Subgroups — #27384477

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	6	yes	maximal
2	#20274867	C₂	1	8	yes	maximal, complemented
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	#16810647	—	2	24	no	—
6	C₂×C₄⋊#88322	C₂²	1	24	yes	—
7	C₂²×C₄²×C₅₃	C₄	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
9	C₄³×C₅₃	C₄	1	3	yes	—
10	C₄³×C₅₃	C₄	1	1	yes	characteristic
11	C₄³×C₅₃	C₂²	1	4	yes	—
12	C₂×#11719917	—	2	24	no	—
13	C₂²×#88322	C₂³	1	8	yes	—
14	C₄⋊#88322	—	2	48	no	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	—	2	16	no	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅₃	D₄	1	12	yes	—
23	C₂×C₄²×C₅₃	Q₈	1	12	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
25	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	24	no	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	24	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
32	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
33	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	24	no	—
35	C₄²×C₅₃	C₄⋊C₄	1	12	yes	—
36	C₄²×C₅₃	C₄⋊C₄	1	12	yes	—
37	C₄²×C₅₃	M₁₆	1	16	yes	complemented
38	C₄²×C₅₃	C₂×D₄	1	12	yes	—
39	C₄²×C₅₃	C₂×Q₈	1	12	yes	—
40	#11719917	—	4	16	no	—
41	C₂×#88322	—	2	24	no	—
42	C₂³×C₅₃	—	2	4	no	—
43	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
44	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	12	no	—
46	C₂×C₄×C₅₃	—	2	48	no	—
47	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
50	C₂×C₄×C₅₃	#127	1	4	yes	—
51	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
52	C₂×C₄×C₅₃	C₂×M₁₆	1	24	yes	—
53	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
54	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
55	#88322	—	4	8	no	—
56	#20791	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	—	2	12	no	—
58	C₂²×C₅₃	#384	1	4	yes	—
59	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
60	C₂²×C₅₃	C₂²×M₁₆	1	4	yes	—
61	C₄×C₅₃	—	2	12	no	—
62	C₄×C₅₃	#384	1	4	yes	—
63	C₄×C₅₃	#423	1	24	yes	complemented
64	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
65	C₄×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
66	C₂×#423	—	53	6	no	—
67	C₂×C₄³	D₅₃	1	1	yes	characteristic
68	#1842	—	53	8	no	—
69	C₂×C₅₃	—	2	4	no	—
70	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#423	1	6	yes	—
72	C₂²×M₁₆	—	53	4	no	—
73	C₂²×C₄²	Dic₅₃	1	4	yes	—
74	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
75	#423	—	106	24	no	—
76	C₂×C₄⋊C₈	—	53	24	no	—
77	C₄³	Dic₅₃	1	3	yes	—
78	C₄³	Dic₅₃	1	1	yes	characteristic
79	C₄³	C₂×D₅₃	1	4	yes	—
80	C₅₃	#20791	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
82	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
83	C₂×M₁₆	—	106	24	no	—
84	C₂²×C₈	—	53	8	no	—
85	C₂×C₄²	—	2	16	no	—
86	C₂×C₄²	Dic₁₀₆	1	12	yes	—
87	C₂×C₄²	D₂₁₂	1	12	yes	—
88	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
89	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
90	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
91	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
92	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
93	C₄⋊C₈	—	106	48	no	—
94	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₄	—	2	24	no	—
96	C₂²×C₄	C₄⋊Dic₅₃	1	24	yes	—
97	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
98	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
99	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
102	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
103	M₁₆	—	212	16	no	—
104	C₂×C₈	—	106	24	no	—
105	C₄²	—	2	24	no	—
106	C₄²	#11719917	1	16	yes	complemented
107	C₄²	C₄⋊Dic₅₃	1	12	yes	—
108	C₄²	C₄⋊Dic₅₃	1	12	yes	—
109	C₄²	C₂×Dic₁₀₆	1	12	yes	—
110	C₄²	C₂×D₂₁₂	1	12	yes	—
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
113	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
119	C₂×C₄	#13133214	1	4	yes	—
120	C₂×C₄	#13133217	1	12	yes	—
121	C₂×C₄	C₂×#11719917	1	24	yes	—
122	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
123	C₂×C₄	C₂²×Dic₁₀₆	1	3	yes	—
124	C₈	—	212	8	no	—
125	C₂²	—	2	12	no	—
126	C₂²	#16810493	1	4	yes	—
127	C₂²	C₂²×C₄⋊Dic₅₃	1	3	yes	—
128	C₂²	C₂²×#11719917	1	4	yes	—
129	C₄	—	2	12	no	—
130	C₄	#16810493	1	4	yes	—
131	C₄	#16810647	1	24	yes	complemented
132	C₄	C₂×#13133214	1	1	yes	characteristic
133	C₄	C₂×#13133217	1	3	yes	—
134	C₂	—	2	4	no	—
135	C₂	C₂×#16810493	1	1	yes	characteristic
136	C₂	C₂×#16810647	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄³×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#20791
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20274867	C₂	semidirect
37. C₄²×C₅₃	M₁₆	semidirect
63. C₄×C₅₃	#423	semidirect
80. C₅₃	#20791	semidirect
106. C₄²	#11719917	semidirect
131. C₄	#16810647	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
#20274867	2
C₂×C₄³×C₅₃	2
#20791	53