Group #20290294 — subgroups

Subgroups — D₈×#88324 #20290294

Order 6784.

196 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₄×#88324	C₂	1	2	yes	maximal, complemented
2	#16811486	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#88324	C₂	1	2	yes	maximal, complemented
4	C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88324	—	2	2	no	—
7	C₄⋊#88324	C₂²	1	2	yes	—
8	C₂²⋊#88324	—	2	2	no	—
9	C₄×#88324	C₂²	1	1	yes	characteristic
10	C₄×#88324	C₂²	1	1	yes	characteristic
11	D₄×Dic₅₃	C₂²	1	2	yes	—
12	#13133151	C₄	1	1	yes	characteristic
13	#13133138	C₄	1	2	yes	—
14	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₂×#88324	—	2	1	no	—
19	C₂×#88324	—	2	2	no	—
20	C₂×#88324	—	4	4	no	—
21	C₂×#88324	D₄	1	1	yes	characteristic
22	C₂×#88324	D₄	1	1	yes	characteristic
23	D₈×C₅₃	C₈	1	4	yes	complemented
24	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
27	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₅₃	—	2	2	no	—
30	D₄×C₅₃	—	2	2	no	—
31	D₄×C₅₃	C₂×C₈	1	4	yes	complemented
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	C₂×Dic₅₃	—	4	2	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₅₃	#41	1	1	yes	characteristic
38	#88324	—	2	2	no	—
39	#88324	—	2	1	no	—
40	#88324	—	4	2	no	—
41	#88324	D₈	1	2	yes	direct product factor
42	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
43	#88322	—	2	1	no	—
44	C₂²×C₅₃	—	2	4	no	—
45	C₂²×C₅₃	—	4	2	no	—
46	C₂²×C₅₃	—	4	2	no	—
47	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
48	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₅₃	#131	1	1	yes	characteristic
50	Dic₅₃	—	2	1	no	—
51	Dic₅₃	—	4	2	no	—
52	Dic₅₃	C₂×D₈	1	1	yes	characteristic
53	Dic₅₃	#135	1	1	yes	characteristic
54	C₈×D₈	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	4	4	no	—
56	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
57	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
58	C₂×C₅₃	#443	1	1	yes	characteristic
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	—	53	1	no	—
64	C₅₃	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	#6157	1	1	yes	characteristic
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	—	53	1	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	—	53	1	no	—
75	C₂×D₄	C₂×#6157	1	2	yes	—
76	C₂²×C₄	—	106	2	no	—
77	D₈	#88324	1	4	yes	direct product factor
78	C₂×C₈	—	53	1	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	106	1	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	—	212	4	no	—
84	C₂×C₈	C₂×#6157	1	1	yes	characteristic
85	C₄⋊C₄	—	53	2	no	—
86	C₂²⋊C₄	—	106	2	no	—
87	C₄²	—	53	1	no	—
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	C₂×#88324	1	4	yes	complemented
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	106	2	no	—
94	C₂×C₄	—	212	2	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
97	C₈	—	53	2	no	—
98	C₈	—	106	1	no	—
99	C₈	—	106	2	no	—
100	C₈	—	106	1	no	—
101	C₈	—	212	2	no	—
102	C₈	C₂×#88324	1	2	yes	complemented
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×#6157	1	1	yes	characteristic
107	C₄	—	53	1	no	—
108	C₄	—	53	1	no	—
109	C₄	—	106	1	no	—
110	C₄	—	212	2	no	—
111	C₄	C₂²×#88324	1	1	yes	characteristic
112	C₄	#13133304	1	1	yes	characteristic
113	C₂	—	4	4	no	—
114	C₂	D₄×#88324	1	1	yes	characteristic
115	C₂	D₈×#6157	1	1	yes	characteristic
116	C₂	#16811665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#88324

C₂

yes

maximal, complemented

#16811486

C₂

yes

maximal, characteristic, complemented

D₄⋊#88324

C₂

yes

maximal, complemented

C₈×#88324

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

—

C₄⋊#88324

C₂²

yes

—

C₂²⋊#88324

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

D₄×Dic₅₃

C₂²

yes

—

#13133151

C₄

yes

characteristic

#13133138

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

—

—

—

—

D₄

yes

characteristic

C₂×#88324

D₄

yes

characteristic

D₈×C₅₃

C₈

yes

complemented

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₈

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

C₂×Dic₅₃

yes

—

—

—

D₈

yes

direct product factor

C₈×C₅₃

C₂×C₈

yes

complemented

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

characteristic

Dic₅₃

#135

yes

characteristic

C₈×D₈

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₄×D₈

yes

characteristic

C₂×C₅₃

#443

yes

characteristic

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

—

C₅₃

C₈×D₈

yes

characteristic, complemented, Sylow

C₂×D₈

#6157

yes

characteristic

C₂²×C₈

—

106

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

C₂×#6157

yes

—

C₂²×C₄

—

106

—

D₈

#88324

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

C₂³

—

D₄

—

D₄

C₂×#88324

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

106

—

100

C₈

—

106

—

101

C₈

—

212

—

102

C₈

C₂×#88324

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×#6157

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

106

—

110

C₄

—

212

—

111

C₄

C₂²×#88324

yes

characteristic

112

C₄

#13133304

yes

characteristic

113

C₂

—

114

C₂

D₄×#88324

yes

characteristic

115

C₂

D₈×#6157

yes

characteristic

116

C₂

#16811665

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×D₈×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×D₈	#6157
hypocenter	C₅₃	C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×D₈×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₂×D₈

#6157

hypocenter

C₅₃

C₈×D₈

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₄×#88324	C₂	semidirect
2. #16811486	C₂	semidirect
3. D₄⋊#88324	C₂	semidirect
4. C₈×#88324	C₂	semidirect
23. D₈×C₅₃	C₈	semidirect
31. D₄×C₅₃	C₂×C₈	semidirect
41. #88324	D₈	direct (N × H)
42. C₈×C₅₃	C₂×C₈	semidirect
64. C₅₃	C₈×D₈	semidirect
77. D₈	#88324	direct (N × H)
90. D₄	C₂×#88324	semidirect
102. C₈	C₂×#88324	semidirect

H ≅ G/N

type

1. D₄×#88324

C₂

semidirect

2. #16811486

C₂

semidirect

3. D₄⋊#88324

C₂

semidirect

4. C₈×#88324

C₂

semidirect

23. D₈×C₅₃

C₈

semidirect

31. D₄×C₅₃

C₂×C₈

semidirect

41. #88324

D₈

direct (N × H)

42. C₈×C₅₃

C₂×C₈

semidirect

64. C₅₃

C₈×D₈

semidirect

77. D₈

#88324

direct (N × H)

90. D₄

C₂×#88324

semidirect

102. C₈

C₂×#88324

semidirect

Subgroups — D₈×#88324 #20290294

Conjugacy classes of subgroups

Complements

Subgroups — D8×#88324 #20290294

Conjugacy classes of subgroups

Complements

Subgroups — D₈×#88324 #20290294