Group #36469177 — subgroups

Subgroups — #36469177

Order 22784.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022532	C₂	1	2	yes	maximal
2	#26022525	C₂	1	2	yes	maximal
3	#26019870	C₂	1	1	yes	maximal, characteristic
4	#26018169	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1710	C₂	1	1	yes	maximal, characteristic
6	#19152003	—	2	1	no	—
7	#19152003	C₂²	1	2	yes	—
8	#19151978	—	2	1	no	—
9	#19151425	—	2	4	no	—
10	#19151423	—	2	4	no	—
11	C₄×#13083549	C₂²	1	2	yes	—
12	#500×C₈₉	C₂²	1	1	yes	characteristic
13	#500×C₈₉	C₂²	1	1	yes	characteristic
14	#487×C₈₉	—	2	1	no	—
15	#487×C₈₉	—	2	1	no	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	4	2	no	—
18	#15668012	—	2	2	no	—
19	C₂×#13083549	D₄	1	4	yes	—
20	#15667946	—	2	2	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	4	2	no	—
23	C₄⋊Q₈×C₈₉	—	2	1	no	—
24	C₄⋊Q₈×C₈₉	—	2	1	no	—
25	#107×C₈₉	—	2	2	no	—
26	#107×C₈₉	—	2	2	no	—
27	#103×C₈₉	—	4	2	no	—
28	#103×C₈₉	—	4	2	no	—
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇₈	—	4	2	no	—
33	Dic₃₅₆	—	8	4	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	#13083549	Q₁₆	1	8	yes	complemented
37	C₂×Q₈×C₈₉	—	4	1	no	—
38	C₂×Q₈×C₈₉	—	4	1	no	—
39	C₂×C₈×C₈₉	D₈	1	2	yes	—
40	C₂×C₈×C₈₉	D₈	1	2	yes	—
41	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₈₉	—	4	1	no	—
43	C₄⋊C₄×C₈₉	—	4	1	no	—
44	C₄⋊C₄×C₈₉	—	4	1	no	—
45	C₄⋊C₄×C₈₉	—	4	1	no	—
46	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₈₉	—	8	2	no	—
48	Q₈×C₈₉	—	8	2	no	—
49	C₂×C₄×C₈₉	—	8	1	no	—
50	C₂×C₄×C₈₉	—	8	1	no	—
51	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₈₉	—	8	2	no	—
55	Dic₁₇₈	—	8	4	no	—
56	C₈×C₈₉	—	2	2	no	—
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₈₉	#493	1	1	yes	characteristic
60	C₄×C₈₉	—	16	1	no	—
61	C₄×C₈₉	—	16	1	no	—
62	C₄×C₈₉	#494	1	2	yes	—
63	C₄×C₈₉	#509	1	2	yes	—
64	C₄×C₈₉	#510	1	2	yes	—
65	Dic₈₉	—	16	2	no	—
66	#12524	—	89	1	no	maximal, Sylow
67	C₂×C₈₉	#1713	1	1	yes	characteristic
68	C₂×C₈₉	#2250	1	1	yes	characteristic
69	C₂×C₈₉	#2251	1	1	yes	characteristic
70	#2254	—	89	2	no	—
71	#2252	—	89	2	no	—
72	#1713	—	89	1	no	—
73	#1710	D₈₉	1	1	yes	characteristic
74	#1311	—	89	1	no	—
75	C₈₉	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	89	2	no	—
77	#500	—	178	1	no	—
78	#500	C₂×D₈₉	1	1	yes	characteristic
79	#500	C₂×D₈₉	1	1	yes	characteristic
80	#494	—	178	1	no	—
81	#487	—	2	1	no	—
82	#487	—	2	1	no	—
83	#367	—	178	4	no	—
84	#366	—	178	4	no	—
85	C₄×C₁₆	—	89	2	no	—
86	C₈²	C₂×D₈₉	1	1	yes	characteristic
87	C₂×Q₁₆	—	356	2	no	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	2	1	no	—
90	C₄⋊Q₈	—	178	2	no	—
91	C₂×C₁₆	—	89	4	no	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	356	2	no	—
95	#107	—	178	2	no	—
96	#107	—	178	2	no	—
97	#103	—	4	2	no	—
98	#103	—	4	2	no	—
99	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	356	2	no	—
105	Q₁₆	—	712	4	no	—
106	C₂×C₈	#13083559	1	2	yes	—
107	C₂×C₈	#13083559	1	2	yes	—
108	C₂×C₈	D₄×D₈₉	1	2	yes	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	356	2	no	—
114	C₄⋊C₄	—	356	2	no	—
115	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
116	C₁₆	—	89	8	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	8	2	no	—
119	Q₈	—	712	4	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	712	2	no	—
123	C₂×C₄	#15668008	1	1	yes	characteristic
124	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
125	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₈₉	1	4	yes	—
129	C₂²	#19151974	1	1	yes	characteristic
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	—	1424	2	no	—
133	C₄	#19151976	1	2	yes	—
134	C₄	#19152033	1	2	yes	—
135	C₄	#19152038	1	2	yes	—
136	C₂	#26019868	1	1	yes	characteristic
137	C₂	#26022519	1	1	yes	characteristic
138	C₂	#26022524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022532

C₂

yes

maximal

#26022525

C₂

yes

maximal

#26019870

C₂

yes

maximal, characteristic

#26018169

C₂

yes

maximal, characteristic

C₈₉×#1710

C₂

yes

maximal, characteristic

#19152003

—

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

—

#500×C₈₉

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

characteristic

#487×C₈₉

—

#487×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

#15668012

—

C₂×#13083549

D₄

yes

—

#15667946

—

#15667946

—

#15667946

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

#103×C₈₉

—

#103×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

Q₁₆

yes

complemented

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×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

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#494

yes

—

C₄×C₈₉

#509

yes

—

C₄×C₈₉

#510

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

D₈₉

yes

characteristic

#1311

—

C₈₉

#12524

yes

characteristic, complemented, Sylow

#500

—

#500

—

178

—

#500

C₂×D₈₉

yes

characteristic

#500

C₂×D₈₉

yes

characteristic

#494

—

178

—

#487

—

#487

—

#367

—

178

—

#366

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

356

—

#107

—

178

—

#107

—

178

—

#103

—

#103

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

100

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

101

C₄×C₈

C₂²×D₈₉

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

356

—

105

Q₁₆

—

712

—

106

C₂×C₈

#13083559

yes

—

107

C₂×C₈

#13083559

yes

—

108

C₂×C₈

D₄×D₈₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

356

—

114

C₄⋊C₄

—

356

—

115

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

116

C₁₆

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

712

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

712

—

123

C₂×C₄

#15668008

yes

characteristic

124

C₂×C₄

C₂×#13083559

yes

characteristic

125

C₂×C₄

C₂×#13083559

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₈₉

yes

—

129

C₂²

#19151974

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

1424

—

133

C₄

#19151976

yes

—

134

C₄

#19152033

yes

—

135

C₄

#19152038

yes

—

136

C₂

#26019868

yes

characteristic

137

C₂

#26022519

yes

characteristic

138

C₂

#26022524

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1710	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1710	D₈₉
hypocenter	C₈₉	#12524
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1710

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
36. #13083549	Q₁₆	semidirect
75. C₈₉	#12524	semidirect

H ≅ G/N

type

36. #13083549

Q₁₆

semidirect

75. C₈₉

#12524

semidirect

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

subgroup	index
#26022532	2
#26022525	2
#26019870	2
#26018169	2
C₈₉×#1710	2
#12524	89

subgroup

index

C₈₉×#1710

Subgroups — #36469177

Conjugacy classes of subgroups

Complements

Maximal subgroups