Subgroups — #28317661

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291617	C₂	1	2	yes	maximal
2	#20290354	C₂	1	2	yes	maximal, complemented
3	#20290338	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	#20290276	C₂	1	1	yes	maximal, characteristic
6	#20290058	C₂	1	2	yes	maximal
7	C₈×#11719936	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811673	—	2	4	no	—
10	#16811673	C₂²	1	4	yes	—
11	Q₈×#88324	C₂²	1	4	yes	—
12	#16811645	—	2	2	no	—
13	#16811645	C₂²	1	2	yes	—
14	#16811642	—	2	2	no	—
15	#16811642	C₂²	1	1	yes	characteristic
16	C₄×#11719936	C₂²	1	1	yes	characteristic
17	#16811486	C₂²	1	2	yes	—
18	#16811484	C₂²	1	2	yes	—
19	#16811473	C₂²	1	4	yes	—
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	#88324⋊C₈	C₂²	1	2	yes	—
22	C₈×#88324	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₅₃	C₂²	1	2	yes	—
25	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
27	#16810717	C₂²	1	1	yes	characteristic
28	C₄×#11719925	C₄	1	2	yes	—
29	#16810715	C₂²	1	2	yes	—
30	#107×D₅₃	C₄	1	1	yes	characteristic
31	#103×D₅₃	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
34	C₂×#11719936	—	2	1	no	—
35	C₄⋊#88324	—	2	4	no	—
36	C₄⋊#88324	—	2	4	no	—
37	C₄⋊#88324	D₄	1	2	yes	—
38	C₄⋊#88324	C₂³	1	4	yes	—
39	#13133274	—	2	2	no	—
40	#13133274	D₄	1	1	yes	characteristic
41	#13133274	D₄	1	1	yes	characteristic
42	#13133272	—	2	2	no	—
43	#13133272	—	2	1	no	—
44	#13133272	—	2	4	no	—
45	C₄×#88324	—	2	4	no	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₄×#88324	C₂³	1	2	yes	—
48	C₂×Q₈×D₅₃	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719925	C₂×C₄	1	2	yes	—
51	Q₁₆×D₅₃	—	2	4	no	—
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₅₃	C₂³	1	2	yes	—
55	C₄×Dic₁₀₆	—	2	2	no	—
56	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
57	#13133182	—	2	2	no	—
58	#13133182	C₂³	1	2	yes	—
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
61	C₄²×D₅₃	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133142	C₂×C₄	1	2	yes	—
66	#13133141	C₂×C₄	1	2	yes	—
67	#13133140	C₂×C₄	1	2	yes	—
68	C₄×#88322	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
73	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
77	Q₈×D₅₃	—	2	4	no	—
78	Q₈×D₅₃	—	4	2	no	—
79	C₂×C₄×D₅₃	—	2	2	no	—
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
84	C₂×Dic₁₀₆	—	2	2	no	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	C₂×#88324	—	2	2	no	—
87	C₂×#88324	—	2	4	no	—
88	C₂×#88324	C₂×D₄	1	2	yes	—
89	C₂×#88324	C₂×D₄	1	2	yes	—
90	#11719936	—	2	4	no	—
91	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	#11719925	—	2	4	no	—
99	C₄⋊Dic₅₃	—	2	2	no	—
100	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
101	#11719919	—	2	2	no	—
102	#11719919	—	2	2	no	—
103	#11719919	—	2	2	no	—
104	C₄×Dic₅₃	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₅₃	#41	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
111	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
112	Dic₂₁₂	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	4	1	no	—
115	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₅₃	—	2	2	no	—
117	Q₈×C₅₃	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
128	C₄×D₅₃	—	2	2	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	4	4	no	—
133	C₄×D₅₃	—	4	4	no	—
134	Dic₁₀₆	—	2	2	no	—
135	Dic₁₀₆	—	2	2	no	—
136	Dic₁₀₆	—	4	4	no	—
137	#88324	—	2	4	no	—
138	#88324	—	2	2	no	—
139	#88324	—	2	2	no	—
140	#88324	—	4	4	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
143	#88322	—	2	1	no	—
144	#88322	—	2	1	no	—
145	#18058	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	#443	1	2	yes	—
150	C₄×C₅₃	—	2	4	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₅₃	#568	1	1	yes	characteristic
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	2	1	no	—
158	Dic₅₃	—	4	2	no	—
159	Dic₅₃	—	4	2	no	—
160	Dic₅₃	#578	1	2	yes	—
161	#2984	—	53	2	no	—
162	C₂×C₄×Q₁₆	—	53	1	no	—
163	#1591	—	53	2	no	—
164	#1587	—	53	4	no	—
165	C₈×Q₁₆	—	53	2	no	—
166	#1568	—	53	1	no	—
167	#1491	—	53	2	no	—
168	C₈×M₁₆	—	53	1	no	—
169	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
170	C₂×C₅₃	#2950	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
172	D₅₃	—	2	2	no	—
173	C₂²×Q₁₆	—	53	1	no	—
174	C₂×C₄×Q₈	—	53	2	no	—
175	#446	—	53	4	no	—
176	#446	—	106	4	no	—
177	C₈×Q₈	—	53	4	no	—
178	C₄×Q₁₆	—	53	1	no	—
179	C₄×Q₁₆	—	53	2	no	—
180	C₄×Q₁₆	—	53	1	no	—
181	C₄×Q₁₆	—	53	2	no	—
182	C₄×Q₁₆	—	53	1	no	—
183	C₄×Q₁₆	#6157	1	1	yes	characteristic
184	#432	—	53	2	no	—
185	#432	—	106	2	no	—
186	#431	—	53	1	no	—
187	#431	—	106	2	no	—
188	C₂×#107	—	53	1	no	—
189	C₂×#103	—	53	2	no	—
190	C₄×M₁₆	—	53	1	no	—
191	C₂×C₄×C₈	—	53	1	no	—
192	#335	—	53	2	no	—
193	#334	—	53	2	no	—
194	Q₈⋊C₈	—	53	4	no	—
195	Q₈⋊C₈	—	53	4	no	—
196	#322	—	53	2	no	—
197	C₈²	—	53	2	no	—
198	C₅₃	#18058	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	53	2	no	—
200	C₂×Q₁₆	—	53	2	no	—
201	C₂×Q₁₆	—	53	1	no	—
202	C₂×Q₁₆	—	106	4	no	—
203	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
204	C₂×M₁₆	—	106	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	53	2	no	—
209	C₄×Q₈	—	53	2	no	—
210	C₄×Q₈	—	106	2	no	—
211	C₄×Q₈	—	106	2	no	—
212	C₄×Q₈	C₂×#6157	1	2	yes	—
213	C₂×C₄⋊C₄	—	53	2	no	—
214	C₂×C₄⋊C₄	—	106	2	no	—
215	C₂×C₄²	—	53	1	no	—
216	C₂×C₄²	—	106	2	no	—
217	#107	—	53	1	no	—
218	#107	—	53	2	no	—
219	#107	C₂×#6157	1	1	yes	characteristic
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	53	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	C₄⋊C₈	—	106	4	no	—
224	#103	—	53	2	no	—
225	#103	—	53	2	no	—
226	#103	—	53	2	no	—
227	#103	C₂×#6157	1	2	yes	—
228	C₂²⋊C₈	—	53	1	no	—
229	C₂²⋊C₈	—	53	1	no	—
230	C₂²⋊C₈	—	106	2	no	—
231	#97	—	106	1	no	—
232	#97	—	106	2	no	—
233	#97	—	106	4	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	106	4	no	—
240	C₄×C₈	C₂×#6157	1	1	yes	characteristic
241	C₂×Q₈	—	53	2	no	—
242	C₂×Q₈	—	106	2	no	—
243	C₂×Q₈	—	106	4	no	—
244	C₂×Q₈	—	212	2	no	—
245	C₂×Q₈	C₂²×#6157	1	2	yes	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	Q₁₆	—	106	4	no	—
252	Q₁₆	—	106	2	no	—
253	Q₁₆	#11719936	1	4	yes	complemented
254	M₁₆	—	106	4	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	212	1	no	—
263	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
264	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	53	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	C₂²×#6157	1	2	yes	—
272	C₄²	—	2	2	no	—
273	C₄²	—	53	1	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	53	1	no	—
276	C₄²	—	106	2	no	—
277	C₄²	—	106	2	no	—
278	C₄²	—	106	2	no	—
279	C₄²	C₂²×#6157	1	1	yes	characteristic
280	C₂³	—	53	1	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	106	2	no	—
283	Q₈	—	106	2	no	—
284	Q₈	—	212	4	no	—
285	Q₈	C₂×#11719936	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	106	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	1	no	—
294	C₂×C₄	—	106	1	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	212	4	no	—
298	C₂×C₄	—	212	4	no	—
299	C₂×C₄	D₄×#6157	1	1	yes	characteristic
300	C₂×C₄	D₄×#6157	1	1	yes	characteristic
301	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	106	2	no	—
304	C₈	—	106	2	no	—
305	C₈	—	106	4	no	—
306	C₈	—	106	1	no	—
307	C₈	—	106	1	no	—
308	C₈	—	212	4	no	—
309	C₈	C₂×#11719936	1	2	yes	—
310	C₂²	—	53	2	no	—
311	C₂²	—	106	1	no	—
312	C₂²	—	106	1	no	—
313	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	53	2	no	—
319	C₄	—	106	1	no	—
320	C₄	—	106	1	no	—
321	C₄	—	212	2	no	—
322	C₄	—	212	2	no	—
323	C₄	C₂²×#11719936	1	1	yes	characteristic
324	C₄	#16811702	1	1	yes	characteristic
325	C₂	—	106	2	no	—
326	C₂	D₄×#11719936	1	1	yes	characteristic
327	C₂	#20291476	1	1	yes	characteristic
328	C₂	D₅₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×Q₁₆	#6157
hypocenter	C₅₃	#18058
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20290354	C₂	semidirect
3. #20290338	C₂	semidirect
4. Q₁₆×#88324	C₂	semidirect
91. Q₁₆×C₅₃	M₁₆	semidirect
198. C₅₃	#18058	semidirect
253. Q₁₆	#11719936	semidirect

Maximal subgroups

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

subgroup	index
#20291617	2
#20290354	2
#20290338	2
Q₁₆×#88324	2
#20290276	2
#20290058	2
C₈×#11719936	2
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