Group #36469844 — subgroups

Subgroups — #36469844

Order 22784.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#19151428	C₂	1	2	yes	maximal
2	C₄×#15667948	C₂	1	2	yes	maximal
3	#26018346	C₂	1	2	yes	maximal
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667948	C₂²	1	4	yes	—
6	C₄×#13083558	—	2	2	no	—
7	#19151428	C₄	1	8	yes	complemented
8	#19151363	—	2	2	no	—
9	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#346×C₈₉	C₄	1	4	yes	—
13	C₂×#13083558	—	2	4	no	—
14	#15667948	—	2	4	no	—
15	#15667948	C₂×C₄	1	8	yes	—
16	#15667923	—	4	2	no	—
17	C₄×#10623494	—	4	2	no	—
18	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	—	2	2	no	—
22	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₈₉	Q₈	1	2	yes	—
27	#13083558	—	2	8	no	—
28	#13083558	—	4	4	no	—
29	C₂×#10623494	—	4	4	no	—
30	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
36	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	#41	1	2	yes	—
38	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
39	C₄²×C₈₉	D₈	1	2	yes	—
40	C₁₆×C₈₉	—	2	2	no	—
41	C₁₆×C₈₉	C₄²	1	4	yes	—
42	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₉	—	4	2	no	—
44	C₂×C₄×C₈₉	#107	1	4	yes	—
45	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#135	1	2	yes	—
49	C₈×C₈₉	—	2	2	no	—
50	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
51	C₈×C₈₉	#117	1	1	yes	characteristic
52	C₈×C₈₉	C₄×Q₈	1	2	yes	—
53	#10623494	—	4	8	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
56	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
57	C₄×C₈₉	—	2	4	no	—
58	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈₉	#427	1	1	yes	characteristic
60	C₄×C₈₉	C₄×D₈	1	2	yes	—
61	#12676	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	2	no	—
63	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
64	C₂×C₈₉	#2161	1	2	yes	—
65	C₂×#368	—	89	2	no	—
66	C₂×#346	D₈₉	1	1	yes	characteristic
67	C₄×#108	—	89	2	no	—
68	#1381	—	89	2	no	—
69	C₈₉	#12676	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
71	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
72	C₂×#108	—	89	4	no	—
73	C₄×M₁₆	—	178	2	no	—
74	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
75	#368	—	89	8	no	—
76	#346	Dic₈₉	1	4	yes	—
77	#341	—	178	2	no	—
78	C₂×M₁₆	—	178	4	no	—
79	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
81	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
84	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
86	#108	—	89	8	no	—
87	#108	—	178	4	no	—
88	#97	—	356	2	no	—
89	C₄×C₈	—	356	2	no	—
90	C₄×C₈	Dic₁₇₈	1	2	yes	—
91	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
92	C₂²×C₄	Dic₃₅₆	1	2	yes	—
93	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
94	M₁₆	—	356	4	no	—
95	M₁₆	—	178	8	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	356	4	no	—
98	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
99	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
100	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
101	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
102	C₂×C₈	#13083574	1	2	yes	—
103	C₄²	C₄⋊Dic₈₉	1	2	yes	—
104	C₄²	D₇₁₂	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×Dic₈₉	1	4	yes	—
107	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#15667946	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
111	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
112	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
113	C₂×C₄	#15668034	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	356	8	no	—
116	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
117	C₈	#15667971	1	1	yes	characteristic
118	C₈	C₄×Dic₁₇₈	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#15667946	1	1	yes	characteristic
121	C₂²	C₄×Dic₃₅₆	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
124	C₄	#19151636	1	1	yes	characteristic
125	C₄	C₄×D₇₁₂	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#15667946	1	1	yes	characteristic
128	C₂	#26022047	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151428

C₂

yes

maximal

C₄×#15667948

C₂

yes

maximal

#26018346

C₂

yes

maximal

C₂×#346×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

C₂²

yes

—

C₄×#13083558

—

#19151428

C₄

yes

complemented

#19151363

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₄

yes

—

C₂×#13083558

—

#15667948

—

#15667948

C₂×C₄

yes

—

#15667923

—

C₄×#10623494

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

Q₁₆

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

D₈

yes

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₄²

yes

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₄²

yes

characteristic

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

C₄×Q₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₈₉

#427

yes

characteristic

C₄×C₈₉

C₄×D₈

yes

—

#12676

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₄×#107

yes

characteristic

C₂×C₈₉

#2161

yes

—

C₂×#368

—

C₂×#346

D₈₉

yes

characteristic

C₄×#108

—

#1381

—

C₈₉

#12676

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#368

—

#346

Dic₈₉

yes

—

#341

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

D₃₅₆

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

—

C₂×C₁₆

C₂×Dic₈₉

yes

—

C₂×C₁₆

C₂×Dic₈₉

yes

—

#108

—

#108

—

178

—

#97

—

356

—

C₄×C₈

—

356

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

Dic₃₅₆

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

M₁₆

—

356

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₂×C₄×D₈₉

yes

—

100

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

101

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

102

C₂×C₈

#13083574

yes

—

103

C₄²

C₄⋊Dic₈₉

yes

—

104

C₄²

D₇₁₂

yes

—

105

C₁₆

—

106

C₁₆

C₄×Dic₈₉

yes

—

107

C₂³

C₂×Dic₃₅₆

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#15667946

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

111

C₂×C₄

C₄×D₃₅₆

yes

—

112

C₂×C₄

C₂×D₇₁₂

yes

characteristic

113

C₂×C₄

#15668034

yes

—

114

C₈

—

115

C₈

—

356

—

116

C₈

C₂×C₄×Dic₈₉

yes

characteristic

117

C₈

#15667971

yes

characteristic

118

C₈

C₄×Dic₁₇₈

yes

—

119

C₂²

—

120

C₂²

C₂×#15667946

yes

characteristic

121

C₂²

C₄×Dic₃₅₆

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊Dic₈₉

yes

characteristic

124

C₄

#19151636

yes

characteristic

125

C₄

C₄×D₇₁₂

yes

—

126

C₂

—

127

C₂

C₄×#15667946

yes

characteristic

128

C₂

#26022047

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12676
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₂

commutator

C₈×C₈₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂²×C₈₉

C₂×#107

prime radical

C₂³×C₈₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
7. #19151428	C₄	semidirect
69. C₈₉	#12676	semidirect

H ≅ G/N

type

7. #19151428

C₄

semidirect

69. C₈₉

#12676

semidirect

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

subgroup	index
C₂×#19151428	2
C₄×#15667948	2
#26018346	2
C₂×#346×C₈₉	2
#12676	89

subgroup

index

C₂×#346×C₈₉

Subgroups — #36469844

Conjugacy classes of subgroups

Complements

Maximal subgroups