Subgroups — #28298041

Order 13568.

352 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290648	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133284	C₂	1	1	yes	maximal, characteristic
3	#20274563	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133284	—	2	1	no	—
5	C₂×#13133284	—	2	1	no	—
6	C₂×#13133284	C₄	1	2	yes	—
7	C₂×C₄⋊#88324	C₄	1	2	yes	—
8	C₂×C₄×#88324	—	2	1	no	—
9	#16811489	—	2	1	no	—
10	C₂×#13133146	—	2	1	no	—
11	C₂×C₈×Dic₅₃	—	2	1	no	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810378	C₄	1	1	yes	characteristic
14	C₅₃×#336	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	2	no	—
16	C₂²×#88324	—	2	2	no	—
17	#13133284	—	2	2	no	—
18	#13133284	—	4	1	no	—
19	#13133284	—	4	1	no	—
20	#13133284	—	4	1	no	—
21	#13133284	—	4	1	no	—
22	#13133284	C₈	1	4	yes	complemented
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	#13133146	—	2	4	no	—
34	C₈×Dic₅₃	—	4	1	no	—
35	C₈×Dic₅₃	—	4	1	no	—
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#88324	—	2	4	no	—
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	—	4	2	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₄²×C₅₃	—	4	1	no	—
55	C₄×Dic₅₃	—	2	1	no	—
56	C₄×Dic₅₃	—	2	1	no	—
57	C₄×Dic₅₃	—	2	2	no	—
58	C₄×Dic₅₃	—	2	2	no	—
59	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
60	C₄×Dic₅₃	M₁₆	1	2	yes	—
61	C₄×Dic₅₃	D₈	1	1	yes	characteristic
62	C₄×Dic₅₃	D₈	1	1	yes	characteristic
63	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
64	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
65	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
66	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
67	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
68	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	4	1	no	—
71	C₂×#88322	—	4	1	no	—
72	C₂³×C₅₃	#95	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	—	2	1	no	—
87	C₂×Dic₅₃	—	2	1	no	—
88	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
89	C₂×Dic₅₃	#100	1	1	yes	characteristic
90	C₂×Dic₅₃	#101	1	1	yes	characteristic
91	C₂×Dic₅₃	#102	1	1	yes	characteristic
92	C₂×Dic₅₃	#102	1	1	yes	characteristic
93	C₂×Dic₅₃	#102	1	1	yes	characteristic
94	C₂×Dic₅₃	#103	1	1	yes	characteristic
95	C₂×Dic₅₃	#103	1	1	yes	characteristic
96	C₂×Dic₅₃	#103	1	1	yes	characteristic
97	C₂×Dic₅₃	#104	1	2	yes	—
98	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₅₃	#106	1	1	yes	characteristic
100	C₂×Dic₅₃	#107	1	1	yes	characteristic
101	C₂×Dic₅₃	#108	1	2	yes	—
102	#88324	—	4	4	no	—
103	#88324	—	8	2	no	—
104	C₈×C₅₃	—	4	2	no	—
105	#88322	—	4	2	no	—
106	#7031	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	#336	1	1	yes	characteristic
108	C₂²×C₅₃	#339	1	1	yes	characteristic
109	C₂²×C₅₃	#340	1	1	yes	characteristic
110	C₂²×C₅₃	#340	1	1	yes	characteristic
111	C₂²×C₅₃	#341	1	1	yes	characteristic
112	C₂²×C₅₃	#343	1	1	yes	characteristic
113	C₂²×C₅₃	#344	1	1	yes	characteristic
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	Dic₅₃	—	4	1	no	—
121	Dic₅₃	D₄⋊C₈	1	2	yes	—
122	Dic₅₃	Q₈⋊C₈	1	2	yes	—
123	Dic₅₃	#331	1	1	yes	characteristic
124	Dic₅₃	#332	1	1	yes	characteristic
125	Dic₅₃	#332	1	1	yes	characteristic
126	Dic₅₃	#333	1	1	yes	characteristic
127	Dic₅₃	#334	1	2	yes	—
128	Dic₅₃	#335	1	2	yes	—
129	#1842	—	53	1	no	—
130	C₄×C₄⋊C₈	—	53	1	no	—
131	C₄×C₄⋊C₈	—	53	1	no	—
132	C₂×C₅₃	#1277	1	1	yes	characteristic
133	C₂×C₅₃	#1279	1	1	yes	characteristic
134	C₂×C₅₃	#1283	1	1	yes	characteristic
135	C₂×C₅₃	#1296	1	1	yes	characteristic
136	C₂×C₅₃	#1299	1	1	yes	characteristic
137	C₂×C₅₃	#1301	1	1	yes	characteristic
138	C₂×C₅₃	#1303	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	53	2	no	—
140	C₂×C₄⋊C₈	—	53	2	no	—
141	C₂×C₄⋊C₈	—	106	1	no	—
142	C₂×C₄⋊C₈	—	106	1	no	—
143	C₂×C₄⋊C₈	—	106	1	no	—
144	C₂×C₄×C₈	—	106	1	no	—
145	C₂×C₄×C₈	—	106	1	no	—
146	C₄³	—	53	1	no	—
147	#336	—	53	1	no	—
148	#336	—	106	1	no	—
149	#336	#6157	1	1	yes	characteristic
150	C₅₃	#7031	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	106	2	no	—
153	C₂²×C₈	—	106	1	no	—
154	C₂²×C₈	—	106	2	no	—
155	C₂×C₄²	—	53	1	no	—
156	C₂×C₄²	—	53	1	no	—
157	C₂×C₄²	—	53	1	no	—
158	C₂×C₄²	—	53	1	no	—
159	C₂×C₄²	—	53	1	no	—
160	C₂×C₄²	—	53	1	no	—
161	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	—	106	4	no	—
165	C₄⋊C₈	—	212	1	no	—
166	C₄⋊C₈	—	212	1	no	—
167	C₄⋊C₈	—	212	1	no	—
168	C₄⋊C₈	—	212	1	no	—
169	C₄⋊C₈	—	212	2	no	—
170	C₄×C₈	—	212	1	no	—
171	C₄×C₈	—	212	1	no	—
172	C₄×C₈	—	212	1	no	—
173	C₄×C₈	—	212	1	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	—	53	1	no	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	1	no	—
178	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	106	2	no	—
185	C₂×C₈	—	106	4	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	—	212	1	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	—	212	2	no	—
192	C₂×C₈	—	212	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	53	2	no	—
195	C₄²	—	53	2	no	—
196	C₄²	—	53	1	no	—
197	C₄²	—	53	1	no	—
198	C₄²	—	53	1	no	—
199	C₄²	—	53	1	no	—
200	C₄²	—	53	1	no	—
201	C₄²	—	53	1	no	—
202	C₄²	—	53	1	no	—
203	C₄²	—	53	1	no	—
204	C₄²	—	106	2	no	—
205	C₄²	—	106	1	no	—
206	C₄²	—	106	2	no	—
207	C₄²	—	106	1	no	—
208	C₂³	(C₂×#6157)⋊C₄	1	1	yes	characteristic
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	53	2	no	—
216	C₂×C₄	—	53	2	no	—
217	C₂×C₄	—	53	2	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	—	53	1	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	1	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	53	1	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	53	1	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	1	no	—
231	C₂×C₄	—	106	1	no	—
232	C₂×C₄	C₈×#6157	1	1	yes	characteristic
233	C₂×C₄	#13133271	1	1	yes	characteristic
234	C₂×C₄	#13133279	1	1	yes	characteristic
235	C₂×C₄	#13133280	1	1	yes	characteristic
236	C₂×C₄	#13133285	1	1	yes	characteristic
237	C₂×C₄	#13133286	1	1	yes	characteristic
238	C₈	—	4	2	no	—
239	C₈	—	424	2	no	—
240	C₈	—	212	2	no	—
241	C₈	—	212	4	no	—
242	C₂²	(C₂×#6157)⋊C₈	1	1	yes	characteristic
243	C₂²	#16811494	1	1	yes	characteristic
244	C₂²	#16811496	1	1	yes	characteristic
245	C₂²	#16811497	1	1	yes	characteristic
246	C₂²	#16811498	1	1	yes	characteristic
247	C₂²	#16811502	1	1	yes	characteristic
248	C₂²	#16811504	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	53	2	no	—
256	C₄	—	53	2	no	—
257	C₄	—	53	2	no	—
258	C₄	—	53	2	no	—
259	C₄	—	53	1	no	—
260	C₄	—	53	1	no	—
261	C₄	—	53	1	no	—
262	C₄	—	53	1	no	—
263	C₄	—	212	1	no	—
264	C₂	#20289544	1	1	yes	characteristic
265	C₂	#20289550	1	1	yes	characteristic
266	C₂	#20289559	1	1	yes	characteristic
267	C₂	#20289594	1	1	yes	characteristic
268	C₂	#20289600	1	1	yes	characteristic
269	C₂	#20289606	1	1	yes	characteristic
270	C₂	#20289613	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#6157)⋊C₄
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₅₃×#336	C₄
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄²×Dic₅₃	C₂²
hypercenter	#336	#6157
hypocenter	C₅₃	#7031
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
22. #13133284	C₈	semidirect
150. C₅₃	#7031	semidirect

Maximal subgroups

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

subgroup	index
#20290648	2
C₄×#13133284	2
#20274563	2
#7031	53