Subgroups — #27273465

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	#16810378	C₄	1	4	yes	—
13	#16810378	C₄	1	4	yes	—
14	#16810345	C₄	1	4	yes	complemented
15	#16810345	C₄	1	4	yes	complemented
16	#16810344	C₄	1	4	yes	complemented
17	C₈×#88322	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#336	C₄	1	4	yes	—
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₄×#88322	C₂×C₄	1	2	yes	—
29	C₄×#88322	C₂×C₄	1	2	yes	—
30	C₄×#88322	C₂×C₄	1	2	yes	—
31	C₄×#88322	C₂×C₄	1	2	yes	—
32	C₄×#88322	C₂×C₄	1	2	yes	—
33	C₄×#88322	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₄²	1	4	yes	—
51	C₂×C₈×C₅₃	C₄²	1	4	yes	—
52	C₄²×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	4	no	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	C₄²	1	4	yes	—
64	C₂×#88322	C₄²	1	4	yes	—
65	C₂×#88322	C₄²	1	4	yes	—
66	C₂×#88322	C₄²	1	4	yes	—
67	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#131	1	4	yes	—
78	C₂×C₄×C₅₃	#131	1	4	yes	—
79	C₂×C₄×C₅₃	#131	1	4	yes	—
80	C₈×C₅₃	—	2	4	no	—
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	4	no	—
83	#88322	—	2	4	no	—
84	#88322	—	2	4	no	—
85	#88322	—	2	4	no	—
86	#7845	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₄³	1	1	yes	characteristic
88	C₂²×C₅₃	#405	1	2	yes	—
89	C₂²×C₅₃	#405	1	2	yes	—
90	C₂²×C₅₃	#405	1	2	yes	—
91	C₄×C₅₃	—	4	4	no	—
92	C₄×C₅₃	#405	1	2	yes	—
93	C₄×C₅₃	#405	1	2	yes	—
94	C₄×C₅₃	#405	1	2	yes	—
95	C₄×C₅₃	#405	1	2	yes	—
96	C₄×C₅₃	#405	1	2	yes	—
97	C₄×C₅₃	#405	1	2	yes	—
98	#1744	—	53	1	no	—
99	#1744	—	53	1	no	—
100	#1744	D₅₃	1	1	yes	characteristic
101	C₂×#322	—	53	1	no	—
102	C₂×#322	—	53	1	no	—
103	C₂×#322	—	53	1	no	—
104	C₂×C₈²	—	53	1	no	—
105	C₂×C₅₃	#1454	1	4	yes	—
106	C₂×C₅₃	#1726	1	1	yes	characteristic
107	C₂×C₅₃	#1726	1	1	yes	characteristic
108	C₂×C₅₃	#1726	1	1	yes	characteristic
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
115	C₄³	C₂×D₅₃	1	1	yes	characteristic
116	#336	—	53	4	no	—
117	#336	—	53	4	no	—
118	#336	Dic₅₃	1	4	yes	—
119	#322	—	53	4	no	—
120	#322	—	53	4	no	—
121	#322	—	53	4	no	—
122	C₈²	—	53	4	no	—
123	C₅₃	#7845	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	—	53	2	no	—
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
129	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
130	C₂×C₄²	C₄×D₅₃	1	2	yes	—
131	C₂×C₄²	C₄×D₅₃	1	2	yes	—
132	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	2	no	—
136	C₄×C₈	—	53	2	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	C₄×D₅₃	1	2	yes	—
143	C₄×C₈	C₄×D₅₃	1	2	yes	—
144	C₄×C₈	C₄×D₅₃	1	2	yes	—
145	C₄×C₈	C₄×D₅₃	1	2	yes	—
146	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
147	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	4	no	—
156	C₂×C₈	—	106	4	no	—
157	C₂×C₈	—	106	4	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
161	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
166	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
167	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
176	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	#13133223	1	4	yes	—
180	C₂×C₄	#13133224	1	4	yes	—
181	C₂×C₄	#13133225	1	4	yes	—
182	C₈	—	2	4	no	—
183	C₈	—	2	4	no	—
184	C₈	—	106	4	no	—
185	C₈	—	106	4	no	—
186	C₈	—	106	4	no	—
187	C₈	—	106	4	no	—
188	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
189	C₂²	#16810578	1	2	yes	—
190	C₂²	#16810579	1	2	yes	—
191	C₂²	#16810581	1	2	yes	—
192	C₄	—	4	4	no	—
193	C₄	#16810577	1	2	yes	—
194	C₄	#16810577	1	2	yes	—
195	C₄	#16810579	1	2	yes	—
196	C₄	#16810579	1	2	yes	—
197	C₄	#16810580	1	2	yes	—
198	C₄	#16810580	1	2	yes	—
199	C₂	#20273015	1	4	yes	—
200	C₂	#20274401	1	1	yes	characteristic
201	C₂	#20274402	1	1	yes	characteristic
202	C₂	#20274403	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7845
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
14. #16810345	C₄	semidirect
15. #16810345	C₄	semidirect
16. #16810344	C₄	semidirect
17. C₈×#88322	C₄	semidirect
123. C₅₃	#7845	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₅₃×#1744	2
#7845	53