Subgroups — #27270704

Order 13568.

332 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	2	yes	maximal
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	—	2	2	no	—
4	C₂×C₄⋊#88322	—	2	2	no	—
5	C₂×C₄⋊#88322	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
7	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	4	2	no	—
11	C₄⋊#88322	—	4	2	no	—
12	C₄⋊#88322	—	4	2	no	—
13	C₄⋊#88322	—	4	2	no	—
14	C₄⋊#88322	—	4	4	no	—
15	C₂²×C₈×C₅₃	—	2	2	no	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	2	yes	—
20	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	—	4	2	no	—
22	C₄⋊C₈×C₅₃	—	4	2	no	—
23	C₄⋊C₈×C₅₃	—	4	2	no	—
24	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	—	4	4	no	—
31	C₄²×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	4	1	no	—
36	C₄²×C₅₃	D₈	1	1	yes	characteristic
37	C₄²×C₅₃	D₈	1	2	yes	—
38	C₄²×C₅₃	SD₁₆	1	2	yes	—
39	C₄²×C₅₃	SD₁₆	1	2	yes	—
40	C₄²×C₅₃	SD₁₆	1	2	yes	—
41	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
42	C₄²×C₅₃	Q₁₆	1	2	yes	—
43	C₂×#88322	—	4	4	no	—
44	C₂×#88322	—	4	4	no	—
45	C₂×#88322	—	4	4	no	—
46	C₂³×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	#100	1	2	yes	—
54	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#101	1	2	yes	—
56	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	2	yes	—
58	C₂×C₄×C₅₃	#102	1	2	yes	—
59	C₂×C₄×C₅₃	#102	1	2	yes	—
60	C₂×C₄×C₅₃	#103	1	2	yes	—
61	C₂×C₄×C₅₃	#103	1	2	yes	—
62	C₂×C₄×C₅₃	#103	1	2	yes	—
63	C₂×C₄×C₅₃	#106	1	2	yes	—
64	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#107	1	2	yes	—
66	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
67	C₈×C₅₃	—	8	2	no	—
68	#88322	—	8	4	no	—
69	#7050	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	#338	1	1	yes	characteristic
71	C₂²×C₅₃	#340	1	2	yes	—
72	C₂²×C₅₃	#340	1	1	yes	characteristic
73	C₂²×C₅₃	#343	1	2	yes	—
74	C₂²×C₅₃	#343	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	4	1	no	—
80	C₄×C₅₃	#331	1	2	yes	—
81	C₄×C₅₃	#331	1	1	yes	characteristic
82	C₄×C₅₃	#332	1	2	yes	—
83	C₄×C₅₃	#332	1	2	yes	—
84	C₄×C₅₃	#332	1	2	yes	—
85	C₄×C₅₃	#333	1	2	yes	—
86	C₄×C₅₃	#333	1	1	yes	characteristic
87	#1842	—	53	2	no	—
88	#1842	D₅₃	1	1	yes	characteristic
89	C₂×C₅₃	#1292	1	2	yes	—
90	C₂×C₅₃	#1292	1	1	yes	characteristic
91	C₂×C₅₃	#1296	1	1	yes	characteristic
92	C₂×C₅₃	#1296	1	2	yes	—
93	C₂×C₅₃	#1309	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	2	2	no	—
95	C₂×C₄⋊C₈	—	53	4	no	—
96	C₂×C₄⋊C₈	—	106	2	no	—
97	C₂×C₄⋊C₈	—	106	2	no	—
98	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
99	C₄³	C₂×D₅₃	1	1	yes	characteristic
100	C₅₃	#7050	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	106	4	no	—
103	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
104	C₂×C₄²	C₄×D₅₃	1	2	yes	—
105	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
106	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	4	2	no	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	212	4	no	—
112	C₄⋊C₈	—	212	2	no	—
113	C₄⋊C₈	—	212	2	no	—
114	C₄⋊C₈	—	212	2	no	—
115	C₄⋊C₈	—	212	2	no	—
116	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₄	#11719919	1	2	yes	—
118	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂²×C₄	#11719921	1	2	yes	—
120	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	4	no	—
123	C₂×C₈	—	212	4	no	—
124	C₂×C₈	—	212	4	no	—
125	C₂×C₈	—	212	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	#11719913	1	2	yes	—
132	C₄²	D₄₂₄	1	1	yes	characteristic
133	C₄²	Dic₂₁₂	1	1	yes	characteristic
134	C₄²	#11719922	1	2	yes	—
135	C₄²	#11719923	1	2	yes	—
136	C₄²	#11719924	1	2	yes	—
137	C₄²	#11719925	1	2	yes	—
138	C₂³	#13133124	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	#13133133	1	2	yes	—
146	C₂×C₄	#13133134	1	1	yes	characteristic
147	C₂×C₄	#13133135	1	2	yes	—
148	C₂×C₄	#13133136	1	1	yes	characteristic
149	C₂×C₄	#13133137	1	2	yes	—
150	C₂×C₄	#13133138	1	2	yes	—
151	C₂×C₄	#13133139	1	2	yes	—
152	C₂×C₄	#13133140	1	2	yes	—
153	C₂×C₄	#13133141	1	2	yes	—
154	C₂×C₄	#13133142	1	2	yes	—
155	C₂×C₄	#13133148	1	2	yes	—
156	C₂×C₄	#13133149	1	1	yes	characteristic
157	C₂×C₄	#13133150	1	2	yes	—
158	C₂×C₄	#13133151	1	1	yes	characteristic
159	C₈	—	8	2	no	—
160	C₈	—	424	4	no	—
161	C₂²	#16810381	1	1	yes	characteristic
162	C₂²	#16810385	1	2	yes	—
163	C₂²	#16810386	1	1	yes	characteristic
164	C₂²	#16810391	1	2	yes	—
165	C₂²	#16810392	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	1	no	—
171	C₄	#16810366	1	2	yes	—
172	C₄	#16810367	1	1	yes	characteristic
173	C₄	#16810368	1	2	yes	—
174	C₄	#16810369	1	2	yes	—
175	C₄	#16810370	1	2	yes	—
176	C₄	#16810371	1	2	yes	—
177	C₄	#16810372	1	1	yes	characteristic
178	C₂	#20272589	1	2	yes	—
179	C₂	#20272590	1	1	yes	characteristic
180	C₂	#20272598	1	1	yes	characteristic
181	C₂	#20272599	1	2	yes	—
182	C₂	#20272628	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#7050
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
100. C₅₃	#7050	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₅₃×#1842	2
#7050	53