Group #20291080 — subgroups

Subgroups — #20291080

Order 6784.

138 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊#11719910	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#11719910	C₂	1	2	yes	maximal
3	#16811511	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810579	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719910	C₂²	1	2	yes	—
7	C₂×#11719910	C₂²	1	2	yes	—
8	C₂×#11719917	C₄	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	#13133128	C₄	1	1	yes	characteristic
11	#13133128	C₄	1	1	yes	characteristic
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄×#88322	C₂²	1	1	yes	characteristic
14	#117×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
17	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#11719917	—	2	2	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂³	1	1	yes	characteristic
26	#11719910	—	2	2	no	—
27	#11719910	—	2	2	no	—
28	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	#88322	—	2	1	no	—
36	#88322	—	2	1	no	—
37	#88322	M₁₆	1	2	yes	—
38	#88322	#41	1	2	yes	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	#117	1	1	yes	characteristic
46	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
47	#2164	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	4	1	no	—
49	C₂×C₅₃	#431	1	1	yes	characteristic
50	C₂×C₅₃	#504	1	2	yes	—
51	#405	—	53	1	no	—
52	C₄⋊C₁₆	—	53	2	no	—
53	C₂²⋊C₁₆	—	53	2	no	—
54	#346	—	53	1	no	—
55	C₄×C₁₆	—	53	1	no	—
56	C₅₃	#2164	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	53	1	no	—
58	C₂²×C₈	—	53	1	no	—
59	#117	#6157	1	1	yes	characteristic
60	C₂×C₁₆	—	53	2	no	—
61	C₂×C₁₆	—	53	2	no	—
62	#97	—	53	1	no	—
63	#97	—	53	1	no	—
64	C₄×C₈	—	53	1	no	—
65	C₄×C₈	—	53	1	no	—
66	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
67	M₁₆	—	106	2	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	53	1	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	106	2	no	—
73	C₄⋊C₄	#88324	1	2	yes	—
74	C₂²⋊C₄	#88324	1	2	yes	—
75	C₄²	C₂×#6157	1	1	yes	characteristic
76	C₄²	C₂×#6157	1	1	yes	characteristic
77	C₁₆	—	106	2	no	—
78	C₁₆	—	106	2	no	—
79	C₂³	C₂×#88324	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×#88324	1	2	yes	—
83	C₂×C₄	C₂×#88324	1	2	yes	—
84	C₂×C₄	C₂×#88324	1	1	yes	characteristic
85	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
86	C₈	—	53	2	no	—
87	C₈	—	53	2	no	—
88	C₈	—	106	1	no	—
89	C₈	—	106	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂²×#88324	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	#13133296	1	1	yes	characteristic
97	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#16811639	1	1	yes	characteristic
100	C₂	#16811681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#11719910

C₂

yes

maximal, complemented

C₂²⋊#11719910

C₂

yes

maximal

#16811511

C₂

yes

maximal, characteristic, complemented

C₄×#11719910

C₂

yes

maximal, characteristic, complemented

#16810579

C₂

yes

maximal, characteristic

C₂×#11719910

C₂²

yes

—

C₂×#11719910

C₂²

yes

—

C₂×#11719917

C₄

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic

#13133128

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

#117×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₈

yes

—

C₂²⋊C₄×C₅₃

C₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719910

—

#11719910

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

—

M₁₆

yes

—

#88322

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

#2164

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#431

yes

characteristic

C₂×C₅₃

#504

yes

—

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₅₃

#2164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

#6157

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

#88324

yes

—

C₂²⋊C₄

#88324

yes

—

C₄²

yes

C₄²

yes

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

C₂×#88324

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#88324

yes

—

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×#88324

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#13133296

yes

characteristic

C₄

D₅₃⋊M₁₆

yes

characteristic

C₂

—

C₂

#16811639

yes

characteristic

100

C₂

#16811681

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	#117×C₅₃	C₄
hypercenter	#117	#6157
hypocenter	C₅₃	#2164
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

#117×C₅₃

C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
1. C₄⋊#11719910	C₂	semidirect
3. #16811511	C₂	semidirect
4. C₄×#11719910	C₂	semidirect
56. C₅₃	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#11719910

C₂

semidirect

3. #16811511

C₂

semidirect

4. C₄×#11719910

C₂

semidirect

56. C₅₃

#2164

semidirect

Subgroups — #20291080

Conjugacy classes of subgroups

Complements