Subgroups — #27403844

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20277670	C₂	1	2	yes	maximal
3	#20275576	C₂	1	2	yes	maximal
4	#20275357	C₂	1	4	yes	maximal
5	#20274875	C₂	1	1	yes	maximal, characteristic
6	#20274721	C₂	1	2	yes	maximal
7	C₄×#13133151	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	C₄×Dic₂₁₂	C₄	1	8	yes	complemented
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	C₂×#13133142	C₂²	1	4	yes	—
16	C₂×#13133142	C₂²	1	2	yes	—
17	#16810529	C₂²	1	4	yes	—
18	#16810510	—	2	4	no	—
19	#16810493	C₂²	1	2	yes	—
20	#16810488	—	2	2	no	—
21	C₄⋊C₄×Dic₅₃	—	2	2	no	—
22	C₄×C₄⋊Dic₅₃	C₂²	1	4	yes	—
23	#16810386	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
30	C₂×Dic₂₁₂	—	2	2	no	—
31	C₂×Dic₂₁₂	—	2	2	no	—
32	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₆	—	2	4	no	—
34	C₄×Dic₁₀₆	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
39	C₂×#11719919	—	2	2	no	—
40	C₂×C₄×Dic₅₃	—	2	2	no	—
41	C₂×C₄×Dic₅₃	—	2	4	no	—
42	#13133151	—	2	4	no	—
43	#13133151	—	4	2	no	—
44	#13133151	C₂×C₄	1	4	yes	—
45	#13133142	—	4	4	no	—
46	#13133142	C₂×C₄	1	8	yes	—
47	#13133124	—	2	4	no	—
48	#13133124	—	2	4	no	—
49	#13133124	—	2	4	no	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
60	#95×C₅₃	D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	2	2	no	—
67	C₂²×Dic₅₃	—	2	2	no	—
68	C₂²×Dic₅₃	—	2	4	no	—
69	C₂×Dic₁₀₆	—	2	2	no	—
70	C₂×Dic₁₀₆	—	2	2	no	—
71	C₂×Dic₁₀₆	—	2	2	no	—
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×Dic₁₀₆	—	2	2	no	—
74	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	#41	1	2	yes	—
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	Q₁₆	1	4	yes	complemented
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	#41	1	2	yes	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	4	no	—
92	C₄⋊Dic₅₃	—	2	4	no	—
93	C₄⋊Dic₅₃	—	4	4	no	—
94	C₄⋊Dic₅₃	—	8	2	no	—
95	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
96	#11719919	—	2	8	no	—
97	C₄×Dic₅₃	—	2	8	no	—
98	C₄×Dic₅₃	—	8	4	no	—
99	Dic₂₁₂	—	2	8	no	—
100	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	4	no	—
121	C₂×Dic₅₃	—	4	2	no	—
122	C₂×Dic₅₃	—	4	2	no	—
123	C₂×Dic₅₃	—	4	2	no	—
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	4	4	no	—
127	C₂×Dic₅₃	—	4	4	no	—
128	Dic₁₀₆	—	2	4	no	—
129	Dic₁₀₆	—	2	4	no	—
130	Dic₁₀₆	—	4	2	no	—
131	Dic₁₀₆	—	4	2	no	—
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	4	2	no	—
134	C₈×C₅₃	#117	1	4	yes	—
135	#22808	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₅₃	#547	1	1	yes	characteristic
138	C₂²×C₅₃	#550	1	1	yes	characteristic
139	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
141	C₂²×C₅₃	#576	1	1	yes	characteristic
142	C₂²×C₅₃	#578	1	1	yes	characteristic
143	C₄×C₅₃	—	2	4	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
147	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₅₃	#520	1	1	yes	characteristic
149	C₄×C₅₃	#555	1	1	yes	characteristic
150	C₄×C₅₃	#556	1	1	yes	characteristic
151	Dic₅₃	—	4	4	no	—
152	Dic₅₃	—	4	4	no	—
153	Dic₅₃	—	8	4	no	—
154	C₂×C₄×Q₁₆	—	53	1	no	—
155	#2355	—	53	2	no	—
156	#1969	—	53	2	no	—
157	#1928	—	53	4	no	—
158	#1845	—	53	1	no	—
159	#1803	—	53	2	no	—
160	C₄×#107	—	53	2	no	—
161	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
162	C₂×C₅₃	#2354	1	1	yes	characteristic
163	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
164	C₂×C₅₃	#2951	1	1	yes	characteristic
165	C₂×C₅₃	#3338	1	1	yes	characteristic
166	C₂×C₅₃	#3342	1	1	yes	characteristic
167	C₂×C₅₃	#3395	1	1	yes	characteristic
168	C₂×C₅₃	#3396	1	1	yes	characteristic
169	C₂²×Q₁₆	—	53	1	no	—
170	C₂×C₄×Q₈	—	53	2	no	—
171	C₄×Q₁₆	—	53	8	no	—
172	C₂×#107	—	53	1	no	—
173	C₂×#107	—	53	2	no	—
174	C₂×#107	—	53	2	no	—
175	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
176	C₂×#103	—	53	2	no	—
177	C₂×#103	—	53	4	no	—
178	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
180	#391	—	53	4	no	—
181	#387	—	106	4	no	—
182	#384	—	53	2	no	—
183	#382	—	106	2	no	—
184	C₄×C₄⋊C₄	—	53	4	no	—
185	C₄×C₄⋊C₄	—	106	2	no	—
186	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
187	#340	—	53	4	no	—
188	#336	C₂×D₅₃	1	2	yes	—
189	C₅₃	#22808	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	53	2	no	—
191	C₂×Q₁₆	—	53	4	no	—
192	C₂×Q₁₆	—	106	2	no	—
193	C₂×Q₁₆	—	106	2	no	—
194	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₄×Q₈	—	53	8	no	—
198	C₄×Q₈	—	106	4	no	—
199	C₂×C₄⋊C₄	—	53	2	no	—
200	C₂×C₄⋊C₄	—	53	4	no	—
201	C₂×C₄⋊C₄	—	106	2	no	—
202	C₂×C₄⋊C₄	—	106	2	no	—
203	C₂×C₄⋊C₄	—	106	2	no	—
204	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
206	C₂×C₄²	—	106	2	no	—
207	C₂×C₄²	—	106	4	no	—
208	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
209	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	#107	—	53	4	no	—
211	#107	—	106	4	no	—
212	#107	—	212	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	53	8	no	—
215	#103	—	212	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₄×D₅₃	1	4	yes	—
218	#95	—	106	4	no	—
219	#95	—	106	4	no	—
220	#95	—	106	4	no	—
221	#95	D₂₁₂	1	2	yes	—
222	C₂×Q₈	—	53	4	no	—
223	C₂×Q₈	—	106	2	no	—
224	C₂×Q₈	—	106	2	no	—
225	C₂×Q₈	—	106	2	no	—
226	C₂×Q₈	—	106	2	no	—
227	C₂×Q₈	—	106	2	no	—
228	C₂²×C₄	—	106	2	no	—
229	C₂²×C₄	—	106	2	no	—
230	C₂²×C₄	—	106	4	no	—
231	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
232	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
234	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
236	Q₁₆	—	106	8	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
244	C₂×C₈	#11719947	1	2	yes	—
245	C₂×C₈	#11719949	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	53	4	no	—
248	C₄⋊C₄	—	106	8	no	—
249	C₄⋊C₄	—	106	2	no	—
250	C₄⋊C₄	—	106	4	no	—
251	C₄⋊C₄	—	106	4	no	—
252	C₄⋊C₄	—	424	2	no	—
253	C₄⋊C₄	—	212	4	no	—
254	C₄⋊C₄	Dic₂₁₂	1	4	yes	complemented
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	106	8	no	—
258	C₄²	—	424	4	no	—
259	C₄²	C₂×C₄×D₅₃	1	2	yes	—
260	C₄²	#11719945	1	2	yes	—
261	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
262	Q₈	—	106	4	no	—
263	Q₈	—	106	4	no	—
264	Q₈	—	212	2	no	—
265	Q₈	—	212	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	106	4	no	—
274	C₂×C₄	—	106	4	no	—
275	C₂×C₄	—	212	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	—	212	4	no	—
278	C₂×C₄	—	212	2	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
283	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
284	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
285	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
286	C₂×C₄	#13133239	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
291	C₂×C₄	#13133263	1	1	yes	characteristic
292	C₂×C₄	#13133264	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#13133175	1	4	yes	—
296	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
297	C₂²	#16811277	1	1	yes	characteristic
298	C₂²	#16811295	1	1	yes	characteristic
299	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
300	C₂²	C₂×#13133239	1	1	yes	characteristic
301	C₂²	#16811428	1	1	yes	characteristic
302	C₂²	#16811440	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	424	4	no	—
307	C₄	—	212	4	no	—
308	C₄	—	212	4	no	—
309	C₄	C₄×Dic₂₁₂	1	4	yes	—
310	C₄	C₂×#13133175	1	1	yes	characteristic
311	C₄	#16811103	1	1	yes	characteristic
312	C₄	#16811338	1	1	yes	characteristic
313	C₄	#16811346	1	1	yes	characteristic
314	C₂	#20277662	1	1	yes	characteristic
315	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
316	C₂	#20283592	1	1	yes	characteristic
317	C₂	#20287108	1	1	yes	characteristic
318	C₂	#20287146	1	1	yes	characteristic
319	C₂	#20287721	1	1	yes	characteristic
320	C₂	#20287728	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#22808
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₁₂	C₄	semidirect
85. C₄⋊C₄×C₅₃	Q₁₆	semidirect
189. C₅₃	#22808	semidirect
254. C₄⋊C₄	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20277670	2
#20275576	2
#20275357	2
#20274875	2
#20274721	2
C₄×#13133151	2
C₈×C₄⋊C₄×C₅₃	2
#22808	53