Group #26021225 — subgroups

Subgroups — #26021225

Order 11392.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151464	C₂	1	1	yes	maximal, characteristic
3	#19151361	C₂	1	4	yes	maximal
4	C₂×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₈	—	2	1	no	—
6	#15668012	—	2	2	no	—
7	#15668012	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₈₉	—	2	1	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#15667919	—	2	1	no	—
12	#15667919	—	2	1	no	—
13	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	#97×C₈₉	—	2	2	no	—
16	C₂×Dic₁₇₈	—	2	4	no	—
17	C₂×Dic₁₇₈	—	4	4	no	—
18	C₂²×Dic₈₉	—	2	2	no	—
19	C₂²×Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	4	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	4	no	—
23	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
30	C₄²×C₈₉	C₂×C₄	1	2	yes	—
31	C₄²×C₈₉	D₄	1	2	yes	—
32	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₈₉	#41	1	2	yes	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	4	no	—
43	C₂×Dic₈₉	—	4	2	no	—
44	C₂×Dic₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	Dic₁₇₈	—	4	8	no	—
47	C₈×C₈₉	—	2	4	no	—
48	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈₉	#124	1	1	yes	characteristic
50	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₈₉	#137	1	2	yes	—
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	C₄×Q₈	1	2	yes	—
56	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
57	Dic₈₉	—	4	4	no	—
58	Dic₈₉	—	8	2	no	—
59	C₂×C₈₉	#391	1	1	yes	characteristic
60	C₂×C₈₉	#418	1	1	yes	characteristic
61	C₂×C₈₉	#419	1	1	yes	characteristic
62	C₂×C₈₉	#489	1	2	yes	—
63	C₂×C₈₉	#501	1	2	yes	—
64	#1993	—	89	1	no	maximal, Sylow
65	C₈₉	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	89	1	no	—
67	C₂×#97	D₈₉	1	1	yes	characteristic
68	#382	—	89	1	no	—
69	#340	—	89	4	no	—
70	C₂²×Q₈	—	178	1	no	—
71	C₂²×C₈	C₂×D₈₉	1	2	yes	—
72	C₄⋊Q₈	—	89	4	no	—
73	C₄⋊Q₈	—	178	2	no	—
74	C₂×C₄⋊C₄	—	89	2	no	—
75	C₂×C₄⋊C₄	—	89	2	no	—
76	C₂×C₄⋊C₄	—	178	1	no	—
77	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	178	1	no	—
80	#95	—	178	1	no	—
81	C₂×Q₈	—	356	4	no	—
82	C₂×Q₈	—	178	4	no	—
83	C₂²×C₄	—	178	2	no	—
84	C₂²×C₄	—	178	2	no	—
85	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
86	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₁₇₈	1	4	yes	complemented
91	C₄⋊C₄	—	89	4	no	—
92	C₄⋊C₄	—	178	4	no	—
93	C₄⋊C₄	—	178	4	no	—
94	C₄⋊C₄	—	178	2	no	—
95	C₄²	C₄×D₈₉	1	2	yes	—
96	C₄²	C₂²⋊D₈₉	1	2	yes	—
97	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
98	Q₈	—	356	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	356	4	no	—
102	C₂×C₄	—	356	2	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	356	2	no	—
105	C₂×C₄	—	178	4	no	—
106	C₂×C₄	#13083551	1	2	yes	—
107	C₂×C₄	#13083551	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
111	C₂×C₄	#13083574	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×#13083551	1	1	yes	characteristic
114	C₂²	#15668000	1	1	yes	characteristic
115	C₂²	#15668012	1	1	yes	characteristic
116	C₂²	#15668040	1	2	yes	—
117	C₂²	#15668045	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	356	4	no	—
121	C₄	—	712	2	no	—
122	C₄	C₄×Dic₁₇₈	1	2	yes	—
123	C₄	#15667991	1	2	yes	—
124	C₂	#19151504	1	1	yes	characteristic
125	C₂	#19151604	1	1	yes	characteristic
126	C₂	#19151608	1	1	yes	characteristic
127	C₂	#19151958	1	2	yes	—
128	C₂	#19152007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

#19151464

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

C₂×#97×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

#15668012

—

#15668012

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₈

yes

complemented

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#124

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×Q₈

yes

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#501

yes

—

#1993

—

maximal, Sylow

C₈₉

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#97

D₈₉

yes

characteristic

#382

—

#340

—

C₂²×Q₈

—

178

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#97

—

#95

—

178

—

#95

—

178

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₇₈

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

C₄×D₈₉

yes

—

C₄²

C₂²⋊D₈₉

yes

—

C₂³

C₂×D₃₅₆

yes

characteristic

Q₈

—

356

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

—

356

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

178

—

106

C₂×C₄

#13083551

yes

—

107

C₂×C₄

#13083551

yes

—

108

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

110

C₂×C₄

C₂×Dic₁₇₈

yes

—

111

C₂×C₄

#13083574

yes

—

112

C₈

—

113

C₂²

C₂×#13083551

yes

characteristic

114

C₂²

#15668000

yes

characteristic

115

C₂²

#15668012

yes

characteristic

116

C₂²

#15668040

yes

—

117

C₂²

#15668045

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

356

—

121

C₄

—

712

—

122

C₄

C₄×Dic₁₇₈

yes

—

123

C₄

#15667991

yes

—

124

C₂

#19151504

yes

characteristic

125

C₂

#19151604

yes

characteristic

126

C₂

#19151608

yes

characteristic

127

C₂

#19151958

yes

—

128

C₂

#19152007

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#97	D₈₉
hypocenter	C₈₉	#1993
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#97×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×#97

D₈₉

hypocenter

C₈₉

#1993

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #15668012	C₄	semidirect
29. C₂×C₈×C₈₉	Q₈	semidirect
65. C₈₉	#1993	semidirect
90. C₂×C₈	Dic₁₇₈	semidirect

H ≅ G/N

type

7. #15668012

C₄

semidirect

29. C₂×C₈×C₈₉

Q₈

semidirect

65. C₈₉

#1993

semidirect

90. C₂×C₈

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15668012	2
#19151464	2
#19151361	2
C₂×#97×C₈₉	2
#1993	89

subgroup

index

C₂×#97×C₈₉

Subgroups — #26021225

Conjugacy classes of subgroups

Complements

Maximal subgroups