Group #26033356 — subgroups

Subgroups — C₂×D₁₆×D₈₉ #26033356

Order 11392.

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

normal subgroups only

This group has 440 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	C₂×D₈×D₈₉	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667953	C₂	1	2	yes	maximal, complemented
4	D₁₆×D₈₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×D₈₉	—	2	2	no	—
8	C₂×D₇₁₂	C₂²	1	2	yes	complemented
9	C₂×#13083559	—	2	2	no	—
10	D₈×D₈₉	—	2	4	no	—
11	D₈×D₈₉	C₂²	1	8	yes	complemented
12	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
13	D₁₄₂₄	C₂²	1	4	yes	complemented
14	#15667953	C₂²	1	8	yes	complemented
15	C₂×#13083549	C₂²	1	1	yes	characteristic, complemented
16	C₁₆×D₈₉	C₂²	1	4	yes	complemented
17	C₂×D₈×C₈₉	C₂²	1	2	yes	complemented
18	D₁₆×C₈₉	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₈₉	—	4	2	no	—
21	C₂×D₃₅₆	—	2	2	no	—
22	C₂×C₂²⋊D₈₉	—	4	2	no	—
23	D₄×D₈₉	—	2	8	no	—
24	D₄×D₈₉	—	4	4	no	—
25	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
26	D₇₁₂	—	2	2	no	—
27	D₇₁₂	C₂³	1	4	yes	complemented
28	#13083559	—	2	8	no	—
29	C₂×#10623494	D₄	1	1	yes	characteristic
30	C₈×D₈₉	D₄	1	2	yes	—
31	C₈×D₈₉	C₂³	1	2	yes	—
32	#13083549	C₂³	1	2	yes	complemented
33	C₂×D₄×C₈₉	—	2	2	no	—
34	D₈×C₈₉	—	2	2	no	—
35	D₈×C₈₉	C₂³	1	4	yes	complemented
36	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₈₉	C₂³	1	2	yes	complemented
38	C₂³×C₈₉	—	4	2	no	—
39	C₂²×D₈₉	—	4	2	no	—
40	C₂²×D₈₉	—	4	8	no	—
41	C₂²×D₈₉	—	8	8	no	—
42	C₂²×D₈₉	D₈	1	1	yes	characteristic
43	D₄×C₈₉	—	2	4	no	—
44	D₄×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊D₈₉	—	4	8	no	—
47	C₂×Dic₈₉	D₈	1	1	yes	characteristic
48	D₃₅₆	—	2	4	no	—
49	D₃₅₆	—	4	2	no	—
50	C₄×D₈₉	D₈	1	2	yes	—
51	C₄×D₈₉	C₂×D₄	1	2	yes	—
52	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₈×C₈₉	C₂⁴	1	1	yes	characteristic
54	#10623494	C₂×D₄	1	2	yes	—
55	C₂²×C₈₉	—	4	4	no	—
56	C₂²×C₈₉	—	8	4	no	—
57	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×D₈₉	—	4	4	no	—
59	C₂×D₈₉	—	8	4	no	—
60	C₂×D₈₉	—	8	16	no	—
61	C₂×D₈₉	D₁₆	1	4	yes	direct product factor
62	C₂×D₈₉	C₂×D₈	1	2	yes	—
63	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
64	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
65	Dic₈₉	C₂×D₈	1	2	yes	—
66	C₂×C₈₉	—	8	4	no	—
67	C₂×C₈₉	C₂×D₁₆	1	2	yes	complemented
68	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
69	D₈₉	—	8	4	no	—
70	D₈₉	C₂×D₁₆	1	4	yes	direct product factor
71	C₂²×D₁₆	—	89	1	no	maximal, Sylow
72	C₈₉	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂²×D₈	—	89	2	no	—
74	C₂×D₁₆	—	89	2	no	—
75	C₂×D₁₆	—	89	8	no	—
76	C₂×D₁₆	—	89	1	no	—
77	C₂×D₁₆	D₈₉	1	1	yes	characteristic, direct product factor
78	C₂²×C₁₆	—	89	1	no	—
79	C₂²×D₄	—	178	2	no	—
80	C₂×D₈	—	89	8	no	—
81	C₂×D₈	—	89	2	no	—
82	C₂×D₈	—	178	4	no	—
83	C₂×D₈	—	178	2	no	—
84	C₂×D₈	C₂×D₈₉	1	2	yes	complemented
85	C₂²×C₈	—	89	1	no	—
86	D₁₆	—	89	8	no	—
87	D₁₆	—	89	4	no	—
88	D₁₆	C₂×D₈₉	1	4	yes	direct product factor
89	C₂×C₁₆	—	89	1	no	—
90	C₂×C₁₆	—	89	4	no	—
91	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
92	C₂⁴	—	356	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	356	4	no	—
95	C₂×D₄	—	356	2	no	—
96	C₂×D₄	—	178	8	no	—
97	C₂×D₄	—	178	2	no	—
98	C₂²×C₄	—	89	1	no	—
99	D₈	—	2	2	no	—
100	D₈	—	89	4	no	—
101	D₈	—	178	8	no	—
102	D₈	—	178	2	no	—
103	D₈	C₂²×D₈₉	1	4	yes	complemented
104	C₂×C₈	—	89	1	no	—
105	C₂×C₈	—	89	2	no	—
106	C₂×C₈	—	89	2	no	—
107	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
108	C₁₆	—	89	2	no	—
109	C₁₆	C₂²×D₈₉	1	2	yes	complemented
110	C₂³	—	4	2	no	—
111	C₂³	—	89	1	no	—
112	C₂³	—	356	8	no	—
113	C₂³	—	356	2	no	—
114	C₂³	—	712	8	no	—
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	D₄	—	356	8	no	—
118	D₄	—	356	2	no	—
119	D₄	—	178	4	no	—
120	C₂×C₄	—	89	1	no	—
121	C₂×C₄	—	89	2	no	—
122	C₂×C₄	—	89	2	no	—
123	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
124	C₈	—	89	2	no	—
125	C₈	D₄×D₈₉	1	1	yes	characteristic
126	C₈	C₂³×D₈₉	1	1	yes	characteristic
127	C₂²	—	4	4	no	—
128	C₂²	—	8	4	no	—
129	C₂²	—	89	4	no	—
130	C₂²	—	89	2	no	—
131	C₂²	—	356	4	no	—
132	C₂²	—	712	16	no	—
133	C₂²	—	712	4	no	—
134	C₂²	D₈×D₈₉	1	1	yes	characteristic
135	C₄	—	89	2	no	—
136	C₄	D₈×D₈₉	1	1	yes	characteristic
137	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
138	C₂	—	8	4	no	—
139	C₂	—	89	4	no	—
140	C₂	—	712	4	no	—
141	C₂	D₁₆×D₈₉	1	2	yes	direct product factor
142	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₈×D₈₉

C₂

yes

maximal, complemented

C₂×D₁₄₂₄

C₂

yes

maximal, characteristic, complemented

C₂×#15667953

C₂

yes

maximal, complemented

D₁₆×D₈₉

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₈₉

—

C₂×D₇₁₂

C₂²

yes

complemented

C₂×#13083559

—

D₈×D₈₉

—

D₈×D₈₉

C₂²

yes

complemented

C₂×C₈×D₈₉

C₂²

yes

characteristic

D₁₄₂₄

C₂²

yes

complemented

#15667953

C₂²

yes

complemented

C₂×#13083549

C₂²

yes

characteristic, complemented

C₁₆×D₈₉

C₂²

yes

complemented

C₂×D₈×C₈₉

C₂²

yes

complemented

D₁₆×C₈₉

C₂²

yes

complemented

C₂×C₁₆×C₈₉

C₂²

yes

characteristic, complemented

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

D₄×D₈₉

—

D₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

characteristic

D₇₁₂

—

D₇₁₂

C₂³

yes

complemented

#13083559

—

C₂×#10623494

D₄

yes

characteristic

C₈×D₈₉

D₄

yes

—

C₈×D₈₉

C₂³

yes

—

#13083549

C₂³

yes

complemented

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

C₂³

yes

complemented

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

C₂³

yes

complemented

C₂³×C₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

D₈

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

D₈

yes

—

C₄×D₈₉

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

C₂⁴

yes

characteristic

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

D₁₆

yes

direct product factor

C₂×D₈₉

C₂×D₈

yes

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂²×D₄

yes

characteristic

Dic₈₉

C₂×D₈

yes

—

C₂×C₈₉

—

C₂×C₈₉

C₂×D₁₆

yes

complemented

C₂×C₈₉

C₂²×D₈

yes

characteristic

D₈₉

—

D₈₉

C₂×D₁₆

yes

direct product factor

C₂²×D₁₆

—

maximal, Sylow

C₈₉

C₂²×D₁₆

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

D₈₉

yes

characteristic, direct product factor

C₂²×C₁₆

—

C₂²×D₄

—

178

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₂×D₈

C₂×D₈₉

yes

complemented

C₂²×C₈

—

D₁₆

—

D₁₆

—

D₁₆

C₂×D₈₉

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic, complemented

C₂⁴

—

356

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

D₈

—

100

D₈

—

101

D₈

—

178

—

102

D₈

—

178

—

103

D₈

C₂²×D₈₉

yes

complemented

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₈₉

yes

characteristic

108

C₁₆

—

109

C₁₆

C₂²×D₈₉

yes

complemented

110

C₂³

—

111

C₂³

—

112

C₂³

—

356

—

113

C₂³

—

356

—

114

C₂³

—

712

—

115

D₄

—

116

D₄

—

117

D₄

—

356

—

118

D₄

—

356

—

119

D₄

—

178

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₈₉

yes

characteristic

124

C₈

—

125

C₈

D₄×D₈₉

yes

characteristic

126

C₈

C₂³×D₈₉

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

356

—

132

C₂²

—

712

—

133

C₂²

—

712

—

134

C₂²

D₈×D₈₉

yes

characteristic

135

C₄

—

136

C₄

D₈×D₈₉

yes

characteristic

137

C₄

C₂×D₄×D₈₉

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

—

712

—

141

C₂

D₁₆×D₈₉

yes

direct product factor

142

C₂

C₂×D₈×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₈×C₈₉	C₂⁴
Frattini	C₈	C₂³×D₈₉
Fitting	C₂×D₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
hypercenter	C₂×D₁₆	D₈₉
hypocenter	C₈₉	C₂²×D₁₆
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₈×C₈₉

C₂⁴

Frattini

C₈

C₂³×D₈₉

Fitting

C₂×D₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

hypercenter

C₂×D₁₆

D₈₉

hypocenter

C₈₉

C₂²×D₁₆

LCS

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. C₂×D₈×D₈₉	C₂	semidirect
2. C₂×D₁₄₂₄	C₂	semidirect
3. C₂×#15667953	C₂	semidirect
4. D₁₆×D₈₉	C₂	direct (N × H)
5. C₂×C₁₆×D₈₉	C₂	semidirect
6. C₂×D₁₆×C₈₉	C₂	semidirect
8. C₂×D₇₁₂	C₂²	semidirect
11. D₈×D₈₉	C₂²	semidirect
13. D₁₄₂₄	C₂²	semidirect
14. #15667953	C₂²	semidirect
15. C₂×#13083549	C₂²	semidirect
16. C₁₆×D₈₉	C₂²	semidirect
17. C₂×D₈×C₈₉	C₂²	semidirect
18. D₁₆×C₈₉	C₂²	semidirect
19. C₂×C₁₆×C₈₉	C₂²	semidirect
27. D₇₁₂	C₂³	semidirect
32. #13083549	C₂³	semidirect
35. D₈×C₈₉	C₂³	semidirect
37. C₁₆×C₈₉	C₂³	semidirect
61. C₂×D₈₉	D₁₆	direct (N × H)
67. C₂×C₈₉	C₂×D₁₆	semidirect
70. D₈₉	C₂×D₁₆	direct (N × H)
72. C₈₉	C₂²×D₁₆	semidirect
77. C₂×D₁₆	D₈₉	direct (N × H)
84. C₂×D₈	C₂×D₈₉	semidirect
88. D₁₆	C₂×D₈₉	direct (N × H)
91. C₂×C₁₆	C₂×D₈₉	semidirect
103. D₈	C₂²×D₈₉	semidirect
109. C₁₆	C₂²×D₈₉	semidirect
141. C₂	D₁₆×D₈₉	direct (N × H)

H ≅ G/N

type

1. C₂×D₈×D₈₉

C₂

semidirect

2. C₂×D₁₄₂₄

C₂

semidirect

3. C₂×#15667953

C₂

semidirect

4. D₁₆×D₈₉

C₂

direct (N × H)

5. C₂×C₁₆×D₈₉

C₂

semidirect

6. C₂×D₁₆×C₈₉

C₂

semidirect

8. C₂×D₇₁₂

C₂²

semidirect

11. D₈×D₈₉

C₂²

semidirect

13. D₁₄₂₄

C₂²

semidirect

14. #15667953

C₂²

semidirect

15. C₂×#13083549

C₂²

semidirect

16. C₁₆×D₈₉

C₂²

semidirect

17. C₂×D₈×C₈₉

C₂²

semidirect

18. D₁₆×C₈₉

C₂²

semidirect

19. C₂×C₁₆×C₈₉

C₂²

semidirect

27. D₇₁₂

C₂³

semidirect

32. #13083549

C₂³

semidirect

35. D₈×C₈₉

C₂³

semidirect

37. C₁₆×C₈₉

C₂³

semidirect

61. C₂×D₈₉

D₁₆

direct (N × H)

67. C₂×C₈₉

C₂×D₁₆

semidirect

70. D₈₉

C₂×D₁₆

direct (N × H)

72. C₈₉

C₂²×D₁₆

semidirect

77. C₂×D₁₆

D₈₉

direct (N × H)

84. C₂×D₈

C₂×D₈₉

semidirect

88. D₁₆

C₂×D₈₉

direct (N × H)

91. C₂×C₁₆

C₂×D₈₉

semidirect

103. D₈

C₂²×D₈₉

semidirect

109. C₁₆

C₂²×D₈₉

semidirect

141. C₂

D₁₆×D₈₉

direct (N × H)

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

subgroup	index
C₂×D₈×D₈₉	2
C₂×D₁₄₂₄	2
C₂×#15667953	2
D₁₆×D₈₉	2
C₂×C₁₆×D₈₉	2
C₂×D₁₆×C₈₉	2
C₂²×D₁₆	89

subgroup

index

C₂×D₈×D₈₉

C₂×D₁₄₂₄

C₂×#15667953

D₁₆×D₈₉

C₂×C₁₆×D₈₉

C₂×D₁₆×C₈₉

C₂²×D₁₆

Subgroups — C₂×D₁₆×D₈₉ #26033356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D16×D89 #26033356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×D₈₉ #26033356