Subgroups — #27289903

Order 13568.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#20277122	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810383	C₂	1	4	yes	maximal
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂×#13133176	C₂²	1	2	yes	—
7	#16810647	—	2	2	no	—
8	#16810647	C₄	1	4	yes	complemented
9	C₂×C₄⋊#88322	—	2	2	no	—
10	#16810493	C₄	1	2	yes	complemented
11	#16810489	C₄	1	2	yes	complemented
12	#16810483	—	2	2	no	—
13	C₄×C₄⋊Dic₅₃	—	2	2	no	—
14	#16810383	—	2	8	no	—
15	C₂²×C₄²×C₅₃	—	2	2	no	—
16	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄³×C₅₃	—	2	2	no	—
19	C₄³×C₅₃	C₄	1	4	yes	complemented
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	2	4	no	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	#13133176	—	2	4	no	—
26	#13133176	—	2	4	no	—
27	C₂×C₄⋊Dic₅₃	—	2	4	no	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	C₄⋊#88322	—	2	8	no	—
32	#13133124	—	2	4	no	—
33	C₂³×C₄×C₅₃	—	2	2	no	—
34	C₂³×C₄×C₅₃	D₄	1	2	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	—	2	16	no	—
37	C₂×C₄²×C₅₃	—	2	8	no	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	4	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	2	yes	—
49	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	16	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	8	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	32	no	—
73	C₄²×C₅₃	—	2	4	no	—
74	C₄²×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	C₄²	1	2	yes	—
77	C₄²×C₅₃	C₄²	1	2	yes	—
78	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
79	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	8	yes	complemented
83	C₂²⋊Dic₅₃	—	4	4	no	—
84	C₄⋊Dic₅₃	—	2	8	no	—
85	C₄⋊Dic₅₃	—	4	8	no	—
86	C₄×Dic₅₃	—	4	2	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	#11719917	—	2	8	no	—
89	#11719917	—	4	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	4	2	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂³×C₅₃	—	2	4	no	—
94	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₅₃	#121	1	2	yes	—
98	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	16	no	—
100	C₂×C₄×C₅₃	—	2	16	no	—
101	C₂×C₄×C₅₃	—	2	16	no	—
102	C₂×C₄×C₅₃	—	2	8	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	8	no	—
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	#95	1	4	yes	—
107	C₂×C₄×C₅₃	#95	1	4	yes	—
108	C₂×C₄×C₅₃	#104	1	8	yes	—
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	#121	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#125	1	2	yes	—
124	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₅₃	—	4	4	no	—
128	C₂×Dic₅₃	—	4	8	no	—
129	#88322	—	4	4	no	—
130	#10839	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₅₃	#380	1	2	yes	—
137	C₂²×C₅₃	#389	1	2	yes	—
138	C₂²×C₅₃	#390	1	1	yes	characteristic
139	C₂²×C₅₃	#391	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#104	1	4	yes	—
141	C₄×C₅₃	—	2	16	no	—
142	C₄×C₅₃	—	2	4	no	—
143	C₄×C₅₃	#339	1	8	yes	complemented
144	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
145	C₄×C₅₃	#382	1	1	yes	characteristic
146	C₄×C₅₃	#384	1	1	yes	characteristic
147	C₄×C₅₃	#389	1	2	yes	—
148	C₄×C₅₃	#390	1	1	yes	characteristic
149	C₄×C₅₃	#391	1	1	yes	characteristic
150	Dic₅₃	—	8	4	no	—
151	C₂×#423	—	53	1	no	—
152	#2285	—	53	1	no	—
153	C₂×C₄³	D₅₃	1	1	yes	characteristic
154	C₂×#339	—	53	4	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#339	1	2	yes	—
159	C₂×C₅₃	#1962	1	2	yes	—
160	C₂×C₅₃	#1994	1	2	yes	—
161	C₂²×M₁₆	—	53	2	no	—
162	C₂×#117	—	53	2	no	—
163	C₂²×C₄²	—	2	2	no	—
164	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
165	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
166	#423	—	53	4	no	—
167	#423	—	106	2	no	—
168	C₂×C₄⋊C₈	—	106	2	no	—
169	#384	—	53	2	no	—
170	#382	—	53	2	no	—
171	#380	—	106	2	no	—
172	C₄×C₄⋊C₄	—	106	2	no	—
173	C₄³	—	2	2	no	—
174	C₄³	Dic₅₃	1	4	yes	complemented
175	#339	—	106	8	no	—
176	C₅₃	#10839	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
179	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
180	C₂×M₁₆	—	53	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	4	no	—
183	C₂×M₁₆	—	106	4	no	—
184	C₂²×C₈	—	106	2	no	—
185	#117	—	106	4	no	—
186	#117	—	106	4	no	—
187	C₂×C₄⋊C₄	—	53	4	no	—
188	C₂×C₄⋊C₄	—	106	4	no	—
189	C₂×C₂²⋊C₄	—	106	2	no	—
190	C₂×C₄²	—	2	16	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	8	no	—
195	C₂×C₄²	—	106	2	no	—
196	C₂×C₄²	Dic₁₀₆	1	4	yes	—
197	C₂×C₄²	Dic₁₀₆	1	2	yes	—
198	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₅₃	1	2	yes	—
201	C₂×C₄²	D₂₁₂	1	4	yes	—
202	C₂×C₄²	D₂₁₂	1	2	yes	—
203	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
204	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
206	C₄⋊C₈	—	106	8	no	—
207	#95	—	106	4	no	—
208	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	106	4	no	—
215	C₂²×C₄	#11719919	1	4	yes	—
216	C₂²×C₄	#11719921	1	2	yes	—
217	C₂²×C₄	#11719921	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₅₃	1	4	yes	—
219	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
224	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
225	C₂²×C₄	#11719945	1	2	yes	—
226	C₂²×C₄	#11719945	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	M₁₆	—	106	8	no	—
230	M₁₆	—	212	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	212	2	no	—
233	C₂×C₈	—	212	2	no	—
234	C₄⋊C₄	—	106	8	no	—
235	C₄⋊C₄	—	212	8	no	—
236	C₂²⋊C₄	—	212	4	no	—
237	C₄²	—	2	32	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	212	2	no	—
242	C₄²	—	212	2	no	—
243	C₄²	C₄×Dic₅₃	1	2	yes	—
244	C₄²	C₄×Dic₅₃	1	2	yes	—
245	C₄²	#11719919	1	2	yes	—
246	C₄²	#11719919	1	2	yes	—
247	C₄²	C₄⋊Dic₅₃	1	4	yes	—
248	C₄²	C₄⋊Dic₅₃	1	8	yes	complemented
249	C₄²	#11719921	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	C₂×#11719921	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
253	C₂³	C₂×#11719919	1	1	yes	characteristic
254	C₂³	#13133191	1	2	yes	—
255	C₂³	#13133196	1	2	yes	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	212	4	no	—
264	C₂×C₄	—	212	8	no	—
265	C₂×C₄	#13133124	1	4	yes	—
266	C₂×C₄	#13133124	1	4	yes	—
267	C₂×C₄	#13133145	1	8	yes	—
268	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	#13133176	1	2	yes	—
274	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
275	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
276	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
277	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
278	C₂×C₄	#13133191	1	2	yes	—
279	C₂×C₄	#13133196	1	2	yes	—
280	C₂×C₄	#13133205	1	1	yes	characteristic
281	C₂×C₄	#13133205	1	1	yes	characteristic
282	C₂×C₄	#13133209	1	2	yes	—
283	C₂×C₄	#13133214	1	1	yes	characteristic
284	C₂×C₄	#13133217	1	2	yes	—
285	C₂×C₄	#13133217	1	1	yes	characteristic
286	C₈	—	212	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₂×#13133124	1	1	yes	characteristic
292	C₂²	#16810483	1	2	yes	—
293	C₂²	#16810521	1	2	yes	—
294	C₂²	#16810525	1	1	yes	characteristic
295	C₂²	#16810529	1	1	yes	characteristic
296	C₂²	C₂×#13133145	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	424	4	no	—
300	C₄	#16810383	1	8	yes	complemented
301	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
302	C₄	#16810489	1	1	yes	characteristic
303	C₄	#16810493	1	1	yes	characteristic
304	C₄	#16810521	1	2	yes	—
305	C₄	#16810525	1	1	yes	characteristic
306	C₄	#16810529	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#20272945	1	1	yes	characteristic
310	C₂	C₂×#16810383	1	2	yes	—
311	C₂	#20275543	1	2	yes	—
312	C₂	#20275692	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#10839
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
8. #16810647	C₄	semidirect
10. #16810493	C₄	semidirect
11. #16810489	C₄	semidirect
19. C₄³×C₅₃	C₄	semidirect
82. C₄²×C₅₃	C₄⋊C₄	semidirect
143. C₄×C₅₃	#339	semidirect
174. C₄³	Dic₅₃	semidirect
176. C₅₃	#10839	semidirect
248. C₄²	C₄⋊Dic₅₃	semidirect
300. C₄	#16810383	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
#20277122	2
C₂×#16810383	2
C₂×C₄³×C₅₃	2
#10839	53