Subgroups — C₅₃×#9796 #27217308

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1965	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#486	—	2	1	no	—
11	C₅₃×#486	—	2	1	no	—
12	C₅₃×#486	—	2	2	no	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#423	C₂²	1	1	yes	characteristic
15	C₅₃×#423	C₂²	1	1	yes	characteristic
16	C₂×#102×C₅₃	C₄	1	4	yes	—
17	C₂×#101×C₅₃	—	2	2	no	—
18	C₂×#100×C₅₃	—	2	2	no	—
19	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#335	—	2	2	no	—
22	C₅₃×#332	—	2	2	no	—
23	C₅₃×#331	—	2	2	no	—
24	C₅₃×#322	—	2	1	no	—
25	C₈²×C₅₃	—	2	1	no	—
26	C₂²×Q₈×C₅₃	—	2	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	2	1	no	—
29	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂×M₁₆×C₅₃	—	2	1	no	—
32	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄⋊Q₈×C₅₃	—	2	1	no	—
35	C₄⋊Q₈×C₅₃	—	2	1	no	—
36	C₄⋊Q₈×C₅₃	—	2	1	no	—
37	C₄⋊Q₈×C₅₃	—	2	1	no	—
38	#127×C₅₃	—	2	1	no	—
39	#127×C₅₃	—	2	1	no	—
40	#127×C₅₃	—	2	1	no	—
41	#127×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₈×C₅₃	—	2	1	no	—
48	C₄⋊C₈×C₅₃	—	2	1	no	—
49	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
50	#102×C₅₃	—	2	4	no	—
51	#102×C₅₃	—	2	4	no	—
52	#101×C₅₃	—	4	4	no	—
53	#100×C₅₃	—	4	4	no	—
54	C₂²⋊C₈×C₅₃	—	2	1	no	—
55	#97×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	2	yes	—
61	C₂⁴×C₅₃	—	2	2	no	—
62	C₂×Q₈×C₅₃	—	4	1	no	—
63	C₂×Q₈×C₅₃	—	4	1	no	—
64	C₂×Q₈×C₅₃	—	4	2	no	—
65	C₂×Q₈×C₅₃	—	4	2	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	4	4	no	—
69	C₂×D₄×C₅₃	—	4	1	no	—
70	C₂×D₄×C₅₃	—	4	1	no	—
71	C₂×D₄×C₅₃	—	4	1	no	—
72	C₂×D₄×C₅₃	—	4	1	no	—
73	C₂×D₄×C₅₃	—	4	2	no	—
74	C₂×D₄×C₅₃	—	4	2	no	—
75	C₂×D₄×C₅₃	—	8	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	M₁₆×C₅₃	—	2	2	no	—
81	M₁₆×C₅₃	—	4	2	no	—
82	M₁₆×C₅₃	—	4	2	no	—
83	M₁₆×C₅₃	D₈	1	4	yes	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	—	4	4	no	—
101	C₂³×C₅₃	—	8	2	no	—
102	C₂³×C₅₃	—	8	2	no	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	8	2	no	—
105	Q₈×C₅₃	—	8	1	no	—
106	Q₈×C₅₃	—	8	1	no	—
107	D₄×C₅₃	—	4	4	no	—
108	D₄×C₅₃	—	8	2	no	—
109	D₄×C₅₃	—	8	2	no	—
110	D₄×C₅₃	—	8	2	no	—
111	D₄×C₅₃	—	8	2	no	—
112	D₄×C₅₃	—	8	1	no	—
113	D₄×C₅₃	—	8	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	2	yes	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	—	8	1	no	—
131	C₈×C₅₃	—	8	1	no	—
132	#9796	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	—	8	2	no	—
137	C₂²×C₅₃	—	8	2	no	—
138	C₂²×C₅₃	—	16	4	no	—
139	C₂²×C₅₃	#390	1	1	yes	characteristic
140	C₂²×C₅₃	C₄×D₈	1	2	yes	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	8	2	no	—
144	C₄×C₅₃	#413	1	2	yes	—
145	C₄×C₅₃	#466	1	2	yes	—
146	C₄×C₅₃	#471	1	2	yes	—
147	C₄×C₅₃	#493	1	1	yes	characteristic
148	C₄×C₅₃	#495	1	1	yes	characteristic
149	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
150	#1966	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#1965	C₂×C₅₃	1	1	yes	characteristic
152	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#1545	C₂×C₅₃	1	1	yes	characteristic
155	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	—	16	2	no	—
158	C₂×C₅₃	#1967	1	1	yes	characteristic
159	C₂×C₅₃	#1970	1	1	yes	characteristic
160	C₂×C₅₃	#1977	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
163	#486	—	2	1	no	—
164	#486	—	2	1	no	—
165	#486	—	2	2	no	—
166	#431	C₂²×C₅₃	1	1	yes	characteristic
167	#423	C₂²×C₅₃	1	1	yes	characteristic
168	#423	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×#102	C₄×C₅₃	1	4	yes	—
170	C₂×#101	—	2	2	no	—
171	C₂×#100	—	2	2	no	—
172	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
174	#335	—	2	2	no	—
175	#332	—	2	2	no	—
176	#331	—	2	2	no	—
177	#322	—	2	1	no	—
178	C₈²	—	2	1	no	—
179	C₅₃	#9796	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
187	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	#127	—	2	1	no	—
193	#127	—	2	1	no	—
194	#127	—	2	1	no	—
195	#127	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
204	#102	—	2	4	no	—
205	#102	—	2	4	no	—
206	#101	—	4	4	no	—
207	#100	—	4	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
213	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
214	C₄×C₈	D₄×C₅₃	1	2	yes	—
215	C₂⁴	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	8	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	D₈×C₅₃	1	4	yes	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
244	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₄²	#41×C₅₃	1	1	yes	characteristic
253	C₂³	—	4	2	no	—
254	C₂³	—	4	4	no	—
255	C₂³	—	8	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
258	Q₈	—	8	1	no	—
259	Q₈	—	8	1	no	—
260	Q₈	—	8	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	1	no	—
265	D₄	—	8	1	no	—
266	D₄	—	8	2	no	—
267	D₄	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
275	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
276	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
279	C₂×C₄	#135×C₅₃	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	8	1	no	—
285	C₈	—	8	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	16	4	no	—
292	C₂²	C₅₃×#390	1	1	yes	characteristic
293	C₂²	C₄×D₈×C₅₃	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	C₅₃×#413	1	2	yes	—
298	C₄	C₅₃×#466	1	2	yes	—
299	C₄	C₅₃×#471	1	2	yes	—
300	C₄	C₅₃×#493	1	1	yes	characteristic
301	C₄	C₅₃×#495	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	C₅₃×#1967	1	1	yes	characteristic
305	C₂	C₅₃×#1970	1	1	yes	characteristic
306	C₂	C₅₃×#1977	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1966	C₂	semidirect
4. C₅₃×#1567	C₂	semidirect
5. C₅₃×#1546	C₂	semidirect
7. C₈×M₁₆×C₅₃	C₂	semidirect
132. #9796	C₅₃	direct (N × H)
150. #1966	C₂×C₅₃	semidirect
152. #1567	C₂×C₅₃	semidirect
153. #1546	C₂×C₅₃	semidirect
155. C₈×M₁₆	C₂×C₅₃	semidirect
179. C₅₃	#9796	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#486	2
C₅₃×#1966	2
C₅₃×#1965	2
C₅₃×#1567	2
C₅₃×#1546	2
C₅₃×#1545	2
C₈×M₁₆×C₅₃	2
#9796	53

Subgroups — C53×#9796 #27217308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9796 #27217308