Group #26019330 — subgroups

Subgroups — #26019330

Order 11392.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19151972	C₂	1	1	yes	maximal, characteristic
2	#19151798	C₂	1	2	yes	maximal
3	#19151347	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#10623494	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083561	—	2	2	no	—
7	C₂×#13083561	—	2	2	no	—
8	#15668009	C₂²	1	1	yes	characteristic
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	#15667932	—	2	2	no	—
11	C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₈₉	—	2	2	no	—
13	C₄⋊Q₈×C₈₉	—	2	2	no	—
14	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₈₉	—	2	2	no	—
16	C₄×Q₈×C₈₉	C₂²	1	2	yes	—
17	C₂×D₃₅₆	—	2	1	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	#13083561	—	4	4	no	—
20	#13083561	—	4	4	no	—
21	C₂×#10623494	—	2	1	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×Q₈×C₈₉	—	2	2	no	—
24	C₂×Q₈×C₈₉	D₄	1	2	yes	—
25	C₂×Q₈×C₈₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₈₉	—	2	4	no	—
27	C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
30	C₄²×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×D₈₉	—	4	1	no	—
34	Q₈×C₈₉	—	2	2	no	—
35	Q₈×C₈₉	—	4	2	no	—
36	Q₈×C₈₉	—	8	2	no	—
37	Q₈×C₈₉	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
43	D₃₅₆	—	4	2	no	—
44	D₃₅₆	—	4	2	no	—
45	D₃₅₆	—	4	2	no	—
46	#10623494	—	2	2	no	—
47	#10623494	—	4	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	4	no	—
52	C₄×C₈₉	—	2	1	no	—
53	C₄×C₈₉	—	4	2	no	—
54	C₄×C₈₉	—	4	2	no	—
55	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
56	C₄×C₈₉	#137	1	2	yes	—
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	D₈₉	1	1	yes	characteristic, complemented
63	#492	—	89	1	no	—
64	#460	—	89	2	no	—
65	#333	—	89	1	no	—
66	Q₈⋊C₈	—	89	2	no	—
67	C₂×SD₁₆	—	178	2	no	—
68	C₂×SD₁₆	—	178	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
72	#127	—	89	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₈₉	1	2	yes	—
75	C₄⋊C₈	—	89	2	no	—
76	#102	—	178	2	no	—
77	C₄×C₈	—	89	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₈₉	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	—	178	2	no	—
86	C₂×C₈	—	178	1	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₈₉	1	1	yes	characteristic
94	C₂³	—	356	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#13083561	1	4	yes	complemented
99	D₄	—	356	2	no	—
100	D₄	—	356	2	no	—
101	D₄	—	356	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
107	C₈	—	356	2	no	—
108	C₈	—	178	2	no	—
109	C₂²	—	712	2	no	—
110	C₂²	—	712	1	no	—
111	C₂²	#15667981	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#13083561	1	2	yes	—
117	C₄	#15668043	1	2	yes	—
118	C₂	—	1424	1	no	—
119	C₂	#19151737	1	2	yes	—
120	C₂	#19151766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151972

C₂

yes

maximal, characteristic

#19151798

C₂

yes

maximal

#19151347

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10623494

C₂

yes

maximal, complemented

Q₈²×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083561

—

C₂×#13083561

—

#15668009

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667932

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

C₂²

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

SD₁₆

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

#137

yes

—

C₂×C₈₉

#448

yes

—

C₂×C₈₉

#453

yes

characteristic

D₈₉

—

#1623

—

maximal, Sylow

C₈₉

#1623

yes

characteristic, complemented, Sylow

Q₈²

D₈₉

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₂×SD₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₉

yes

—

C₄⋊C₈

—

#102

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

SD₁₆

—

356

—

SD₁₆

—

356

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

356

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#13083561

yes

complemented

D₄

—

356

—

100

D₄

—

356

—

101

D₄

—

356

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

107

C₈

—

356

—

108

C₈

—

178

—

109

C₂²

—

712

—

110

C₂²

—

712

—

111

C₂²

#15667981

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#13083561

yes

—

117

C₄

#15668043

yes

—

118

C₂

—

1424

—

119

C₂

#19151737

yes

—

120

C₂

#19151766

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	Q₈²×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#15668009	C₂²
hypercenter	Q₈²	D₈₉
hypocenter	C₈₉	#1623
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

Q₈²×C₈₉

C₂

C₂²×C₈₉

C₂²

Q₈²

D₈₉

hypocenter

C₈₉

#1623

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #19151347	C₂	semidirect
4. Q₈⋊#10623494	C₂	semidirect
5. Q₈²×C₈₉	C₂	semidirect
37. Q₈×C₈₉	SD₁₆	semidirect
61. C₈₉	#1623	semidirect
62. Q₈²	D₈₉	semidirect
98. Q₈	#13083561	semidirect

H ≅ G/N

type

3. #19151347

C₂

semidirect

4. Q₈⋊#10623494

C₂

semidirect

5. Q₈²×C₈₉

C₂

semidirect

37. Q₈×C₈₉

SD₁₆

semidirect

61. C₈₉

#1623

semidirect

62. Q₈²

D₈₉

semidirect

98. Q₈

#13083561

semidirect

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

subgroup	index
#19151972	2
#19151798	2
#19151347	2
Q₈⋊#10623494	2
Q₈²×C₈₉	2
#1623	89

subgroup

index

Q₈²×C₈₉

Subgroups — #26019330

Conjugacy classes of subgroups

Complements

Maximal subgroups