Group #26018986 — subgroups

Subgroups — #26018986

Order 11392.

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

normal subgroups only

This group has 254 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	#19152149	C₂	1	1	yes	maximal, characteristic
2	#19151620	C₂	1	2	yes	maximal, complemented
3	#19151347	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151341	C₂	1	2	yes	maximal, complemented
5	#333×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083575	C₄	1	2	yes	—
7	C₂×#13083557	—	2	2	no	—
8	#15668009	C₂²	1	1	yes	characteristic
9	#15668001	C₄	1	2	yes	—
10	#15667987	—	2	2	no	—
11	#15667974	—	2	2	no	—
12	C₄²×D₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊#10623494	C₂²	1	2	yes	—
14	#15667941	—	2	2	no	—
15	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	#13083575	—	4	4	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₈₉	—	2	2	no	—
22	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₈₉	D₄	1	2	yes	—
24	#13083557	—	4	4	no	—
25	C₂×#10623494	—	2	2	no	—
26	#13083551	—	2	4	no	—
27	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₈₉	D₄	1	2	yes	—
29	C₂×Q₈×C₈₉	C₂×C₄	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	—	2	2	no	—
34	C₂²×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₈₉	—	4	2	no	—
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
39	C₂×Dic₈₉	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	D₃₅₆	—	4	2	no	—
42	D₃₅₆	—	4	4	no	—
43	D₃₅₆	—	4	2	no	—
44	D₃₅₆	—	4	2	no	—
45	C₄×D₈₉	—	2	2	no	—
46	C₄×D₈₉	—	2	2	no	—
47	C₄×D₈₉	—	2	2	no	—
48	C₄×D₈₉	—	4	4	no	—
49	C₈×C₈₉	—	4	2	no	—
50	#10623494	—	4	2	no	—
51	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₈₉	—	2	1	no	—
53	C₂×D₈₉	—	2	2	no	—
54	C₂×D₈₉	—	4	2	no	—
55	C₂×D₈₉	—	4	4	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	4	2	no	—
58	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₈₉	#137	1	2	yes	—
60	Dic₈₉	—	2	2	no	—
61	Dic₈₉	#100	1	2	yes	—
62	C₂×C₈₉	C₂×#100	1	1	yes	characteristic
63	C₂×C₈₉	#417	1	2	yes	—
64	D₈₉	—	4	1	no	—
65	D₈₉	—	8	2	no	—
66	#1560	—	89	1	no	maximal, Sylow
67	C₈₉	#1560	1	1	yes	characteristic, complemented, Sylow
68	#532	—	89	1	no	—
69	#423	—	89	2	no	—
70	#333	—	89	1	no	—
71	#333	D₈₉	1	1	yes	characteristic, complemented
72	#331	—	89	2	no	—
73	C₂×#41	—	89	2	no	—
74	C₂×M₁₆	—	178	2	no	—
75	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
76	#127	—	89	1	no	—
77	#124	—	89	2	no	—
78	#121	—	178	2	no	—
79	C₄×D₄	—	178	2	no	—
80	C₂×C₄²	—	89	1	no	—
81	C₄⋊C₈	—	89	2	no	—
82	C₄⋊C₈	—	178	2	no	—
83	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
84	#41	—	356	4	no	—
85	C₂×Q₈	C₄×D₈₉	1	2	yes	—
86	C₂×D₄	—	89	2	no	—
87	C₂×D₄	—	178	2	no	—
88	C₂×D₄	—	178	2	no	—
89	C₂²×C₄	—	89	1	no	—
90	C₂²×C₄	—	89	2	no	—
91	C₂²×C₄	—	178	2	no	—
92	M₁₆	—	356	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	178	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	178	4	no	—
97	C₄²	—	89	1	no	—
98	C₄²	—	89	2	no	—
99	C₄²	C₂²×D₈₉	1	1	yes	characteristic
100	C₂³	—	89	1	no	—
101	C₂³	—	178	2	no	—
102	Q₈	—	4	2	no	—
103	D₄	—	356	2	no	—
104	D₄	—	356	4	no	—
105	D₄	—	356	2	no	—
106	D₄	—	356	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	89	1	no	—
109	C₂×C₄	—	89	2	no	—
110	C₂×C₄	—	356	4	no	—
111	C₂×C₄	—	178	2	no	—
112	C₂×C₄	—	178	2	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₂×C₄	D₄×D₈₉	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	356	2	no	—
118	C₂²	—	356	2	no	—
119	C₂²	—	356	4	no	—
120	C₂²	—	178	1	no	—
121	C₂²	—	178	2	no	—
122	C₂²	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	89	2	no	—
126	C₄	—	178	2	no	—
127	C₄	#15668039	1	2	yes	—
128	C₄	#15668044	1	2	yes	—
129	C₂	—	356	1	no	—
130	C₂	—	712	2	no	—
131	C₂	#100×D₈₉	1	1	yes	characteristic
132	C₂	#19151598	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152149

C₂

yes

maximal, characteristic

#19151620

C₂

yes

maximal, complemented

#19151347

C₂

yes

maximal, characteristic, complemented

#19151341

C₂

yes

maximal, complemented

#333×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083575

C₄

yes

—

C₂×#13083557

—

#15668009

C₂²

yes

characteristic

#15668001

C₄

yes

—

#15667987

—

#15667974

—

C₄²×D₈₉

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667941

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

#13083575

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

—

#13083557

—

C₂×#10623494

—

#13083551

—

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×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

C₂²⋊C₄

yes

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

#100

yes

—

C₂×C₈₉

C₂×#100

yes

characteristic

C₂×C₈₉

#417

yes

—

D₈₉

—

D₈₉

—

#1560

—

maximal, Sylow

C₈₉

#1560

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

#331

—

C₂×#41

—

C₂×M₁₆

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

—

—

178

—

C₄×D₄

—

178

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#41

—

356

—

C₂×Q₈

C₄×D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

100

C₂³

—

101

C₂³

—

178

—

102

Q₈

—

103

D₄

—

356

—

104

D₄

—

356

—

105

D₄

—

356

—

106

D₄

—

356

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

356

—

111

C₂×C₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

115

C₂×C₄

D₄×D₈₉

yes

—

116

C₈

—

117

C₈

—

356

—

118

C₂²

—

356

—

119

C₂²

—

356

—

120

C₂²

—

178

—

121

C₂²

—

178

—

122

C₂²

C₂²⋊C₄×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

178

—

127

C₄

#15668039

yes

—

128

C₄

#15668044

yes

—

129

C₂

—

356

—

130

C₂

—

712

—

131

C₂

#100×D₈₉

yes

characteristic

132

C₂

#19151598

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	#19152149	C₂
cosocle	C₄²×C₈₉	C₂³
hypercenter	#333	D₈₉
hypocenter	C₈₉	#1560
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

#19152149

C₂

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #19151620	C₂	semidirect
3. #19151347	C₂	semidirect
4. #19151341	C₂	semidirect
5. #333×C₈₉	C₂	semidirect
67. C₈₉	#1560	semidirect
71. #333	D₈₉	semidirect

H ≅ G/N

type

2. #19151620

C₂

semidirect

3. #19151347

C₂

semidirect

4. #19151341

C₂

semidirect

5. #333×C₈₉

C₂

semidirect

67. C₈₉

#1560

semidirect

71. #333

D₈₉

semidirect

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

subgroup	index
#19152149	2
#19151620	2
#19151347	2
#19151341	2
#333×C₈₉	2
#1560	89

subgroup

index

#333×C₈₉

Subgroups — #26018986

Conjugacy classes of subgroups

Complements

Maximal subgroups