Subgroups — #28311424

Order 13568.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719910	C₂	1	6	yes	maximal
2	#20274402	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	C₂²	1	4	yes	—
4	C₄⋊#11719910	—	2	12	no	—
5	C₂×#13133128	C₄	1	3	yes	—
6	C₂×C₄×#88322	C₂²	1	3	yes	—
7	C₅₃×#376	C₄	1	1	yes	characteristic
8	C₂×#11719910	—	2	12	no	—
9	C₂²×#88322	C₂×C₄	1	3	yes	—
10	C₂²×#88322	C₂³	1	1	yes	characteristic
11	#13133128	—	2	6	no	—
12	C₄×#88322	D₄	1	6	yes	—
13	C₄×#88322	Q₈	1	6	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
15	#95×C₅₃	C₈	1	4	yes	—
16	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
19	C₄²×C₅₃	M₁₆	1	6	yes	—
20	C₂×#88322	—	2	6	no	—
21	C₂×#88322	C₄⋊C₄	1	6	yes	—
22	C₂×#88322	C₂×D₄	1	3	yes	—
23	C₂×#88322	C₂×Q₈	1	3	yes	—
24	#11719910	—	4	4	no	—
25	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	12	no	—
27	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
30	#88322	—	2	6	no	—
31	#88322	#127	1	1	yes	characteristic
32	#88322	C₄⋊Q₈	1	3	yes	—
33	#12899	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
35	C₂²×C₅₃	#504	1	4	yes	—
36	C₄×C₅₃	—	2	12	no	—
37	C₄×C₅₃	#384	1	1	yes	characteristic
38	C₄×C₅₃	#423	1	3	yes	—
39	C₂×C₄⋊C₁₆	—	53	6	no	—
40	#1726	—	53	1	no	—
41	C₂×C₅₃	#1842	1	1	yes	characteristic
42	C₂×C₅₃	#2151	1	6	yes	—
43	C₂²×C₁₆	—	53	4	no	—
44	C₂×#97	—	53	3	no	—
45	C₂×C₄×C₈	—	53	3	no	—
46	#376	#6157	1	1	yes	characteristic
47	C₄⋊C₁₆	—	106	12	no	—
48	C₅₃	#12899	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	53	1	no	—
50	C₂²×C₈	—	53	3	no	—
51	C₂×C₄²	C₂×#6157	1	3	yes	—
52	C₂×C₁₆	—	106	12	no	—
53	#97	—	106	6	no	—
54	C₄×C₈	—	53	6	no	—
55	C₄×C₈	—	53	6	no	—
56	#95	#88324	1	4	yes	—
57	C₂²×C₄	C₂×#88324	1	6	yes	—
58	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
59	C₂×C₈	—	53	3	no	—
60	C₂×C₈	—	53	3	no	—
61	C₂×C₈	—	53	6	no	—
62	C₂×C₈	—	106	6	no	—
63	C₄²	#11719936	1	6	yes	—
64	C₄²	C₄⋊#6157	1	6	yes	—
65	C₁₆	—	212	4	no	—
66	C₂³	C₂²×#88324	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#88324	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
70	C₂×C₄	C₂×#11719936	1	3	yes	—
71	C₈	—	53	1	no	—
72	C₈	—	53	3	no	—
73	C₈	—	106	6	no	—
74	C₂²	C₂×C₄⋊#88324	1	3	yes	—
75	C₂²	#16811681	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#16811569	1	1	yes	characteristic
78	C₄	#16811627	1	3	yes	—
79	C₂	#20290647	1	1	yes	characteristic
80	C₂	#20291050	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#376	C₄
cosocle	C₂²×#88322	C₂³
hypercenter	#376	#6157
hypocenter	C₅₃	#12899
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
48. C₅₃	#12899	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊#11719910	2
#20274402	2
#12899	53