Group #36469115 — subgroups

Subgroups — #36469115

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	#26022477	C₂	1	4	yes	maximal, complemented
2	#26019867	C₂	1	1	yes	maximal, characteristic
3	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151975	—	2	1	no	—
6	#19151975	—	2	1	no	—
7	#19151975	C₂²	1	2	yes	—
8	#19151398	—	2	8	no	—
9	C₄×#13083549	C₂²	1	2	yes	—
10	#500×C₈₉	—	2	1	no	—
11	#500×C₈₉	C₂²	1	2	yes	—
12	#494×C₈₉	—	2	1	no	—
13	C₈²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×D₇₁₂	—	2	4	no	—
15	C₂×D₇₁₂	—	4	2	no	—
16	C₂×D₇₁₂	—	4	2	no	—
17	#15668009	—	2	2	no	—
18	C₂×#13083549	D₄	1	4	yes	—
19	C₂×Q₁₆×C₈₉	—	4	2	no	—
20	C₄⋊Q₈×C₈₉	—	2	2	no	—
21	#107×C₈₉	—	2	4	no	—
22	#107×C₈₉	—	4	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×D₃₅₆	—	4	2	no	—
27	C₂×D₃₅₆	—	4	2	no	—
28	C₂×D₃₅₆	—	4	2	no	—
29	D₇₁₂	—	4	8	no	—
30	D₇₁₂	—	8	4	no	—
31	D₇₁₂	—	8	4	no	—
32	#13083549	—	2	4	no	—
33	C₂×Q₈×C₈₉	—	4	2	no	—
34	Q₁₆×C₈₉	—	8	4	no	—
35	C₂×C₈×C₈₉	D₈	1	2	yes	—
36	C₂×C₈×C₈₉	D₈	1	2	yes	—
37	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₈₉	—	4	2	no	—
39	C₄⋊C₄×C₈₉	—	4	2	no	—
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₈₉	—	8	2	no	—
42	Q₈×C₈₉	—	8	4	no	—
43	C₂×C₄×C₈₉	—	8	2	no	—
44	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	D₃₅₆	—	8	4	no	—
48	D₃₅₆	—	8	4	no	—
49	D₃₅₆	—	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	#493	1	1	yes	characteristic
54	C₂×D₈₉	—	16	2	no	—
55	C₂×D₈₉	—	16	2	no	—
56	C₂×D₈₉	—	16	2	no	—
57	C₄×C₈₉	—	16	2	no	—
58	C₄×C₈₉	#495	1	2	yes	—
59	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
60	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
61	#12509	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	#1709	1	1	yes	characteristic
63	C₂×C₈₉	#2246	1	1	yes	characteristic
64	C₂×C₈₉	#2248	1	1	yes	characteristic
65	D₈₉	—	32	2	no	—
66	#2240	—	89	4	no	—
67	#1713	D₈₉	1	1	yes	characteristic, complemented
68	#1712	—	89	1	no	—
69	C₈×C₁₆	—	89	1	no	—
70	C₈₉	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	2	1	no	—
72	#500	C₂×D₈₉	1	2	yes	—
73	#494	—	2	1	no	—
74	#493	—	89	2	no	—
75	#493	—	178	1	no	—
76	#493	—	178	1	no	—
77	#357	—	178	8	no	—
78	C₄×C₁₆	—	89	2	no	—
79	C₈²	C₂×D₈₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×D₈	—	356	2	no	—
82	C₂×D₈	—	356	2	no	—
83	C₂×D₈	—	178	4	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	178	2	no	—
86	C₂×C₁₆	—	89	4	no	—
87	#107	—	2	4	no	—
88	#107	—	4	2	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	—	4	2	no	—
93	C₂×D₄	—	356	2	no	—
94	C₂×D₄	—	356	2	no	—
95	C₂×D₄	—	356	2	no	—
96	Q₁₆	—	8	4	no	—
97	D₈	—	356	8	no	—
98	D₈	—	712	4	no	—
99	D₈	—	712	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	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
106	C₁₆	—	178	4	no	—
107	C₂³	—	712	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	712	4	no	—
110	D₄	—	712	4	no	—
111	D₄	—	712	4	no	—
112	C₂×C₄	—	8	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	#15667955	1	4	yes	—
118	C₈	#15668035	1	4	yes	—
119	C₂²	—	1424	2	no	—
120	C₂²	—	1424	2	no	—
121	C₂²	—	1424	2	no	—
122	C₂²	#19151974	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	#19151983	1	2	yes	—
125	C₄	C₂×#15667953	1	2	yes	—
126	C₄	C₂×#15667955	1	2	yes	—
127	C₂	—	2848	2	no	—
128	C₂	#26019859	1	1	yes	characteristic
129	C₂	#26022501	1	1	yes	characteristic
130	C₂	#26022509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022477

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1713

C₂

yes

maximal, characteristic, complemented

#19151975

—

#19151975

—

#19151975

C₂²

yes

—

#19151398

—

C₄×#13083549

C₂²

yes

—

#500×C₈₉

—

#500×C₈₉

C₂²

yes

—

#494×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

C₂×#13083549

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

D₁₆

yes

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2240

—

#1713

D₈₉

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₈₉

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₈₉

yes

—

#494

—

#493

—

#493

—

178

—

#493

—

178

—

#357

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

356

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

178

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

D₈

—

356

—

D₈

—

712

—

D₈

—

712

—

100

C₂×C₈

#13083559

yes

—

101

C₂×C₈

#13083559

yes

—

102

C₂×C₈

D₄×D₈₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

106

C₁₆

—

178

—

107

C₂³

—

712

—

108

Q₈

—

109

D₄

—

712

—

110

D₄

—

712

—

111

D₄

—

712

—

112

C₂×C₄

—

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

#15667955

yes

—

118

C₈

#15668035

yes

—

119

C₂²

—

1424

—

120

C₂²

—

1424

—

121

C₂²

—

1424

—

122

C₂²

#19151974

yes

characteristic

123

C₄

—

124

C₄

#19151983

yes

—

125

C₄

C₂×#15667953

yes

—

126

C₄

C₂×#15667955

yes

—

127

C₂

—

2848

—

128

C₂

#26019859

yes

characteristic

129

C₂

#26022501

yes

characteristic

130

C₂

#26022509

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂²×C₈₉	#493
prime radical	#26019867	C₂
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#12509
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1713

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #26022477	C₂	semidirect
3. C₈×#13083549	C₂	semidirect
4. C₈₉×#1713	C₂	semidirect
67. #1713	D₈₉	semidirect
70. C₈₉	#12509	semidirect

H ≅ G/N

type

1. #26022477

C₂

semidirect

3. C₈×#13083549

C₂

semidirect

4. C₈₉×#1713

C₂

semidirect

67. #1713

D₈₉

semidirect

70. C₈₉

#12509

semidirect

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

subgroup	index
#26022477	2
#26019867	2
C₈×#13083549	2
C₈₉×#1713	2
#12509	89

subgroup

index

C₈₉×#1713

Subgroups — #36469115

Conjugacy classes of subgroups

Complements

Maximal subgroups