Group #20291099 — subgroups

Subgroups — #20291099

Order 6784.

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

normal subgroups only

This group has 162 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	C₂×#13133291	C₂	1	2	yes	maximal, complemented
2	C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#11719910	C₂	1	2	yes	maximal, complemented
4	C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#88322	C₂	1	1	yes	maximal, characteristic
6	#13133291	—	2	4	no	—
7	C₂×#11719910	C₂²	1	2	yes	—
8	C₂×#11719910	C₂²	1	1	yes	characteristic
9	C₂×#11719910	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	C₄	1	1	yes	characteristic
12	C₂²⋊#88322	C₄	1	2	yes	—
13	C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
15	C₂×D₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#88322	—	2	2	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	2	yes	—
24	C₂×#88322	C₂³	1	1	yes	characteristic
25	#11719910	—	2	2	no	—
26	#11719910	—	2	1	no	—
27	#11719910	D₄	1	2	yes	complemented
28	C₂³×C₅₃	C₂×C₈	1	2	yes	—
29	D₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	#88322	—	2	2	no	—
37	#88322	—	2	1	no	—
38	#88322	C₂×D₄	1	1	yes	characteristic
39	#88322	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	—	2	2	no	—
43	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	M₃₂	1	2	yes	complemented
47	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
48	C₄×C₅₃	#131	1	1	yes	characteristic
49	#2169	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	4	2	no	—
51	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
52	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
53	C₂×C₅₃	#504	1	1	yes	characteristic
54	C₂×M₃₂	—	53	2	no	—
55	C₈×D₄	—	53	1	no	—
56	C₄⋊C₁₆	—	53	1	no	—
57	C₂²⋊C₁₆	—	53	2	no	—
58	C₄×C₁₆	—	53	1	no	—
59	C₅₃	#2169	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	2	no	—
61	C₄×D₄	#6157	1	1	yes	characteristic
62	M₃₂	—	106	4	no	—
63	C₂×C₁₆	—	53	1	no	—
64	C₂×C₁₆	—	53	1	no	—
65	C₂×C₁₆	—	53	2	no	—
66	C₄⋊C₈	—	53	1	no	—
67	C₂²⋊C₈	—	53	2	no	—
68	C₄×C₈	—	53	1	no	—
69	C₂×D₄	#88324	1	1	yes	characteristic
70	C₂²×C₄	C₂×#6157	1	2	yes	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	—	53	2	no	—
74	C₂×C₈	—	106	2	no	—
75	C₂×C₈	—	106	2	no	—
76	C₄⋊C₄	#88324	1	1	yes	characteristic
77	C₂²⋊C₄	#88324	1	2	yes	—
78	C₄²	C₂×#6157	1	1	yes	characteristic
79	C₁₆	—	53	2	no	—
80	C₁₆	—	106	2	no	—
81	C₁₆	—	106	1	no	—
82	C₂³	C₂×#88324	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#88324	1	1	yes	characteristic
87	C₂×C₄	C₂×#88324	1	2	yes	—
88	C₂×C₄	C₂×#88324	1	1	yes	characteristic
89	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
90	C₈	—	53	1	no	—
91	C₈	—	53	1	no	—
92	C₈	—	106	2	no	—
93	C₈	—	106	1	no	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#88324	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	#13133291	1	2	yes	complemented
101	C₄	D₄×#6157	1	1	yes	characteristic
102	C₄	#13133304	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×#88324	1	1	yes	characteristic
105	C₂	C₂×#13133291	1	1	yes	characteristic
106	C₂	#16811681	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133291

C₂

yes

maximal, complemented

C₄⋊#11719910

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11719910

C₂

yes

maximal, complemented

C₄×#11719910

C₂

yes

maximal, characteristic, complemented

D₄×#88322

C₂

yes

maximal, characteristic

#13133291

—

C₂×#11719910

C₂²

yes

—

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₄

yes

characteristic

C₂²⋊#88322

C₄

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₈

yes

characteristic

C₂²⋊C₄×C₅₃

C₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂³

yes

—

—

D₄

yes

complemented

C₂³×C₅₃

C₂×C₈

yes

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₃₂

yes

complemented

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

#504

yes

characteristic

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₅₃

#2169

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄×D₄

#6157

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

#88324

yes

characteristic

C₂²×C₄

C₂×#6157

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

#88324

yes

characteristic

C₂²⋊C₄

#88324

yes

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

C₂×#88324

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#88324

yes

characteristic

C₂×C₄

C₂×#88324

yes

—

C₂×C₄

C₂×#88324

yes

characteristic

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#88324

yes

characteristic

C₄

—

C₄

—

100

C₄

#13133291

yes

complemented

101

C₄

D₄×#6157

yes

characteristic

102

C₄

#13133304

yes

characteristic

103

C₂

—

104

C₂

D₄×#88324

yes

characteristic

105

C₂

C₂×#13133291

yes

characteristic

106

C₂

#16811681

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×D₄×C₅₃	C₄
hypercenter	C₄×D₄	#6157
hypocenter	C₅₃	#2169
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₄×D₄×C₅₃

C₄

C₄×D₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#13133291	C₂	semidirect
2. C₄⋊#11719910	C₂	semidirect
3. C₂²⋊#11719910	C₂	semidirect
4. C₄×#11719910	C₂	semidirect
27. #11719910	D₄	semidirect
46. C₄×C₅₃	M₃₂	semidirect
59. C₅₃	#2169	semidirect
100. C₄	#13133291	semidirect

H ≅ G/N

type

1. C₂×#13133291

C₂

semidirect

2. C₄⋊#11719910

C₂

semidirect

3. C₂²⋊#11719910

C₂

semidirect

4. C₄×#11719910

C₂

semidirect

27. #11719910

D₄

semidirect

46. C₄×C₅₃

M₃₂

semidirect

59. C₅₃

#2169

semidirect

100. C₄

#13133291

semidirect

Subgroups — #20291099

Conjugacy classes of subgroups

Complements