Group #20273511 — subgroups

Subgroups — C₈×D₄₂₄ #20273511

Order 6784.

196 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
3	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810352	C₂	1	2	yes	maximal, complemented
5	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	—	2	2	no	—
8	C₄×D₂₁₂	C₂²	1	2	yes	—
9	#13133151	C₄	1	1	yes	characteristic
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133139	C₄	1	2	yes	—
12	#13133129	—	2	2	no	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719921	—	2	2	no	—
25	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×#88322	—	2	2	no	—
27	D₄₂₄	C₈	1	4	yes	direct product factor
28	C₈×D₅₃	—	4	4	no	—
29	C₂²×D₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	D₂₁₂	—	2	2	no	—
35	D₂₁₂	C₂×C₈	1	4	yes	complemented
36	C₄×D₅₃	—	4	2	no	—
37	C₄×D₅₃	—	4	2	no	—
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	—	2	1	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
42	C₈×C₅₃	D₈	1	2	yes	complemented
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
45	C₂×D₅₃	—	2	4	no	—
46	C₂×D₅₃	—	4	2	no	—
47	C₂×D₅₃	—	4	2	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₅₃	#131	1	1	yes	characteristic
51	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₃	#135	1	1	yes	characteristic
53	Dic₅₃	—	4	2	no	—
54	C₈×D₈	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
56	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
57	C₂×C₅₃	#443	1	1	yes	characteristic
58	D₅₃	—	4	4	no	—
59	C₄×D₈	—	53	1	no	—
60	C₈×D₄	—	53	2	no	—
61	#335	—	53	1	no	—
62	D₄⋊C₈	—	53	2	no	—
63	C₈²	D₅₃	1	1	yes	characteristic, complemented
64	C₅₃	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	—	53	1	no	—
66	C₂²×C₈	—	106	2	no	—
67	C₄×D₄	—	53	2	no	—
68	#107	—	53	1	no	—
69	C₄⋊C₈	—	53	2	no	—
70	#102	—	53	2	no	—
71	C₂²⋊C₈	—	106	2	no	—
72	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂×D₄	—	53	2	no	—
76	C₂²×C₄	—	106	2	no	—
77	D₈	—	53	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	106	2	no	—
80	C₂×C₈	—	212	4	no	—
81	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
82	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
83	C₂×C₈	D₂₁₂	1	1	yes	characteristic
84	C₂×C₈	D₂₁₂	1	1	yes	characteristic
85	C₄⋊C₄	—	53	2	no	—
86	C₂²⋊C₄	—	106	2	no	—
87	C₄²	C₂²×D₅₃	1	1	yes	characteristic
88	C₂³	—	106	2	no	—
89	D₄	—	53	4	no	—
90	D₄	—	106	2	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	—	212	2	no	—
93	C₂×C₄	—	212	2	no	—
94	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
96	C₂×C₄	#11719945	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	212	2	no	—
101	C₈	C₈×D₅₃	1	2	yes	complemented
102	C₈	D₄₂₄	1	2	yes	direct product factor
103	C₂²	—	106	4	no	—
104	C₂²	—	212	2	no	—
105	C₂²	—	212	2	no	—
106	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	212	2	no	—
109	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
110	C₄	#13133223	1	1	yes	characteristic
111	C₄	C₂×D₄₂₄	1	1	yes	characteristic
112	C₄	#13133239	1	1	yes	characteristic
113	C₂	—	212	4	no	—
114	C₂	C₈×D₂₁₂	1	1	yes	characteristic
115	C₂	C₄×D₄₂₄	1	1	yes	characteristic
116	C₂	#16810741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₂₄

C₂

yes

maximal, characteristic

C₈×D₂₁₂

C₂

yes

maximal, complemented

#16810376

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, complemented

C₈²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₄

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133139

C₄

yes

—

#13133129

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

D₄₂₄

C₈

yes

direct product factor

C₈×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×C₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

complemented

C₈×C₅₃

D₈

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

C₈×D₈

—

maximal, Sylow

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₄×D₈

yes

characteristic

C₂×C₅₃

#443

yes

characteristic

D₅₃

—

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

D₅₃

yes

characteristic, complemented

C₅₃

C₈×D₈

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

106

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

212

—

101

C₈

C₈×D₅₃

yes

complemented

102

C₈

D₄₂₄

yes

direct product factor

103

C₂²

—

106

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

C₄×D₂₁₂

yes

characteristic

107

C₄

—

108

C₄

—

212

—

109

C₄

C₂×C₈×D₅₃

yes

characteristic

110

C₄

#13133223

yes

characteristic

111

C₄

C₂×D₄₂₄

yes

characteristic

112

C₄

#13133239

yes

characteristic

113

C₂

—

212

—

114

C₂

C₈×D₂₁₂

yes

characteristic

115

C₂

C₄×D₄₂₄

yes

characteristic

116

C₂

#16810741

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

D₂₁₂

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈²×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₈²

D₅₃

hypocenter

C₅₃

C₈×D₈

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. C₈×D₂₁₂	C₂	semidirect
3. #16810376	C₂	semidirect
4. #16810352	C₂	semidirect
5. C₈²×C₅₃	C₂	semidirect
27. D₄₂₄	C₈	direct (N × H)
35. D₂₁₂	C₂×C₈	semidirect
41. C₈×C₅₃	C₂×C₈	semidirect
42. C₈×C₅₃	D₈	semidirect
63. C₈²	D₅₃	semidirect
64. C₅₃	C₈×D₈	semidirect
101. C₈	C₈×D₅₃	semidirect
102. C₈	D₄₂₄	direct (N × H)

H ≅ G/N

type

2. C₈×D₂₁₂

C₂

semidirect

3. #16810376

C₂

semidirect

4. #16810352

C₂

semidirect

5. C₈²×C₅₃

C₂

semidirect

27. D₄₂₄

C₈

direct (N × H)

35. D₂₁₂

C₂×C₈

semidirect

41. C₈×C₅₃

C₂×C₈

semidirect

42. C₈×C₅₃

D₈

semidirect

63. C₈²

D₅₃

semidirect

64. C₅₃

C₈×D₈

semidirect

101. C₈

C₈×D₅₃

semidirect

102. C₈

D₄₂₄

direct (N × H)

Subgroups — C₈×D₄₂₄ #20273511

Conjugacy classes of subgroups

Complements

Subgroups — C8×D424 #20273511

Conjugacy classes of subgroups

Complements

Subgroups — C₈×D₄₂₄ #20273511