Subgroups — #28328481

Order 13568.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	2	yes	maximal, complemented
2	#20291098	C₂	1	4	yes	maximal, complemented
3	#20291094	C₂	1	2	yes	maximal, complemented
4	#20291079	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291079	C₂	1	2	yes	maximal, complemented
6	C₄×#13133292	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290916	C₂	1	2	yes	maximal, complemented
8	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133292	C₂²	1	2	yes	—
10	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	—
11	C₄⋊#11719910	C₂²	1	2	yes	—
12	C₄⋊#11719910	C₂²	1	4	yes	—
13	#16811518	C₂²	1	4	yes	—
14	#16811518	C₂²	1	2	yes	—
15	#16811512	C₂²	1	2	yes	—
16	#16811512	C₂²	1	1	yes	characteristic
17	#16811512	C₂²	1	1	yes	characteristic
18	#16811511	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₄×#11719949	C₄	1	1	yes	characteristic
21	Q₈×#88322	C₂²	1	1	yes	characteristic
22	C₈×D₂₁₂	C₄	1	2	yes	—
23	C₈×D₂₁₂	C₄	1	1	yes	characteristic
24	#16810688	C₂²	1	1	yes	characteristic
25	#16810688	C₂²	1	2	yes	—
26	#16810675	C₄	1	1	yes	characteristic
27	#16810675	C₄	1	2	yes	—
28	C₄×C₈×D₅₃	C₂²	1	2	yes	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
31	#13133292	—	2	4	no	—
32	D₅₃⋊C₁₆	—	2	8	no	—
33	C₂×#11719910	C₂³	1	4	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂×#11719949	C₈	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂³	1	2	yes	—
40	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
41	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
44	#13133179	C₈	1	2	yes	—
45	#13133179	C₈	1	1	yes	characteristic
46	#13133175	C₈	1	2	yes	—
47	#13133175	C₈	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	2	yes	—
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
52	#13133129	C₂×C₄	1	2	yes	—
53	#13133129	C₂×C₄	1	2	yes	—
54	#13133129	C₂×C₄	1	2	yes	—
55	C₄×#88322	C₂³	1	1	yes	characteristic
56	C₄×#88322	C₂³	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
66	#11719949	—	2	4	no	—
67	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
68	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	#11719921	C₂×C₈	1	2	yes	—
83	#11719921	C₂×C₈	1	2	yes	—
84	#11719921	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
93	C₂×#88322	C₂²×C₄	1	2	yes	—
94	C₂×#88322	C₂⁴	1	1	yes	characteristic
95	C₈×D₅₃	—	2	4	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	4	no	—
98	C₈×D₅₃	—	2	2	no	—
99	#11719910	—	2	4	no	—
100	#11719910	—	2	2	no	—
101	#11719910	#41	1	4	yes	complemented
102	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
104	Q₈×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	D₂₁₂	—	2	4	no	—
115	D₂₁₂	M₃₂	1	4	yes	complemented
116	C₄×D₅₃	—	2	4	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	1	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	1	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	4	2	no	—
123	C₄×D₅₃	—	4	1	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	#131	1	2	yes	—
127	#88322	—	2	1	no	—
128	#88322	—	2	2	no	—
129	#88322	C₂×#41	1	2	yes	—
130	#33484	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	2	1	no	—
135	C₂×D₅₃	—	2	1	no	—
136	C₂×D₅₃	—	2	1	no	—
137	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
142	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
144	Dic₅₃	—	2	1	no	—
145	Dic₅₃	—	2	1	no	—
146	Dic₅₃	—	2	2	no	—
147	Dic₅₃	—	2	1	no	—
148	C₈×#41	—	53	1	no	—
149	#2183	—	53	2	no	—
150	#2168	—	53	4	no	—
151	D₄×C₁₆	—	53	2	no	—
152	#2163	—	53	1	no	—
153	#2163	—	53	2	no	—
154	C₄×M₃₂	—	53	1	no	—
155	C₂×#346	—	53	2	no	—
156	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
157	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
158	C₂×C₅₃	#3407	1	1	yes	characteristic
159	D₅₃	—	2	4	no	—
160	D₅₃	—	4	1	no	—
161	C₄×#41	—	53	1	no	—
162	C₂×M₃₂	—	53	2	no	—
163	C₂²×C₁₆	—	53	4	no	—
164	C₈×Q₈	—	53	1	no	—
165	C₈×Q₈	#6157	1	1	yes	characteristic
166	C₈×D₄	—	53	1	no	—
167	C₈×D₄	—	53	2	no	—
168	C₈×D₄	—	53	2	no	—
169	C₈×D₄	—	53	1	no	—
170	#431	—	53	1	no	—
171	#431	—	53	2	no	—
172	C₂×C₄×C₈	—	53	2	no	—
173	C₂×C₄×C₈	—	53	1	no	—
174	C₄⋊C₁₆	—	53	2	no	—
175	C₄⋊C₁₆	—	53	4	no	—
176	C₂²⋊C₁₆	—	53	4	no	—
177	C₂²⋊C₁₆	—	53	2	no	—
178	#346	—	53	1	no	—
179	#346	—	53	2	no	—
180	#346	—	53	4	no	—
181	#346	—	53	1	no	—
182	C₄×C₁₆	—	53	2	no	—
183	C₅₃	#33484	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	53	1	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	1	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₄×Q₈	—	53	1	no	—
190	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
191	C₄×D₄	—	53	1	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	—	53	1	no	—
194	C₄×D₄	—	53	2	no	—
195	#117	—	53	2	no	—
196	#117	—	53	1	no	—
197	C₂×C₄²	—	53	2	no	—
198	C₂×C₄²	—	53	1	no	—
199	M₃₂	—	106	4	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	53	2	no	—
202	C₂×C₁₆	—	53	4	no	—
203	C₂×C₁₆	—	106	8	no	—
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	53	1	no	—
206	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
207	C₄⋊C₈	C₂×#6157	1	2	yes	—
208	C₂²⋊C₈	—	53	2	no	—
209	C₂²⋊C₈	—	53	2	no	—
210	C₂²⋊C₈	—	53	2	no	—
211	C₄×C₈	—	53	2	no	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	1	no	—
214	C₄×C₈	—	53	1	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	C₂×#6157	1	2	yes	—
217	C₄×C₈	C₂×#6157	1	1	yes	characteristic
218	#41	—	106	4	no	—
219	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
220	C₂×D₄	—	53	1	no	—
221	C₂×D₄	—	53	2	no	—
222	C₂²×C₄	—	53	2	no	—
223	C₂²×C₄	—	53	1	no	—
224	C₂²×C₄	—	53	2	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	C₂²×#6157	1	2	yes	—
236	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
237	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	53	1	no	—
240	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
241	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
242	C₂²⋊C₄	—	53	2	no	—
243	C₂²⋊C₄	—	53	2	no	—
244	C₂²⋊C₄	—	53	2	no	—
245	C₄²	—	53	2	no	—
246	C₄²	—	53	1	no	—
247	C₄²	—	53	2	no	—
248	C₄²	—	53	1	no	—
249	C₄²	—	53	1	no	—
250	C₄²	C₂²×#6157	1	2	yes	—
251	C₄²	C₂²×#6157	1	1	yes	characteristic
252	C₁₆	—	53	4	no	—
253	C₁₆	—	106	4	no	—
254	C₁₆	—	106	2	no	—
255	C₂³	—	53	2	no	—
256	C₂³	—	53	1	no	—
257	Q₈	—	2	2	no	—
258	D₄	—	53	4	no	—
259	D₄	—	106	4	no	—
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	106	4	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	1	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	106	1	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	—	212	1	no	—
272	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
273	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
275	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	53	2	no	—
280	C₈	—	106	1	no	—
281	C₈	—	106	2	no	—
282	C₈	#13133305	1	2	yes	—
283	C₂²	—	53	4	no	—
284	C₂²	—	106	2	no	—
285	C₂²	—	106	2	no	—
286	C₂²	—	106	1	no	—
287	C₂²	—	106	1	no	—
288	C₂²	—	106	1	no	—
289	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	106	1	no	—
294	C₄	—	106	1	no	—
295	C₄	—	106	2	no	—
296	C₄	—	106	1	no	—
297	C₄	D₅₃⋊M₃₂	1	2	yes	—
298	C₄	#41×#6157	1	1	yes	characteristic
299	C₄	C₂×#13133305	1	1	yes	characteristic
300	C₂	—	106	4	no	—
301	C₂	—	212	1	no	—
302	C₂	#20291521	1	1	yes	characteristic
303	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
304	C₂	#20291683	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₈×Q₈×C₅₃	C₄
prime radical	C₂×#11719949	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#33484
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
2. #20291098	C₂	semidirect
3. #20291094	C₂	semidirect
4. #20291079	C₂	semidirect
5. #20291079	C₂	semidirect
6. C₄×#13133292	C₂	semidirect
7. #20290916	C₂	semidirect
101. #11719910	#41	semidirect
115. D₂₁₂	M₃₂	semidirect
183. C₅₃	#33484	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291098	2
#20291094	2
#20291079	2
C₄×#13133292	2
#20290916	2
C₈×#11719949	2
#33484	53