Group #22559005 — subgroups

Subgroups — C₈×D₅₃₆ #22559005

Order 8576.

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×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₈×D₂₆₈	C₂	1	2	yes	maximal, complemented
3	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863125	C₂	1	2	yes	maximal, complemented
5	C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₃₆	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₇	—	2	2	no	—
8	C₄×D₂₆₈	C₂²	1	2	yes	—
9	#14523977	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#14523965	C₄	1	2	yes	—
12	(C₂×D₆₇)⋊C₈	—	2	2	no	—
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₆₇	—	2	2	no	—
18	D₅₃₆	C₈	1	4	yes	direct product factor
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₈×D₆₇	—	4	4	no	—
21	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
22	(C₂×D₆₇)⋊C₄	—	2	2	no	—
23	C₂×C₈×C₆₇	—	2	1	no	—
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×D₆₇	—	2	2	no	—
30	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
33	C₂×Dic₆₇	—	2	2	no	—
34	D₂₆₈	—	2	2	no	—
35	D₂₆₈	C₂×C₈	1	4	yes	complemented
36	C₄×D₆₇	—	4	2	no	—
37	C₄×D₆₇	—	4	2	no	—
38	C₈×C₆₇	—	2	2	no	—
39	C₈×C₆₇	—	2	1	no	—
40	C₈×C₆₇	—	2	1	no	—
41	C₈×C₆₇	C₂×C₈	1	2	yes	complemented
42	C₈×C₆₇	D₈	1	2	yes	complemented
43	C₆₇⋊C₈	—	4	2	no	—
44	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
45	C₂×D₆₇	—	2	4	no	—
46	C₂×D₆₇	—	4	2	no	—
47	C₂×D₆₇	—	4	2	no	—
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₆₇	#131	1	1	yes	characteristic
51	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₆₇	#135	1	1	yes	characteristic
53	Dic₆₇	—	4	2	no	—
54	C₂×C₆₇	C₈×D₄	1	1	yes	characteristic
55	C₂×C₆₇	C₄×D₈	1	1	yes	characteristic
56	C₂×C₆₇	#443	1	1	yes	characteristic
57	D₆₇	—	4	4	no	—
58	C₈×D₈	—	67	1	no	maximal, Sylow
59	C₆₇	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
60	C₄×D₈	—	67	1	no	—
61	C₈×D₄	—	67	2	no	—
62	#335	—	67	1	no	—
63	D₄⋊C₈	—	67	2	no	—
64	C₈²	D₆₇	1	1	yes	characteristic, complemented
65	C₂×D₈	—	67	1	no	—
66	C₂²×C₈	—	134	2	no	—
67	C₄×D₄	—	67	2	no	—
68	#107	—	67	1	no	—
69	C₄⋊C₈	—	67	2	no	—
70	#102	—	67	2	no	—
71	C₂²⋊C₈	—	134	2	no	—
72	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
74	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
75	C₂×D₄	—	67	2	no	—
76	C₂²×C₄	—	134	2	no	—
77	D₈	—	67	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	268	4	no	—
80	C₂×C₈	—	134	2	no	—
81	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
82	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
83	C₂×C₈	D₂₆₈	1	1	yes	characteristic
84	C₂×C₈	D₂₆₈	1	1	yes	characteristic
85	C₄⋊C₄	—	67	2	no	—
86	C₂²⋊C₄	—	134	2	no	—
87	C₄²	C₂²×D₆₇	1	1	yes	characteristic
88	C₂³	—	134	2	no	—
89	D₄	—	67	4	no	—
90	D₄	—	134	2	no	—
91	C₂×C₄	—	268	2	no	—
92	C₂×C₄	—	268	2	no	—
93	C₂×C₄	—	134	2	no	—
94	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
96	C₂×C₄	#11761628	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	268	2	no	—
101	C₈	C₈×D₆₇	1	2	yes	complemented
102	C₈	D₅₃₆	1	2	yes	direct product factor
103	C₂²	—	268	2	no	—
104	C₂²	—	268	2	no	—
105	C₂²	—	134	4	no	—
106	C₂²	C₄×D₂₆₈	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	268	2	no	—
109	C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
110	C₄	#14524049	1	1	yes	characteristic
111	C₄	C₂×D₅₃₆	1	1	yes	characteristic
112	C₄	#14524065	1	1	yes	characteristic
113	C₂	—	268	4	no	—
114	C₂	C₈×D₂₆₈	1	1	yes	characteristic
115	C₂	C₄×D₅₃₆	1	1	yes	characteristic
116	C₂	#16863514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃₆

C₂

yes

maximal, characteristic

C₈×D₂₆₈

C₂

yes

maximal, complemented

#16863149

C₂

yes

maximal, characteristic, complemented

#16863125

C₂

yes

maximal, complemented

C₈²×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

C₄

yes

characteristic

C₂×C₈×D₆₇

—

C₄×D₂₆₈

C₂²

yes

—

#14523977

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523965

C₄

yes

—

(C₂×D₆₇)⋊C₈

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

C₂×C₄

yes

—

C₂×C₄×D₆₇

—

D₅₃₆

C₈

yes

direct product factor

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

C₂×C₈

yes

complemented

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₈

yes

complemented

C₈×C₆₇

D₈

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

#131

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

#135

yes

characteristic

Dic₆₇

—

C₂×C₆₇

C₈×D₄

yes

characteristic

C₂×C₆₇

C₄×D₈

yes

characteristic

C₂×C₆₇

#443

yes

characteristic

D₆₇

—

C₈×D₈

—

maximal, Sylow

C₆₇

C₈×D₈

yes

characteristic, complemented, Sylow

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

D₆₇

yes

characteristic, complemented

C₂×D₈

—

C₂²×C₈

—

134

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

134

—

D₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

134

—

D₄

—

D₄

—

134

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

#11761628

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

268

—

101

C₈

C₈×D₆₇

yes

complemented

102

C₈

D₅₃₆

yes

direct product factor

103

C₂²

—

268

—

104

C₂²

—

268

—

105

C₂²

—

134

—

106

C₂²

C₄×D₂₆₈

yes

characteristic

107

C₄

—

108

C₄

—

268

—

109

C₄

C₂×C₈×D₆₇

yes

characteristic

110

C₄

#14524049

yes

characteristic

111

C₄

C₂×D₅₃₆

yes

characteristic

112

C₄

#14524065

yes

characteristic

113

C₂

—

268

—

114

C₂

C₈×D₂₆₈

yes

characteristic

115

C₂

C₄×D₅₃₆

yes

characteristic

116

C₂

#16863514

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₈
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈²×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	C₈²	D₆₇
hypocenter	C₆₇	C₈×D₈
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

D₂₆₈

commutator

C₄×C₆₇

C₂²×C₈

Frattini

C₄²

C₂²×D₆₇

Fitting

C₈²×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

hypercenter

C₈²

D₆₇

hypocenter

C₆₇

C₈×D₈

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. C₈×D₂₆₈	C₂	semidirect
3. #16863149	C₂	semidirect
4. #16863125	C₂	semidirect
5. C₈²×C₆₇	C₂	semidirect
18. D₅₃₆	C₈	direct (N × H)
35. D₂₆₈	C₂×C₈	semidirect
41. C₈×C₆₇	C₂×C₈	semidirect
42. C₈×C₆₇	D₈	semidirect
59. C₆₇	C₈×D₈	semidirect
64. C₈²	D₆₇	semidirect
101. C₈	C₈×D₆₇	semidirect
102. C₈	D₅₃₆	direct (N × H)

H ≅ G/N

type

2. C₈×D₂₆₈

C₂

semidirect

3. #16863149

C₂

semidirect

4. #16863125

C₂

semidirect

5. C₈²×C₆₇

C₂

semidirect

18. D₅₃₆

C₈

direct (N × H)

35. D₂₆₈

C₂×C₈

semidirect

41. C₈×C₆₇

C₂×C₈

semidirect

42. C₈×C₆₇

D₈

semidirect

59. C₆₇

C₈×D₈

semidirect

64. C₈²

D₆₇

semidirect

101. C₈

C₈×D₆₇

semidirect

102. C₈

D₅₃₆

direct (N × H)

Subgroups — C₈×D₅₃₆ #22559005

Conjugacy classes of subgroups

Complements

Subgroups — C8×D536 #22559005

Conjugacy classes of subgroups

Complements

Subgroups — C₈×D₅₃₆ #22559005