Subgroups — #28328375

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291477	C₂	1	2	yes	maximal, complemented
2	#20291111	C₂	1	4	yes	maximal, complemented
3	Q₃₂×#6157	C₂	1	2	yes	maximal, complemented
4	D₁₆×#6157	C₂	1	2	yes	maximal, complemented
5	C₂×#16811549	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291023	C₂	1	2	yes	maximal, complemented
7	#20290923	C₂	1	1	yes	maximal, characteristic, complemented
8	#508×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811692	—	2	2	no	—
10	D₅₃⋊M₃₂	C₂²	1	1	yes	characteristic
11	#16811662	—	2	4	no	—
12	Q₁₆×#6157	C₂²	1	4	yes	—
13	D₈×#6157	C₂²	1	4	yes	—
14	C₂×#13133286	C₂²	1	2	yes	—
15	#16811614	—	2	2	no	—
16	#16811586	C₂²	1	1	yes	characteristic
17	#16811549	C₂²	1	2	yes	—
18	#16811549	C₂²	1	2	yes	—
19	#16811536	C₂²	1	4	yes	—
20	#16811535	C₂²	1	4	yes	—
21	#16811510	C₂²	1	2	yes	—
22	C₁₆×#6157	C₂²	1	2	yes	—
23	#135×D₅₃	C₂²	1	2	yes	—
24	#16811056	C₄	1	1	yes	characteristic, complemented
25	#16811055	C₄	1	1	yes	characteristic, complemented
26	#16811054	C₄	1	2	yes	complemented
27	#16811053	C₄	1	2	yes	complemented
28	#16811052	C₄	1	1	yes	characteristic, complemented
29	Q₃₂×D₅₃	C₂²	1	1	yes	characteristic
30	SD₃₂×D₅₃	C₂²	1	2	yes	—
31	D₁₆×D₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#508	C₄	1	1	yes	characteristic, complemented
34	D₅₃⋊M₁₆	D₄	1	1	yes	characteristic
35	Q₈×#6157	—	2	4	no	—
36	D₄×#6157	—	2	4	no	—
37	D₄×#6157	—	4	4	no	—
38	#13133296	—	4	2	no	—
39	#13133296	D₄	1	1	yes	characteristic
40	C₂×C₄⋊#6157	—	2	2	no	—
41	C₂×C₄⋊#6157	—	4	2	no	—
42	#13133292	—	2	1	no	—
43	#13133291	—	2	1	no	—
44	D₅₃⋊C₁₆	C₂³	1	2	yes	—
45	#13133286	—	2	2	no	—
46	#13133286	C₂³	1	4	yes	—
47	#13133280	—	2	4	no	—
48	#13133279	—	2	4	no	—
49	#13133271	D₄	1	2	yes	—
50	C₈×#6157	C₂³	1	2	yes	—
51	#41×D₅₃	—	2	2	no	—
52	#13133240	C₂×C₄	1	2	yes	complemented
53	#13133239	C₂×C₄	1	2	yes	complemented
54	#13133238	—	2	2	no	—
55	#13133237	—	2	2	no	—
56	#13133236	C₂×C₄	1	2	yes	complemented
57	Q₁₆×D₅₃	C₂³	1	2	yes	—
58	SD₁₆×D₅₃	—	2	2	no	—
59	D₈×D₅₃	C₂³	1	2	yes	—
60	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
61	Dic₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
62	#13133163	C₂×C₄	1	2	yes	complemented
63	#13133162	C₂×C₄	1	2	yes	complemented
64	#13133161	C₂×C₄	1	2	yes	complemented
65	#13133160	C₂×C₄	1	2	yes	complemented
66	D₈₄₈	C₂×C₄	1	1	yes	characteristic, complemented
67	#13133158	C₂×C₄	1	2	yes	complemented
68	C₂×#11719908	C₂×C₄	1	1	yes	characteristic, complemented
69	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
70	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
71	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
72	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
73	#135×C₅₃	C₂×C₄	1	2	yes	complemented
74	Q₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
75	SD₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
76	D₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂²×#6157	—	4	2	no	—
79	C₂²×#6157	—	4	4	no	—
80	#41×C₅₃	—	2	2	no	—
81	#11719949	—	2	2	no	—
82	Q₈×D₅₃	—	2	2	no	—
83	#11719947	—	2	2	no	—
84	#11719947	—	4	2	no	—
85	D₄×D₅₃	—	2	2	no	—
86	D₄×D₅₃	—	4	2	no	—
87	#11719945	—	2	2	no	—
88	#11719945	—	4	2	no	—
89	C₂×C₄×D₅₃	—	4	2	no	—
90	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊#6157	—	4	4	no	—
92	C₂²⋊#6157	D₈	1	2	yes	—
93	#11719940	—	2	1	no	—
94	C₄⋊#6157	—	2	4	no	—
95	C₄⋊#6157	—	4	4	no	—
96	C₄⋊#6157	—	4	2	no	—
97	C₄⋊#6157	D₈	1	1	yes	characteristic
98	C₄⋊#6157	D₈	1	1	yes	characteristic
99	C₄×#6157	—	4	4	no	—
100	C₄×#6157	C₂×D₄	1	2	yes	—
101	#11719936	—	2	1	no	—
102	D₅₃⋊C₈	C₂×D₄	1	2	yes	—
103	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₅₃	—	2	2	no	—
105	D₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	#11719925	—	2	2	no	—
108	#11719924	—	2	2	no	—
109	#11719923	—	2	2	no	—
110	#11719922	—	2	2	no	—
111	C₂×#88322	#41	1	1	yes	characteristic
112	Dic₂₁₂	C₂²×C₄	1	2	yes	—
113	D₄₂₄	C₂²×C₄	1	2	yes	—
114	#11719913	—	2	2	no	—
115	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
117	C₈×D₅₃	#41	1	1	yes	characteristic
118	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
119	#11719910	—	2	2	no	—
120	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	#11719908	C₂²×C₄	1	1	yes	characteristic
123	#11719908	C₂²×C₄	1	1	yes	characteristic
124	C₂²×D₅₃	—	4	2	no	—
125	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×#6157	—	4	4	no	—
127	C₂×#6157	—	16	4	no	—
128	C₂×#6157	—	16	2	no	—
129	C₂×#6157	C₂×D₈	1	2	yes	—
130	C₂×#6157	C₂×D₈	1	2	yes	—
131	Q₈×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂²⋊D₅₃	—	4	2	no	—
137	C₂²⋊D₅₃	—	4	2	no	—
138	C₂²⋊D₅₃	—	4	2	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	#135	1	1	yes	characteristic
141	D₂₁₂	—	2	2	no	—
142	D₂₁₂	—	4	2	no	—
143	C₄×D₅₃	—	4	2	no	—
144	C₄×D₅₃	—	4	2	no	—
145	C₄×D₅₃	—	4	2	no	—
146	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
147	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
148	C₄×D₅₃	#135	1	1	yes	characteristic
149	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
150	Dic₁₀₆	—	2	2	no	—
151	Dic₁₀₆	—	4	2	no	—
152	#88324	—	2	2	no	—
153	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	#88322	C₄×D₄	1	1	yes	characteristic
156	#88322	C₂×#41	1	1	yes	characteristic
157	#33459	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	4	2	no	—
159	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
160	C₂×D₅₃	—	2	1	no	—
161	C₂×D₅₃	—	2	1	no	—
162	C₂×D₅₃	—	4	2	no	—
163	C₂×D₅₃	—	8	2	no	—
164	C₂×D₅₃	—	8	2	no	—
165	C₂×D₅₃	C₄×D₈	1	1	yes	characteristic
166	C₂×D₅₃	C₂²×D₈	1	1	yes	characteristic
167	#6157	—	2	2	no	—
168	#6157	—	2	2	no	—
169	#6157	—	8	4	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
172	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
173	Dic₅₃	—	4	2	no	—
174	Dic₅₃	C₄×D₈	1	1	yes	characteristic
175	Dic₅₃	C₂×#135	1	1	yes	characteristic
176	C₂×#508	—	53	1	no	—
177	#2951	—	53	2	no	—
178	#2175	—	53	4	no	—
179	C₄×Q₃₂	—	53	2	no	—
180	C₄×D₁₆	—	53	2	no	—
181	C₂×#366	—	53	1	no	—
182	#2139	—	53	2	no	—
183	#2108	—	53	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	8	2	no	—
186	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
187	D₅₃	—	2	1	no	—
188	D₅₃	—	8	2	no	—
189	D₅₃	#3415	1	1	yes	characteristic
190	D₅₃	#3417	1	1	yes	characteristic
191	C₂×#135	—	53	2	no	—
192	#520	—	106	2	no	—
193	#508	—	53	2	no	—
194	#508	—	53	1	no	—
195	#508	—	53	2	no	—
196	#508	—	53	1	no	—
197	#508	—	53	1	no	—
198	#508	#6157	1	1	yes	characteristic, complemented
199	C₂×Q₃₂	—	53	1	no	—
200	C₂×SD₃₂	—	53	2	no	—
201	C₂×D₁₆	—	53	1	no	—
202	C₂×M₃₂	—	53	1	no	—
203	C₂²×C₁₆	—	53	1	no	—
204	#440	—	106	4	no	—
205	C₄×Q₁₆	—	53	4	no	—
206	C₄×D₈	—	53	4	no	—
207	C₂×#107	—	53	2	no	—
208	#417	—	106	2	no	—
209	#405	—	53	1	no	—
210	#366	—	53	2	no	—
211	#366	—	53	2	no	—
212	#358	—	53	4	no	—
213	#357	—	53	4	no	—
214	#346	—	53	2	no	—
215	C₄×C₁₆	—	53	2	no	—
216	C₅₃	#33459	1	1	yes	characteristic, complemented, Sylow
217	C₂×#41	—	106	2	no	—
218	#135	—	53	2	no	—
219	#135	—	53	2	no	—
220	#135	—	53	2	no	—
221	#135	—	106	2	no	—
222	#135	—	106	2	no	—
223	#135	C₂×#6157	1	2	yes	complemented
224	C₂×Q₁₆	—	53	2	no	—
225	C₂×SD₁₆	—	106	2	no	—
226	C₂×D₈	—	53	2	no	—
227	C₂×M₁₆	—	53	1	no	—
228	C₂²×C₈	—	53	1	no	—
229	C₄×Q₈	—	106	4	no	—
230	C₄×D₄	—	106	4	no	—
231	C₄×D₄	—	212	4	no	—
232	#117	—	53	1	no	—
233	#117	—	212	2	no	—
234	C₂×C₄⋊C₄	—	106	2	no	—
235	C₂×C₄⋊C₄	—	212	2	no	—
236	Q₃₂	—	53	2	no	—
237	Q₃₂	—	53	1	no	—
238	Q₃₂	C₂×#6157	1	1	yes	characteristic, complemented
239	SD₃₂	—	53	2	no	—
240	SD₃₂	—	53	2	no	—
241	SD₃₂	—	53	2	no	—
242	SD₃₂	C₂×#6157	1	2	yes	complemented
243	D₁₆	—	53	2	no	—
244	D₁₆	—	53	1	no	—
245	D₁₆	C₂×#6157	1	1	yes	characteristic, complemented
246	M₃₂	—	106	1	no	—
247	M₃₂	—	106	1	no	—
248	C₂×C₁₆	—	53	1	no	—
249	C₂×C₁₆	—	53	1	no	—
250	C₂×C₁₆	—	53	1	no	—
251	C₂×C₁₆	—	53	1	no	—
252	C₂×C₁₆	—	53	1	no	—
253	C₂×C₁₆	—	53	2	no	—
254	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic, complemented
255	#107	—	53	4	no	—
256	#107	—	106	2	no	—
257	#103	—	106	4	no	—
258	#102	—	106	4	no	—
259	#97	—	53	2	no	—
260	C₄×C₈	—	53	2	no	—
261	#41	—	2	2	no	—
262	#41	—	106	2	no	—
263	#41	—	106	2	no	—
264	#41	—	106	2	no	—
265	#41	—	212	2	no	—
266	#41	—	212	2	no	—
267	C₂×Q₈	—	106	2	no	—
268	C₂×D₄	—	106	2	no	—
269	C₂×D₄	—	212	2	no	—
270	C₂²×C₄	—	53	1	no	—
271	C₂²×C₄	—	212	2	no	—
272	C₂²×C₄	—	212	4	no	—
273	C₂²×C₄	—	212	2	no	—
274	Q₁₆	—	53	2	no	—
275	Q₁₆	—	106	2	no	—
276	Q₁₆	C₂²×#6157	1	2	yes	—
277	SD₁₆	—	2	2	no	—
278	SD₁₆	—	106	2	no	—
279	SD₁₆	—	106	2	no	—
280	SD₁₆	—	106	2	no	—
281	D₈	—	53	2	no	—
282	D₈	—	106	2	no	—
283	D₈	C₂²×#6157	1	2	yes	—
284	M₁₆	—	106	1	no	—
285	M₁₆	—	106	1	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	1	no	—
289	C₂×C₈	—	53	1	no	—
290	C₂×C₈	—	53	1	no	—
291	C₂×C₈	—	53	2	no	—
292	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
293	C₄⋊C₄	—	53	1	no	—
294	C₄⋊C₄	—	53	1	no	—
295	C₄⋊C₄	—	106	4	no	—
296	C₄⋊C₄	—	212	2	no	—
297	C₄⋊C₄	—	212	4	no	—
298	C₂²⋊C₄	—	53	2	no	—
299	C₂²⋊C₄	—	212	4	no	—
300	C₄²	—	53	2	no	—
301	C₄²	—	212	4	no	—
302	C₁₆	—	53	1	no	—
303	C₁₆	—	53	1	no	—
304	C₁₆	—	106	2	no	—
305	C₁₆	C₂²×#6157	1	1	yes	characteristic
306	C₁₆	C₂²×#6157	1	1	yes	characteristic
307	C₂³	—	53	1	no	—
308	C₂³	—	212	2	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	106	2	no	—
311	Q₈	—	212	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	106	2	no	—
315	D₄	—	212	2	no	—
316	D₄	—	212	2	no	—
317	D₄	—	212	2	no	—
318	D₄	—	212	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	1	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	53	1	no	—
324	C₂×C₄	—	53	1	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	53	2	no	—
327	C₂×C₄	—	848	4	no	—
328	C₂×C₄	—	848	2	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	—	212	2	no	—
332	C₂×C₄	—	212	4	no	—
333	C₂×C₄	—	212	2	no	—
334	C₂×C₄	D₄×#6157	1	1	yes	characteristic
335	C₈	—	53	1	no	—
336	C₈	—	53	1	no	—
337	C₈	—	106	2	no	—
338	C₈	D₄×#6157	1	1	yes	characteristic
339	C₈	C₂³×#6157	1	1	yes	characteristic
340	C₂²	—	4	2	no	—
341	C₂²	—	53	1	no	—
342	C₂²	—	53	1	no	—
343	C₂²	—	106	1	no	—
344	C₂²	—	106	1	no	—
345	C₂²	—	424	2	no	—
346	C₂²	—	424	2	no	—
347	C₂²	—	212	2	no	—
348	C₂²	D₈×#6157	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	53	1	no	—
351	C₄	—	53	1	no	—
352	C₄	—	106	2	no	—
353	C₄	—	106	2	no	—
354	C₄	—	424	4	no	—
355	C₄	—	212	2	no	—
356	C₄	D₈×#6157	1	1	yes	characteristic
357	C₄	C₂×D₄×#6157	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	8	2	no	—
360	C₂	—	53	1	no	—
361	C₂	—	53	1	no	—
362	C₂	—	106	1	no	—
363	C₂	—	424	2	no	—
364	C₂	C₂×D₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6157
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₈	C₂³×#6157
Fitting	C₅₃×#508	C₄
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#508×D₅₃	C₂
cosocle	C₈×D₅₃	C₂⁴
hypercenter	#508	#6157
hypocenter	C₅₃	#33459
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #20291477	C₂	semidirect
2. #20291111	C₂	semidirect
3. Q₃₂×#6157	C₂	semidirect
4. D₁₆×#6157	C₂	semidirect
5. C₂×#16811549	C₂	semidirect
6. #20291023	C₂	semidirect
7. #20290923	C₂	semidirect
24. #16811056	C₄	semidirect
25. #16811055	C₄	semidirect
26. #16811054	C₄	semidirect
27. #16811053	C₄	semidirect
28. #16811052	C₄	semidirect
33. C₅₃×#508	C₄	semidirect
52. #13133240	C₂×C₄	semidirect
53. #13133239	C₂×C₄	semidirect
56. #13133236	C₂×C₄	semidirect
61. Dic₄₂₄	C₂×C₄	semidirect
62. #13133163	C₂×C₄	semidirect
63. #13133162	C₂×C₄	semidirect
64. #13133161	C₂×C₄	semidirect
65. #13133160	C₂×C₄	semidirect
66. D₈₄₈	C₂×C₄	semidirect
67. #13133158	C₂×C₄	semidirect
68. C₂×#11719908	C₂×C₄	semidirect
69. C₁₆×D₅₃	C₂×C₄	semidirect
70. C₁₆×D₅₃	C₂×C₄	semidirect
73. #135×C₅₃	C₂×C₄	semidirect
74. Q₃₂×C₅₃	C₂×C₄	semidirect
75. SD₃₂×C₅₃	C₂×C₄	semidirect
76. D₁₆×C₅₃	C₂×C₄	semidirect
77. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
198. #508	#6157	semidirect
216. C₅₃	#33459	semidirect
223. #135	C₂×#6157	semidirect
238. Q₃₂	C₂×#6157	semidirect
242. SD₃₂	C₂×#6157	semidirect
245. D₁₆	C₂×#6157	semidirect
254. C₂×C₁₆	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
#20291477	2
#20291111	2
Q₃₂×#6157	2
D₁₆×#6157	2
C₂×#16811549	2
#20291023	2
#20290923	2
#508×D₅₃	2
#33459	53