Group #20273793 — subgroups

Subgroups — #20273793

Order 6784.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16810997	C₂	1	1	yes	maximal, characteristic
2	#16810823	C₂	1	2	yes	maximal
3	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#88322	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719924	—	2	2	no	—
7	C₂×#11719924	—	2	2	no	—
8	#13133214	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133137	—	2	2	no	—
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₅₃	—	2	2	no	—
13	C₄⋊Q₈×C₅₃	—	2	2	no	—
14	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₅₃	—	2	2	no	—
16	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂×Q₈×C₅₃	D₄	1	2	yes	—
20	C₂×D₂₁₂	—	2	2	no	—
21	C₂×D₂₁₂	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	2	4	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#11719924	—	4	4	no	—
30	#11719924	—	4	4	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂²×D₅₃	—	4	1	no	—
34	Q₈×C₅₃	—	2	2	no	—
35	Q₈×C₅₃	—	4	2	no	—
36	Q₈×C₅₃	—	8	2	no	—
37	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	D₂₁₂	—	4	2	no	—
44	D₂₁₂	—	4	2	no	—
45	D₂₁₂	—	4	2	no	—
46	#88322	—	2	2	no	—
47	#88322	—	4	2	no	—
48	C₂²×C₅₃	#120	1	1	yes	characteristic
49	C₂×D₅₃	—	8	2	no	—
50	C₂×D₅₃	—	8	1	no	—
51	C₄×C₅₃	—	2	4	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	4	2	no	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
56	C₄×C₅₃	#137	1	2	yes	—
57	#1623	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#448	1	2	yes	—
59	C₂×C₅₃	#453	1	1	yes	characteristic
60	D₅₃	—	16	1	no	—
61	Q₈²	D₅₃	1	1	yes	characteristic, complemented
62	#492	—	53	1	no	—
63	#460	—	53	2	no	—
64	#333	—	53	1	no	—
65	Q₈⋊C₈	—	53	2	no	—
66	C₅₃	#1623	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	106	2	no	—
68	C₂×SD₁₆	—	106	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
72	#127	—	53	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₅₃	1	2	yes	—
75	C₄⋊C₈	—	53	2	no	—
76	#102	—	106	2	no	—
77	C₄×C₈	—	53	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
81	C₂×D₄	—	106	2	no	—
82	C₂×D₄	—	106	1	no	—
83	SD₁₆	—	212	4	no	—
84	SD₁₆	—	212	4	no	—
85	C₂×C₈	—	106	1	no	—
86	C₂×C₈	—	106	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₅₃	1	1	yes	characteristic
94	C₂³	—	212	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#11719924	1	4	yes	complemented
99	D₄	—	212	2	no	—
100	D₄	—	212	2	no	—
101	D₄	—	212	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
107	C₈	—	106	2	no	—
108	C₈	—	212	2	no	—
109	C₂²	—	424	2	no	—
110	C₂²	—	424	1	no	—
111	C₂²	#13133186	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#11719924	1	2	yes	—
117	C₄	#13133248	1	2	yes	—
118	C₂	—	848	1	no	—
119	C₂	#16810762	1	2	yes	—
120	C₂	#16810791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810997

C₂

yes

maximal, characteristic

#16810823

C₂

yes

maximal

#16810372

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, complemented

Q₈²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

C₂×#11719924

—

#13133214

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133137

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

#88322

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

#137

yes

—

#1623

—

maximal, Sylow

C₂×C₅₃

#448

yes

—

C₂×C₅₃

#453

yes

characteristic

D₅₃

—

Q₈²

D₅₃

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₅₃

#1623

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄⋊C₈

—

#102

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

SD₁₆

—

212

—

SD₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

212

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#11719924

yes

complemented

D₄

—

212

—

100

D₄

—

212

—

101

D₄

—

212

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

107

C₈

—

106

—

108

C₈

—

212

—

109

C₂²

—

424

—

110

C₂²

—

424

—

111

C₂²

#13133186

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#11719924

yes

—

117

C₄

#13133248

yes

—

118

C₂

—

848

—

119

C₂

#16810762

yes

—

120

C₂

#16810791

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	Q₈²×C₅₃	C₂
socle	C₂²×C₅₃	#120
hypercenter	Q₈²	D₅₃
hypocenter	C₅₃	#1623
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

Q₈²×C₅₃

C₂

socle

C₂²×C₅₃

#120

hypercenter

Q₈²

D₅₃

hypocenter

C₅₃

#1623

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #16810372	C₂	semidirect
4. Q₈⋊#88322	C₂	semidirect
5. Q₈²×C₅₃	C₂	semidirect
37. Q₈×C₅₃	SD₁₆	semidirect
61. Q₈²	D₅₃	semidirect
66. C₅₃	#1623	semidirect
98. Q₈	#11719924	semidirect

H ≅ G/N

type

3. #16810372

C₂

semidirect

4. Q₈⋊#88322

C₂

semidirect

5. Q₈²×C₅₃

C₂

semidirect

37. Q₈×C₅₃

SD₁₆

semidirect

61. Q₈²

D₅₃

semidirect

66. C₅₃

#1623

semidirect

98. Q₈

#11719924

semidirect

Subgroups — #20273793

Conjugacy classes of subgroups

Complements