Subgroups — #27537178

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	#20276521	C₂	1	8	yes	maximal, complemented
3	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276325	C₂	1	2	yes	maximal
5	M₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719912	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133174	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	2	yes	—
11	#16810577	C₄	1	4	yes	—
12	#16810577	C₂²	1	4	yes	—
13	C₂×#13133128	C₄	1	1	yes	characteristic
14	C₂×#13133127	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊#11719908	C₂²	1	4	yes	complemented
17	#16810402	C₂²	1	8	yes	—
18	#16810398	C₂²	1	4	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719912	—	2	4	no	—
25	C₂×#11719912	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	—	2	2	no	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	C₂³	1	2	yes	—
32	#13133174	C₈	1	4	yes	—
33	#13133174	C₂×C₄	1	4	yes	—
34	C₂×#11719919	C₈	1	2	yes	—
35	C₂×#11719921	C₈	1	2	yes	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	#13133157	—	2	4	no	—
38	C₂×#11719908	C₂³	1	4	yes	—
39	#13133128	C₂×C₄	1	4	yes	—
40	#13133127	C₂×C₄	1	4	yes	—
41	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₅₃	—	2	4	no	—
47	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₅₃	—	2	2	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	#41	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×C₈	1	2	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	#11719921	C₂×C₈	1	8	yes	—
68	#11719919	C₂×C₈	1	8	yes	—
69	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
70	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	M₁₆	1	4	yes	—
73	C₂×#88322	C₂²×C₄	1	2	yes	—
74	C₂×#88322	C₂²×C₄	1	2	yes	—
75	#11719912	—	2	8	no	—
76	C₈×D₅₃	—	2	4	no	—
77	C₈×D₅₃	—	2	2	no	—
78	C₈×D₅₃	—	2	2	no	—
79	C₁₆×C₅₃	—	2	4	no	—
80	#11719908	—	2	4	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	2	no	—
83	C₂²×D₅₃	—	4	2	no	—
84	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	#117	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
94	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
95	C₄×D₅₃	—	2	2	no	—
96	C₄×D₅₃	—	2	2	no	—
97	C₄×D₅₃	—	2	4	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	#117	1	2	yes	—
100	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
102	#88322	—	2	2	no	—
103	#88322	C₂×M₁₆	1	4	yes	—
104	#33419	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	#431	1	2	yes	—
107	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
108	C₂×D₅₃	—	2	4	no	—
109	C₂×D₅₃	—	2	2	no	—
110	C₂×D₅₃	—	4	4	no	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	#431	1	2	yes	—
113	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
114	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
115	Dic₅₃	—	2	4	no	—
116	Dic₅₃	—	2	2	no	—
117	C₂×#405	—	53	1	no	—
118	#2164	—	53	8	no	—
119	#2150	—	53	1	no	—
120	#2150	D₅₃	1	1	yes	characteristic, complemented
121	#2112	—	53	2	no	—
122	C₄×M₃₂	—	53	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
125	C₂×C₅₃	#3407	1	2	yes	—
126	D₅₃	—	4	2	no	—
127	C₂²×M₁₆	—	53	1	no	—
128	C₂³×C₈	—	53	1	no	—
129	C₂×#117	—	53	1	no	—
130	C₂×M₃₂	—	53	2	no	—
131	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
132	#405	—	53	4	no	—
133	#405	—	53	4	no	—
134	C₂×#97	—	53	1	no	—
135	C₂×#97	—	53	1	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₄⋊C₁₆	—	53	4	no	—
139	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
140	C₂²⋊C₁₆	—	53	8	no	—
141	#346	—	53	4	no	—
142	C₄×C₁₆	—	53	4	no	—
143	C₅₃	#33419	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	—	53	1	no	—
145	C₂×M₁₆	—	53	4	no	—
146	C₂×M₁₆	—	106	4	no	—
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	2	no	—
150	C₂²×C₈	—	53	2	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	—	106	2	no	—
153	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
155	#117	—	53	4	no	—
156	#117	—	53	4	no	—
157	C₂×C₄⋊C₄	—	53	2	no	—
158	C₂×C₂²⋊C₄	—	53	2	no	—
159	C₂×C₄²	—	53	1	no	—
160	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
161	M₃₂	—	2	4	no	—
162	M₃₂	—	106	4	no	—
163	C₂×C₁₆	—	53	4	no	—
164	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
165	#97	—	53	4	no	—
166	#97	—	53	4	no	—
167	C₄×C₈	—	53	2	no	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	C₄×D₅₃	1	2	yes	—
170	C₄×C₈	C₂²×D₅₃	1	2	yes	—
171	C₂⁴	—	53	1	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	53	2	no	—
175	C₂²×C₄	—	53	2	no	—
176	C₂²×C₄	—	106	2	no	—
177	C₂²×C₄	—	106	2	no	—
178	C₂²×C₄	C₈×D₅₃	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	M₁₆	—	106	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	53	4	no	—
183	C₂×C₈	—	53	2	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	106	2	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	—	106	2	no	—
188	C₂×C₈	—	106	2	no	—
189	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₂×C₈	#11719947	1	2	yes	—
192	C₂×C₈	#11719949	1	2	yes	—
193	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
194	C₄⋊C₄	—	53	8	no	—
195	C₂²⋊C₄	—	53	8	no	—
196	C₄²	—	53	2	no	—
197	C₄²	—	53	2	no	—
198	C₄²	C₈×D₅₃	1	2	yes	—
199	C₄²	C₂×C₄×D₅₃	1	2	yes	—
200	C₁₆	—	2	4	no	—
201	C₁₆	—	106	4	no	—
202	C₂³	—	53	2	no	—
203	C₂³	—	106	2	no	—
204	C₂³	—	212	2	no	—
205	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	53	4	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	—	106	2	no	—
211	C₂×C₄	—	106	2	no	—
212	C₂×C₄	—	106	4	no	—
213	C₂×C₄	—	212	2	no	—
214	C₂×C₄	#13133175	1	2	yes	—
215	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
216	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
217	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
218	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	53	4	no	—
221	C₈	—	106	2	no	—
222	C₈	#13133175	1	2	yes	—
223	C₈	C₂×#11719947	1	1	yes	characteristic
224	C₈	C₂×#11719949	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	106	4	no	—
227	C₂²	—	106	2	no	—
228	C₂²	—	212	4	no	—
229	C₂²	#16810675	1	2	yes	—
230	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	106	4	no	—
233	C₄	—	106	2	no	—
234	C₄	#16810675	1	2	yes	—
235	C₄	C₂×#13133175	1	1	yes	characteristic
236	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	212	2	no	—
239	C₂	C₂×#16810675	1	1	yes	characteristic
240	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#33419
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #20276521	C₂	semidirect
3. #20276471	C₂	semidirect
5. M₃₂×Dic₅₃	C₂	semidirect
6. C₅₃×#2150	C₂	semidirect
16. C₄⋊#11719908	C₂²	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
22. C₅₃×C₄⋊C₁₆	C₂²	semidirect
120. #2150	D₅₃	semidirect
139. C₄⋊C₁₆	C₂×D₅₃	semidirect
143. C₅₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
#20276521	2
#20276471	2
#20276325	2
M₃₂×Dic₅₃	2
C₅₃×#2150	2
#33419	53