
69	C₄⋊Q₈	—	89	2	no	—
70	C₄⋊Q₈	—	89	1	no	—
71	C₄⋊Q₈	—	178	2	no	—
72	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	178	2	no	—
74	C₂×C₄²	—	89	1	no	—
75	C₄⋊C₈	—	89	2	no	—
76	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
77	C₂²⋊C₈	—	178	2	no	—
78	#97	—	178	2	no	—
79	C₂×Q₈	—	89	2	no	—
80	C₂×Q₈	—	356	4	no	—
81	C₂×Q₈	—	178	2	no	—
82	C₂×Q₈	C₄×D₈₉	1	2	yes	—
83	C₂²×C₄	—	89	1	no	—
84	C₂²×C₄	—	89	2	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	178	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	178	4	no	—
90	C₄⋊C₄	—	178	2	no	—
91	C₄²	—	89	1	no	—
92	C₄²	—	89	2	no	—
93	C₄²	C₂²×D₈₉	1	1	yes	characteristic
94	C₂³	—	89	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	356	2	no	—
97	Q₈	—	356	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	89	2	no	—
100	C₂×C₄	—	89	1	no	—
101	C₂×C₄	—	356	4	no	—
102	C₂×C₄	—	178	2	no	—
103	C₂×C₄	—	178	2	no	—
104	C₂×C₄	—	178	2	no	—
105	C₂×C₄	—	178	2	no	—
106	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
107	C₂×C₄	D₄×D₈₉	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	356	2	no	—
110	C₂²	—	89	2	no	—
111	C₂²	—	178	2	no	—
112	C₂²	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	356	2	no	—
116	C₄	—	178	1	no	—
117	C₄	—	178	2	no	—
118	C₄	#15668041	1	2	yes	—
119	C₄	#15668044	1	2	yes	—
120	C₂	—	178	2	no	—
121	C₂	#100×D₈₉	1	1	yes	characteristic
122	C₂	#19151605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₈₉

C₂

yes

maximal, characteristic

#19151654

C₂

yes

maximal

#19151347

C₂

yes

maximal, characteristic, complemented

#19151346

C₂

yes

maximal, complemented

#333×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

C₄

yes

—

#15668012

C₂²

yes

characteristic

#15668011

—

#15668010

C₄

yes

—

C₄⋊C₄×D₈₉

—

C₄²×D₈₉

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667924

—

#15667922

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

Q₈×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂²⋊C₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²⋊C₄

yes

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

#100

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#137

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#100

yes

characteristic

C₂×C₈₉

#419

yes

—

D₈₉

—

#1558

—

maximal, Sylow

C₈₉

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₂²⋊C₈

—

178

—

#97

—

178

—

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂×Q₈

C₄×D₈₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

356

—

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

178

—

103

C₂×C₄

—

178

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

178

—

106

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

107

C₂×C₄

D₄×D₈₉

yes

—

108

C₈

—

109

C₈

—

356

—

110

C₂²

—

111

C₂²

—

178

—

112

C₂²

C₂²⋊C₄×D₈₉

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

356

—

116

C₄

—

178

—

117

C₄

—

178

—

118

C₄

#15668041

yes

—

119

C₄

#15668044

yes

—

120

C₂

—

178

—

121

C₂

#100×D₈₉

yes

characteristic

122

C₂

#19151605

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#333×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₈₉	C₂²⋊C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#333	D₈₉
hypocenter	C₈₉	#1558
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#333×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₈₉

C₂²⋊C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #19151347	C₂	semidirect
4. #19151346	C₂	semidirect
5. #333×C₈₉	C₂	semidirect
62. C₈₉	#1558	semidirect
67. #333	D₈₉	semidirect

H ≅ G/N

type

3. #19151347

C₂

semidirect

4. #19151346

C₂

semidirect

5. #333×C₈₉

C₂

semidirect

62. C₈₉

#1558

semidirect

67. #333

D₈₉

semidirect

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

subgroup	index
C₄⋊Q₈×D₈₉	2
#19151654	2
#19151347	2
#19151346	2
#333×C₈₉	2
#1558	89

subgroup

index

C₄⋊Q₈×D₈₉

#333×C₈₉

Subgroups — #26018976

Conjugacy classes of subgroups

Complements

Maximal subgroups