Group #26018529 — subgroups

Subgroups — C₈×#13083557 #26018529

Order 11392.

224 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19151649	C₂	1	1	yes	maximal, characteristic
2	C₄×#13083557	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083557	C₄	1	2	yes	—
9	C₂×C₈×D₈₉	C₄	1	2	yes	—
10	C₄²×D₈₉	C₂²	1	1	yes	characteristic
11	#15667941	C₄	1	2	yes	—
12	#15667924	C₄	1	2	yes	—
13	#15667922	C₄	1	2	yes	—
14	C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₈×Dic₈₉	C₄	1	2	yes	—
18	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
24	#13083557	C₈	1	8	yes	direct product factor
25	C₂×#10623494	C₂×C₄	1	2	yes	—
26	C₂×#10623494	C₂×C₄	1	2	yes	—
27	C₂×#10623494	C₂×C₄	1	2	yes	—
28	C₈×D₈₉	—	2	4	no	—
29	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×D₈₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₈₉	C₄²	1	1	yes	characteristic
41	C₂×Dic₈₉	C₄²	1	1	yes	characteristic
42	C₂×Dic₈₉	C₄²	1	1	yes	characteristic
43	C₄×D₈₉	—	2	2	no	—
44	C₄×D₈₉	—	2	2	no	—
45	C₄×D₈₉	C₂×C₈	1	4	yes	—
46	C₈×C₈₉	—	2	2	no	—
47	C₈×C₈₉	C₂×C₈	1	4	yes	complemented
48	C₈×C₈₉	M₁₆	1	4	yes	complemented
49	#10623494	—	2	2	no	—
50	#10623494	—	2	2	no	—
51	#10623494	C₂×C₈	1	4	yes	complemented
52	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₈₉	—	2	1	no	—
54	C₂×D₈₉	—	2	1	no	—
55	C₂×D₈₉	C₄×C₈	1	2	yes	—
56	C₄×C₈₉	C₂²×C₈	1	2	yes	—
57	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
58	C₄×C₈₉	#131	1	2	yes	—
59	Dic₈₉	—	2	1	no	—
60	Dic₈₉	—	2	1	no	—
61	Dic₈₉	C₄×C₈	1	2	yes	—
62	C₂×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂×C₈₉	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₈₉	#405	1	1	yes	characteristic
65	D₈₉	—	2	2	no	—
66	C₈×M₁₆	—	89	1	no	maximal, Sylow
67	C₈₉	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	#431	—	89	1	no	—
69	C₄×M₁₆	—	89	1	no	—
70	C₂×C₄×C₈	—	89	1	no	—
71	#322	—	89	1	no	—
72	#322	—	89	1	no	—
73	C₈²	—	89	1	no	—
74	C₈²	D₈₉	1	1	yes	characteristic, complemented
75	C₂×M₁₆	—	89	2	no	—
76	C₂²×C₈	—	89	2	no	—
77	C₂×C₄²	—	89	1	no	—
78	C₄⋊C₈	—	89	2	no	—
79	C₂²⋊C₈	—	89	2	no	—
80	#97	—	89	2	no	—
81	C₄×C₈	—	89	2	no	—
82	C₄×C₈	—	89	1	no	—
83	C₄×C₈	—	89	1	no	—
84	C₄×C₈	—	89	1	no	—
85	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
87	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂²×C₄	—	89	1	no	—
89	C₂²×C₄	—	89	1	no	—
90	C₂²×C₄	—	89	1	no	—
91	M₁₆	—	89	8	no	—
92	C₂×C₈	—	89	2	no	—
93	C₂×C₈	—	89	2	no	—
94	C₂×C₈	—	89	2	no	—
95	C₂×C₈	—	178	4	no	—
96	C₂×C₈	C₄×D₈₉	1	2	yes	—
97	C₂×C₈	C₄×D₈₉	1	2	yes	—
98	C₂×C₈	C₄×D₈₉	1	2	yes	—
99	C₄²	—	89	1	no	—
100	C₄²	—	89	1	no	—
101	C₄²	—	89	1	no	—
102	C₄²	C₂²×D₈₉	1	1	yes	characteristic
103	C₂³	—	89	1	no	—
104	C₂×C₄	—	89	1	no	—
105	C₂×C₄	—	89	4	no	—
106	C₂×C₄	—	89	1	no	—
107	C₂×C₄	—	89	1	no	—
108	C₂×C₄	—	178	2	no	—
109	C₂×C₄	—	178	2	no	—
110	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	89	4	no	—
115	C₈	—	178	2	no	—
116	C₈	—	178	2	no	—
117	C₈	C₈×D₈₉	1	4	yes	complemented
118	C₈	#13083557	1	4	yes	direct product factor
119	C₂²	—	89	2	no	—
120	C₂²	—	178	1	no	—
121	C₂²	—	178	1	no	—
122	C₂²	C₄²×D₈₉	1	1	yes	characteristic
123	C₄	—	89	2	no	—
124	C₄	—	178	1	no	—
125	C₄	—	178	1	no	—
126	C₄	C₂×C₈×D₈₉	1	2	yes	—
127	C₄	C₂×#13083557	1	2	yes	—
128	C₄	#15668018	1	2	yes	—
129	C₂	—	178	2	no	—
130	C₂	C₄×C₈×D₈₉	1	1	yes	characteristic
131	C₂	C₄×#13083557	1	1	yes	characteristic
132	C₂	#19151552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151649

C₂

yes

maximal, characteristic

C₄×#13083557

C₂

yes

maximal, characteristic

C₄×C₈×D₈₉

C₂

yes

maximal, characteristic

#19151320

C₂

yes

maximal, characteristic, complemented

#19151319

C₂

yes

maximal, characteristic, complemented

C₈×#10623494

C₂

yes

maximal, characteristic, complemented

C₈²×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083557

C₄

yes

—

C₂×C₈×D₈₉

C₄

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667941

C₄

yes

—

#15667924

C₄

yes

—

#15667922

C₄

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₈×Dic₈₉

C₄

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

#13083557

C₈

yes

direct product factor

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₈×D₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

C₄²

yes

characteristic

C₂×Dic₈₉

C₄²

yes

characteristic

C₂×Dic₈₉

C₄²

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

C₂×C₈

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₈

yes

complemented

C₈×C₈₉

M₁₆

yes

—

—

C₂×C₈

yes

complemented

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₄×C₈

yes

—

C₄×C₈₉

C₂²×C₈

yes

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

#131

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₄×C₈

yes

—

C₂×C₈₉

C₂×C₄×C₈

yes

characteristic

C₂×C₈₉

C₄×M₁₆

yes

characteristic

C₂×C₈₉

#405

yes

characteristic

D₈₉

—

C₈×M₁₆

—

maximal, Sylow

C₈₉

C₈×M₁₆

yes

characteristic, complemented, Sylow

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₈²

—

C₈²

D₈₉

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₈₉

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

111

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

112

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

178

—

116

C₈

—

178

—

117

C₈

C₈×D₈₉

yes

complemented

118

C₈

#13083557

yes

direct product factor

119

C₂²

—

120

C₂²

—

178

—

121

C₂²

—

178

—

122

C₂²

C₄²×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

178

—

125

C₄

—

178

—

126

C₄

C₂×C₈×D₈₉

yes

—

127

C₄

C₂×#13083557

yes

—

128

C₄

#15668018

yes

—

129

C₂

—

178

—

130

C₂

C₄×C₈×D₈₉

yes

characteristic

131

C₂

C₄×#13083557

yes

characteristic

132

C₂

#19151552

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₈₉
Fitting	C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄²
prime radical	C₂²×D₈₉	C₄²
cosocle	C₄²×C₈₉	C₂³
hypercenter	C₈²	D₈₉
hypocenter	C₈₉	C₈×M₁₆
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×D₈₉

commutator

C₂×C₈₉

C₂×C₄×C₈

Frattini

C₄²

C₂²×D₈₉

Fitting

C₈²×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄²

prime radical

C₂²×D₈₉

C₄²

cosocle

C₄²×C₈₉

C₂³

hypercenter

C₈²

D₈₉

hypocenter

C₈₉

C₈×M₁₆

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
4. #19151320	C₂	semidirect
5. #19151319	C₂	semidirect
6. C₈×#10623494	C₂	semidirect
7. C₈²×C₈₉	C₂	semidirect
24. #13083557	C₈	direct (N × H)
47. C₈×C₈₉	C₂×C₈	semidirect
48. C₈×C₈₉	M₁₆	semidirect
51. #10623494	C₂×C₈	semidirect
67. C₈₉	C₈×M₁₆	semidirect
74. C₈²	D₈₉	semidirect
117. C₈	C₈×D₈₉	semidirect
118. C₈	#13083557	direct (N × H)

H ≅ G/N

type

4. #19151320

C₂

semidirect

5. #19151319

C₂

semidirect

6. C₈×#10623494

C₂

semidirect

7. C₈²×C₈₉

C₂

semidirect

24. #13083557

C₈

direct (N × H)

47. C₈×C₈₉

C₂×C₈

semidirect

48. C₈×C₈₉

M₁₆

semidirect

51. #10623494

C₂×C₈

semidirect

67. C₈₉

C₈×M₁₆

semidirect

74. C₈²

D₈₉

semidirect

117. C₈

C₈×D₈₉

semidirect

118. C₈

#13083557

direct (N × H)

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

subgroup	index
#19151649	2
C₄×#13083557	2
C₄×C₈×D₈₉	2
#19151320	2
#19151319	2
C₈×#10623494	2
C₈²×C₈₉	2
C₈×M₁₆	89

subgroup

index

C₄×C₈×D₈₉

C₈²×C₈₉

C₈×M₁₆

Subgroups — C₈×#13083557 #26018529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#13083557 #26018529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13083557 #26018529