Subgroups — #28310237

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#20289760	C₂	1	2	yes	maximal, complemented
3	#20289759	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133292	—	2	2	no	—
6	D₅₃⋊M₃₂	—	2	4	no	—
7	D₅₃⋊M₃₂	C₄	1	8	yes	complemented
8	C₂×#13133291	C₂²	1	4	yes	—
9	C₂×D₅₃⋊C₁₆	—	2	2	no	—
10	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	—	2	4	no	—
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
18	#13133292	—	2	8	no	—
19	#13133291	—	2	4	no	—
20	#13133291	C₂×C₄	1	8	yes	complemented
21	D₅₃⋊C₁₆	—	2	8	no	—
22	C₂×#11719910	—	2	4	no	—
23	C₂²×C₄×D₅₃	—	2	1	no	—
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂³	1	1	yes	characteristic
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	C₈	1	2	yes	—
30	C₂×C₈×D₅₃	C₈	1	2	yes	—
31	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₅₃	D₄	1	4	yes	—
33	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×D₅₃	—	2	2	no	—
40	C₄²×D₅₃	C₈	1	4	yes	—
41	C₄×#88322	—	2	2	no	—
42	C₈×Dic₅₃	—	2	2	no	—
43	C₈×Dic₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	1	no	—
52	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	8	no	—
54	C₂×C₄×D₅₃	—	2	4	no	—
55	C₂×C₄×D₅₃	C₄²	1	2	yes	—
56	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	C₄²	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂×C₈	1	2	yes	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	—	2	4	no	—
74	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	C₂×C₈	1	2	yes	—
78	C₂×#88322	C₂²×C₄	1	2	yes	—
79	C₂×#88322	C₂×D₄	1	1	yes	characteristic
80	C₂×#88322	C₂×D₄	1	2	yes	—
81	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
82	C₈×D₅₃	—	2	4	no	—
83	C₈×D₅₃	—	2	4	no	—
84	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
85	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
86	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
87	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
88	#11719910	—	2	8	no	—
89	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂²×D₅₃	—	2	4	no	—
91	C₂²×D₅₃	#97	1	2	yes	—
92	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
93	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
94	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	4	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	#97	1	2	yes	—
106	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
107	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
108	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
109	C₄×D₅₃	—	2	16	no	—
110	C₄×D₅₃	#95	1	2	yes	—
111	C₄×D₅₃	#95	1	2	yes	—
112	C₄×D₅₃	C₄×C₈	1	2	yes	—
113	C₄×D₅₃	C₄×C₈	1	2	yes	—
114	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
115	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
116	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
117	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
118	C₈×C₅₃	—	2	2	no	—
119	C₈×C₅₃	#95	1	2	yes	—
120	C₈×C₅₃	#95	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	C₂×C₂²⋊C₄	1	2	yes	—
123	#88322	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×#1376	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂×D₅₃	—	2	2	no	—
130	C₂×D₅₃	—	2	4	no	—
131	C₂×D₅₃	—	2	4	no	—
132	C₂×D₅₃	#336	1	2	yes	—
133	C₂×D₅₃	#336	1	2	yes	—
134	C₂×D₅₃	#364	1	4	yes	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
140	Dic₅₃	—	2	4	no	—
141	Dic₅₃	#336	1	2	yes	—
142	Dic₅₃	#336	1	2	yes	—
143	C₂²×M₃₂	—	53	2	no	—
144	C₂²×C₄×C₈	—	53	1	no	—
145	#1376	—	53	2	no	—
146	#1376	—	53	2	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#364	1	2	yes	—
151	D₅₃	—	2	2	no	—
152	D₅₃	#1376	1	4	yes	complemented
153	C₂³×C₈	—	53	1	no	—
154	C₂³×C₈	—	53	1	no	—
155	C₂²×C₄²	—	53	1	no	—
156	C₂×M₃₂	—	53	4	no	—
157	C₂×M₃₂	—	53	8	no	—
158	C₂×M₃₂	—	106	4	no	—
159	C₂×M₃₂	—	106	2	no	—
160	C₂²×C₁₆	—	106	2	no	—
161	C₂×C₄×C₈	—	53	1	no	—
162	C₂×C₄×C₈	—	53	1	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₂×C₄×C₈	—	106	4	no	—
165	C₂×C₄×C₈	#6157	1	1	yes	characteristic
166	C₅₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	53	1	no	—
168	C₂³×C₄	—	106	1	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	53	1	no	—
175	C₂²×C₈	—	53	4	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	—	106	2	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
182	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
183	C₂×C₄²	—	53	1	no	—
184	C₂×C₄²	—	53	4	no	—
185	C₂×C₄²	—	106	2	no	—
186	C₂×C₄²	—	106	1	no	—
187	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
188	M₃₂	—	53	8	no	—
189	M₃₂	—	106	4	no	—
190	M₃₂	—	106	8	no	—
191	C₂×C₁₆	—	106	4	no	—
192	C₂×C₁₆	—	106	8	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	106	2	no	—
195	C₄×C₈	—	106	2	no	—
196	C₄×C₈	—	106	2	no	—
197	C₂⁴	—	53	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	53	2	no	—
201	C₂²×C₄	—	53	2	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	53	1	no	—
207	C₂²×C₄	—	53	2	no	—
208	C₂²×C₄	—	106	4	no	—
209	C₂²×C₄	—	106	8	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	4	no	—
219	C₂×C₈	—	53	4	no	—
220	C₂×C₈	—	53	4	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	106	4	no	—
224	C₂×C₈	—	106	4	no	—
225	C₂×C₈	—	106	1	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
228	C₂×C₈	C₄×#6157	1	2	yes	—
229	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
230	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊#6157	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	53	2	no	—
234	C₄²	—	106	4	no	—
235	C₄²	—	106	1	no	—
236	C₄²	D₅₃⋊C₈	1	2	yes	—
237	C₁₆	—	106	8	no	—
238	C₂³	—	53	1	no	—
239	C₂³	—	53	2	no	—
240	C₂³	—	53	2	no	—
241	C₂³	—	53	1	no	—
242	C₂³	—	106	4	no	—
243	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	53	2	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	53	2	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	106	16	no	—
259	C₂×C₄	—	106	4	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
264	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
265	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
266	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	53	2	no	—
269	C₈	—	53	2	no	—
270	C₈	—	106	2	no	—
271	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
272	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	53	2	no	—
275	C₂²	—	53	4	no	—
276	C₂²	—	53	2	no	—
277	C₂²	—	106	4	no	—
278	C₂²	—	106	4	no	—
279	C₂²	—	106	2	no	—
280	C₂²	#16811582	1	1	yes	characteristic
281	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
282	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	53	2	no	—
285	C₄	—	53	2	no	—
286	C₄	—	106	4	no	—
287	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
288	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
289	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
290	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	53	4	no	—
294	C₂	—	106	2	no	—
295	C₂	#20290376	1	1	yes	characteristic
296	C₂	D₅₃⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#1376
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289760	C₂	semidirect
3. #20289759	C₂	semidirect
7. D₅₃⋊M₃₂	C₄	semidirect
20. #13133291	C₂×C₄	semidirect
152. D₅₃	#1376	semidirect
166. C₅₃	C₂×#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊M₃₂	2
#20289760	2
#20289759	2
C₂×C₄×C₈×D₅₃	2
C₂×#1376	53