Group #36475424 — subgroups

Subgroups — #36475424

Order 22784.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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×Dic₁₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151438	C₂	1	2	yes	maximal
3	#26022503	C₂	1	2	yes	maximal
4	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₂²	1	2	yes	—
7	C₂×#15667958	—	2	2	no	—
8	#19151976	—	2	2	no	—
9	Dic₁₄₂₄	—	2	2	no	—
10	#19151438	—	2	4	no	—
11	C₂×#15667918	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
14	Q₆₄×C₈₉	—	2	2	no	—
15	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	—	2	2	no	—
17	C₂×#13083564	—	4	2	no	—
18	#15668010	—	4	2	no	—
19	Dic₇₁₂	—	2	2	no	—
20	Dic₇₁₂	—	2	2	no	—
21	#15667958	—	4	4	no	—
22	C₂×#13083549	D₄	1	1	yes	characteristic
23	C₈×Dic₈₉	D₄	1	1	yes	characteristic
24	#15667918	D₄	1	2	yes	—
25	C₂×Q₁₆×C₈₉	—	2	2	no	—
26	Q₃₂×C₈₉	—	2	2	no	—
27	Q₃₂×C₈₉	—	2	2	no	—
28	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
29	C₃₂×C₈₉	D₄	1	2	yes	—
30	C₂×Dic₁₇₈	—	4	2	no	—
31	Dic₃₅₆	—	4	2	no	—
32	Dic₃₅₆	—	4	2	no	—
33	#13083564	—	8	4	no	—
34	C₂×#10623494	D₈	1	1	yes	characteristic
35	#13083553	—	8	4	no	—
36	C₄×Dic₈₉	D₈	1	1	yes	characteristic
37	#13083549	D₈	1	2	yes	—
38	C₂×Q₈×C₈₉	—	4	2	no	—
39	Q₁₆×C₈₉	—	4	2	no	—
40	Q₁₆×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₈₉	—	8	2	no	—
45	Q₈×C₈₉	—	8	2	no	—
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₉	—	8	2	no	—
49	C₂×Dic₈₉	D₁₆	1	2	yes	—
50	Dic₁₇₈	—	8	2	no	—
51	Dic₁₇₈	—	8	2	no	—
52	C₈×C₈₉	#127	1	1	yes	characteristic
53	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
54	#10623494	D₁₆	1	2	yes	—
55	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₈₉	—	16	2	no	—
57	C₄×C₈₉	#493	1	1	yes	characteristic
58	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
59	Dic₈₉	—	16	2	no	—
60	Dic₈₉	Q₆₄	1	4	yes	complemented
61	#13726	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	#2246	1	1	yes	characteristic
63	C₂×C₈₉	C₂×Q₆₄	1	2	yes	—
64	C₂×Q₆₄	—	89	2	no	—
65	C₂×Q₆₄	—	89	1	no	—
66	C₂×Q₆₄	D₈₉	1	1	yes	characteristic
67	#2247	—	89	2	no	—
68	C₄×C₃₂	—	89	1	no	—
69	C₈₉	#13726	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₃₂	—	89	2	no	—
71	C₂×Q₃₂	—	178	2	no	—
72	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
73	#494	—	178	2	no	—
74	Q₆₄	—	2	2	no	—
75	Q₆₄	—	178	4	no	—
76	Q₆₄	—	178	2	no	—
77	C₂×C₃₂	—	89	1	no	—
78	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
79	C₄×C₁₆	—	89	1	no	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	356	2	no	—
82	C₂×Q₁₆	—	178	2	no	—
83	C₄⋊Q₈	—	356	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	356	4	no	—
87	Q₃₂	—	178	2	no	—
88	Q₃₂	—	178	2	no	—
89	C₂×C₁₆	—	89	1	no	—
90	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
91	C₄×C₈	—	89	1	no	—
92	C₃₂	—	89	2	no	—
93	C₃₂	C₂²⋊D₈₉	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	356	2	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	Q₁₆	—	356	2	no	—
99	Q₁₆	—	356	2	no	—
100	Q₁₆	—	712	4	no	—
101	C₂×C₈	—	89	1	no	—
102	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
103	C₄⋊C₄	—	712	4	no	—
104	C₄²	—	89	1	no	—
105	C₁₆	—	89	2	no	—
106	C₁₆	D₄×D₈₉	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
108	Q₈	—	8	2	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	712	2	no	—
111	Q₈	—	712	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	89	2	no	—
114	C₂×C₄	—	712	2	no	—
115	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
116	C₈	—	89	2	no	—
117	C₈	#15668008	1	1	yes	characteristic
118	C₈	D₈×D₈₉	1	1	yes	characteristic
119	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	89	4	no	—
122	C₄	—	1424	2	no	—
123	C₄	#19151973	1	1	yes	characteristic
124	C₄	D₁₆×D₈₉	1	1	yes	characteristic
125	C₂	#26022500	1	1	yes	characteristic
126	C₂	Q₆₄×D₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₄₂₄

C₂

yes

maximal, characteristic

C₂×#19151438

C₂

yes

maximal

#26022503

C₂

yes

maximal

C₃₂×Dic₈₉

C₂

yes

maximal, characteristic

C₂×Q₆₄×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₂²

yes

—

C₂×#15667958

—

#19151976

—

Dic₁₄₂₄

—

#19151438

—

C₂×#15667918

C₂²

yes

characteristic

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

C₂²

yes

—

Q₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×#13083564

—

#15668010

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667958

—

C₂×#13083549

D₄

yes

characteristic

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

D₄

yes

—

C₂×Q₁₆×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

C₂×#10623494

D₈

yes

characteristic

#13083553

—

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

D₈

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₁₆

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₂×D₈

yes

characteristic

#10623494

D₁₆

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#493

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

Q₆₄

yes

—

C₂×C₈₉

yes

C₂×C₈₉

C₂×Q₆₄

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

D₈₉

yes

characteristic

#2247

—

C₄×C₃₂

—

C₈₉

#13726

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

C₂×Q₃₂

C₂×D₈₉

yes

—

#494

—

178

—

Q₆₄

—

Q₆₄

—

178

—

Q₆₄

—

178

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

356

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

356

—

Q₃₂

—

178

—

Q₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

356

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

356

—

Q₁₆

—

356

—

100

Q₁₆

—

712

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₈₉

yes

characteristic

103

C₄⋊C₄

—

712

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

D₄×D₈₉

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

712

—

111

Q₈

—

712

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

712

—

115

C₂×C₄

D₈×D₈₉

yes

characteristic

116

C₈

—

117

C₈

#15668008

yes

characteristic

118

C₈

D₈×D₈₉

yes

characteristic

119

C₂²

D₁₆×D₈₉

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

1424

—

123

C₄

#19151973

yes

characteristic

124

C₄

D₁₆×D₈₉

yes

characteristic

125

C₂

#26022500

yes

characteristic

126

C₂

Q₆₄×D₈₉

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂×Q₆₄×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	C₂×Q₆₄	D₈₉
hypocenter	C₈₉	#13726
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₈₉

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂×Q₆₄×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

hypercenter

C₂×Q₆₄

D₈₉

hypocenter

C₈₉

#13726

LCS

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
60. Dic₈₉	Q₆₄	semidirect
69. C₈₉	#13726	semidirect

H ≅ G/N

type

60. Dic₈₉

Q₆₄

semidirect

69. C₈₉

#13726

semidirect

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

subgroup	index
C₂×Dic₁₄₂₄	2
C₂×#19151438	2
#26022503	2
C₃₂×Dic₈₉	2
C₂×Q₆₄×C₈₉	2
#13726	89

subgroup

index

C₂×Dic₁₄₂₄

C₂×#19151438

#26022503

C₃₂×Dic₈₉

C₂×Q₆₄×C₈₉

#13726

Subgroups — #36475424

Conjugacy classes of subgroups

Complements

Maximal subgroups