Subgroups — C₅₃×#13795 #27221307

Order 13568.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₅₃	C₂	1	2	yes	maximal
2	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊#95	C₂	1	8	yes	maximal, complemented
4	C₅₃×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#117×C₅₃	—	2	8	no	—
8	C₂×#117×C₅₃	—	2	8	no	—
9	C₂×#117×C₅₃	C₄	1	16	yes	complemented
10	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
11	C₂²×C₄⋊C₄×C₅₃	C₄	1	8	yes	complemented
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
16	C₂×#95×C₅₃	C₂²	1	16	yes	—
17	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	—	2	8	no	—
19	C₂³×C₄×C₅₃	—	4	8	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₅₃	D₄	1	8	yes	—
24	C₂³×C₄×C₅₃	D₄	1	4	yes	—
25	C₂³×C₄×C₅₃	Q₈	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
29	#117×C₅₃	—	2	64	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	32	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
37	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
39	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
40	C₂×C₄²×C₅₃	—	2	8	no	—
41	C₂×C₄²×C₅₃	—	2	8	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	complemented
43	#95×C₅₃	—	4	16	no	—
44	C₂⁴×C₅₃	—	2	4	no	—
45	C₂⁴×C₅₃	—	2	4	no	—
46	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
49	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
50	C₂⁴×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	16	no	—
53	C₂²×C₄×C₅₃	—	2	8	no	—
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	16	no	—
57	C₂²×C₄×C₅₃	—	2	16	no	—
58	C₂²×C₄×C₅₃	—	2	16	no	—
59	C₂²×C₄×C₅₃	—	4	16	no	—
60	C₂²×C₄×C₅₃	—	4	32	no	—
61	C₂²×C₄×C₅₃	C₄²	1	16	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	16	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	32	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	16	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	32	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
75	C₄⋊C₄×C₅₃	—	2	32	no	—
76	C₄⋊C₄×C₅₃	—	2	32	no	—
77	C₂²⋊C₄×C₅₃	—	2	32	no	—
78	C₄²×C₅₃	—	2	32	no	—
79	C₂³×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	2	16	no	—
82	C₂³×C₅₃	—	2	8	no	—
83	C₂³×C₅₃	—	2	8	no	—
84	C₂³×C₅₃	—	2	8	no	—
85	C₂³×C₅₃	—	2	8	no	—
86	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
87	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
91	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
93	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
94	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
97	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
98	C₂×C₄×C₅₃	—	2	8	no	—
99	C₂×C₄×C₅₃	—	2	16	no	—
100	C₂×C₄×C₅₃	—	2	16	no	—
101	C₂×C₄×C₅₃	—	2	32	no	—
102	C₂×C₄×C₅₃	—	4	32	no	—
103	C₂×C₄×C₅₃	—	4	8	no	—
104	C₂×C₄×C₅₃	—	4	16	no	—
105	C₂×C₄×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	—	4	16	no	—
107	C₂×C₄×C₅₃	#95	1	64	yes	complemented
108	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
115	#13795	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	—	2	4	no	—
118	C₂²×C₅₃	—	2	16	no	—
119	C₂²×C₅₃	—	2	8	no	—
120	C₂²×C₅₃	—	2	16	no	—
121	C₂²×C₅₃	—	2	8	no	—
122	C₂²×C₅₃	C₂×#95	1	8	yes	—
123	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
124	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
126	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
128	C₂²×C₅₃	#518	1	2	yes	—
129	C₂²×C₅₃	#519	1	2	yes	—
130	C₂²×C₅₃	#520	1	8	yes	—
131	C₂²×C₅₃	#535	1	4	yes	—
132	C₂²×C₅₃	#536	1	4	yes	—
133	C₂²×C₅₃	#537	1	4	yes	—
134	C₂²×C₅₃	#544	1	4	yes	—
135	C₄×C₅₃	—	4	8	no	—
136	C₄×C₅₃	—	4	16	no	—
137	C₄×C₅₃	C₂×#95	1	16	yes	—
138	C₂²×#117	C₂×C₅₃	1	2	yes	—
139	C₂³×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
140	C₄⋊#95	C₂×C₅₃	1	8	yes	complemented
141	C₂²⋊#95	C₂×C₅₃	1	4	yes	—
142	C₂×C₅₃	—	2	8	no	—
143	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
144	C₂×C₅₃	#2274	1	2	yes	—
145	C₂×C₅₃	#2309	1	1	yes	characteristic
146	C₂×C₅₃	#2310	1	1	yes	characteristic
147	C₂×C₅₃	#2311	1	4	yes	—
148	C₂×C₅₃	#2316	1	2	yes	—
149	C₂×C₅₃	#2317	1	4	yes	—
150	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
151	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×#117	—	2	8	no	—
153	C₂×#117	—	2	8	no	—
154	C₂×#117	C₄×C₅₃	1	16	yes	complemented
155	C₂²×C₄⋊C₄	—	2	8	no	—
156	C₂²×C₄⋊C₄	C₄×C₅₃	1	8	yes	complemented
157	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
159	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
160	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
161	C₂×#95	C₂²×C₅₃	1	16	yes	—
162	C₅₃	#13795	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
164	C₂³×C₄	—	2	8	no	—
165	C₂³×C₄	—	4	8	no	—
166	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	—
167	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
168	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	—
169	C₂³×C₄	D₄×C₅₃	1	8	yes	—
170	C₂³×C₄	D₄×C₅₃	1	4	yes	—
171	C₂³×C₄	Q₈×C₅₃	1	4	yes	—
172	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
173	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
174	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
175	#117	—	2	64	no	—
176	C₂×C₄⋊C₄	—	2	8	no	—
177	C₂×C₄⋊C₄	—	2	8	no	—
178	C₂×C₄⋊C₄	—	2	8	no	—
179	C₂×C₄⋊C₄	—	2	8	no	—
180	C₂×C₄⋊C₄	—	2	32	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	complemented
183	C₂×C₂²⋊C₄	—	2	8	no	—
184	C₂×C₂²⋊C₄	—	2	8	no	—
185	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	8	no	—
188	C₂×C₄²	C₂×C₄×C₅₃	1	16	yes	complemented
189	#95	—	4	16	no	—
190	C₂⁴	—	2	4	no	—
191	C₂⁴	—	2	4	no	—
192	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
193	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
194	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
195	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
196	C₂⁴	C₂×Q₈×C₅₃	1	2	yes	—
197	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	16	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	—	2	16	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	4	16	no	—
206	C₂²×C₄	—	4	32	no	—
207	C₂²×C₄	C₄²×C₅₃	1	16	yes	—
208	C₂²×C₄	C₂²⋊C₄×C₅₃	1	16	yes	—
209	C₂²×C₄	C₂²⋊C₄×C₅₃	1	32	yes	—
210	C₂²×C₄	C₄⋊C₄×C₅₃	1	16	yes	—
211	C₂²×C₄	C₄⋊C₄×C₅₃	1	32	yes	—
212	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
220	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
221	C₄⋊C₄	—	2	32	no	—
222	C₄⋊C₄	—	2	32	no	—
223	C₂²⋊C₄	—	2	32	no	—
224	C₄²	—	2	32	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	2	8	no	—
231	C₂³	—	2	8	no	—
232	C₂³	C₂×C₄²×C₅₃	1	4	yes	—
233	C₂³	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
234	C₂³	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
235	C₂³	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
236	C₂³	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
237	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
238	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
239	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
240	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
241	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
242	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
243	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
244	C₂×C₄	—	2	16	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	32	no	—
248	C₂×C₄	—	4	32	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	4	16	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	#95×C₅₃	1	64	yes	complemented
254	C₂×C₄	C₂×C₄²×C₅₃	1	8	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	16	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	16	no	—
267	C₂²	C₂×#95×C₅₃	1	8	yes	—
268	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
270	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	2	yes	—
271	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
272	C₂²	C₂²×C₄⋊C₄×C₅₃	1	2	yes	—
273	C₂²	C₅₃×#518	1	2	yes	—
274	C₂²	C₅₃×#519	1	2	yes	—
275	C₂²	C₅₃×#520	1	8	yes	—
276	C₂²	C₅₃×#535	1	4	yes	—
277	C₂²	C₅₃×#536	1	4	yes	—
278	C₂²	C₅₃×#537	1	4	yes	—
279	C₂²	C₅₃×#544	1	4	yes	—
280	C₄	—	4	8	no	—
281	C₄	—	4	16	no	—
282	C₄	C₂×#95×C₅₃	1	16	yes	—
283	C₂	—	2	8	no	—
284	C₂	C₂²×#95×C₅₃	1	1	yes	characteristic
285	C₂	C₅₃×#2274	1	2	yes	—
286	C₂	C₅₃×#2309	1	1	yes	characteristic
287	C₂	C₅₃×#2310	1	1	yes	characteristic
288	C₂	C₅₃×#2311	1	4	yes	—
289	C₂	C₅₃×#2316	1	2	yes	—
290	C₂	C₅₃×#2317	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×C₄⋊#95	C₂	semidirect
9. C₂×#117×C₅₃	C₄	semidirect
11. C₂²×C₄⋊C₄×C₅₃	C₄	semidirect
36. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
42. C₂×C₄²×C₅₃	C₂×C₄	semidirect
107. C₂×C₄×C₅₃	#95	semidirect
115. #13795	C₅₃	direct (N × H)
140. C₄⋊#95	C₂×C₅₃	semidirect
154. C₂×#117	C₄×C₅₃	semidirect
156. C₂²×C₄⋊C₄	C₄×C₅₃	semidirect
162. C₅₃	#13795	direct (N × H)
182. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
188. C₂×C₄²	C₂×C₄×C₅₃	semidirect
253. C₂×C₄	#95×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₅₃×C₄⋊#95	2
C₅₃×C₂²⋊#95	2
#13795	53

Subgroups — C53×#13795 #27221307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13795 #27221307