Subgroups — #27284100

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	C₂×C₄×#88322	—	2	1	no	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
13	C₅₃×#391	—	2	1	no	—
14	C₅₃×#389	—	2	1	no	—
15	C₅₃×#389	C₄	1	2	yes	—
16	C₅₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	—	4	2	no	—
24	C₄⋊#88322	—	4	2	no	—
25	C₄×#88322	—	2	4	no	—
26	C₄×#88322	—	2	4	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	Q₈	1	4	yes	complemented
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
32	C₄×Q₈×C₅₃	—	4	2	no	—
33	C₄×Q₈×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₂×C₈	1	2	yes	—
75	C₄²×C₅₃	M₁₆	1	2	yes	—
76	C₄²×C₅₃	D₈	1	2	yes	—
77	C₄²×C₅₃	SD₁₆	1	2	yes	—
78	C₄²×C₅₃	SD₁₆	1	2	yes	—
79	C₄²×C₅₃	Q₁₆	1	2	yes	—
80	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₅₃	#41	1	2	yes	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₅₃	—	4	8	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₅₃	#102	1	2	yes	—
105	C₂×C₄×C₅₃	#102	1	2	yes	—
106	C₂×C₄×C₅₃	#103	1	2	yes	—
107	C₂×C₄×C₅₃	#103	1	2	yes	—
108	C₂×C₄×C₅₃	#104	1	2	yes	—
109	C₂×C₄×C₅₃	#104	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#131	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
121	#88322	—	2	8	no	—
122	#88322	—	8	2	no	—
123	#9870	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	#391	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₃	#408	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	—	8	2	no	—
136	C₄×C₅₃	D₄⋊C₈	1	4	yes	—
137	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
138	C₄×C₅₃	C₈×Q₈	1	2	yes	—
139	C₄×C₅₃	#446	1	2	yes	—
140	C₄×C₅₃	#486	1	1	yes	characteristic
141	C₄×C₅₃	#487	1	1	yes	characteristic
142	C₄×C₅₃	#498	1	1	yes	characteristic
143	C₄×C₅₃	#500	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
145	#1842	—	53	1	no	—
146	C₄²×C₈	—	53	1	no	—
147	#1279	—	53	4	no	—
148	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
149	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
150	C₂×C₅₃	#1492	1	1	yes	characteristic
151	C₂×C₅₃	#1493	1	1	yes	characteristic
152	C₂×C₅₃	#1984	1	1	yes	characteristic
153	C₂×C₅₃	#1990	1	1	yes	characteristic
154	C₂×C₅₃	#1994	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
156	C₂×C₄×Q₈	—	2	1	no	—
157	C₂×C₄⋊C₈	—	53	2	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×C₄×C₈	—	53	2	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	C₂×C₄×C₈	—	106	1	no	—
163	#391	—	2	1	no	—
164	#389	—	2	1	no	—
165	#389	Dic₅₃	1	2	yes	—
166	#384	Dic₅₃	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
169	C₄³	C₂×D₅₃	1	1	yes	characteristic
170	C₅₃	#9870	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	#88322	1	4	yes	complemented
176	C₄×Q₈	—	4	2	no	—
177	C₄×Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	212	2	no	—
194	C₄⋊C₈	—	212	2	no	—
195	C₄×C₈	—	53	4	no	—
196	C₄×C₈	—	106	4	no	—
197	C₄×C₈	—	106	4	no	—
198	C₄×C₈	—	106	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	—	212	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	C₂×#88322	1	4	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×#88322	1	2	yes	—
232	C₄²	#11719917	1	2	yes	—
233	C₄²	#11719922	1	2	yes	—
234	C₄²	#11719923	1	2	yes	—
235	C₄²	#11719924	1	2	yes	—
236	C₄²	#11719925	1	2	yes	—
237	C₄²	Q₈×D₅₃	1	2	yes	—
238	C₄²	#11719949	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²⋊#88322	1	2	yes	—
254	C₂×C₄	C₂²⋊#88322	1	2	yes	—
255	C₂×C₄	#13133138	1	2	yes	—
256	C₂×C₄	#13133138	1	2	yes	—
257	C₂×C₄	#13133141	1	2	yes	—
258	C₂×C₄	#13133141	1	2	yes	—
259	C₂×C₄	#13133144	1	2	yes	—
260	C₂×C₄	#13133144	1	2	yes	—
261	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
262	C₂×C₄	#13133193	1	2	yes	—
263	C₂×C₄	#13133206	1	1	yes	characteristic
264	C₂×C₄	#13133215	1	1	yes	characteristic
265	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
267	C₂×C₄	#13133224	1	2	yes	—
268	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
272	C₈	—	106	8	no	—
273	C₈	—	424	2	no	—
274	C₂²	#16810530	1	1	yes	characteristic
275	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
276	C₂²	#16810590	1	1	yes	characteristic
277	C₂²	C₂×#13133138	1	1	yes	characteristic
278	C₂²	C₂×#13133141	1	1	yes	characteristic
279	C₂²	C₂×#13133144	1	1	yes	characteristic
280	C₂²	C₂×#13133144	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	D₄⋊#88322	1	4	yes	—
287	C₄	Q₈⋊#88322	1	4	yes	—
288	C₄	Q₈×#88322	1	2	yes	—
289	C₄	#16810754	1	2	yes	—
290	C₄	#16810971	1	1	yes	characteristic
291	C₄	#16810975	1	1	yes	characteristic
292	C₄	#16811019	1	1	yes	characteristic
293	C₄	#16811026	1	1	yes	characteristic
294	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
295	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
296	C₂	#20273166	1	1	yes	characteristic
297	C₂	#20273170	1	1	yes	characteristic
298	C₂	#20275652	1	1	yes	characteristic
299	C₂	#20275677	1	1	yes	characteristic
300	C₂	#20275693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#9870
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₄×#88322	Q₈	semidirect
31. C₄⋊Q₈×C₅₃	C₈	semidirect
170. C₅₃	#9870	semidirect
175. C₄⋊Q₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄²×#88322	2
#20272556	2
C₄×C₄⋊Q₈×C₅₃	2
#9870	53