Subgroups — #27277850

Order 13568.

362 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal
4	C₂×#16810373	C₂	1	2	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810378	—	2	2	no	—
10	#16810378	C₄	1	2	yes	—
11	#16810378	C₄	1	2	yes	—
12	#16810375	—	2	4	no	—
13	#16810373	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₄×#88322	D₄	1	2	yes	—
24	C₄×#88322	Q₈	1	2	yes	—
25	C₄×#88322	Q₈	1	2	yes	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	4	4	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	4	no	—
44	C₄²×C₅₃	—	4	2	no	—
45	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	4	no	—
55	C₂×#88322	C₄⋊C₄	1	4	yes	—
56	C₂×#88322	C₄⋊C₄	1	4	yes	—
57	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#131	1	2	yes	—
72	C₂×C₄×C₅₃	#131	1	2	yes	—
73	C₂×C₄×C₅₃	#135	1	4	yes	—
74	C₈×C₅₃	—	8	2	no	—
75	#88322	—	2	4	no	—
76	#88322	—	2	4	no	—
77	#88322	—	8	2	no	—
78	#8811	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	#384	1	1	yes	characteristic
80	C₂²×C₅₃	#424	1	1	yes	characteristic
81	C₂²×C₅₃	#424	1	1	yes	characteristic
82	C₂²×C₅₃	#427	1	2	yes	—
83	C₂²×C₅₃	#430	1	2	yes	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	—	4	2	no	—
88	C₄×C₅₃	#424	1	2	yes	—
89	C₄×C₅₃	#424	1	2	yes	—
90	C₄×C₅₃	#495	1	2	yes	—
91	C₄×C₅₃	#499	1	2	yes	—
92	#1831	—	53	1	no	—
93	#1831	D₅₃	1	1	yes	characteristic
94	#1744	—	53	1	no	—
95	C₂×#335	—	53	2	no	—
96	C₂×#334	—	53	2	no	—
97	C₂×C₅₃	#1572	1	2	yes	—
98	C₂×C₅₃	#1572	1	2	yes	—
99	C₂×C₅₃	#1843	1	1	yes	characteristic
100	C₂×C₅₃	#1846	1	1	yes	characteristic
101	C₂×C₅₃	#1849	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	53	2	no	—
103	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₄³	C₂×D₅₃	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	#336	—	53	2	no	—
109	#336	—	53	2	no	—
110	#336	—	106	2	no	—
111	#335	—	106	4	no	—
112	#334	—	106	4	no	—
113	C₅₃	#8811	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	—	106	2	no	—
118	C₂×C₄²	Dic₁₀₆	1	2	yes	—
119	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
122	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	106	2	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	53	2	no	—
131	C₂²×C₄	#11719919	1	2	yes	—
132	C₂²×C₄	#11719919	1	2	yes	—
133	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	53	4	no	—
139	C₂×C₈	—	53	4	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	—	212	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₄×D₅₃	1	1	yes	characteristic
151	C₄²	D₄×D₅₃	1	1	yes	characteristic
152	C₄²	Q₈×D₅₃	1	1	yes	characteristic
153	C₄²	Q₈×D₅₃	1	1	yes	characteristic
154	C₂³	C₂×#11719919	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	#13133213	1	1	yes	characteristic
165	C₂×C₄	#13133215	1	1	yes	characteristic
166	C₂×C₄	#13133216	1	1	yes	characteristic
167	C₂×C₄	#13133216	1	1	yes	characteristic
168	C₂×C₄	#13133223	1	2	yes	—
169	C₂×C₄	#13133225	1	2	yes	—
170	C₂×C₄	#13133237	1	4	yes	—
171	C₈	—	8	2	no	—
172	C₈	—	106	4	no	—
173	C₈	—	106	4	no	—
174	C₈	—	424	2	no	—
175	C₂²	#16810492	1	1	yes	characteristic
176	C₂²	#16810649	1	1	yes	characteristic
177	C₂²	#16810651	1	1	yes	characteristic
178	C₂²	#16810659	1	2	yes	—
179	C₂²	#16810671	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#16810648	1	2	yes	—
185	C₄	#16810648	1	2	yes	—
186	C₄	#16811005	1	2	yes	—
187	C₄	#16811022	1	2	yes	—
188	C₂	#20273493	1	2	yes	—
189	C₂	#20273496	1	2	yes	—
190	C₂	#20274868	1	1	yes	characteristic
191	C₂	#20274877	1	1	yes	characteristic
192	C₂	#20274889	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8811
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
113. C₅₃	#8811	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274461	2
C₂×#16810375	2
C₂×#16810373	2
C₅₃×#1831	2
#8811	53