Group #20273459 — subgroups

Subgroups — C₂×#16810376 #20273459

Order 6784.

200 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊#88322	C₂	1	2	yes	maximal
2	#16810376	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	—	2	2	no	—
5	C₄⋊#88322	—	2	2	no	—
6	C₄⋊#88322	C₂²	1	4	yes	—
7	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	C₈	1	4	yes	complemented
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄²×C₅₃	Q₈	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×#88322	—	2	4	no	—
24	C₂×#88322	—	4	4	no	—
25	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
32	C₂×C₄×C₅₃	D₈	1	2	yes	—
33	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₅₃	—	2	2	no	—
38	C₈×C₅₃	C₂×C₈	1	4	yes	complemented
39	#88322	—	4	4	no	—
40	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂²×C₅₃	#107	1	2	yes	—
42	C₂²×C₅₃	#108	1	2	yes	—
43	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
47	C₄×C₅₃	#107	1	2	yes	—
48	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×#335	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#335	1	4	yes	complemented
54	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
56	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	53	2	no	—
58	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
59	#335	—	53	4	no	—
60	C₅₃	C₂×#335	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₈	—	106	2	no	—
62	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
63	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₄⋊C₈	—	53	4	no	—
66	C₄⋊C₈	—	106	2	no	—
67	C₄×C₈	Dic₅₃	1	2	yes	—
68	C₄×C₈	C₂×D₅₃	1	2	yes	—
69	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
70	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	106	4	no	—
74	C₂×C₈	—	212	4	no	—
75	C₂×C₈	#88322	1	4	yes	complemented
76	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
78	C₄²	Dic₁₀₆	1	1	yes	characteristic
79	C₄²	D₂₁₂	1	1	yes	characteristic
80	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
81	C₄²	C₂²×D₅₃	1	1	yes	characteristic
82	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	D₄₂₄	1	2	yes	—
85	C₂×C₄	Dic₂₁₂	1	2	yes	—
86	C₂×C₄	C₂×#88322	1	2	yes	—
87	C₂×C₄	#11719917	1	2	yes	—
88	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
93	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	212	4	no	—
96	C₈	C₂×#88322	1	4	yes	complemented
97	C₂²	C₄⋊#88322	1	2	yes	—
98	C₂²	#13133151	1	2	yes	—
99	C₂²	#13133153	1	2	yes	—
100	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	C₄⋊#88322	1	2	yes	—
104	C₄	#13133151	1	2	yes	—
105	C₄	C₂²×#88322	1	1	yes	characteristic
106	C₄	C₂×#11719917	1	1	yes	characteristic
107	C₄	C₂×D₄₂₄	1	1	yes	characteristic
108	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
109	C₂	#16810376	1	4	yes	direct product factor
110	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic
111	C₂	C₂×#13133151	1	1	yes	characteristic
112	C₂	C₂×#13133153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal

#16810376

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₈

yes

complemented

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

complemented

#88322

—

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

#107

yes

—

C₂²×C₅₃

#108

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#335

—

C₅₃

C₂×#335

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

Dic₅₃

yes

characteristic

C₂²×C₈

Dic₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

#88322

yes

complemented

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

D₄₂₄

yes

—

C₂×C₄

Dic₂₁₂

yes

—

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

C₂×#88322

yes

complemented

C₂²

C₄⋊#88322

yes

—

C₂²

#13133151

yes

—

C₂²

#13133153

yes

—

100

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₄⋊#88322

yes

—

104

C₄

#13133151

yes

—

105

C₄

C₂²×#88322

yes

characteristic

106

C₄

C₂×#11719917

yes

characteristic

107

C₄

C₂×D₄₂₄

yes

characteristic

108

C₄

C₂×Dic₂₁₂

yes

characteristic

109

C₂

#16810376

yes

direct product factor

110

C₂

C₂×C₄⋊#88322

yes

characteristic

111

C₂

C₂×#13133151

yes

characteristic

112

C₂

C₂×#13133153

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#335
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₁₂

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₂×#335

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810376	C₂	direct (N × H)
16. C₂×C₈×C₅₃	C₈	semidirect
38. C₈×C₅₃	C₂×C₈	semidirect
53. C₂×C₅₃	#335	semidirect
60. C₅₃	C₂×#335	semidirect
75. C₂×C₈	#88322	semidirect
96. C₈	C₂×#88322	semidirect
109. C₂	#16810376	direct (N × H)

H ≅ G/N

type

2. #16810376

C₂

direct (N × H)

16. C₂×C₈×C₅₃

C₈

semidirect

38. C₈×C₅₃

C₂×C₈

semidirect

53. C₂×C₅₃

#335

semidirect

60. C₅₃

C₂×#335

semidirect

75. C₂×C₈

#88322

semidirect

96. C₈

C₂×#88322

semidirect

109. C₂

#16810376

direct (N × H)

Subgroups — C₂×#16810376 #20273459

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810376 #20273459

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810376 #20273459