Group #26033363 — subgroups

Subgroups — C₂×Q₃₂×D₈₉ #26033363

Order 11392.

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

normal subgroups only

This group has 360 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×Q₁₆×D₈₉	C₂	1	2	yes	maximal
2	C₂×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667958	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₈₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₈₉	—	2	2	no	—
8	C₂×Dic₃₅₆	C₂²	1	2	yes	—
9	C₂×#13083564	—	2	2	no	—
10	Q₁₆×D₈₉	—	2	4	no	—
11	Q₁₆×D₈₉	C₂²	1	8	yes	—
12	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
13	Dic₇₁₂	C₂²	1	4	yes	complemented
14	#15667958	C₂²	1	8	yes	complemented
15	C₂×#13083549	C₂²	1	1	yes	characteristic
16	C₁₆×D₈₉	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
18	Q₃₂×C₈₉	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	4	2	no	—
22	Q₈×D₈₉	—	2	8	no	—
23	Q₈×D₈₉	—	4	4	no	—
24	C₂×C₄×D₈₉	—	4	2	no	—
25	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
26	Dic₃₅₆	—	2	2	no	—
27	Dic₃₅₆	C₂³	1	4	yes	—
28	#13083564	—	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	—
33	C₂×Q₈×C₈₉	—	2	2	no	—
34	Q₁₆×C₈₉	—	2	2	no	—
35	Q₁₆×C₈₉	C₂³	1	4	yes	—
36	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₈₉	C₂³	1	2	yes	—
38	C₂²×D₈₉	D₈	1	1	yes	characteristic
39	Q₈×C₈₉	—	2	4	no	—
40	Q₈×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₈₉	—	4	2	no	—
44	C₂×Dic₈₉	D₈	1	1	yes	characteristic
45	C₄×D₈₉	—	4	8	no	—
46	C₄×D₈₉	D₈	1	2	yes	—
47	C₄×D₈₉	C₂×D₄	1	2	yes	—
48	Dic₁₇₈	—	2	4	no	—
49	Dic₁₇₈	—	4	2	no	—
50	Dic₁₇₈	—	4	8	no	—
51	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₈×C₈₉	C₂⁴	1	1	yes	characteristic
53	#10623494	C₂×D₄	1	2	yes	—
54	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₂×D₈₉	Q₃₂	1	4	yes	direct product factor
56	C₂×D₈₉	C₂×D₈	1	2	yes	—
57	C₄×C₈₉	—	4	4	no	—
58	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
60	Dic₈₉	—	4	4	no	—
61	Dic₈₉	C₂×D₈	1	2	yes	—
62	C₂×C₈₉	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
64	D₈₉	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	89	1	no	maximal, Sylow
66	C₈₉	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	89	2	no	—
68	C₂×Q₃₂	—	89	2	no	—
69	C₂×Q₃₂	—	89	8	no	—
70	C₂×Q₃₂	—	89	1	no	—
71	C₂×Q₃₂	D₈₉	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	89	1	no	—
73	C₂²×Q₈	—	178	2	no	—
74	C₂×Q₁₆	—	89	8	no	—
75	C₂×Q₁₆	—	89	2	no	—
76	C₂×Q₁₆	—	178	4	no	—
77	C₂×Q₁₆	—	178	2	no	—
78	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
79	C₂²×C₈	—	89	1	no	—
80	Q₃₂	—	89	8	no	—
81	Q₃₂	—	89	4	no	—
82	Q₃₂	C₂×D₈₉	1	4	yes	direct product factor
83	C₂×C₁₆	—	89	1	no	—
84	C₂×C₁₆	—	89	4	no	—
85	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	356	2	no	—
88	C₂×Q₈	—	356	4	no	—
89	C₂×Q₈	—	178	8	no	—
90	C₂×Q₈	—	178	2	no	—
91	C₂²×C₄	—	89	1	no	—
92	C₂²×C₄	—	356	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	89	4	no	—
95	Q₁₆	—	178	8	no	—
96	Q₁₆	—	178	2	no	—
97	Q₁₆	C₂²×D₈₉	1	4	yes	—
98	C₂×C₈	—	89	1	no	—
99	C₂×C₈	—	89	2	no	—
100	C₂×C₈	—	89	2	no	—
101	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	C₁₆	—	89	2	no	—
103	C₁₆	C₂²×D₈₉	1	2	yes	—
104	C₂³	—	89	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	356	8	no	—
108	Q₈	—	356	2	no	—
109	Q₈	—	178	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	89	1	no	—
112	C₂×C₄	—	89	2	no	—
113	C₂×C₄	—	89	2	no	—
114	C₂×C₄	—	356	8	no	—
115	C₂×C₄	—	356	2	no	—
116	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
117	C₈	—	89	2	no	—
118	C₈	D₄×D₈₉	1	1	yes	characteristic
119	C₈	C₂³×D₈₉	1	1	yes	characteristic
120	C₂²	—	89	4	no	—
121	C₂²	—	89	2	no	—
122	C₂²	D₈×D₈₉	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	89	2	no	—
125	C₄	—	356	4	no	—
126	C₄	D₈×D₈₉	1	1	yes	characteristic
127	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
128	C₂	—	89	4	no	—
129	C₂	Q₃₂×D₈₉	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₈₉

C₂

yes

maximal

C₂×Dic₇₁₂

C₂

yes

maximal, characteristic, complemented

C₂×#15667958

C₂

yes

maximal, complemented

Q₃₂×D₈₉

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₈₉

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

—

C₂×Dic₃₅₆

C₂²

yes

—

C₂×#13083564

—

Q₁₆×D₈₉

—

Q₁₆×D₈₉

C₂²

yes

—

C₂×C₈×D₈₉

C₂²

yes

characteristic

Dic₇₁₂

C₂²

yes

complemented

#15667958

C₂²

yes

complemented

C₂×#13083549

C₂²

yes

characteristic

C₁₆×D₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

—

Q₃₂×C₈₉

C₂²

yes

complemented

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

Q₈×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

characteristic

Dic₃₅₆

—

Dic₃₅₆

C₂³

yes

—

#13083564

—

C₂×#10623494

D₄

yes

characteristic

C₈×D₈₉

D₄

yes

—

C₈×D₈₉

C₂³

yes

—

#13083549

C₂³

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

C₂³

yes

—

C₂²×D₈₉

D₈

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

D₈

yes

—

C₄×D₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

C₂⁴

yes

characteristic

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂×D₈₉

Q₃₂

yes

direct product factor

C₂×D₈₉

C₂×D₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂²×D₄

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂×D₈

yes

—

C₂×C₈₉

C₂×Q₃₂

yes

complemented

C₂×C₈₉

C₂²×D₈

yes

characteristic

D₈₉

C₂×Q₃₂

yes

direct product factor

C₂²×Q₃₂

—

maximal, Sylow

C₈₉

C₂²×Q₃₂

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₈₉

yes

characteristic, direct product factor

C₂²×C₁₆

—

C₂²×Q₈

—

178

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

C₂×D₈₉

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

356

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

178

—

Q₁₆

—

178

—

Q₁₆

C₂²×D₈₉

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₈₉

yes

characteristic

102

C₁₆

—

103

C₁₆

C₂²×D₈₉

yes

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

356

—

108

Q₈

—

356

—

109

Q₈

—

178

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

356

—

115

C₂×C₄

—

356

—

116

C₂×C₄

D₄×D₈₉

yes

characteristic

117

C₈

—

118

C₈

D₄×D₈₉

yes

characteristic

119

C₈

C₂³×D₈₉

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

D₈×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

356

—

126

C₄

D₈×D₈₉

yes

characteristic

127

C₄

C₂×D₄×D₈₉

yes

characteristic

128

C₂

—

129

C₂

Q₃₂×D₈₉

yes

direct product factor

130

C₂

C₂×D₈×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₈×C₈₉	C₂⁴
Frattini	C₈	C₂³×D₈₉
Fitting	C₂×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
prime radical	C₂²×D₈₉	D₈
hypercenter	C₂×Q₃₂	D₈₉
hypocenter	C₈₉	C₂²×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₈×C₈₉

C₂⁴

Frattini

C₈

C₂³×D₈₉

Fitting

C₂×Q₃₂×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

prime radical

C₂²×D₈₉

D₈

hypercenter

C₂×Q₃₂

D₈₉

hypocenter

C₈₉

C₂²×Q₃₂

LCS

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×Dic₇₁₂	C₂	semidirect
3. C₂×#15667958	C₂	semidirect
4. Q₃₂×D₈₉	C₂	direct (N × H)
6. C₂×Q₃₂×C₈₉	C₂	semidirect
13. Dic₇₁₂	C₂²	semidirect
14. #15667958	C₂²	semidirect
18. Q₃₂×C₈₉	C₂²	semidirect
55. C₂×D₈₉	Q₃₂	direct (N × H)
62. C₂×C₈₉	C₂×Q₃₂	semidirect
64. D₈₉	C₂×Q₃₂	direct (N × H)
66. C₈₉	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₈₉	direct (N × H)
82. Q₃₂	C₂×D₈₉	direct (N × H)
129. C₂	Q₃₂×D₈₉	direct (N × H)

H ≅ G/N

type

2. C₂×Dic₇₁₂

C₂

semidirect

3. C₂×#15667958

C₂

semidirect

4. Q₃₂×D₈₉

C₂

direct (N × H)

6. C₂×Q₃₂×C₈₉

C₂

semidirect

13. Dic₇₁₂

C₂²

semidirect

14. #15667958

C₂²

semidirect

18. Q₃₂×C₈₉

C₂²

semidirect

55. C₂×D₈₉

Q₃₂

direct (N × H)

62. C₂×C₈₉

C₂×Q₃₂

semidirect

64. D₈₉

C₂×Q₃₂

direct (N × H)

66. C₈₉

C₂²×Q₃₂

semidirect

71. C₂×Q₃₂

D₈₉

direct (N × H)

82. Q₃₂

C₂×D₈₉

direct (N × H)

129. C₂

Q₃₂×D₈₉

direct (N × H)

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

subgroup	index
C₂×Q₁₆×D₈₉	2
C₂×Dic₇₁₂	2
C₂×#15667958	2
Q₃₂×D₈₉	2
C₂×C₁₆×D₈₉	2
C₂×Q₃₂×C₈₉	2
C₂²×Q₃₂	89

subgroup

index

C₂×Q₁₆×D₈₉

C₂×Dic₇₁₂

C₂×#15667958

Q₃₂×D₈₉

C₂×C₁₆×D₈₉

C₂×Q₃₂×C₈₉

C₂²×Q₃₂

Subgroups — C₂×Q₃₂×D₈₉ #26033363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Q32×D89 #26033363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×D₈₉ #26033363