Group #26016552 — subgroups

Subgroups — C₈₉×#2078 #26016552

Order 11392.

198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 76 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×#107×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#417×C₈₉	C₂	1	2	yes	maximal
3	#408×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#343×C₈₉	C₂	1	2	yes	maximal, complemented
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
11	#107×C₈₉	—	2	2	no	—
12	#103×C₈₉	—	2	2	no	—
13	#102×C₈₉	—	2	2	no	—
14	C₂²⋊C₈×C₈₉	—	2	2	no	—
15	#41×C₈₉	—	2	2	no	—
16	#41×C₈₉	—	2	2	no	—
17	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
18	C₂×D₄×C₈₉	—	2	1	no	—
19	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₉	—	2	1	no	—
21	C₂²×C₄×C₈₉	—	2	2	no	—
22	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₈₉	—	2	2	no	—
24	M₁₆×C₈₉	—	2	2	no	—
25	M₁₆×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	—	2	1	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	D₄	1	2	yes	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂³×C₈₉	—	2	1	no	—
33	C₂³×C₈₉	#41	1	1	yes	characteristic
34	Q₈×C₈₉	—	2	2	no	—
35	D₄×C₈₉	—	2	2	no	—
36	D₄×C₈₉	—	4	2	no	—
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	4	1	no	—
39	C₂×C₄×C₈₉	—	4	1	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	—	8	2	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₄×C₈₉	#41	1	2	yes	—
46	C₈×C₈₉	—	2	2	no	—
47	C₈×C₈₉	—	4	2	no	—
48	C₈×C₈₉	—	4	1	no	—
49	C₂²×C₈₉	—	2	1	no	—
50	C₂²×C₈₉	—	2	1	no	—
51	C₂²×C₈₉	—	4	1	no	—
52	C₂²×C₈₉	—	4	1	no	—
53	C₂²×C₈₉	—	4	1	no	—
54	C₂²×C₈₉	#121	1	1	yes	characteristic
55	C₂²×C₈₉	#126	1	2	yes	—
56	C₄×C₈₉	—	4	1	no	—
57	C₄×C₈₉	—	8	2	no	—
58	C₄×C₈₉	#123	1	1	yes	characteristic
59	C₄×C₈₉	#123	1	2	yes	—
60	C₄×C₈₉	#124	1	1	yes	characteristic
61	C₂×C₈₉	—	2	2	no	—
62	C₂×C₈₉	—	8	1	no	—
63	C₂×C₈₉	#397	1	1	yes	characteristic
64	C₂×C₈₉	#472	1	1	yes	characteristic
65	C₂×C₈₉	#473	1	1	yes	characteristic
66	#2078	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₈₉	#2078	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#107	C₂×C₈₉	1	1	yes	characteristic, complemented
69	#417	C₂×C₈₉	1	2	yes	—
70	#408	C₂×C₈₉	1	1	yes	characteristic
71	#343	C₂×C₈₉	1	2	yes	complemented
72	#341	C₂×C₈₉	1	1	yes	characteristic, complemented
73	C₂×#41	C₂²×C₈₉	1	1	yes	characteristic
74	C₂×M₁₆	C₂²×C₈₉	1	2	yes	—
75	C₂×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₈₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
78	#107	—	2	2	no	—
79	#103	—	2	2	no	—
80	#102	—	2	2	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	D₄×C₈₉	1	1	yes	characteristic
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	D₄×C₈₉	1	1	yes	characteristic
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂³×C₈₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	D₄×C₈₉	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	#41×C₈₉	1	1	yes	characteristic
101	Q₈	—	2	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	4	2	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
111	C₂×C₄	#41×C₈₉	1	2	yes	—
112	C₂×C₄	#41×C₈₉	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#121×C₈₉	1	1	yes	characteristic
122	C₂²	#126×C₈₉	1	2	yes	—
123	C₄	—	4	1	no	—
124	C₄	—	8	2	no	—
125	C₄	#123×C₈₉	1	1	yes	characteristic
126	C₄	#123×C₈₉	1	2	yes	—
127	C₄	#124×C₈₉	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	8	1	no	—
130	C₂	#397×C₈₉	1	1	yes	characteristic
131	C₂	#472×C₈₉	1	1	yes	characteristic
132	C₂	#473×C₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₈₉

C₂

yes

maximal, characteristic, complemented

#417×C₈₉

C₂

yes

maximal

#408×C₈₉

C₂

yes

maximal, characteristic

#343×C₈₉

C₂

yes

maximal, complemented

#341×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

#107×C₈₉

—

#103×C₈₉

—

#102×C₈₉

—

C₂²⋊C₈×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#126

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

characteristic

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#124

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#2078

yes

characteristic, direct product factor, Sylow

C₂×#107

C₂×C₈₉

yes

characteristic, complemented

#417

C₂×C₈₉

yes

—

#408

C₂×C₈₉

yes

characteristic

#343

C₂×C₈₉

yes

complemented

#341

C₂×C₈₉

yes

characteristic, complemented

C₂×#41

C₂²×C₈₉

yes

characteristic

C₂×M₁₆

C₂²×C₈₉

yes

—

C₂×M₁₆

C₂²×C₈₉

yes

characteristic

C₂²×C₈

C₂²×C₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₈₉

yes

—

#107

—

#103

—

#102

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₄×C₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

D₄×C₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

100

C₂³

#41×C₈₉

yes

characteristic

101

Q₈

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

110

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

111

C₂×C₄

#41×C₈₉

yes

—

112

C₂×C₄

#41×C₈₉

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#121×C₈₉

yes

characteristic

122

C₂²

#126×C₈₉

yes

—

123

C₄

—

124

C₄

—

125

C₄

#123×C₈₉

yes

characteristic

126

C₄

#123×C₈₉

yes

—

127

C₄

#124×C₈₉

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#397×C₈₉

yes

characteristic

131

C₂

#472×C₈₉

yes

characteristic

132

C₂

#473×C₈₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₈₉
prime radical	C₂×#41×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₈₉

prime radical

C₂×#41×C₈₉

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

N	H ≅ G/N	type
1. C₂×#107×C₈₉	C₂	semidirect
4. #343×C₈₉	C₂	semidirect
5. #341×C₈₉	C₂	semidirect
66. #2078	C₈₉	direct (N × H)
67. C₈₉	#2078	direct (N × H)
68. C₂×#107	C₂×C₈₉	semidirect
71. #343	C₂×C₈₉	semidirect
72. #341	C₂×C₈₉	semidirect

H ≅ G/N

type

1. C₂×#107×C₈₉

C₂

semidirect

4. #343×C₈₉

C₂

semidirect

5. #341×C₈₉

C₂

semidirect

66. #2078

C₈₉

direct (N × H)

67. C₈₉

#2078

direct (N × H)

68. C₂×#107

C₂×C₈₉

semidirect

71. #343

C₂×C₈₉

semidirect

72. #341

C₂×C₈₉

semidirect

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

subgroup	index
C₂×#107×C₈₉	2
#417×C₈₉	2
#408×C₈₉	2
#343×C₈₉	2
#341×C₈₉	2
#2078	89

subgroup

index

C₂×#107×C₈₉

#417×C₈₉

#408×C₈₉

#343×C₈₉

#341×C₈₉

#2078

Subgroups — C₈₉×#2078 #26016552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#2078 #26016552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#2078 #26016552