Group #36474940 — subgroups

Subgroups — #36474940

Order 22784.

178 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19151431	C₂	1	1	yes	maximal, characteristic
2	#368×D₈₉	C₂	1	2	yes	maximal
3	#26018456	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018455	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
7	#108×D₈₉	—	2	2	no	—
8	#19151431	C₄	1	4	yes	—
9	C₂×#15667918	C₂²	1	1	yes	characteristic
10	#19151428	C₂²	1	2	yes	—
11	#19151427	—	2	2	no	—
12	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
13	#368×C₈₉	C₂²	1	2	yes	—
14	M₁₆×D₈₉	—	4	2	no	—
15	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
16	C₂×#13083549	D₄	1	1	yes	characteristic
17	C₁₆×D₈₉	C₂×C₄	1	2	yes	—
18	C₁₆×D₈₉	Q₈	1	2	yes	—
19	#15667948	—	2	2	no	—
20	#15667947	—	4	2	no	—
21	#15667918	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
23	#108×C₈₉	—	2	2	no	—
24	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
26	#13083558	—	4	2	no	—
27	#13083557	—	8	2	no	—
28	C₂×#10623494	D₈	1	1	yes	characteristic
29	C₈×D₈₉	—	8	2	no	—
30	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
31	C₈×D₈₉	Q₁₆	1	2	yes	—
32	#13083549	C₄⋊C₄	1	2	yes	—
33	M₁₆×C₈₉	—	4	2	no	—
34	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×D₈₉	Q₃₂	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
39	C₂×Dic₈₉	Q₃₂	1	1	yes	characteristic
40	C₄×D₈₉	#107	1	2	yes	—
41	C₄×D₈₉	D₁₆	1	2	yes	—
42	C₈×C₈₉	—	8	2	no	—
43	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
45	#10623494	—	8	2	no	—
46	#10623494	#107	1	2	yes	—
47	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
48	C₂×D₈₉	—	4	1	no	—
49	C₂×D₈₉	#366	1	2	yes	—
50	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
51	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
52	Dic₈₉	#366	1	2	yes	—
53	#13636	—	89	1	no	maximal, Sylow
54	C₂×C₈₉	—	2	1	no	—
55	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
56	D₈₉	—	2	2	no	—
57	C₂×M₆₄	—	89	1	no	—
58	C₂×#368	—	89	2	no	—
59	#1425	—	89	1	no	—
60	#1425	—	89	2	no	—
61	#1425	D₈₉	1	1	yes	characteristic, complemented
62	C₈₉	#13636	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₁₆	—	89	1	no	—
64	C₂×#108	—	178	2	no	—
65	M₆₄	—	89	4	no	—
66	C₂×C₃₂	—	89	1	no	—
67	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
68	#368	—	89	2	no	—
69	#368	—	178	2	no	—
70	#368	C₂×D₈₉	1	2	yes	—
71	C₂×M₁₆	—	356	2	no	—
72	C₂²×C₈	—	89	1	no	—
73	C₂×C₁₆	—	89	1	no	—
74	C₂×C₁₆	—	89	2	no	—
75	C₂×C₁₆	—	89	2	no	—
76	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
77	#108	—	2	2	no	—
78	#108	—	356	2	no	—
79	#108	—	178	2	no	—
80	C₃₂	—	89	2	no	—
81	C₃₂	C₄×D₈₉	1	2	yes	—
82	C₂²×C₄	—	89	1	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	356	2	no	—
85	M₁₆	—	712	2	no	—
86	C₂×C₈	—	89	1	no	—
87	C₂×C₈	—	89	2	no	—
88	C₂×C₈	—	89	2	no	—
89	C₂×C₈	—	712	2	no	—
90	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
91	C₁₆	—	89	2	no	—
92	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
93	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
94	C₂³	—	89	1	no	—
95	C₂×C₄	—	89	1	no	—
96	C₂×C₄	—	89	2	no	—
97	C₂×C₄	—	89	2	no	—
98	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
99	C₈	—	8	2	no	—
100	C₈	—	89	2	no	—
101	C₈	—	712	2	no	—
102	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
103	C₈	Q₁₆×D₈₉	1	1	yes	characteristic
104	C₂²	—	89	2	no	—
105	C₂²	—	356	1	no	—
106	C₂²	Q₃₂×D₈₉	1	1	yes	characteristic
107	C₄	—	89	2	no	—
108	C₄	#107×D₈₉	1	1	yes	characteristic
109	C₄	D₁₆×D₈₉	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	178	2	no	—
112	C₂	#366×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151431

C₂

yes

maximal, characteristic

#368×D₈₉

C₂

yes

maximal

#26018456

C₂

yes

maximal, characteristic, complemented

#26018455

C₂

yes

maximal, complemented

C₈₉×#1425

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₈₉

C₂²

yes

characteristic

#108×D₈₉

—

#19151431

C₄

yes

—

C₂×#15667918

C₂²

yes

characteristic

#19151428

C₂²

yes

—

#19151427

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#368×C₈₉

C₂²

yes

—

M₁₆×D₈₉

—

C₂×C₈×D₈₉

D₄

yes

characteristic

C₂×#13083549

D₄

yes

characteristic

C₁₆×D₈₉

C₂×C₄

yes

—

C₁₆×D₈₉

Q₈

yes

—

#15667948

—

#15667947

—

#15667918

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

C₃₂×C₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

D₈

yes

—

—

D₈

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

C₄⋊C₄

yes

—

C₈×D₈₉

Q₁₆

yes

—

#13083549

C₄⋊C₄

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₂²×D₈₉

Q₃₂

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

Q₃₂

yes

characteristic

C₄×D₈₉

#107

yes

—

C₄×D₈₉

D₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₂×Q₁₆

yes

—

yes

—

C₂²×C₈₉

C₂×Q₃₂

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

#366

yes

—

C₄×C₈₉

C₂×#107

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

#366

yes

—

#13636

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#366

yes

characteristic

D₈₉

—

C₂×M₆₄

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#13636

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

178

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#368

—

#368

—

178

—

#368

C₂×D₈₉

yes

—

C₂×M₁₆

—

356

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#108

—

#108

—

356

—

#108

—

178

—

C₃₂

—

C₃₂

C₄×D₈₉

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

356

—

M₁₆

—

712

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

712

—

C₂×C₈

D₄×D₈₉

yes

characteristic

C₁₆

—

C₁₆

C₂×C₄×D₈₉

yes

characteristic

C₁₆

Q₈×D₈₉

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×D₈₉

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

712

—

102

C₈

C₄⋊C₄×D₈₉

yes

characteristic

103

C₈

Q₁₆×D₈₉

yes

characteristic

104

C₂²

—

105

C₂²

—

356

—

106

C₂²

Q₃₂×D₈₉

yes

characteristic

107

C₄

—

108

C₄

#107×D₈₉

yes

characteristic

109

C₄

D₁₆×D₈₉

yes

characteristic

110

C₂

—

111

C₂

—

178

—

112

C₂

#366×D₈₉

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1425	C₂
socle	C₂×C₈₉	C₂×#366
prime radical	C₂²×D₈₉	Q₃₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1425	D₈₉
hypocenter	C₈₉	#13636
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₁₆×D₈₉

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1425

C₂

socle

C₂×C₈₉

C₂×#366

prime radical

C₂²×D₈₉

Q₃₂

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

N	H ≅ G/N	type
3. #26018456	C₂	semidirect
4. #26018455	C₂	semidirect
5. C₈₉×#1425	C₂	semidirect
61. #1425	D₈₉	semidirect
62. C₈₉	#13636	semidirect

H ≅ G/N

type

3. #26018456

C₂

semidirect

4. #26018455

C₂

semidirect

5. C₈₉×#1425

C₂

semidirect

61. #1425

D₈₉

semidirect

62. C₈₉

#13636

semidirect

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

subgroup	index
C₂×#19151431	2
#368×D₈₉	2
#26018456	2
#26018455	2
C₈₉×#1425	2
#13636	89

subgroup

index

#368×D₈₉

C₈₉×#1425

Subgroups — #36474940

Conjugacy classes of subgroups

Complements

Maximal subgroups