Subgroups — #27536660

Order 13568.

830 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	C₂×#16811022	C₂	1	4	yes	maximal
2	#20283078	C₂	1	1	yes	maximal, characteristic
3	#20274877	C₂	1	4	yes	maximal
4	#20274874	C₂	1	2	yes	maximal
5	#20274871	C₂	1	2	yes	maximal
6	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133209	C₂²	1	2	yes	—
9	#16811022	—	2	16	no	—
10	C₂×#13133150	C₂²	1	8	yes	—
11	C₂×#13133148	C₂²	1	8	yes	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	#16810521	—	2	4	no	—
17	#16810493	C₂²	1	2	yes	—
18	#16810489	C₂²	1	2	yes	—
19	C₂×#125×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#389	—	2	4	no	—
21	C₅₃×#384	C₂²	1	2	yes	—
22	C₅₃×#382	C₂²	1	2	yes	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	C₂³	1	2	yes	—
25	C₂²×#88322	C₂³	1	2	yes	—
26	#13133209	—	2	8	no	—
27	C₂×C₄⋊Dic₅₃	—	2	4	no	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
29	C₂×#11719919	—	2	4	no	—
30	#13133150	—	2	16	no	—
31	#13133148	—	2	16	no	—
32	C₄×#88322	D₄	1	8	yes	—
33	C₄×#88322	Q₈	1	4	yes	—
34	C₄×#88322	Q₈	1	4	yes	—
35	C₄×#88322	Q₈	1	8	yes	—
36	#13133124	—	2	4	no	—
37	#125×C₅₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	D₄	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#95×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₄⋊C₄×C₅₃	—	4	8	no	—
54	C₄⋊C₄×C₅₃	—	4	8	no	—
55	C₄²×C₅₃	—	2	8	no	—
56	C₄²×C₅₃	C₂×D₄	1	4	yes	—
57	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
58	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
59	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
60	C₄⋊Dic₅₃	—	2	8	no	—
61	C₄⋊Dic₅₃	—	4	8	no	—
62	#11719919	—	4	8	no	—
63	C₂×#88322	C₂×D₄	1	4	yes	—
64	C₂×#88322	C₂×D₄	1	4	yes	—
65	C₂×#88322	C₂×Q₈	1	4	yes	—
66	C₂×#88322	C₂×Q₈	1	4	yes	—
67	C₂×#88322	C₂×Q₈	1	4	yes	—
68	C₂×#88322	C₂×Q₈	1	4	yes	—
69	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	4	8	no	—
74	C₂×C₄×C₅₃	#127	1	4	yes	—
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
77	C₂×C₄×C₅₃	#135	1	8	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
82	C₂×Dic₅₃	—	4	4	no	—
83	C₂×Dic₅₃	—	4	8	no	—
84	#88322	C₄⋊Q₈	1	8	yes	—
85	#88322	C₄⋊Q₈	1	8	yes	—
86	#33311	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×#135	1	4	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	4	no	—
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#499	1	8	yes	complemented
95	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
96	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
97	Dic₅₃	—	8	4	no	—
98	C₂×#499	—	53	4	no	—
99	#2859	—	53	1	no	—
100	#2859	D₅₃	1	1	yes	characteristic
101	#1846	—	53	4	no	—
102	#1845	—	53	2	no	—
103	#1844	—	53	2	no	—
104	C₄²×C₈	—	53	1	no	—
105	C₂×C₅₃	#2868	1	1	yes	characteristic
106	C₂×C₅₃	#3147	1	2	yes	—
107	C₂×C₅₃	C₂×#499	1	2	yes	—
108	C₂×C₅₃	#3160	1	2	yes	—
109	C₂×#125	—	53	2	no	—
110	C₂×#125	C₂×D₅₃	1	2	yes	—
111	#499	—	106	16	no	—
112	C₂×#107	—	53	8	no	—
113	C₂×#106	—	53	8	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	#389	—	2	4	no	—
119	#389	—	106	4	no	—
120	#384	—	53	2	no	—
121	#384	C₂×D₅₃	1	2	yes	—
122	#382	—	53	2	no	—
123	#382	C₂×D₅₃	1	2	yes	—
124	C₄³	C₂×D₅₃	1	1	yes	characteristic
125	C₅₃	#33311	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	53	2	no	—
128	#125	—	2	8	no	—
129	#125	—	106	8	no	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	53	4	no	—
133	C₂×C₄⋊C₄	—	106	4	no	—
134	C₂×C₄⋊C₄	—	106	4	no	—
135	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
136	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
137	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	106	16	no	—
141	#106	—	106	16	no	—
142	C₄×C₈	—	53	8	no	—
143	C₄×C₈	—	53	4	no	—
144	C₄×C₈	—	53	4	no	—
145	C₄×C₈	—	53	8	no	—
146	#95	—	2	4	no	—
147	#95	—	106	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	106	4	no	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
153	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	4	no	—
156	C₂×C₈	—	53	4	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	53	4	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	8	no	—
163	C₄⋊C₄	—	106	8	no	—
164	C₄⋊C₄	—	212	8	no	—
165	C₄⋊C₄	—	212	8	no	—
166	C₄²	—	2	8	no	—
167	C₄²	D₄×D₅₃	1	4	yes	—
168	C₄²	Q₈×D₅₃	1	4	yes	—
169	C₄²	Q₈×D₅₃	1	2	yes	—
170	C₄²	Q₈×D₅₃	1	2	yes	—
171	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	212	8	no	—
177	C₂×C₄	—	212	4	no	—
178	C₂×C₄	#13133213	1	4	yes	—
179	C₂×C₄	#13133215	1	4	yes	—
180	C₂×C₄	#13133215	1	4	yes	—
181	C₂×C₄	#13133237	1	8	yes	—
182	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
183	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
186	C₈	—	53	8	no	—
187	C₈	—	53	8	no	—
188	C₂²	C₂×#13133213	1	1	yes	characteristic
189	C₂²	C₂×#13133215	1	1	yes	characteristic
190	C₂²	C₂×#13133215	1	1	yes	characteristic
191	C₂²	C₂×#13133237	1	4	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	8	4	no	—
195	C₄	—	424	4	no	—
196	C₄	#16811022	1	8	yes	complemented
197	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
198	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
199	C₂	#20283111	1	1	yes	characteristic
200	C₂	#20285141	1	2	yes	—
201	C₂	C₂×#16811022	1	2	yes	—
202	C₂	#20285225	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#33311
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
94. C₄×C₅₃	#499	semidirect
125. C₅₃	#33311	semidirect
196. C₄	#16811022	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811022	2
#20283078	2
#20274877	2
#20274874	2
#20274871	2
C₄²×#88322	2
C₅₃×#2859	2
#33311	53