Subgroups — #27289965

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810393	C₂	1	4	yes	maximal
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂²×#11719917	C₂²	1	2	yes	—
7	#16810647	—	2	1	no	—
8	#16810647	—	2	1	no	—
9	#16810647	—	2	2	no	—
10	#16810647	—	2	2	no	—
11	#16810647	C₄	1	2	yes	complemented
12	#16810647	C₄	1	2	yes	complemented
13	C₂×C₄⋊#88322	—	2	2	no	—
14	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
15	#16810393	—	2	8	no	—
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₅₃	C₄	1	4	yes	—
20	C₅₃×#376	—	2	1	no	—
21	C₄³×C₅₃	—	2	1	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	4	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	—	2	4	no	—
29	C₂×#11719917	C₂×C₄	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₄⋊#88322	—	2	4	no	—
33	C₄⋊#88322	—	2	4	no	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	4	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	D₄	1	4	yes	—
41	#117×C₅₃	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	D₄	1	2	yes	—
53	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
55	#95×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	#41	1	2	yes	—
74	C₂²×C₄×C₅₃	#41	1	2	yes	—
75	C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
79	C₄²×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	—	2	4	no	—
81	C₄²×C₅₃	—	4	8	no	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	#11719917	—	2	8	no	—
93	#11719917	—	4	4	no	—
94	#11719917	—	4	4	no	—
95	#11719917	—	4	2	no	—
96	#11719917	—	4	2	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	2	8	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂×#88322	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₅₃	#117	1	2	yes	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	8	no	—
116	C₂×C₄×C₅₃	#95	1	2	yes	—
117	C₂×C₄×C₅₃	#95	1	2	yes	—
118	C₂×C₄×C₅₃	#95	1	2	yes	—
119	C₂×C₄×C₅₃	#95	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₅₃	#121	1	2	yes	—
129	C₂×C₄×C₅₃	#121	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#125	1	2	yes	—
135	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
138	#88322	—	4	4	no	—
139	#88322	—	4	4	no	—
140	#10856	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
147	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
148	C₂²×C₅₃	#380	1	2	yes	—
149	C₂²×C₅₃	#382	1	1	yes	characteristic
150	C₂²×C₅₃	#384	1	1	yes	characteristic
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	4	8	no	—
155	C₄×C₅₃	#344	1	4	yes	complemented
156	C₄×C₅₃	#389	1	2	yes	—
157	C₄×C₅₃	#389	1	2	yes	—
158	C₄×C₅₃	#390	1	1	yes	characteristic
159	C₄×C₅₃	#390	1	1	yes	characteristic
160	C₄×C₅₃	#391	1	1	yes	characteristic
161	C₄×C₅₃	#391	1	1	yes	characteristic
162	C₂×#423	—	53	1	no	—
163	C₂×#423	—	53	1	no	—
164	C₄×#117	D₅₃	1	1	yes	characteristic
165	C₂×#344	—	53	4	no	—
166	C₂×C₅₃	—	2	2	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	1	no	—
170	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#344	1	2	yes	—
172	C₂²×M₁₆	—	53	2	no	—
173	C₂²×M₁₆	—	53	2	no	—
174	C₂×#117	C₂×D₅₃	1	2	yes	—
175	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
176	#423	—	53	2	no	—
177	#423	—	53	2	no	—
178	#423	—	106	2	no	—
179	#423	—	106	1	no	—
180	#423	—	106	1	no	—
181	#423	—	106	2	no	—
182	C₂×C₄⋊C₈	—	53	4	no	—
183	C₂×C₄⋊C₈	—	106	2	no	—
184	C₄×C₄⋊C₄	—	2	2	no	—
185	C₄×C₂²⋊C₄	Dic₅₃	1	4	yes	—
186	#376	—	2	1	no	—
187	C₄³	—	2	1	no	—
188	#344	—	106	8	no	—
189	C₅₃	#10856	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
191	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
192	C₂×M₁₆	—	53	4	no	—
193	C₂×M₁₆	—	106	4	no	—
194	C₂×M₁₆	—	106	2	no	—
195	C₂×M₁₆	—	106	4	no	—
196	C₂×M₁₆	—	106	2	no	—
197	C₂×M₁₆	—	106	2	no	—
198	C₂×M₁₆	—	106	4	no	—
199	C₂×M₁₆	—	106	4	no	—
200	C₂²×C₈	—	53	4	no	—
201	C₂²×C₈	—	106	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	Dic₁₀₆	1	4	yes	—
205	#117	D₂₁₂	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
211	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₅₃	1	2	yes	—
213	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
215	C₂×C₄²	D₂₁₂	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
219	C₄⋊C₈	—	106	4	no	—
220	C₄⋊C₈	—	106	4	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
228	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
230	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
233	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
234	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
235	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
239	C₂²×C₄	#11719945	1	2	yes	—
240	C₂²×C₄	#11719945	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	M₁₆	—	106	8	no	—
244	M₁₆	—	212	4	no	—
245	M₁₆	—	212	4	no	—
246	M₁₆	—	212	2	no	—
247	M₁₆	—	212	2	no	—
248	C₂×C₈	—	106	4	no	—
249	C₂×C₈	—	106	4	no	—
250	C₂×C₈	—	106	8	no	—
251	C₂×C₈	—	212	2	no	—
252	C₂×C₈	—	212	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	4	8	no	—
260	C₄²	#11719919	1	2	yes	—
261	C₄²	#11719919	1	1	yes	characteristic
262	C₄²	#11719919	1	1	yes	characteristic
263	C₄²	#11719919	1	1	yes	characteristic
264	C₄²	#11719919	1	2	yes	—
265	C₄²	#11719919	1	1	yes	characteristic
266	C₄²	#11719921	1	2	yes	—
267	C₄²	#11719921	1	2	yes	—
268	C₄²	#11719921	1	2	yes	—
269	C₄²	#11719921	1	2	yes	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
272	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
273	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
274	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
275	C₂³	#13133176	1	2	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	#13133124	1	2	yes	—
285	C₂×C₄	#13133124	1	2	yes	—
286	C₂×C₄	#13133124	1	2	yes	—
287	C₂×C₄	#13133124	1	2	yes	—
288	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
291	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
292	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
293	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
294	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
295	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
296	C₂×C₄	#13133191	1	2	yes	—
297	C₂×C₄	#13133191	1	2	yes	—
298	C₂×C₄	#13133196	1	2	yes	—
299	C₂×C₄	#13133196	1	2	yes	—
300	C₂×C₄	#13133205	1	1	yes	characteristic
301	C₂×C₄	#13133205	1	1	yes	characteristic
302	C₂×C₄	#13133209	1	2	yes	—
303	C₂×C₄	#13133214	1	1	yes	characteristic
304	C₂×C₄	#13133217	1	2	yes	—
305	C₂×C₄	#13133217	1	1	yes	characteristic
306	C₈	—	212	4	no	—
307	C₈	—	212	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×#13133124	1	1	yes	characteristic
314	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
315	C₂²	#16810483	1	2	yes	—
316	C₂²	#16810489	1	1	yes	characteristic
317	C₂²	#16810493	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	8	no	—
322	C₄	#16810393	1	4	yes	complemented
323	C₄	#16810521	1	2	yes	—
324	C₄	#16810521	1	2	yes	—
325	C₄	#16810525	1	1	yes	characteristic
326	C₄	#16810525	1	1	yes	characteristic
327	C₄	#16810529	1	1	yes	characteristic
328	C₄	#16810529	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	#20272945	1	1	yes	characteristic
334	C₂	C₂×#16810393	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#10856
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
11. #16810647	C₄	semidirect
12. #16810647	C₄	semidirect
14. C₂×C₄⋊#88322	C₄	semidirect
155. C₄×C₅₃	#344	semidirect
189. C₅₃	#10856	semidirect
322. C₄	#16810393	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×#16810393	2
C₄×#117×C₅₃	2
#10856	53