Group #36469113 — subgroups

Subgroups — #36469113

Order 22784.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#26022475	C₂	1	4	yes	maximal, complemented
2	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1712	C₂	1	1	yes	maximal, characteristic
5	#19152003	—	2	1	no	—
6	#19152003	C₂²	1	2	yes	—
7	#19151978	—	2	1	no	—
8	#19151399	—	2	8	no	—
9	C₄×#13083549	C₂²	1	2	yes	—
10	#493×C₈₉	—	2	1	no	—
11	#493×C₈₉	—	2	1	no	—
12	#493×C₈₉	C₂²	1	2	yes	—
13	C₈²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₆	—	4	2	no	—
15	#15668012	—	2	2	no	—
16	C₂×#13083549	D₄	1	4	yes	—
17	#15667946	—	2	4	no	—
18	#15667946	—	4	2	no	—
19	C₂×D₈×C₈₉	—	2	4	no	—
20	C₂×D₈×C₈₉	—	4	2	no	—
21	C₂×D₈×C₈₉	—	4	2	no	—
22	#127×C₈₉	—	2	2	no	—
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₇₈	—	4	2	no	—
27	Dic₃₅₆	—	8	4	no	—
28	C₄⋊Dic₈₉	—	4	2	no	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	#13083549	—	2	4	no	—
31	C₂×D₄×C₈₉	—	4	2	no	—
32	C₂×D₄×C₈₉	—	4	2	no	—
33	C₂×D₄×C₈₉	—	4	2	no	—
34	D₈×C₈₉	—	4	8	no	—
35	D₈×C₈₉	—	8	4	no	—
36	D₈×C₈₉	—	8	4	no	—
37	C₂×C₈×C₈₉	D₈	1	2	yes	—
38	C₂×C₈×C₈₉	D₈	1	2	yes	—
39	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂³×C₈₉	—	8	2	no	—
42	D₄×C₈₉	—	8	4	no	—
43	D₄×C₈₉	—	8	4	no	—
44	D₄×C₈₉	—	8	4	no	—
45	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₉	—	8	2	no	—
49	Dic₁₇₈	—	8	4	no	—
50	C₈×C₈₉	D₁₆	1	4	yes	—
51	C₈×C₈₉	SD₃₂	1	4	yes	—
52	C₈×C₈₉	#135	1	4	yes	—
53	C₂²×C₈₉	—	16	2	no	—
54	C₂²×C₈₉	—	16	2	no	—
55	C₂²×C₈₉	—	16	2	no	—
56	C₂²×C₈₉	#493	1	1	yes	characteristic
57	C₄×C₈₉	#495	1	2	yes	—
58	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
59	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
60	Dic₈₉	—	16	2	no	—
61	#12509	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	32	2	no	—
63	C₂×C₈₉	#1709	1	1	yes	characteristic
64	C₂×C₈₉	#2246	1	1	yes	characteristic
65	C₂×C₈₉	#2248	1	1	yes	characteristic
66	#2240	—	89	4	no	—
67	#1713	—	89	1	no	—
68	#1712	D₈₉	1	1	yes	characteristic
69	C₈×C₁₆	—	89	1	no	—
70	C₈₉	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	89	2	no	—
72	#500	—	178	1	no	—
73	#494	—	178	1	no	—
74	#493	—	2	1	no	—
75	#493	—	2	1	no	—
76	#493	C₂×D₈₉	1	2	yes	—
77	#357	—	178	8	no	—
78	C₄×C₁₆	—	89	2	no	—
79	C₈²	C₂×D₈₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	356	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	178	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	89	4	no	—
87	#107	—	356	2	no	—
88	#107	—	178	4	no	—
89	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	356	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	712	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#13083559	1	2	yes	—
101	C₂×C₈	#13083559	1	2	yes	—
102	C₂×C₈	D₄×D₈₉	1	2	yes	—
103	C₄⋊C₄	—	356	2	no	—
104	C₄⋊C₄	—	356	2	no	—
105	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
106	C₁₆	—	178	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	712	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	712	2	no	—
113	C₂×C₄	#15668008	1	1	yes	characteristic
114	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
115	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
116	C₈	#15667953	1	4	yes	—
117	C₈	#15667956	1	4	yes	—
118	C₈	#15668031	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#19151974	1	1	yes	characteristic
123	C₄	—	1424	2	no	—
124	C₄	#19151981	1	2	yes	—
125	C₄	C₂×#15667953	1	2	yes	—
126	C₄	C₂×#15667956	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#26019857	1	1	yes	characteristic
129	C₂	#26022501	1	1	yes	characteristic
130	C₂	#26022507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022475

C₂

yes

maximal, complemented

#26019870

C₂

yes

maximal, characteristic, complemented

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1712

C₂

yes

maximal, characteristic

#19152003

—

#19152003

C₂²

yes

—

#19151978

—

#19151399

—

C₄×#13083549

C₂²

yes

—

#493×C₈₉

—

#493×C₈₉

—

#493×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

#15668012

—

C₂×#13083549

D₄

yes

—

#15667946

—

#15667946

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#127×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

D₁₆

yes

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

#495

yes

—

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

Dic₈₉

—

#12509

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

D₈₉

yes

characteristic

C₈×C₁₆

—

C₈₉

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

—

178

—

#494

—

178

—

#493

—

#493

—

#493

C₂×D₈₉

yes

—

#357

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

178

—

#127

—

C₂×C₁₆

—

#107

—

356

—

#107

—

178

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

712

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#13083559

yes

—

101

C₂×C₈

#13083559

yes

—

102

C₂×C₈

D₄×D₈₉

yes

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

106

C₁₆

—

178

—

107

C₂³

—

108

Q₈

—

712

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

712

—

113

C₂×C₄

#15668008

yes

characteristic

114

C₂×C₄

C₂×#13083559

yes

characteristic

115

C₂×C₄

C₂×#13083559

yes

characteristic

116

C₈

#15667953

yes

—

117

C₈

#15667956

yes

—

118

C₈

#15668031

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#19151974

yes

characteristic

123

C₄

—

1424

—

124

C₄

#19151981

yes

—

125

C₄

C₂×#15667953

yes

—

126

C₄

C₂×#15667956

yes

—

127

C₂

—

128

C₂

#26019857

yes

characteristic

129

C₂

#26022501

yes

characteristic

130

C₂

#26022507

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting, prime radical	C₈₉×#1712	C₂
socle	C₂²×C₈₉	#493
hypercenter	#1712	D₈₉
hypocenter	C₈₉	#12509
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting, prime radical

C₈₉×#1712

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #26022475	C₂	semidirect
2. #26019870	C₂	semidirect
3. C₈×#13083549	C₂	semidirect
70. C₈₉	#12509	semidirect

H ≅ G/N

type

1. #26022475

C₂

semidirect

2. #26019870

C₂

semidirect

3. C₈×#13083549

C₂

semidirect

70. C₈₉

#12509

semidirect

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

subgroup	index
#26022475	2
#26019870	2
C₈×#13083549	2
C₈₉×#1712	2
#12509	89

subgroup

index

C₈₉×#1712

Subgroups — #36469113

Conjugacy classes of subgroups

Complements

Maximal subgroups