Subgroups — #28298040

Order 13568.

352 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	#20290647	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88324	—	2	1	no	—
5	C₂×C₄⋊#88324	—	2	1	no	—
6	C₂×C₄⋊#88324	—	2	1	no	—
7	C₂×C₄⋊#88324	C₄	1	2	yes	—
8	C₂×C₄×#88324	—	2	1	no	—
9	#16811489	C₄	1	2	yes	—
10	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	—	2	1	no	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810377	—	2	1	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	2	no	—
16	C₂²×#88324	—	2	2	no	—
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₄⋊#88324	—	4	1	no	—
20	C₄⋊#88324	—	4	1	no	—
21	C₄⋊#88324	—	4	1	no	—
22	C₄⋊#88324	—	4	1	no	—
23	C₄⋊#88324	—	4	2	no	—
24	C₄×#88324	—	4	1	no	—
25	C₄×#88324	—	4	1	no	—
26	C₂²×#88322	—	2	1	no	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	C₈	1	2	yes	complemented
35	C₈×Dic₅₃	—	4	1	no	—
36	C₈×Dic₅₃	—	4	1	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	1	no	—
40	C₄⋊C₈×C₅₃	C₈	1	2	yes	complemented
41	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	—	4	2	no	—
52	C₂×#88324	—	4	2	no	—
53	C₂×#88324	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
60	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	1	no	—
63	C₄×Dic₅₃	—	2	1	no	—
64	C₄×Dic₅₃	—	2	1	no	—
65	C₄×Dic₅₃	—	2	1	no	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	—	4	1	no	—
68	C₄×Dic₅₃	D₈	1	1	yes	characteristic
69	C₄×Dic₅₃	D₈	1	1	yes	characteristic
70	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
71	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
72	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
73	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
74	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
75	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	4	1	no	—
78	C₂×#88322	—	4	1	no	—
79	C₂³×C₅₃	#95	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	#100	1	1	yes	characteristic
104	C₂×Dic₅₃	#101	1	1	yes	characteristic
105	C₂×Dic₅₃	#102	1	1	yes	characteristic
106	C₂×Dic₅₃	#102	1	1	yes	characteristic
107	C₂×Dic₅₃	#102	1	1	yes	characteristic
108	C₂×Dic₅₃	#103	1	1	yes	characteristic
109	C₂×Dic₅₃	#103	1	1	yes	characteristic
110	C₂×Dic₅₃	#103	1	1	yes	characteristic
111	C₂×Dic₅₃	#106	1	1	yes	characteristic
112	C₂×Dic₅₃	#107	1	1	yes	characteristic
113	#88324	—	4	4	no	—
114	#88324	—	8	2	no	—
115	C₈×C₅₃	—	4	2	no	—
116	#88322	—	4	2	no	—
117	#7031	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	#336	1	1	yes	characteristic
119	C₂²×C₅₃	#339	1	1	yes	characteristic
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	#340	1	1	yes	characteristic
122	C₂²×C₅₃	#341	1	1	yes	characteristic
123	C₂²×C₅₃	#343	1	1	yes	characteristic
124	C₂²×C₅₃	#344	1	1	yes	characteristic
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
128	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
129	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
130	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
131	C₄×C₅₃	#334	1	1	yes	characteristic
132	C₄×C₅₃	#334	1	1	yes	characteristic
133	C₄×C₅₃	#335	1	1	yes	characteristic
134	C₄×C₅₃	#335	1	1	yes	characteristic
135	Dic₅₃	—	2	1	no	—
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	—	2	1	no	—
139	Dic₅₃	—	4	1	no	—
140	Dic₅₃	#331	1	1	yes	characteristic
141	Dic₅₃	#332	1	1	yes	characteristic
142	Dic₅₃	#332	1	1	yes	characteristic
143	Dic₅₃	#333	1	1	yes	characteristic
144	#1842	—	53	1	no	—
145	C₄×C₄⋊C₈	—	53	1	no	—
146	C₄×C₄⋊C₈	—	53	1	no	—
147	C₂×C₅₃	#1277	1	1	yes	characteristic
148	C₂×C₅₃	#1279	1	1	yes	characteristic
149	C₂×C₅₃	#1283	1	1	yes	characteristic
150	C₂×C₅₃	#1296	1	1	yes	characteristic
151	C₂×C₅₃	#1299	1	1	yes	characteristic
152	C₂×C₅₃	#1301	1	1	yes	characteristic
153	C₂×C₅₃	#1303	1	1	yes	characteristic
154	C₂×C₄⋊C₈	—	53	1	no	—
155	C₂×C₄⋊C₈	—	53	2	no	—
156	C₂×C₄⋊C₈	—	106	1	no	—
157	C₂×C₄⋊C₈	—	106	1	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
160	C₂×C₄×C₈	—	106	1	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	C₄³	—	53	1	no	—
163	#336	—	53	2	no	—
164	#336	—	106	1	no	—
165	C₅₃	#7031	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	—	106	1	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂×C₄²	—	53	1	no	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	1	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	53	2	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₄⋊C₈	—	106	1	no	—
182	C₄⋊C₈	—	212	1	no	—
183	C₄⋊C₈	—	212	1	no	—
184	C₄⋊C₈	—	212	1	no	—
185	C₄⋊C₈	—	212	1	no	—
186	C₄⋊C₈	—	212	2	no	—
187	C₄⋊C₈	#88324	1	2	yes	complemented
188	C₄×C₈	—	212	1	no	—
189	C₄×C₈	—	212	1	no	—
190	C₄×C₈	—	212	1	no	—
191	C₄×C₈	—	212	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	106	4	no	—
203	C₂×C₈	—	106	2	no	—
204	C₂×C₈	—	212	2	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₂×C₈	—	212	1	no	—
210	C₂×C₈	—	212	1	no	—
211	C₄²	—	53	1	no	—
212	C₄²	—	53	1	no	—
213	C₄²	—	53	1	no	—
214	C₄²	—	53	1	no	—
215	C₄²	—	53	1	no	—
216	C₄²	—	53	1	no	—
217	C₄²	—	53	1	no	—
218	C₄²	—	53	1	no	—
219	C₄²	—	106	1	no	—
220	C₄²	—	106	1	no	—
221	C₄²	—	106	1	no	—
222	C₄²	—	106	1	no	—
223	C₄²	—	106	1	no	—
224	C₄²	—	106	1	no	—
225	C₄²	—	212	1	no	—
226	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
227	C₄²	#11719936	1	1	yes	characteristic
228	C₄²	C₂×#88324	1	1	yes	characteristic
229	C₄²	#11719940	1	1	yes	characteristic
230	C₂³	(C₂×#6157)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	1	no	—
236	C₂×C₄	—	53	1	no	—
237	C₂×C₄	—	53	1	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	1	no	—
240	C₂×C₄	—	53	1	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	1	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	106	1	no	—
247	C₂×C₄	—	106	1	no	—
248	C₂×C₄	—	106	1	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	C₄×#88324	1	1	yes	characteristic
251	C₂×C₄	#13133272	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
253	C₂×C₄	#13133274	1	1	yes	characteristic
254	C₂×C₄	#13133279	1	1	yes	characteristic
255	C₂×C₄	#13133280	1	1	yes	characteristic
256	C₂×C₄	#13133281	1	1	yes	characteristic
257	C₂×C₄	#13133282	1	1	yes	characteristic
258	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
259	C₂×C₄	#13133284	1	1	yes	characteristic
260	C₂×C₄	#13133285	1	1	yes	characteristic
261	C₂×C₄	#13133286	1	1	yes	characteristic
262	C₂×C₄	#13133287	1	1	yes	characteristic
263	C₂×C₄	#13133288	1	1	yes	characteristic
264	C₈	—	4	2	no	—
265	C₈	—	424	2	no	—
266	C₈	—	212	4	no	—
267	C₈	—	212	2	no	—
268	C₂²	#16811489	1	1	yes	characteristic
269	C₂²	#16811493	1	1	yes	characteristic
270	C₂²	#16811496	1	1	yes	characteristic
271	C₂²	#16811497	1	1	yes	characteristic
272	C₂²	#16811499	1	1	yes	characteristic
273	C₂²	#16811502	1	1	yes	characteristic
274	C₂²	#16811506	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	53	1	no	—
278	C₄	—	53	1	no	—
279	C₄	—	53	1	no	—
280	C₄	—	53	1	no	—
281	C₄	—	106	1	no	—
282	C₄	—	106	1	no	—
283	C₄	—	106	1	no	—
284	C₄	—	106	1	no	—
285	C₄	—	212	1	no	—
286	C₄	D₄⋊#88324	1	1	yes	characteristic
287	C₄	#16811471	1	1	yes	characteristic
288	C₄	Q₈⋊#88324	1	1	yes	characteristic
289	C₄	#16811473	1	1	yes	characteristic
290	C₄	#16811484	1	1	yes	characteristic
291	C₄	#16811485	1	1	yes	characteristic
292	C₄	#16811486	1	1	yes	characteristic
293	C₄	#16811487	1	1	yes	characteristic
294	C₂	#20289545	1	1	yes	characteristic
295	C₂	#20289551	1	1	yes	characteristic
296	C₂	#20289561	1	1	yes	characteristic
297	C₂	#20289594	1	1	yes	characteristic
298	C₂	#20289601	1	1	yes	characteristic
299	C₂	#20289607	1	1	yes	characteristic
300	C₂	#20289612	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#6157)⋊C₄
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄²×Dic₅₃	C₂²
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#7031
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
34. C₄⋊#88322	C₈	semidirect
40. C₄⋊C₈×C₅₃	C₈	semidirect
165. C₅₃	#7031	semidirect
187. C₄⋊C₈	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290647	2
C₄×C₄⋊#88324	2
C₄⋊C₈×Dic₅₃	2
#7031	53