Subgroups — #27537264

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811040	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276541	C₂	1	4	yes	maximal, complemented
4	D₄×#11719908	C₂	1	4	yes	maximal, complemented
5	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276329	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276326	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
10	#16811040	—	2	4	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂×#13133157	C₂²	1	2	yes	complemented
14	C₂²×#11719908	—	2	4	no	—
15	C₂²×#11719908	C₂²	1	2	yes	—
16	C₂²×#11719908	C₂²	1	2	yes	—
17	C₂²×#11719908	C₂²	1	1	yes	characteristic
18	C₂²×#11719908	C₂²	1	1	yes	characteristic
19	C₂²×#11719908	C₂²	1	2	yes	complemented
20	C₄⋊#11719908	C₂²	1	4	yes	complemented
21	C₂²⋊#11719908	C₂²	1	4	yes	—
22	C₂²⋊#11719908	C₂²	1	4	yes	complemented
23	#16810399	C₂²	1	2	yes	complemented
24	C₄×#11719908	C₂²	1	2	yes	complemented
25	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#525	C₄	1	1	yes	characteristic
28	C₅₃×#435	C₄	1	4	yes	—
29	C₈×D₄×C₅₃	C₂²	1	4	yes	—
30	C₅₃×#424	C₄	1	1	yes	characteristic
31	C₅₃×#408	C₄	1	1	yes	characteristic
32	C₅₃×#407	C₄	1	1	yes	characteristic
33	C₅₃×#405	C₂²	1	1	yes	characteristic
34	#13133157	—	2	2	no	—
35	#13133157	—	2	2	no	—
36	#13133157	—	2	4	no	—
37	C₂×#11719908	—	2	2	no	—
38	C₂×#11719908	—	2	4	no	—
39	C₂×#11719908	—	2	4	no	—
40	C₂×#11719908	—	2	4	no	—
41	C₂×#11719908	D₄	1	4	yes	complemented
42	C₂×#11719908	C₂³	1	2	yes	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₂×#11719908	C₂³	1	2	yes	—
45	C₂×#11719908	C₂³	1	1	yes	characteristic
46	C₂×#11719908	C₂³	1	1	yes	characteristic
47	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	#131×C₅₃	—	2	2	no	—
50	#131×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₂²×C₈×C₅₃	—	2	2	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	#123×C₅₃	C₈	1	2	yes	—
62	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
63	#121×C₅₃	C₈	1	2	yes	—
64	#120×C₅₃	C₈	1	2	yes	—
65	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
66	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	#97×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂³	1	2	yes	—
72	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
73	#41×C₅₃	—	2	2	no	—
74	#41×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
77	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
78	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₅₃	#41	1	2	yes	—
101	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
104	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
106	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
108	#11719908	—	2	2	no	—
109	#11719908	—	2	2	no	—
110	#11719908	—	2	2	no	—
111	#11719908	C₂×D₄	1	2	yes	—
112	#11719908	C₂×D₄	1	2	yes	—
113	C₂³×C₅₃	—	2	2	no	—
114	C₂³×C₅₃	—	2	4	no	—
115	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
118	Q₈×C₅₃	—	2	2	no	—
119	D₄×C₅₃	—	2	4	no	—
120	D₄×C₅₃	—	2	8	no	—
121	D₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#131	1	2	yes	—
134	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₈×C₅₃	—	2	2	no	—
136	C₈×C₅₃	—	2	2	no	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	C₄×D₄	1	2	yes	—
139	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
141	#33432	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	8	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	4	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	C₈×D₄	1	2	yes	—
149	C₂²×C₅₃	#504	1	4	yes	complemented
150	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	C₈×D₄	1	2	yes	—
155	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
157	C₂×#504	—	53	1	no	—
158	C₂³×C₁₆	—	53	1	no	—
159	#2929	D₅₃	1	1	yes	characteristic
160	#2168	—	53	4	no	—
161	D₄×C₁₆	—	53	4	no	—
162	#2151	—	53	1	no	—
163	#2113	—	53	1	no	—
164	#2112	—	53	1	no	—
165	#2108	—	53	1	no	—
166	C₂×C₅₃	—	2	2	no	—
167	C₂×C₅₃	—	2	4	no	—
168	C₂×C₅₃	—	4	2	no	—
169	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#504	1	2	yes	—
171	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
172	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
173	#525	Dic₅₃	1	1	yes	characteristic
174	#504	—	106	4	no	—
175	C₂×M₃₂	—	53	1	no	—
176	C₂×M₃₂	—	53	1	no	—
177	C₂×M₃₂	—	53	2	no	—
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	—	53	1	no	—
180	C₂²×C₁₆	—	53	2	no	—
181	C₂²×C₁₆	—	53	2	no	—
182	C₂²×C₁₆	—	53	2	no	—
183	C₂²×C₁₆	—	106	4	no	—
184	#435	Dic₅₃	1	4	yes	—
185	C₈×D₄	C₂×D₅₃	1	4	yes	—
186	#424	Dic₅₃	1	1	yes	characteristic
187	#408	Dic₅₃	1	1	yes	characteristic
188	#407	Dic₅₃	1	1	yes	characteristic
189	#405	C₂×D₅₃	1	1	yes	characteristic
190	C₄⋊C₁₆	—	53	4	no	—
191	C₂²⋊C₁₆	—	53	4	no	—
192	C₂²⋊C₁₆	—	53	4	no	—
193	#346	—	53	2	no	—
194	C₄×C₁₆	—	53	2	no	—
195	C₅₃	#33432	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
202	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
208	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	#123	#88322	1	2	yes	—
211	C₂²⋊Q₈	#88322	1	2	yes	—
212	#121	#88322	1	2	yes	—
213	#120	#88322	1	2	yes	—
214	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
215	#117	C₂×Dic₅₃	1	1	yes	characteristic
216	M₃₂	—	106	4	no	—
217	M₃₂	—	106	2	no	—
218	M₃₂	—	106	2	no	—
219	C₂×C₁₆	—	53	4	no	—
220	C₂×C₁₆	—	53	1	no	—
221	C₂×C₁₆	—	53	1	no	—
222	C₂×C₁₆	—	53	2	no	—
223	C₂×C₁₆	—	53	2	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	106	4	no	—
226	C₂×C₁₆	—	106	4	no	—
227	C₂×C₁₆	—	106	4	no	—
228	C₂×C₁₆	—	106	2	no	—
229	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
230	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
231	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
232	#97	C₂×Dic₅₃	1	2	yes	—
233	C₄×C₈	C₂²×D₅₃	1	2	yes	—
234	C₂⁴	C₂×#88322	1	1	yes	characteristic
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
238	C₂×D₄	C₂×#88322	1	4	yes	—
239	C₂×D₄	C₂×#88322	1	2	yes	—
240	C₂×D₄	C₂×#88322	1	1	yes	characteristic
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×#88322	1	2	yes	—
244	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	D₄×D₅₃	1	2	yes	—
258	C₂×C₈	#11719947	1	2	yes	—
259	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
260	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
262	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#88322	1	2	yes	—
265	C₄⋊C₄	C₂×#88322	1	4	yes	—
266	C₂²⋊C₄	C₂×#88322	1	2	yes	—
267	C₂²⋊C₄	C₂×#88322	1	4	yes	—
268	C₂²⋊C₄	C₂×#88322	1	4	yes	—
269	C₄²	C₂²×Dic₅₃	1	2	yes	—
270	C₁₆	—	53	2	no	—
271	C₁₆	—	53	2	no	—
272	C₁₆	—	106	2	no	—
273	C₁₆	—	106	2	no	—
274	C₁₆	—	106	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂²×#88322	1	2	yes	—
278	C₂³	C₂²×#88322	1	2	yes	—
279	C₂³	C₂²×#88322	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	8	no	—
282	D₄	—	2	4	no	—
283	D₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
291	C₂×C₄	C₂²×#88322	1	4	yes	—
292	C₂×C₄	C₂²×#88322	1	2	yes	—
293	C₂×C₄	C₂²×#88322	1	2	yes	—
294	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
295	C₂×C₄	#13133224	1	2	yes	—
296	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	D₄×Dic₅₃	1	2	yes	—
301	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
302	C₈	C₂×#11719947	1	1	yes	characteristic
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	D₄×#88322	1	2	yes	—
310	C₂²	#16811040	1	4	yes	complemented
311	C₂²	C₂³×#88322	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	D₄×#88322	1	2	yes	—
316	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
317	C₄	C₂×#13133224	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	—	2	2	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×D₄×#88322	1	1	yes	characteristic
322	C₂	C₂×#16811040	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2929	C₂
prime radical	C₅₃×#525	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2929	D₅₃
hypocenter	C₅₃	#33432
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. C₂×#16811040	C₂	semidirect
2. C₂³×#11719908	C₂	semidirect
3. #20276541	C₂	semidirect
4. D₄×#11719908	C₂	semidirect
5. #20276474	C₂	semidirect
6. #20276329	C₂	semidirect
7. #20276326	C₂	semidirect
8. #20276312	C₂	semidirect
13. C₂×#13133157	C₂²	semidirect
19. C₂²×#11719908	C₂²	semidirect
20. C₄⋊#11719908	C₂²	semidirect
22. C₂²⋊#11719908	C₂²	semidirect
23. #16810399	C₂²	semidirect
24. C₄×#11719908	C₂²	semidirect
41. C₂×#11719908	D₄	semidirect
149. C₂²×C₅₃	#504	semidirect
195. C₅₃	#33432	semidirect
310. C₂²	#16811040	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811040	2
C₂³×#11719908	2
#20276541	2
D₄×#11719908	2
#20276474	2
#20276329	2
#20276326	2
#20276312	2
C₅₃×#2929	2
#33432	53