Subgroups — #27349723

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20273007	C₂	1	2	yes	maximal, complemented
3	#20273004	C₂	1	4	yes	maximal, complemented
4	#20272998	C₂	1	2	yes	maximal, complemented
5	C₈×#11719917	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16810680	C₄	1	4	yes	—
8	#16810680	C₄	1	4	yes	—
9	#16810676	C₂²	1	4	yes	—
10	#16810676	C₂²	1	4	yes	—
11	#16810647	C₄	1	4	yes	—
12	#16810587	C₄	1	4	yes	—
13	C₄×#11719917	C₂²	1	4	yes	—
14	#16810345	C₂²	1	8	yes	complemented
15	#16810344	C₂²	1	4	yes	complemented
16	C₈×#88322	C₂²	1	4	yes	complemented
17	C₂³×C₈×C₅₃	C₄	1	2	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
20	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	C₂×C₄	1	8	yes	—
26	C₄⋊#88322	C₂×C₄	1	8	yes	—
27	C₄⋊#88322	C₂×C₄	1	8	yes	—
28	C₂²⋊#88322	C₂×C₄	1	8	yes	—
29	C₂²⋊#88322	C₂×C₄	1	8	yes	—
30	#13133128	C₂×C₄	1	8	yes	—
31	C₄×#88322	C₂³	1	8	yes	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	8	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	8	no	—
68	C₂×C₈×C₅₃	—	2	8	no	—
69	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
70	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₄²	1	2	yes	—
74	C₄²×C₅₃	C₄²	1	1	yes	characteristic
75	C₄²×C₅₃	C₄²	1	2	yes	—
76	C₄²×C₅₃	C₄²	1	1	yes	characteristic
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
83	#11719917	—	2	16	no	—
84	C₂×#88322	C₂²×C₄	1	16	yes	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	2	4	no	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
103	C₂×C₄×C₅₃	#131	1	8	yes	—
104	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
108	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
109	#88322	—	2	16	no	—
110	#17323	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	8	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
116	C₂²×C₅₃	#405	1	4	yes	—
117	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
122	C₄×C₅₃	#405	1	4	yes	—
123	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
124	C₄×C₅₃	C₂×#131	1	4	yes	—
125	#2920	—	53	2	no	—
126	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
127	#1452	—	53	2	no	—
128	#1451	—	53	4	no	—
129	#1450	—	53	2	no	—
130	C₈×M₁₆	—	53	4	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	2	4	no	—
133	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#405	1	2	yes	—
135	C₂³×C₈	Dic₅₃	1	2	yes	—
136	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	#432	—	53	4	no	—
138	#432	—	53	4	no	—
139	#431	—	53	4	no	—
140	#431	—	53	4	no	—
141	#423	—	53	4	no	—
142	#407	—	53	4	no	—
143	C₄×M₁₆	—	53	4	no	—
144	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
145	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
149	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
150	#322	—	53	4	no	—
151	#322	—	53	8	no	—
152	C₈²	—	53	4	no	—
153	C₅₃	#17323	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	C₄×D₅₃	1	2	yes	—
155	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂×M₁₆	—	53	8	no	—
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
160	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
161	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
162	C₂×C₄²	C₄×D₅₃	1	2	yes	—
163	C₂×C₄²	C₄×D₅₃	1	2	yes	—
164	C₂×C₄²	C₄×D₅₃	1	2	yes	—
165	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄⋊C₈	—	53	8	no	—
170	C₄⋊C₈	—	53	8	no	—
171	C₂²⋊C₈	—	53	8	no	—
172	C₂²⋊C₈	—	53	8	no	—
173	#97	—	53	8	no	—
174	C₄×C₈	—	53	8	no	—
175	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
176	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
177	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
178	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
179	C₄×C₈	C₂²×D₅₃	1	2	yes	—
180	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
188	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
189	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
192	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
195	M₁₆	—	106	16	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	53	16	no	—
200	C₂×C₈	#11719917	1	8	yes	complemented
201	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
202	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
203	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
204	C₄²	C₄×Dic₅₃	1	2	yes	—
205	C₄²	C₄×Dic₅₃	1	2	yes	—
206	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
207	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
208	C₄²	C₂×C₄×D₅₃	1	2	yes	—
209	C₄²	C₂×C₄×D₅₃	1	2	yes	—
210	C₄²	C₂×C₄×D₅₃	1	2	yes	—
211	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
212	C₄²	C₂²×Dic₅₃	1	2	yes	—
213	C₄²	C₂³×D₅₃	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
217	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
227	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
231	C₂×C₄	C₂×#11719917	1	4	yes	—
232	C₂×C₄	#13133223	1	8	yes	—
233	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
234	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	—	106	16	no	—
237	C₈	C₂×#11719917	1	4	yes	—
238	C₈	C₂×#11719917	1	4	yes	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₄×#11719917	1	2	yes	—
244	C₂²	#16810578	1	4	yes	—
245	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	C₄×#11719917	1	2	yes	—
250	C₄	#16810578	1	4	yes	—
251	C₄	C₂²×#11719917	1	2	yes	—
252	C₄	C₂×#13133223	1	4	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	4	no	—
255	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
256	C₂	C₂×#16810578	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#17323
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20273007	C₂	semidirect
3. #20273004	C₂	semidirect
4. #20272998	C₂	semidirect
5. C₈×#11719917	C₂	semidirect
14. #16810345	C₂²	semidirect
15. #16810344	C₂²	semidirect
16. C₈×#88322	C₂²	semidirect
69. C₂×C₈×C₅₃	M₁₆	semidirect
153. C₅₃	#17323	semidirect
200. C₂×C₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20273007	2
#20273004	2
#20272998	2
C₈×#11719917	2
C₂²×C₄×C₈×C₅₃	2
#17323	53