Subgroups — #27540969

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287064	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276943	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276914	C₂	1	2	yes	maximal, complemented
5	#20276857	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276810	C₂	1	2	yes	maximal, complemented
7	#20276760	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276630	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	Q₈×D₂₁₂	—	2	1	no	—
14	#16811278	—	2	1	no	—
15	C₂×#13133161	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	#16811028	C₂²	1	1	yes	characteristic
18	#16811003	C₂²	1	1	yes	characteristic
19	#16810935	—	2	2	no	—
20	#16810903	—	2	1	no	—
21	#16810865	C₂²	1	2	yes	—
22	#16810856	C₂²	1	2	yes	—
23	#16810833	—	2	1	no	—
24	#16810780	—	2	2	no	—
25	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
26	#107×D₅₃	C₂²	1	2	yes	—
27	#16810450	C₂²	1	2	yes	—
28	C₄⋊#11719908	C₂²	1	1	yes	characteristic
29	#16810428	C₂²	1	2	yes	—
30	#16810428	C₂²	1	2	yes	—
31	#16810427	C₂²	1	2	yes	—
32	#16810426	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	1	yes	characteristic
39	C₅₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	4	1	no	—
42	C₂×Q₈×D₅₃	—	2	2	no	—
43	C₂×Dic₂₁₂	—	2	1	no	—
44	C₂×Dic₂₁₂	C₂³	1	2	yes	—
45	C₂×#11719925	—	2	2	no	—
46	Q₁₆×D₅₃	—	2	4	no	—
47	Q₁₆×D₅₃	—	4	2	no	—
48	C₂×C₈×D₅₃	C₂³	1	2	yes	—
49	#13133217	—	2	1	no	—
50	#13133217	—	2	1	no	—
51	#13133217	—	4	1	no	—
52	#13133199	—	4	2	no	—
53	#13133197	—	2	2	no	—
54	#13133197	—	4	2	no	—
55	#13133196	—	4	1	no	—
56	#13133188	—	2	2	no	—
57	C₄×D₂₁₂	—	4	1	no	—
58	C₄×D₂₁₂	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	#13133161	—	4	2	no	—
61	C₂×#11719908	C₂³	1	2	yes	—
62	C₁₆×D₅₃	—	4	2	no	—
63	#13133151	—	2	1	no	—
64	#13133151	C₂³	1	2	yes	—
65	#13133150	—	2	2	no	—
66	C₄⋊#88322	D₄	1	1	yes	characteristic
67	#13133140	—	2	2	no	—
68	#13133138	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	SD₃₂×C₅₃	—	2	2	no	—
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#107×C₅₃	C₂³	1	1	yes	characteristic
79	#103×C₅₃	—	2	1	no	—
80	#102×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	—	4	1	no	—
85	C₂×C₂²⋊D₅₃	—	4	2	no	—
86	Q₈×D₅₃	—	4	4	no	—
87	Q₈×D₅₃	—	8	2	no	—
88	#11719945	—	8	4	no	—
89	C₂×D₂₁₂	D₈	1	1	yes	characteristic
90	C₂×C₄×D₅₃	—	4	2	no	—
91	C₂×C₄×D₅₃	—	4	2	no	—
92	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
93	C₂×Dic₁₀₆	—	2	2	no	—
94	C₂×Dic₁₀₆	—	4	1	no	—
95	C₂×Dic₁₀₆	—	4	2	no	—
96	Q₁₆×C₅₃	—	2	1	no	—
97	Q₁₆×C₅₃	#41	1	2	yes	complemented
98	D₈×C₅₃	—	2	1	no	—
99	D₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₄⋊C₄×C₅₃	—	2	1	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₂²⋊C₄×C₅₃	—	4	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²⋊Dic₅₃	—	4	2	no	—
109	#11719925	—	4	4	no	—
110	#11719921	—	4	2	no	—
111	#11719921	D₈	1	2	yes	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	C₄⋊Dic₅₃	—	4	1	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	—	4	1	no	—
116	C₄⋊Dic₅₃	—	4	2	no	—
117	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
118	#11719919	—	4	2	no	—
119	C₂×#88322	C₂×D₄	1	2	yes	—
120	Dic₂₁₂	—	2	2	no	—
121	Dic₂₁₂	—	2	2	no	—
122	Dic₂₁₂	—	8	1	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₈×D₅₃	—	2	2	no	—
125	C₁₆×C₅₃	—	2	1	no	—
126	C₁₆×C₅₃	—	2	1	no	—
127	#11719908	—	2	2	no	—
128	C₂³×C₅₃	—	4	1	no	—
129	C₂²×D₅₃	C₂×D₈	1	2	yes	—
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	D₄×C₅₃	—	4	1	no	—
133	D₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	8	2	no	—
138	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₅₃	—	8	4	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	C₂×Dic₅₃	—	4	2	no	—
144	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
145	D₂₁₂	SD₃₂	1	4	yes	complemented
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	8	4	no	—
149	C₄×D₅₃	—	8	4	no	—
150	Dic₁₀₆	—	4	2	no	—
151	Dic₁₀₆	—	4	2	no	—
152	Dic₁₀₆	—	8	4	no	—
153	Dic₁₀₆	—	8	2	no	—
154	C₈×C₅₃	—	2	1	no	—
155	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
157	#88322	—	2	2	no	—
158	#33865	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	8	1	no	—
160	C₂²×C₅₃	—	8	1	no	—
161	C₂²×C₅₃	—	8	1	no	—
162	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	C₂×SD₃₂	1	4	yes	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	—	8	1	no	—
169	C₄×C₅₃	—	8	1	no	—
170	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
171	C₄×C₅₃	#546	1	1	yes	characteristic
172	C₄×C₅₃	#577	1	1	yes	characteristic
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	8	2	no	—
175	Dic₅₃	—	8	2	no	—
176	#3334	—	53	1	no	—
177	D₄×Q₁₆	—	53	1	no	—
178	#2241	—	53	1	no	—
179	#2235	—	53	2	no	—
180	#2225	—	53	1	no	—
181	#2217	—	53	2	no	—
182	#2208	—	53	1	no	—
183	#2185	—	53	2	no	—
184	C₄×SD₃₂	D₅₃	1	1	yes	characteristic, complemented
185	D₄×C₁₆	—	53	1	no	—
186	C₂×#358	—	53	2	no	—
187	C₂×C₅₃	—	16	1	no	—
188	C₂×C₅₃	#3294	1	1	yes	characteristic
189	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
190	C₂×C₅₃	#3417	1	1	yes	characteristic
191	D₅₃	—	2	4	no	—
192	C₂²×Q₁₆	—	53	2	no	—
193	D₄×Q₈	—	106	1	no	—
194	#547	—	106	1	no	—
195	C₂×SD₃₂	—	53	2	no	—
196	C₂×SD₃₂	C₂×D₅₃	1	1	yes	characteristic
197	C₂²×C₁₆	—	53	2	no	—
198	#500	—	53	1	no	—
199	#494	—	53	1	no	—
200	#480	—	106	2	no	—
201	#474	—	106	1	no	—
202	#467	—	53	2	no	—
203	#466	—	53	2	no	—
204	#462	—	106	1	no	—
205	#451	—	106	2	no	—
206	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
208	C₈×D₄	—	53	1	no	—
209	C₂×#107	—	53	2	no	—
210	#367	—	53	2	no	—
211	#367	C₂×D₅₃	1	1	yes	characteristic
212	C₄⋊C₁₆	—	53	1	no	—
213	#358	—	53	2	no	—
214	#358	—	53	2	no	—
215	#358	—	53	2	no	—
216	#358	—	53	2	no	—
217	#358	C₂×D₅₃	1	1	yes	characteristic
218	#357	C₂×D₅₃	1	1	yes	characteristic
219	C₂²⋊C₁₆	—	53	2	no	—
220	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
221	C₅₃	#33865	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	212	1	no	—
223	C₂²×Q₈	—	106	2	no	—
224	C₂×Q₁₆	—	53	2	no	—
225	C₂×Q₁₆	—	106	2	no	—
226	C₂×Q₁₆	—	106	4	no	—
227	C₂×Q₁₆	—	106	1	no	—
228	C₂×Q₁₆	—	212	2	no	—
229	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
230	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
231	C₂²×C₈	—	53	2	no	—
232	C₄⋊Q₈	—	106	1	no	—
233	C₄⋊Q₈	—	106	1	no	—
234	C₄⋊Q₈	—	212	1	no	—
235	#123	—	212	2	no	—
236	C₂²⋊Q₈	—	106	2	no	—
237	C₂²⋊Q₈	—	212	2	no	—
238	C₂²⋊Q₈	—	212	1	no	—
239	#121	—	106	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	53	1	no	—
243	C₄×D₄	—	212	1	no	—
244	C₂×C₄⋊C₄	—	106	2	no	—
245	SD₃₂	—	2	2	no	—
246	SD₃₂	—	212	2	no	—
247	C₂×C₁₆	—	53	2	no	—
248	C₂×C₁₆	—	212	2	no	—
249	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
251	#107	—	53	2	no	—
252	#107	—	106	2	no	—
253	#107	—	106	1	no	—
254	#107	C₂²×D₅₃	1	1	yes	characteristic
255	#107	C₂²×D₅₃	1	1	yes	characteristic
256	C₄⋊C₈	—	53	1	no	—
257	#103	—	2	1	no	—
258	#103	—	106	2	no	—
259	#102	—	2	1	no	—
260	#102	—	106	2	no	—
261	C₂²⋊C₈	—	53	2	no	—
262	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	#41	—	424	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	106	2	no	—
266	C₂×Q₈	—	424	2	no	—
267	C₂×Q₈	—	212	2	no	—
268	C₂×Q₈	—	212	4	no	—
269	C₂×Q₈	—	212	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	53	1	no	—
272	C₂×D₄	—	212	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	53	2	no	—
275	C₂²×C₄	—	212	2	no	—
276	C₂²×C₄	—	212	2	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	106	2	no	—
279	Q₁₆	—	106	2	no	—
280	Q₁₆	—	424	1	no	—
281	Q₁₆	—	212	4	no	—
282	Q₁₆	#11719945	1	2	yes	complemented
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	53	1	no	—
294	C₄⋊C₄	—	106	2	no	—
295	C₄⋊C₄	—	212	2	no	—
296	C₄⋊C₄	—	212	2	no	—
297	C₄⋊C₄	—	212	1	no	—
298	C₄⋊C₄	—	212	1	no	—
299	C₄⋊C₄	—	212	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	53	2	no	—
302	C₂²⋊C₄	—	212	2	no	—
303	C₂²⋊C₄	—	212	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	D₄×D₅₃	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	106	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	53	2	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	424	4	no	—
314	Q₈	—	424	2	no	—
315	Q₈	—	212	2	no	—
316	Q₈	—	212	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	53	4	no	—
320	D₄	—	424	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	424	4	no	—
329	C₂×C₄	—	424	4	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	—	212	2	no	—
332	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
333	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	106	2	no	—
337	C₈	C₂×#11719945	1	1	yes	characteristic
338	C₈	#13133263	1	1	yes	characteristic
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	53	4	no	—
343	C₂²	—	106	2	no	—
344	C₂²	—	106	2	no	—
345	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	106	2	no	—
351	C₄	—	424	2	no	—
352	C₄	—	424	2	no	—
353	C₄	SD₃₂×D₅₃	1	2	yes	—
354	C₄	#16811269	1	1	yes	characteristic
355	C₄	#16811430	1	1	yes	characteristic
356	C₂	—	16	1	no	—
357	C₂	—	106	4	no	—
358	C₂	#20286530	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic
360	C₂	#20287865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287064	C₂
hypercenter	C₄×SD₃₂	D₅₃
hypocenter	C₅₃	#33865
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₂₁₂	C₂	semidirect
3. #20276943	C₂	semidirect
4. #20276914	C₂	semidirect
5. #20276857	C₂	semidirect
6. #20276810	C₂	semidirect
7. #20276760	C₂	semidirect
8. #20276630	C₂	semidirect
9. C₁₆×D₂₁₂	C₂	semidirect
10. #358×D₅₃	C₂	semidirect
11. C₄×SD₃₂×C₅₃	C₂	semidirect
97. Q₁₆×C₅₃	#41	semidirect
145. D₂₁₂	SD₃₂	semidirect
184. C₄×SD₃₂	D₅₃	semidirect
221. C₅₃	#33865	semidirect
282. Q₁₆	#11719945	semidirect

Maximal subgroups

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

subgroup	index
#20287064	2
Q₁₆×D₂₁₂	2
#20276943	2
#20276914	2
#20276857	2
#20276810	2
#20276760	2
#20276630	2
C₁₆×D₂₁₂	2
#358×D₅₃	2
C₄×SD₃₂×C₅₃	2
#33865	53