Group #36469141 — subgroups

Subgroups — #36469141

Order 22784.

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

normal subgroups only

This group has 310 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26022504	C₂	1	2	yes	maximal
3	#26019868	C₂	1	2	yes	maximal
4	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	2	no	—
7	C₂×#15667958	—	2	4	no	—
8	#19152001	—	2	4	no	—
9	#19151978	C₂²	1	2	yes	—
10	C₄×#13083549	C₂²	1	1	yes	characteristic
11	C₈×#10623494	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₆	—	2	2	no	—
16	C₂×Dic₃₅₆	—	2	2	no	—
17	#15668012	—	2	2	no	—
18	Dic₇₁₂	—	4	4	no	—
19	#15667958	—	4	8	no	—
20	C₂×#13083549	D₄	1	2	yes	—
21	#15667945	—	4	8	no	—
22	C₄×#10623494	D₄	1	2	yes	—
23	C₂×Q₁₆×C₈₉	—	2	2	no	—
24	C₂×Q₁₆×C₈₉	—	2	2	no	—
25	C₄⋊Q₈×C₈₉	—	2	2	no	—
26	Q₃₂×C₈₉	—	4	4	no	—
27	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	4	2	no	—
30	Dic₃₅₆	—	4	4	no	—
31	Dic₃₅₆	—	4	4	no	—
32	C₂×#10623494	D₈	1	4	yes	—
33	C₄⋊Dic₈₉	—	4	4	no	—
34	#13083549	D₈	1	4	yes	—
35	C₂×Q₈×C₈₉	—	4	2	no	—
36	Q₁₆×C₈₉	—	4	4	no	—
37	Q₁₆×C₈₉	—	4	4	no	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	4	4	no	—
41	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	D₈	1	4	yes	—
43	Q₈×C₈₉	—	8	2	no	—
44	Q₈×C₈₉	—	8	2	no	—
45	C₂×C₄×C₈₉	—	8	2	no	—
46	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
48	C₂×Dic₈₉	—	8	2	no	—
49	Dic₁₇₈	—	8	2	no	—
50	Dic₁₇₈	—	8	2	no	—
51	C₈×C₈₉	C₂×D₈	1	2	yes	—
52	C₈×C₈₉	C₂×D₈	1	2	yes	—
53	#10623494	Q₃₂	1	8	yes	complemented
54	C₂²×C₈₉	#493	1	1	yes	characteristic
55	C₄×C₈₉	—	16	2	no	—
56	C₄×C₈₉	#493	1	1	yes	characteristic
57	C₄×C₈₉	#493	1	1	yes	characteristic
58	C₄×C₈₉	C₂×Q₃₂	1	4	yes	—
59	Dic₈₉	—	16	2	no	—
60	#12516	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	#1712	1	1	yes	characteristic
62	C₂×C₈₉	#2247	1	2	yes	—
63	#2247	—	89	2	no	—
64	#2247	—	89	1	no	—
65	#2247	D₈₉	1	1	yes	characteristic
66	#1713	—	89	2	no	—
67	C₈×C₁₆	—	89	1	no	—
68	C₈₉	#12516	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	2	2	no	—
70	C₂×Q₃₂	—	178	4	no	—
71	C₂×Q₃₂	—	178	2	no	—
72	#500	—	178	4	no	—
73	#494	—	89	2	no	—
74	#494	C₂×D₈₉	1	2	yes	—
75	C₄×C₁₆	—	89	1	no	—
76	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
77	C₈²	—	89	1	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	178	2	no	—
81	C₂×Q₁₆	—	178	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	178	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	356	8	no	—
86	Q₃₂	—	356	4	no	—
87	C₂×C₁₆	—	89	2	no	—
88	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
89	#107	—	356	8	no	—
90	C₄×C₈	—	89	2	no	—
91	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	356	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	356	4	no	—
97	Q₁₆	—	356	4	no	—
98	C₂×C₈	—	89	4	no	—
99	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	356	4	no	—
103	C₄²	D₄×D₈₉	1	1	yes	characteristic
104	C₁₆	—	89	4	no	—
105	C₁₆	#13083559	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	712	2	no	—
109	Q₈	—	712	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	712	2	no	—
112	C₂×C₄	#15668008	1	1	yes	characteristic
113	C₂×C₄	D₈×D₈₉	1	2	yes	—
114	C₈	—	89	8	no	—
115	C₈	D₈×D₈₉	1	2	yes	—
116	C₈	C₂×#13083559	1	2	yes	—
117	C₂²	#19151973	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1424	2	no	—
120	C₄	#19151973	1	1	yes	characteristic
121	C₄	#19151974	1	1	yes	characteristic
122	C₄	Q₃₂×D₈₉	1	4	yes	—
123	C₂	#26019866	1	1	yes	characteristic
124	C₂	#26022503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26022504

C₂

yes

maximal

#26019868

C₂

yes

maximal

C₁₆×#10623494

C₂

yes

maximal, characteristic

C₈₉×#2247

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×#15667958

—

#19152001

—

#19151978

C₂²

yes

—

C₄×#13083549

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

—

#494×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

#15667958

—

C₂×#13083549

D₄

yes

—

#15667945

—

C₄×#10623494

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

D₈

yes

—

C₄⋊Dic₈₉

—

#13083549

D₈

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

D₈

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₂×D₈

yes

—

#10623494

Q₃₂

yes

complemented

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2247

—

#2247

—

#2247

D₈₉

yes

characteristic

#1713

—

C₈×C₁₆

—

C₈₉

#12516

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

C₂×Q₃₂

—

178

—

#500

—

178

—

#494

—

#494

C₂×D₈₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

Q₃₂

—

Q₃₂

—

356

—

Q₃₂

—

356

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

—

#107

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

356

—

Q₁₆

—

356

—

C₂×C₈

—

C₂×C₈

D₄×D₈₉

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

356

—

103

C₄²

D₄×D₈₉

yes

characteristic

104

C₁₆

—

105

C₁₆

#13083559

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

712

—

109

Q₈

—

712

—

110

C₂×C₄

—

111

C₂×C₄

—

712

—

112

C₂×C₄

#15668008

yes

characteristic

113

C₂×C₄

D₈×D₈₉

yes

—

114

C₈

—

115

C₈

D₈×D₈₉

yes

—

116

C₈

C₂×#13083559

yes

—

117

C₂²

#19151973

yes

characteristic

118

C₄

—

119

C₄

—

1424

—

120

C₄

#19151973

yes

characteristic

121

C₄

#19151974

yes

characteristic

122

C₄

Q₃₂×D₈₉

yes

—

123

C₂

#26019866

yes

characteristic

124

C₂

#26022503

yes

—

	subgroup	G/H
center	C₂²	#19151973
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2247	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#2247	D₈₉
hypocenter	C₈₉	#12516
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#19151973

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2247

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. #10623494	Q₃₂	semidirect
68. C₈₉	#12516	semidirect

H ≅ G/N

type

53. #10623494

Q₃₂

semidirect

68. C₈₉

#12516

semidirect

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

subgroup	index
#26022505	2
#26022504	2
#26019868	2
C₁₆×#10623494	2
C₈₉×#2247	2
#12516	89

subgroup

index

#26022505

#26022504

#26019868

C₁₆×#10623494

C₈₉×#2247

#12516

Subgroups — #36469141

Conjugacy classes of subgroups

Complements

Maximal subgroups