Subgroups — #20291050

Order 6784.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133291	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719910	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	C₂	1	4	yes	maximal, complemented
4	#16810579	C₂	1	1	yes	maximal, characteristic
5	#13133291	—	2	2	no	—
6	C₂×#11719910	—	2	2	no	—
7	C₂×#11719910	C₂²	1	2	yes	—
8	C₂×#11719910	C₂²	1	2	yes	—
9	C₂×#11719917	C₄	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133128	C₄	1	2	yes	—
12	C₄×#88322	C₂²	1	2	yes	—
13	#117×C₅₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
16	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
17	C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	#11719917	—	2	2	no	—
19	C₂×#88322	C₂×C₄	1	2	yes	—
20	C₂×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×#88322	D₄	1	2	yes	—
22	C₂×#88322	Q₈	1	2	yes	—
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	#11719910	—	2	2	no	—
25	#11719910	—	2	2	no	—
26	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	#88322	—	2	2	no	—
33	#88322	C₄⋊C₄	1	2	yes	—
34	#88322	C₂×D₄	1	1	yes	characteristic
35	#88322	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₅₃	—	2	4	no	—
40	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
43	#2151	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	#504	1	2	yes	—
47	C₂×M₃₂	—	53	1	no	—
48	C₂²×C₁₆	—	53	1	no	—
49	#405	—	53	1	no	—
50	C₄⋊C₁₆	—	53	4	no	—
51	C₅₃	#2151	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₁₆	—	53	1	no	—
53	C₂²×C₈	—	53	1	no	—
54	#117	#6157	1	1	yes	characteristic
55	M₃₂	—	106	2	no	—
56	C₂×C₁₆	—	53	2	no	—
57	C₂×C₁₆	—	53	2	no	—
58	C₂×C₁₆	—	106	2	no	—
59	#97	—	53	2	no	—
60	C₄×C₈	—	53	2	no	—
61	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
62	M₁₆	—	106	2	no	—
63	C₂×C₈	—	53	2	no	—
64	C₂×C₈	—	53	2	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	53	1	no	—
67	C₂×C₈	—	53	2	no	—
68	C₄⋊C₄	#88324	1	2	yes	—
69	C₂²⋊C₄	#88324	1	2	yes	—
70	C₄²	C₂×#6157	1	2	yes	—
71	C₁₆	—	106	2	no	—
72	C₁₆	—	106	2	no	—
73	C₂³	C₂×#88324	1	1	yes	characteristic
74	C₂×C₄	#11719936	1	2	yes	—
75	C₂×C₄	C₄⋊#6157	1	2	yes	—
76	C₂×C₄	C₂×#88324	1	4	yes	—
77	C₂×C₄	C₂×#88324	1	1	yes	characteristic
78	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
79	C₈	—	53	2	no	—
80	C₈	—	53	1	no	—
81	C₈	—	53	1	no	—
82	C₈	—	106	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#88324	1	2	yes	—
85	C₂²	C₂²×#88324	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#88324	1	2	yes	—
88	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
89	C₄	C₂×#11719936	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#88324	1	1	yes	characteristic
92	C₂	#16811681	1	2	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	#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊#11719910	C₂	semidirect
51. C₅₃	#2151	semidirect