Subgroups — #27283045

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	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	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	1	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	Q₈⋊#88322	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#391	C₄	1	2	yes	—
15	C₅₃×#389	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#384	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	D₄	1	4	yes	—
26	C₄×#88322	—	2	4	no	—
27	C₄×#88322	—	4	2	no	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
31	C₄×Q₈×C₅₃	—	2	4	no	—
32	C₄×Q₈×C₅₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	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	C₂³	1	1	yes	characteristic
47	#95×C₅₃	—	2	2	no	—
48	#95×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
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	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	—	4	4	no	—
74	C₄²×C₅₃	C₂×C₈	1	2	yes	—
75	C₄²×C₅₃	M₁₆	1	2	yes	—
76	C₄²×C₅₃	SD₁₆	1	2	yes	—
77	C₄²×C₅₃	Q₁₆	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₄²×C₅₃	#41	1	2	yes	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	8	no	—
88	Q₈×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	2	1	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	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	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	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²⋊C₈	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	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	#121	1	2	yes	—
111	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#131	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#137	1	2	yes	—
119	#88322	—	4	4	no	—
120	#88322	—	4	4	no	—
121	#9673	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	#390	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₅₃	#408	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
126	C₂²×C₅₃	#419	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
128	C₂²×C₅₃	#421	1	1	yes	characteristic
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	4	4	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	—	8	2	no	—
136	C₄×C₅₃	Q₈⋊C₈	1	4	yes	complemented
137	C₄×C₅₃	C₈×D₄	1	2	yes	—
138	C₄×C₅₃	#436	1	2	yes	—
139	C₄×C₅₃	#461	1	2	yes	—
140	C₄×C₅₃	#462	1	2	yes	—
141	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
142	C₄×C₄⋊C₈	—	53	1	no	—
143	C₄×C₄⋊C₈	—	53	1	no	—
144	C₂×Q₈⋊C₈	—	53	4	no	—
145	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₅₃	#1479	1	1	yes	characteristic
147	C₂×C₅₃	#1481	1	1	yes	characteristic
148	C₂×C₅₃	#1485	1	1	yes	characteristic
149	C₂×C₅₃	#1954	1	1	yes	characteristic
150	C₂×C₅₃	#1980	1	1	yes	characteristic
151	C₂×C₅₃	#1981	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
154	C₂×C₄⋊C₈	—	53	2	no	—
155	C₂×C₄⋊C₈	—	106	1	no	—
156	C₂×C₄×C₈	—	53	2	no	—
157	C₂×C₄×C₈	—	106	1	no	—
158	#391	Dic₅₃	1	2	yes	—
159	#389	—	2	1	no	—
160	#389	—	2	1	no	—
161	#384	Dic₅₃	1	1	yes	characteristic
162	C₄×C₄⋊C₄	—	2	1	no	—
163	C₄×C₄⋊C₄	—	2	1	no	—
164	C₄³	C₂×D₅₃	1	1	yes	characteristic
165	#336	—	106	1	no	—
166	#336	—	106	1	no	—
167	Q₈⋊C₈	—	106	8	no	—
168	C₅₃	#9673	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
170	C₂²×C₈	—	106	2	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	#88322	1	4	yes	complemented
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₅₃	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	C₄⋊C₈	—	53	4	no	—
191	C₄⋊C₈	—	106	4	no	—
192	C₄⋊C₈	—	106	2	no	—
193	C₄×C₈	—	106	4	no	—
194	C₄×C₈	—	212	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂×#88322	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	212	2	no	—
213	C₂×C₈	—	212	2	no	—
214	C₂×C₈	—	212	2	no	—
215	C₂×C₈	—	212	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂×#88322	1	2	yes	—
229	C₄²	#11719917	1	2	yes	—
230	C₄²	#11719924	1	2	yes	—
231	C₄²	#11719925	1	2	yes	—
232	C₄²	D₄×D₅₃	1	2	yes	—
233	C₄²	#11719947	1	2	yes	—
234	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	4	4	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	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	C₂²⋊#88322	1	2	yes	—
253	C₂×C₄	C₂²⋊#88322	1	2	yes	—
254	C₂×C₄	#13133141	1	2	yes	—
255	C₂×C₄	#13133141	1	2	yes	—
256	C₂×C₄	#13133144	1	2	yes	—
257	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
258	C₂×C₄	#13133188	1	2	yes	—
259	C₂×C₄	#13133204	1	1	yes	characteristic
260	C₂×C₄	#13133213	1	1	yes	characteristic
261	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
263	C₂×C₄	#13133224	1	2	yes	—
264	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
266	C₂×C₄	#13133248	1	2	yes	—
267	C₈	—	212	4	no	—
268	C₈	—	212	4	no	—
269	C₂²	#16810526	1	1	yes	characteristic
270	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
271	C₂²	#16810590	1	1	yes	characteristic
272	C₂²	C₂×#13133141	1	1	yes	characteristic
273	C₂²	#16810632	1	1	yes	characteristic
274	C₂²	C₂×#13133144	1	1	yes	characteristic
275	C₂²	#16810640	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	4	no	—
281	C₄	—	4	1	no	—
282	C₄	—	8	2	no	—
283	C₄	Q₈⋊#88322	1	4	yes	complemented
284	C₄	D₄×#88322	1	2	yes	—
285	C₄	#16810701	1	2	yes	—
286	C₄	#16810825	1	2	yes	—
287	C₄	#16810830	1	2	yes	—
288	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
289	C₂	#20273110	1	1	yes	characteristic
290	C₂	#20273118	1	1	yes	characteristic
291	C₂	#20273136	1	1	yes	characteristic
292	C₂	#20275510	1	1	yes	characteristic
293	C₂	#20275635	1	1	yes	characteristic
294	C₂	#20275639	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	#9673
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
30. C₄⋊Q₈×C₅₃	C₈	semidirect
136. C₄×C₅₃	Q₈⋊C₈	semidirect
168. C₅₃	#9673	semidirect
173. C₄⋊Q₈	#88322	semidirect
283. 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
C₄×C₄⋊#88322	2
C₂×Q₈⋊#88322	2
C₄×C₄⋊Q₈×C₅₃	2
#9673	53