Subgroups — #28311410

Order 13568.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289753	C₂	1	6	yes	maximal
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	C₂²	1	4	yes	—
4	C₂×C₄×#88322	C₄	1	3	yes	—
5	C₂×C₄×#88322	C₂²	1	3	yes	—
6	C₄³×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719910	—	2	12	no	—
8	C₂²×#88322	C₂×C₄	1	3	yes	—
9	C₂²×#88322	C₂³	1	1	yes	characteristic
10	C₄×#88322	—	2	6	no	—
11	C₄×#88322	—	2	6	no	—
12	C₂×C₄²×C₅₃	C₈	1	3	yes	—
13	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	—	2	6	no	—
19	C₄²×C₅₃	—	2	3	no	—
20	C₄²×C₅₃	—	2	3	no	—
21	C₄²×C₅₃	C₁₆	1	4	yes	complemented
22	C₂×#88322	—	2	3	no	—
23	C₂×#88322	—	2	3	no	—
24	C₂×#88322	M₁₆	1	6	yes	—
25	C₂×#88322	#41	1	6	yes	—
26	#11719910	—	4	4	no	—
27	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	3	no	—
29	C₂×C₄×C₅₃	—	2	3	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₅₃	M₃₂	1	6	yes	—
34	C₂×C₄×C₅₃	#117	1	3	yes	—
35	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
36	#88322	—	2	6	no	—
37	#88322	#126	1	4	yes	—
38	#12894	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	#431	1	3	yes	—
40	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₅₃	C₂×M₃₂	1	3	yes	—
42	C₄×C₅₃	—	2	6	no	—
43	C₄×C₅₃	—	2	6	no	—
44	C₄×C₅₃	#383	1	1	yes	characteristic
45	C₄×C₅₃	#432	1	3	yes	—
46	C₄²×C₈	—	53	1	no	—
47	#1374	—	53	6	no	—
48	C₂×C₅₃	#1839	1	1	yes	characteristic
49	C₂×C₅₃	#2162	1	3	yes	—
50	C₂×C₅₃	#2163	1	3	yes	—
51	C₂²×C₁₆	—	53	4	no	—
52	C₂×C₄×C₈	—	53	3	no	—
53	C₂×C₄×C₈	—	53	3	no	—
54	C₄³	#6157	1	1	yes	characteristic
55	C₅₃	#12894	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	1	no	—
57	C₂²×C₈	—	53	3	no	—
58	C₂×C₄²	#88324	1	3	yes	—
59	C₂×C₄²	#88324	1	1	yes	characteristic
60	C₂×C₄²	C₂×#6157	1	3	yes	—
61	C₂×C₁₆	—	106	12	no	—
62	C₄×C₈	—	106	6	no	—
63	C₄×C₈	—	106	6	no	—
64	C₂²×C₄	C₂×#88324	1	3	yes	—
65	C₂²×C₄	C₂×#88324	1	3	yes	—
66	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
67	C₂×C₈	—	53	6	no	—
68	C₂×C₈	—	53	6	no	—
69	C₂×C₈	—	106	3	no	—
70	C₂×C₈	—	106	3	no	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#11719910	1	4	yes	complemented
75	C₁₆	—	212	4	no	—
76	C₂³	C₂²×#88324	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#11719910	1	6	yes	—
82	C₂×C₄	#13133291	1	6	yes	—
83	C₂×C₄	#13133296	1	3	yes	—
84	C₂×C₄	D₅₃⋊M₁₆	1	3	yes	—
85	C₈	—	53	4	no	—
86	C₈	—	106	6	no	—
87	C₂²	#16811639	1	3	yes	—
88	C₂²	C₂²×#11719910	1	1	yes	characteristic
89	C₂²	C₂×#13133291	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#16811568	1	1	yes	characteristic
93	C₄	#16811646	1	3	yes	—
94	C₂	#20290642	1	1	yes	characteristic
95	C₂	#20291072	1	3	yes	—
96	C₂	#20291076	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₄³×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₄³	#6157
hypocenter	C₅₃	#12894
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
21. C₄²×C₅₃	C₁₆	semidirect
55. C₅₃	#12894	semidirect
74. C₄²	#11719910	semidirect

Maximal subgroups

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

subgroup	index
#20289753	2
C₄²×#88322	2
#12894	53