Subgroups — C₂×C₅₃×#3150 #27268147

Order 13568.

3326 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₅₃	C₂	1	4	yes	maximal, complemented
2	C₂×C₅₃×#532	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3150	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#497	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#496	C₂	1	4	yes	maximal, complemented
6	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₅₃	C₂²	1	4	yes	—
8	C₂×#135×C₅₃	—	2	32	no	—
9	C₂×#135×C₅₃	C₂²	1	32	yes	complemented
10	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	—
11	C₂²×SD₁₆×C₅₃	C₂²	1	8	yes	—
12	C₂²×D₈×C₅₃	C₂²	1	4	yes	—
13	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#532	—	2	8	no	—
15	C₅₃×#532	C₂²	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#127×C₅₃	C₂²	1	2	yes	—
18	C₂×#124×C₅₃	C₂²	1	4	yes	—
19	C₂×#121×C₅₃	—	2	4	no	—
20	C₂×C₄×D₄×C₅₃	—	2	4	no	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#497	C₂²	1	32	yes	complemented
23	C₅₃×#496	C₂²	1	32	yes	complemented
24	C₂×#97×C₅₃	C₂²	1	4	yes	—
25	C₂×#97×C₅₃	C₂²	1	8	yes	complemented
26	C₂×#41×C₅₃	—	2	8	no	—
27	C₂×#41×C₅₃	—	2	32	no	—
28	C₂×#41×C₅₃	C₂³	1	16	yes	—
29	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
30	C₂²×D₄×C₅₃	—	2	4	no	—
31	C₂²×D₄×C₅₃	—	2	4	no	—
32	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₅₃	—	2	4	no	—
34	C₂³×C₄×C₅₃	D₄	1	2	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	#135×C₅₃	—	2	128	no	—
37	C₂×Q₁₆×C₅₃	—	2	16	no	—
38	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
39	C₂×SD₁₆×C₅₃	—	2	32	no	—
40	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	—
41	C₂×D₈×C₅₃	—	2	16	no	—
42	C₂×D₈×C₅₃	C₂³	1	16	yes	—
43	C₂²×C₈×C₅₃	D₄	1	16	yes	complemented
44	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
46	C₄⋊Q₈×C₅₃	—	2	4	no	—
47	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
48	#127×C₅₃	—	2	4	no	—
49	#127×C₅₃	C₂³	1	8	yes	—
50	#124×C₅₃	—	2	8	no	—
51	#124×C₅₃	C₂³	1	16	yes	—
52	#121×C₅₃	—	2	32	no	—
53	C₄×D₄×C₅₃	—	2	32	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
56	C₂×C₄²×C₅₃	D₄	1	4	yes	—
57	C₂×C₄²×C₅₃	D₄	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
60	#97×C₅₃	C₂³	1	16	yes	—
61	C₂⁴×C₅₃	—	2	4	no	—
62	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
63	#41×C₅₃	—	2	64	no	—
64	#41×C₅₃	—	4	32	no	—
65	C₂×Q₈×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	—	2	16	no	—
67	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
68	C₂×D₄×C₅₃	—	2	16	no	—
69	C₂×D₄×C₅₃	—	2	16	no	—
70	C₂×D₄×C₅₃	—	2	16	no	—
71	C₂×D₄×C₅₃	—	2	4	no	—
72	C₂×D₄×C₅₃	—	4	16	no	—
73	C₂×D₄×C₅₃	—	4	16	no	—
74	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	—	2	16	no	—
77	C₂²×C₄×C₅₃	—	4	4	no	—
78	C₂²×C₄×C₅₃	—	4	16	no	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
86	Q₁₆×C₅₃	—	2	32	no	—
87	SD₁₆×C₅₃	—	2	64	no	—
88	D₈×C₅₃	—	2	32	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	32	yes	complemented
91	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
92	C₄⋊C₄×C₅₃	—	2	16	no	—
93	C₂²⋊C₄×C₅₃	—	2	32	no	—
94	C₄²×C₅₃	C₂×D₄	1	8	yes	—
95	C₄²×C₅₃	C₂×D₄	1	4	yes	—
96	C₄²×C₅₃	C₂⁴	1	4	yes	—
97	C₂³×C₅₃	—	2	4	no	—
98	C₂³×C₅₃	—	2	8	no	—
99	C₂³×C₅₃	—	4	8	no	—
100	C₂³×C₅₃	—	4	16	no	—
101	C₂³×C₅₃	#127	1	4	yes	—
102	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
103	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	16	no	—
105	Q₈×C₅₃	—	4	8	no	—
106	D₄×C₅₃	—	2	16	no	—
107	D₄×C₅₃	—	4	32	no	—
108	D₄×C₅₃	—	4	32	no	—
109	D₄×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	—	2	8	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	4	8	no	—
113	C₂×C₄×C₅₃	—	4	8	no	—
114	C₂×C₄×C₅₃	—	4	32	no	—
115	C₂×C₄×C₅₃	#127	1	4	yes	—
116	C₂×C₄×C₅₃	#127	1	8	yes	—
117	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
122	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
123	C₈×C₅₃	C₂²×D₄	1	16	yes	—
124	C₂×#3150	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₅₃	—	2	8	no	—
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	4	8	no	—
128	C₂²×C₅₃	—	4	16	no	—
129	C₂²×C₅₃	—	8	16	no	—
130	C₂²×C₅₃	C₂×#127	1	4	yes	—
131	C₂²×C₅₃	C₂×#127	1	2	yes	—
132	C₂²×C₅₃	#575	1	4	yes	—
133	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₅₃	—	2	8	no	—
135	C₄×C₅₃	—	4	8	no	—
136	C₄×C₅₃	C₂×#127	1	4	yes	—
137	C₄×C₅₃	C₂×#127	1	2	yes	—
138	C₄×C₅₃	C₂³×D₄	1	2	yes	—
139	C₂²×#135	C₂×C₅₃	1	4	yes	complemented
140	C₂×#532	C₂×C₅₃	1	2	yes	complemented
141	#3150	C₂×C₅₃	1	16	yes	direct product factor
142	C₂×#497	C₂×C₅₃	1	4	yes	complemented
143	C₂×#496	C₂×C₅₃	1	4	yes	complemented
144	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	—	8	8	no	—
147	C₂×C₅₃	#3150	1	4	yes	direct product factor
148	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#575	1	2	yes	—
150	C₂²×#41	C₂²×C₅₃	1	4	yes	—
151	C₂×#135	—	2	32	no	—
152	C₂×#135	C₂²×C₅₃	1	32	yes	complemented
153	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	—
154	C₂²×SD₁₆	C₂²×C₅₃	1	8	yes	—
155	C₂²×D₈	C₂²×C₅₃	1	4	yes	—
156	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
157	#532	—	2	8	no	—
158	#532	C₂²×C₅₃	1	16	yes	complemented
159	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
160	C₂×#127	C₂²×C₅₃	1	2	yes	—
161	C₂×#124	C₂²×C₅₃	1	4	yes	—
162	C₂×#121	—	2	4	no	—
163	C₂×C₄×D₄	—	2	4	no	—
164	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
165	#497	C₂²×C₅₃	1	32	yes	complemented
166	#496	C₂²×C₅₃	1	32	yes	complemented
167	C₂×#97	C₂²×C₅₃	1	4	yes	—
168	C₂×#97	C₂²×C₅₃	1	8	yes	complemented
169	C₅₃	C₂×#3150	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#41	—	2	8	no	—
171	C₂×#41	—	2	32	no	—
172	C₂×#41	C₂³×C₅₃	1	16	yes	—
173	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
174	C₂²×D₄	—	2	4	no	—
175	C₂²×D₄	—	2	4	no	—
176	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	D₄×C₅₃	1	2	yes	—
179	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	#135	—	2	128	no	—
181	C₂×Q₁₆	—	2	16	no	—
182	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	—
183	C₂×SD₁₆	—	2	32	no	—
184	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	—
185	C₂×D₈	—	2	16	no	—
186	C₂×D₈	C₂³×C₅₃	1	16	yes	—
187	C₂²×C₈	D₄×C₅₃	1	16	yes	complemented
188	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
192	#127	—	2	4	no	—
193	#127	C₂³×C₅₃	1	8	yes	—
194	#124	—	2	8	no	—
195	#124	C₂³×C₅₃	1	16	yes	—
196	#121	—	2	32	no	—
197	C₄×D₄	—	2	32	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₂²⋊C₄	—	2	8	no	—
200	C₂×C₄²	D₄×C₅₃	1	2	yes	—
201	C₂×C₄²	D₄×C₅₃	1	4	yes	—
202	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
204	#97	C₂³×C₅₃	1	16	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
207	#41	—	2	64	no	—
208	#41	—	4	32	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	16	no	—
211	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
212	C₂×D₄	—	2	16	no	—
213	C₂×D₄	—	2	16	no	—
214	C₂×D₄	—	2	16	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	4	16	no	—
217	C₂×D₄	—	4	16	no	—
218	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	16	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
228	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
230	Q₁₆	—	2	32	no	—
231	SD₁₆	—	2	64	no	—
232	D₈	—	2	32	no	—
233	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	—
234	C₂×C₈	C₂×D₄×C₅₃	1	32	yes	complemented
235	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
236	C₄⋊C₄	—	2	16	no	—
237	C₂²⋊C₄	—	2	32	no	—
238	C₄²	C₂×D₄×C₅₃	1	8	yes	—
239	C₄²	C₂×D₄×C₅₃	1	4	yes	—
240	C₄²	C₂⁴×C₅₃	1	4	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	16	no	—
245	C₂³	#127×C₅₃	1	4	yes	—
246	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
247	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
248	Q₈	—	2	16	no	—
249	Q₈	—	4	8	no	—
250	D₄	—	2	16	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	32	no	—
253	D₄	—	4	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	32	no	—
259	C₂×C₄	#127×C₅₃	1	4	yes	—
260	C₂×C₄	#127×C₅₃	1	8	yes	—
261	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
262	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
263	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
265	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
266	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₅₃	1	16	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	4	16	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	8	16	no	—
273	C₂²	C₂×#127×C₅₃	1	4	yes	—
274	C₂²	C₂×#127×C₅₃	1	2	yes	—
275	C₂²	C₅₃×#575	1	4	yes	—
276	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
277	C₄	—	2	8	no	—
278	C₄	—	4	8	no	—
279	C₄	C₂×#127×C₅₃	1	4	yes	—
280	C₄	C₂×#127×C₅₃	1	2	yes	—
281	C₄	C₂³×D₄×C₅₃	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	8	8	no	—
284	C₂	C₅₃×#3150	1	4	yes	direct product factor
285	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
286	C₂	C₂×C₅₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₅₃	C₂	semidirect
2. C₂×C₅₃×#532	C₂	semidirect
3. C₅₃×#3150	C₂	direct (N × H)
4. C₂×C₅₃×#497	C₂	semidirect
5. C₂×C₅₃×#496	C₂	semidirect
6. C₂²×#97×C₅₃	C₂	semidirect
9. C₂×#135×C₅₃	C₂²	semidirect
15. C₅₃×#532	C₂²	semidirect
22. C₅₃×#497	C₂²	semidirect
23. C₅₃×#496	C₂²	semidirect
25. C₂×#97×C₅₃	C₂²	semidirect
43. C₂²×C₈×C₅₃	D₄	semidirect
90. C₂×C₈×C₅₃	C₂×D₄	semidirect
124. C₂×#3150	C₅₃	direct (N × H)
139. C₂²×#135	C₂×C₅₃	semidirect
140. C₂×#532	C₂×C₅₃	semidirect
141. #3150	C₂×C₅₃	direct (N × H)
142. C₂×#497	C₂×C₅₃	semidirect
143. C₂×#496	C₂×C₅₃	semidirect
144. C₂²×#97	C₂×C₅₃	semidirect
147. C₂×C₅₃	#3150	direct (N × H)
152. C₂×#135	C₂²×C₅₃	semidirect
158. #532	C₂²×C₅₃	semidirect
165. #497	C₂²×C₅₃	semidirect
166. #496	C₂²×C₅₃	semidirect
168. C₂×#97	C₂²×C₅₃	semidirect
169. C₅₃	C₂×#3150	direct (N × H)
187. C₂²×C₈	D₄×C₅₃	semidirect
234. C₂×C₈	C₂×D₄×C₅₃	semidirect
284. C₂	C₅₃×#3150	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₅₃	2
C₂×C₅₃×#532	2
C₅₃×#3150	2
C₂×C₅₃×#497	2
C₂×C₅₃×#496	2
C₂²×#97×C₅₃	2
C₂×#3150	53

Subgroups — C2×C53×#3150 #27268147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3150 #27268147