Subgroups — #27293125

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	2	yes	maximal
2	#20272633	C₂	1	2	yes	maximal, complemented
3	#20272632	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	C₄	1	4	yes	—
6	C₂²⋊#11719908	C₄	1	4	yes	—
7	C₄×#11719908	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₈²×C₅₃	C₄	1	1	yes	characteristic
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719908	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₅₃	C₈	1	2	yes	—
17	C₂²×C₈×C₅₃	C₈	1	2	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₈	1	2	yes	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₄²	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₃	C₄²	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	#11719908	—	2	8	no	—
46	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	M₃₂	1	4	yes	—
61	C₈×C₅₃	#131	1	4	yes	—
62	#11411	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	#346	1	2	yes	—
66	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	#346	1	2	yes	—
70	C₄×C₅₃	#405	1	2	yes	—
71	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
72	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
73	#2162	—	53	2	no	—
74	C₂×C₈²	D₅₃	1	1	yes	characteristic
75	#1312	—	53	2	no	—
76	#1311	—	53	2	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1450	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
80	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
82	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₄⋊C₁₆	—	53	4	no	—
85	C₂²⋊C₁₆	—	53	4	no	—
86	C₄×C₁₆	—	53	4	no	—
87	C₈²	Dic₅₃	1	1	yes	characteristic
88	C₈²	Dic₅₃	1	1	yes	characteristic
89	C₈²	C₂×D₅₃	1	1	yes	characteristic
90	C₈²	C₂×D₅₃	1	1	yes	characteristic
91	C₅₃	#11411	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	#88322	1	2	yes	—
93	C₂²×C₈	#88322	1	2	yes	—
94	C₂²×C₈	C₄×D₅₃	1	2	yes	—
95	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂×C₁₆	—	53	8	no	—
97	C₄×C₈	#88322	1	2	yes	—
98	C₄×C₈	#88322	1	2	yes	—
99	C₄×C₈	C₄×D₅₃	1	2	yes	—
100	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
107	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
108	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	C₂×#88322	1	2	yes	—
113	C₂×C₈	C₂×#88322	1	2	yes	—
114	C₂×C₈	C₂×#88322	1	2	yes	—
115	C₂×C₈	C₂×#88322	1	2	yes	—
116	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
118	C₄²	C₂×#88322	1	1	yes	characteristic
119	C₄²	C₂×#88322	1	1	yes	characteristic
120	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
121	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
122	C₁₆	—	106	8	no	—
123	C₂³	C₄×#88322	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×#88322	1	1	yes	characteristic
127	C₂×C₄	C₄×#88322	1	1	yes	characteristic
128	C₂×C₄	C₄×#88322	1	1	yes	characteristic
129	C₂×C₄	#13133157	1	4	yes	—
130	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
132	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	#13133157	1	4	yes	—
138	C₈	#13133223	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16810399	1	2	yes	—
142	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#16810399	1	2	yes	—
146	C₄	#16810578	1	2	yes	—
147	C₄	C₂×#13133157	1	2	yes	—
148	C₄	C₂×#13133157	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#20272998	1	1	yes	characteristic
151	C₂	C₂×#16810399	1	1	yes	characteristic
152	C₂	C₄×#13133157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₈²×C₅₃	C₂
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#11411
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20272633	C₂	semidirect
3. #20272632	C₂	semidirect
91. C₅₃	#11411	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
#20272633	2
#20272632	2
C₂×C₈²×C₅₃	2
#11411	53