Group #36441155 — subgroups

Subgroups — #36441155

Order 22784.

136 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 53 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	#26019849	C₂	1	1	yes	maximal, characteristic
2	#26018316	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1370	C₂	1	1	yes	maximal, characteristic
4	#19152003	C₄	1	1	yes	characteristic
5	#19151996	—	2	1	no	—
6	#19151978	C₄	1	1	yes	characteristic
7	#19151949	—	2	1	no	—
8	C₄⋊#13083549	—	2	1	no	—
9	C₄⋊C₁₆×C₈₉	—	2	1	no	—
10	C₈²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₅₆	—	4	1	no	—
12	#15668012	—	2	1	no	—
13	#15668012	—	2	1	no	—
14	C₂×#13083549	—	4	1	no	—
15	#15667946	—	4	1	no	—
16	#15667944	—	4	1	no	—
17	#15667944	—	4	1	no	—
18	#15667937	—	4	1	no	—
19	#15667937	—	4	1	no	—
20	C₂×C₁₆×C₈₉	—	4	1	no	—
21	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₇₈	—	4	1	no	—
25	C₂×Dic₁₇₈	—	4	1	no	—
26	Dic₃₅₆	—	8	2	no	—
27	C₄⋊Dic₈₉	—	4	1	no	—
28	C₄⋊Dic₈₉	—	4	1	no	—
29	C₄⋊Dic₈₉	—	4	1	no	—
30	C₄⋊Dic₈₉	—	4	1	no	—
31	#13083549	—	8	1	no	—
32	C₂×C₈×C₈₉	—	2	1	no	—
33	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
34	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
37	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₁₆×C₈₉	—	8	1	no	—
39	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
42	C₂×Dic₈₉	—	8	1	no	—
43	C₂×Dic₈₉	—	8	1	no	—
44	Dic₁₇₈	—	8	2	no	—
45	Dic₁₇₈	—	8	1	no	—
46	Dic₁₇₈	—	8	1	no	—
47	C₈×C₈₉	—	2	2	no	—
48	C₈×C₈₉	—	2	1	no	—
49	C₈×C₈₉	—	2	1	no	—
50	C₈×C₈₉	—	2	1	no	—
51	C₈×C₈₉	SD₃₂	1	1	yes	characteristic
52	C₈×C₈₉	Q₃₂	1	1	yes	characteristic
53	C₂²×C₈₉	#331	1	1	yes	characteristic
54	C₄×C₈₉	—	2	1	no	—
55	C₄×C₈₉	#358	1	1	yes	characteristic
56	C₄×C₈₉	#359	1	1	yes	characteristic
57	C₄×C₈₉	#360	1	1	yes	characteristic
58	C₄×C₈₉	#362	1	1	yes	characteristic
59	Dic₈₉	—	16	1	no	—
60	Dic₈₉	—	16	1	no	—
61	#7095	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	#1359	1	1	yes	characteristic
63	C₂×C₈₉	#1360	1	1	yes	characteristic
64	C₂×C₈₉	#1362	1	1	yes	characteristic
65	#1707	—	89	1	no	—
66	#1370	D₈₉	1	1	yes	characteristic
67	#1367	—	89	1	no	—
68	C₈₉	#7095	1	1	yes	characteristic, complemented, Sylow
69	#500	—	89	1	no	—
70	#498	—	178	1	no	—
71	#494	—	89	1	no	—
72	#487	—	178	1	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	178	1	no	—
75	C₈²	C₂×D₈₉	1	1	yes	characteristic
76	C₂×Q₁₆	—	356	1	no	—
77	C₄⋊Q₈	—	178	1	no	—
78	C₄⋊Q₈	—	178	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	356	1	no	—
81	#107	—	356	1	no	—
82	#106	—	356	1	no	—
83	#106	—	356	1	no	—
84	#103	—	356	1	no	—
85	#103	—	356	1	no	—
86	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
87	C₄×C₈	D₃₅₆	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
89	C₂×Q₈	—	356	1	no	—
90	C₂×Q₈	—	356	1	no	—
91	Q₁₆	—	712	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	#13083559	1	1	yes	characteristic
94	C₂×C₈	D₇₁₂	1	1	yes	characteristic
95	C₂×C₈	#13083561	1	1	yes	characteristic
96	C₂×C₈	#13083563	1	1	yes	characteristic
97	C₄⋊C₄	—	356	1	no	—
98	C₄⋊C₄	—	356	1	no	—
99	C₄⋊C₄	—	356	1	no	—
100	C₄⋊C₄	—	356	1	no	—
101	C₄²	#13083551	1	1	yes	characteristic
102	C₁₆	—	8	1	no	—
103	C₁₆	—	712	1	no	—
104	Q₈	—	712	2	no	—
105	Q₈	—	712	1	no	—
106	Q₈	—	712	1	no	—
107	C₂×C₄	—	712	1	no	—
108	C₂×C₄	—	712	1	no	—
109	C₂×C₄	#15667928	1	1	yes	characteristic
110	C₂×C₄	#15667932	1	1	yes	characteristic
111	C₂×C₄	#15667934	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	#15667956	1	1	yes	characteristic
117	C₈	#15667958	1	1	yes	characteristic
118	C₂²	#19151341	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	1424	1	no	—
121	C₄	—	1424	1	no	—
122	C₄	#19151401	1	1	yes	characteristic
123	C₄	#19151406	1	1	yes	characteristic
124	C₄	#19151409	1	1	yes	characteristic
125	C₄	#19151413	1	1	yes	characteristic
126	C₂	#26018292	1	1	yes	characteristic
127	C₂	#26018296	1	1	yes	characteristic
128	C₂	#26018300	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019849

C₂

yes

maximal, characteristic

#26018316

C₂

yes

maximal, characteristic

C₈₉×#1370

C₂

yes

maximal, characteristic

#19152003

C₄

yes

characteristic

#19151996

—

#19151978

C₄

yes

characteristic

#19151949

—

C₄⋊#13083549

—

C₄⋊C₁₆×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₁₆×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₃₂

yes

characteristic

C₈×C₈₉

Q₃₂

yes

characteristic

C₂²×C₈₉

#331

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1367

—

C₈₉

#7095

yes

characteristic, complemented, Sylow

#500

—

#498

—

178

—

#494

—

#487

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

356

—

#107

—

356

—

#106

—

356

—

#106

—

356

—

#103

—

356

—

#103

—

356

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

Q₁₆

—

712

—

C₂×C₈

—

C₂×C₈

#13083559

yes

characteristic

C₂×C₈

D₇₁₂

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

100

C₄⋊C₄

—

356

—

101

C₄²

#13083551

yes

characteristic

102

C₁₆

—

103

C₁₆

—

712

—

104

Q₈

—

712

—

105

Q₈

—

712

—

106

Q₈

—

712

—

107

C₂×C₄

—

712

—

108

C₂×C₄

—

712

—

109

C₂×C₄

#15667928

yes

characteristic

110

C₂×C₄

#15667932

yes

characteristic

111

C₂×C₄

#15667934

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#15667956

yes

characteristic

117

C₈

#15667958

yes

characteristic

118

C₂²

#19151341

yes

characteristic

119

C₄

—

120

C₄

—

1424

—

121

C₄

—

1424

—

122

C₄

#19151401

yes

characteristic

123

C₄

#19151406

yes

characteristic

124

C₄

#19151409

yes

characteristic

125

C₄

#19151413

yes

characteristic

126

C₂

#26018292

yes

characteristic

127

C₂

#26018296

yes

characteristic

128

C₂

#26018300

yes

characteristic

	subgroup	G/H
center	C₂²	#19151341
commutator	C₄×C₈×C₈₉	C₂×C₄
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1370	C₂
socle, prime radical	C₂²×C₈₉	#331
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1370	D₈₉
hypocenter	C₈₉	#7095
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#19151341

commutator

C₄×C₈×C₈₉

C₂×C₄

Frattini

C₈²

C₂×D₈₉

Fitting

C₈₉×#1370

C₂

socle, prime radical

C₂²×C₈₉

#331

cosocle

C₈²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. C₈₉	#7095	semidirect

H ≅ G/N

type

68. C₈₉

#7095

semidirect

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

subgroup	index
#26019849	2
#26018316	2
C₈₉×#1370	2
#7095	89

subgroup

index

#26019849

#26018316

C₈₉×#1370

#7095

Subgroups — #36441155

Conjugacy classes of subgroups

Complements

Maximal subgroups