Subgroups — #27300050

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276343	C₂	1	2	yes	maximal
4	#20272801	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16810444	—	2	1	no	—
9	#16810444	—	2	1	no	—
10	#16810444	—	2	1	no	—
11	#16810444	—	2	1	no	—
12	#16810407	—	2	4	no	—
13	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#405	—	2	1	no	—
17	C₅₃×#405	—	2	1	no	—
18	C₅₃×#405	C₄	1	1	yes	characteristic
19	C₅₃×#405	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	—	2	1	no	—
21	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	—	2	2	no	—
26	#13133157	—	2	2	no	—
27	#13133157	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	1	no	—
33	#131×C₅₃	—	2	1	no	—
34	#131×C₅₃	C₈	1	4	yes	—
35	#131×C₅₃	D₄	1	2	yes	—
36	#131×C₅₃	Q₈	1	2	yes	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	—	2	1	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	—	2	1	no	—
48	#117×C₅₃	—	2	1	no	—
49	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	—	2	1	no	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#97×C₅₃	—	2	1	no	—
53	#97×C₅₃	—	2	1	no	—
54	#97×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	#41×C₅₃	—	2	1	no	—
63	#41×C₅₃	—	2	1	no	—
64	#41×C₅₃	C₄⋊C₄	1	2	yes	—
65	#41×C₅₃	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	1	no	—
74	M₁₆×C₅₃	—	2	1	no	—
75	M₁₆×C₅₃	—	2	4	no	—
76	M₁₆×C₅₃	—	2	2	no	—
77	M₁₆×C₅₃	—	2	2	no	—
78	M₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
79	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₂²⋊C₄×C₅₃	—	2	2	no	—
98	C₂²⋊C₄×C₅₃	M₁₆	1	2	yes	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	1	no	—
101	C₄²×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
104	#11719908	—	4	2	no	—
105	#11719908	—	4	2	no	—
106	C₂³×C₅₃	—	2	1	no	—
107	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	Q₈×C₅₃	—	2	1	no	—
109	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
110	D₄×C₅₃	—	2	2	no	—
111	D₄×C₅₃	—	2	2	no	—
112	D₄×C₅₃	—	2	1	no	—
113	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
131	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
132	C₈×C₅₃	#121	1	1	yes	characteristic
133	C₈×C₅₃	#121	1	1	yes	characteristic
134	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
136	#12792	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	—	4	2	no	—
140	C₂²×C₅₃	#407	1	1	yes	characteristic
141	C₂²×C₅₃	#423	1	1	yes	characteristic
142	C₂²×C₅₃	#431	1	1	yes	characteristic
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	—	4	1	no	—
147	C₄×C₅₃	#380	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₅₃	#432	1	1	yes	characteristic
151	C₄×#131	D₅₃	1	1	yes	characteristic
152	C₂×#364	—	53	1	no	—
153	C₂×#364	—	53	1	no	—
154	#2117	—	53	2	no	—
155	#1378	—	53	2	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	4	2	no	—
160	C₂×C₅₃	#1797	1	1	yes	characteristic
161	C₂×#131	Dic₅₃	1	1	yes	characteristic
162	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
163	C₄×#41	Dic₅₃	1	1	yes	characteristic
164	C₂×M₃₂	—	53	2	no	—
165	C₂×M₃₂	—	53	2	no	—
166	#405	—	2	1	no	—
167	#405	—	2	1	no	—
168	#405	Dic₅₃	1	1	yes	characteristic
169	#405	C₂×D₅₃	1	1	yes	characteristic
170	C₄×M₁₆	—	2	1	no	—
171	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
172	C₂×C₄×C₈	—	2	1	no	—
173	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
174	#364	—	106	1	no	—
175	#364	—	106	1	no	—
176	#364	—	106	1	no	—
177	#364	—	106	1	no	—
178	#350	—	106	4	no	—
179	C₅₃	#12792	1	1	yes	characteristic, complemented, Sylow
180	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
181	#131	—	2	1	no	—
182	#131	—	2	1	no	—
183	#131	—	2	2	no	—
184	#131	#88322	1	4	yes	—
185	#131	Dic₁₀₆	1	2	yes	—
186	#131	D₂₁₂	1	2	yes	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
190	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₄×Q₈	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	2	no	—
198	#117	—	2	1	no	—
199	#117	C₂×Dic₅₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
202	M₃₂	—	106	2	no	—
203	M₃₂	—	106	2	no	—
204	M₃₂	—	106	2	no	—
205	M₃₂	—	106	2	no	—
206	C₂×C₁₆	—	106	2	no	—
207	C₂×C₁₆	—	106	2	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
218	#41	—	2	1	no	—
219	#41	—	2	1	no	—
220	#41	C₂×#88322	1	2	yes	—
221	#41	C₄⋊Dic₅₃	1	2	yes	—
222	C₂×Q₈	#11719917	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	#11719917	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
229	M₁₆	—	2	1	no	—
230	M₁₆	—	2	1	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	C₂×#88322	1	2	yes	—
235	M₁₆	C₄⋊Dic₅₃	1	2	yes	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×#88322	1	2	yes	—
242	C₂×C₈	#11719917	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
247	C₂×C₈	#11719945	1	1	yes	characteristic
248	C₂×C₈	#11719945	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	#11719917	1	2	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
259	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
260	C₁₆	—	212	2	no	—
261	C₁₆	—	212	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	C₂×#11719917	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₄⋊#88322	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	C₄⋊#88322	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₂×C₄	#13133176	1	1	yes	characteristic
281	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂²⋊#88322	1	2	yes	—
287	C₈	C₂²⋊#88322	1	2	yes	—
288	C₈	#13133191	1	1	yes	characteristic
289	C₈	#13133191	1	1	yes	characteristic
290	C₈	#13133196	1	1	yes	characteristic
291	C₈	#13133196	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#16810587	1	1	yes	characteristic
296	C₂²	#16810647	1	1	yes	characteristic
297	C₂²	#16810676	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	#16810483	1	1	yes	characteristic
303	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
304	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
305	C₄	#16810680	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#20274695	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂×C₅₃	#1797
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#12792
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16810444	C₂	semidirect
2. C₂×#16810444	C₂	semidirect
4. #20272801	C₂	semidirect
179. C₅₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20276343	2
#20272801	2
C₄×#131×C₅₃	2
#12792	53