Subgroups — C₂²×#16811280 #28291028

Order 13568.

12142 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811280	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133204	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133196	C₂	1	3	yes	maximal, complemented
5	C₂²×#13133188	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×C₂²⋊D₅₃	C₂	1	3	yes	maximal, complemented
8	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Dic₁₀₆	C₂²	1	3	yes	—
10	C₂²×Q₈×D₅₃	—	2	16	no	—
11	C₂²×Q₈×D₅₃	C₂²	1	24	yes	complemented
12	C₂³×C₂²⋊D₅₃	C₂²	1	3	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	3	yes	—
14	#16811280	C₂²	1	256	yes	direct product factor
15	C₂×#13133204	C₂²	1	72	yes	complemented
16	C₂×#13133196	C₂²	1	72	yes	complemented
17	C₂×#13133188	C₂²	1	72	yes	complemented
18	C₂×Q₈×Dic₅₃	C₂²	1	24	yes	complemented
19	C₂×C₄×C₂²⋊D₅₃	C₂²	1	72	yes	complemented
20	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
21	C₂²×#11719919	C₂²	1	3	yes	—
22	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
23	C₂²×C₂²⋊Dic₅₃	C₂²	1	6	yes	—
24	C₂²×#11719921	C₂²	1	1	yes	characteristic
25	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
26	C₂²×#41×C₅₃	—	2	16	no	—
27	C₂²×#41×C₅₃	C₂²	1	24	yes	complemented
28	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
30	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
31	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₀₆	—	2	24	no	—
33	C₂²×Dic₁₀₆	C₂³	1	36	yes	—
34	C₂×Q₈×D₅₃	—	2	192	no	—
35	C₂×Q₈×D₅₃	C₂³	1	64	yes	complemented
36	C₂²×C₂²⋊D₅₃	—	2	24	no	—
37	C₂²×C₂²⋊D₅₃	C₂³	1	36	yes	—
38	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂³×Dic₅₃	C₂³	1	3	yes	—
40	C₂³×Dic₅₃	C₂³	1	3	yes	—
41	C₂²×C₄×D₅₃	—	2	24	no	—
42	C₂²×C₄×D₅₃	C₂³	1	36	yes	—
43	#13133204	C₂³	1	192	yes	complemented
44	#13133196	C₂³	1	192	yes	complemented
45	#13133188	C₂³	1	192	yes	complemented
46	Q₈×Dic₅₃	C₂³	1	64	yes	complemented
47	C₄×C₂²⋊D₅₃	C₂³	1	192	yes	complemented
48	C₂×C₄⋊Dic₅₃	C₂³	1	36	yes	—
49	C₂×#11719919	C₂³	1	36	yes	—
50	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	—
51	C₂×C₂²⋊Dic₅₃	C₂³	1	72	yes	—
52	C₂×#11719921	C₂³	1	12	yes	—
53	C₂×C₄×Dic₅₃	C₂³	1	36	yes	—
54	C₂⁵×C₅₃	C₂³	1	3	yes	—
55	C₂×#41×C₅₃	—	2	192	no	—
56	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
57	C₂²×Q₈×C₅₃	D₄	1	16	yes	complemented
58	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
59	C₂²×D₄×C₅₃	—	2	24	no	—
60	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
61	C₂³×C₄×C₅₃	—	2	24	no	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
65	C₂⁴×C₅₃	—	2	12	no	—
66	C₂⁴×C₅₃	—	2	12	no	—
67	C₂⁴×C₅₃	—	2	12	no	—
68	C₂⁴×C₅₃	C₂⁴	1	18	yes	—
69	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂³×D₅₃	—	2	4	no	—
71	C₂³×D₅₃	—	2	4	no	—
72	C₂³×D₅₃	#41	1	8	yes	complemented
73	C₂³×D₅₃	C₂⁴	1	6	yes	—
74	#41×C₅₃	—	2	256	no	—
75	C₂×Q₈×C₅₃	C₂×D₄	1	96	yes	complemented
76	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
77	C₂×D₄×C₅₃	—	2	144	no	—
78	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
79	C₂²×C₄×C₅₃	—	2	4	no	—
80	C₂²×C₄×C₅₃	—	2	144	no	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
84	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
85	C₂×C₂²⋊D₅₃	—	2	144	no	—
86	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
87	C₂²×Dic₅₃	—	2	12	no	—
88	C₂²×Dic₅₃	—	2	12	no	—
89	C₂²×Dic₅₃	—	2	4	no	—
90	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
91	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
92	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
93	Q₈×D₅₃	—	2	256	no	—
94	C₂×C₄×D₅₃	—	2	144	no	—
95	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
96	C₂×Dic₁₀₆	—	2	144	no	—
97	C₂×Dic₁₀₆	C₂⁴	1	48	yes	—
98	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
99	C₂²⋊Dic₅₃	C₂⁴	1	96	yes	—
100	#11719921	C₂⁴	1	16	yes	—
101	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
102	#11719919	C₂⁴	1	48	yes	—
103	C₄×Dic₅₃	C₂⁴	1	48	yes	—
104	C₂³×C₅₃	—	2	36	no	—
105	C₂³×C₅₃	—	2	36	no	—
106	C₂³×C₅₃	—	2	36	no	—
107	C₂³×C₅₃	—	4	48	no	—
108	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
109	C₂³×C₅₃	C₂×#41	1	4	yes	—
110	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
111	C₂³×C₅₃	C₂⁵	1	12	yes	—
112	C₂³×C₅₃	C₂⁵	1	3	yes	—
113	C₂²×D₅₃	—	2	32	no	—
114	C₂²×D₅₃	—	2	12	no	—
115	C₂²×D₅₃	—	2	12	no	—
116	C₂²×D₅₃	C₂×#41	1	24	yes	complemented
117	C₂²×D₅₃	C₂⁵	1	4	yes	—
118	Q₈×C₅₃	C₂²×D₄	1	64	yes	complemented
119	D₄×C₅₃	—	2	96	no	—
120	C₂×C₄×C₅₃	—	2	12	no	—
121	C₂×C₄×C₅₃	—	2	96	no	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	—
123	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
124	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
125	C₂²⋊D₅₃	—	2	96	no	—
126	C₂×Dic₅₃	—	2	36	no	—
127	C₂×Dic₅₃	—	2	36	no	—
128	C₂×Dic₅₃	—	2	12	no	—
129	C₂×Dic₅₃	C₂⁵	1	12	yes	—
130	C₂×Dic₅₃	C₂⁵	1	12	yes	—
131	C₂×Dic₅₃	C₂⁵	1	4	yes	—
132	C₄×D₅₃	—	2	96	no	—
133	Dic₁₀₆	—	2	96	no	—
134	C₂²×#548	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	12	no	—
136	C₂²×C₅₃	—	2	12	no	—
137	C₂²×C₅₃	—	2	12	no	—
138	C₂²×C₅₃	—	4	72	no	—
139	C₂²×C₅₃	#548	1	16	yes	complemented
140	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
141	C₂²×C₅₃	C₂²×#41	1	6	yes	—
142	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
143	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
144	C₂×D₅₃	—	2	4	no	—
145	C₂×D₅₃	—	2	4	no	—
146	C₂×D₅₃	—	2	48	no	—
147	C₂×D₅₃	C₂²×#41	1	8	yes	complemented
148	C₄×C₅₃	—	2	4	no	—
149	C₄×C₅₃	C₂³×D₄	1	24	yes	—
150	Dic₅₃	—	2	12	no	—
151	Dic₅₃	—	2	12	no	—
152	Dic₅₃	—	2	4	no	—
153	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
154	C₂⁴×Q₈	—	53	1	no	—
155	C₂×#548	—	53	48	no	—
156	C₂²×#124	—	53	3	no	—
157	C₂²×C₂²⋊Q₈	—	53	3	no	—
158	C₂²×#121	—	53	3	no	—
159	C₂²×C₄×Q₈	—	53	1	no	—
160	C₂²×C₄×D₄	—	53	3	no	—
161	C₂×C₅₃	—	4	12	no	—
162	C₂×C₅₃	C₂×#548	1	12	yes	complemented
163	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
164	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
165	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
166	D₅₃	—	2	8	no	—
167	C₂²×#41	—	2	16	no	—
168	C₂²×#41	C₂×D₅₃	1	24	yes	complemented
169	C₂³×Q₈	—	53	24	no	—
170	C₂³×Q₈	—	53	3	no	—
171	C₂³×Q₈	—	106	16	no	—
172	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂³×D₄	—	53	3	no	—
174	C₂³×D₄	C₂×D₅₃	1	3	yes	—
175	C₂⁴×C₄	—	53	3	no	—
176	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
177	#548	—	53	256	no	—
178	C₂×#124	—	53	72	no	—
179	C₂×C₂²⋊Q₈	—	53	72	no	—
180	C₂×#121	—	53	72	no	—
181	C₂×C₄×Q₈	—	53	24	no	—
182	C₂×C₄×D₄	—	53	72	no	—
183	C₂²×C₄⋊C₄	—	53	3	no	—
184	C₂²×C₄⋊C₄	—	53	3	no	—
185	C₂²×C₂²⋊C₄	—	53	3	no	—
186	C₂²×C₂²⋊C₄	—	53	6	no	—
187	C₂²×C₂²⋊C₄	—	53	1	no	—
188	C₂²×C₄²	—	53	3	no	—
189	C₅₃	C₂²×#548	1	1	yes	characteristic, complemented, Sylow
190	C₂⁵	—	53	1	no	—
191	C₂⁵	C₂²×D₅₃	1	3	yes	—
192	C₂×#41	—	2	192	no	—
193	C₂×#41	C₂²×D₅₃	1	64	yes	complemented
194	C₂²×Q₈	—	53	64	no	—
195	C₂²×Q₈	—	53	36	no	—
196	C₂²×Q₈	—	106	192	no	—
197	C₂²×Q₈	—	106	24	no	—
198	C₂²×Q₈	C₂²⋊D₅₃	1	16	yes	complemented
199	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
200	C₂²×D₄	—	2	24	no	—
201	C₂²×D₄	—	53	36	no	—
202	C₂²×D₄	—	106	24	no	—
203	C₂²×D₄	C₂²×D₅₃	1	36	yes	—
204	C₂³×C₄	—	2	24	no	—
205	C₂³×C₄	—	53	1	no	—
206	C₂³×C₄	—	53	3	no	—
207	C₂³×C₄	—	53	3	no	—
208	C₂³×C₄	—	53	36	no	—
209	C₂³×C₄	—	106	24	no	—
210	C₂³×C₄	C₂²×D₅₃	1	36	yes	—
211	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
213	#124	—	53	192	no	—
214	C₂²⋊Q₈	—	53	192	no	—
215	#121	—	53	192	no	—
216	C₄×Q₈	—	53	64	no	—
217	C₄×D₄	—	53	192	no	—
218	C₂×C₄⋊C₄	—	53	36	no	—
219	C₂×C₄⋊C₄	—	53	36	no	—
220	C₂×C₂²⋊C₄	—	53	36	no	—
221	C₂×C₂²⋊C₄	—	53	72	no	—
222	C₂×C₂²⋊C₄	—	53	12	no	—
223	C₂×C₄²	—	53	36	no	—
224	C₂⁴	—	2	12	no	—
225	C₂⁴	—	2	12	no	—
226	C₂⁴	—	2	12	no	—
227	C₂⁴	—	53	8	no	—
228	C₂⁴	—	53	6	no	—
229	C₂⁴	—	106	4	no	—
230	C₂⁴	—	106	4	no	—
231	C₂⁴	C₂³×D₅₃	1	18	yes	—
232	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
233	#41	—	2	256	no	—
234	C₂×Q₈	—	53	48	no	—
235	C₂×Q₈	—	106	256	no	—
236	C₂×Q₈	—	106	144	no	—
237	C₂×Q₈	C₂×C₂²⋊D₅₃	1	96	yes	complemented
238	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
239	C₂×D₄	—	2	144	no	—
240	C₂×D₄	—	53	48	no	—
241	C₂×D₄	—	106	144	no	—
242	C₂×D₄	C₂³×D₅₃	1	48	yes	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	144	no	—
245	C₂²×C₄	—	53	6	no	—
246	C₂²×C₄	—	53	18	no	—
247	C₂²×C₄	—	53	18	no	—
248	C₂²×C₄	—	53	48	no	—
249	C₂²×C₄	—	106	12	no	—
250	C₂²×C₄	—	106	12	no	—
251	C₂²×C₄	—	106	4	no	—
252	C₂²×C₄	—	106	144	no	—
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
254	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
255	C₂²×C₄	C₂³×D₅₃	1	48	yes	—
256	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
257	C₄⋊C₄	—	53	48	no	—
258	C₄⋊C₄	—	53	48	no	—
259	C₂²⋊C₄	—	53	48	no	—
260	C₂²⋊C₄	—	53	96	no	—
261	C₂²⋊C₄	—	53	16	no	—
262	C₄²	—	53	48	no	—
263	C₂³	—	2	36	no	—
264	C₂³	—	2	36	no	—
265	C₂³	—	2	36	no	—
266	C₂³	—	4	48	no	—
267	C₂³	—	53	24	no	—
268	C₂³	—	53	4	no	—
269	C₂³	—	106	32	no	—
270	C₂³	—	106	12	no	—
271	C₂³	—	106	12	no	—
272	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
273	C₂³	#41×D₅₃	1	4	yes	—
274	C₂³	#13133264	1	4	yes	—
275	C₂³	C₂⁴×D₅₃	1	12	yes	—
276	C₂³	C₂⁴×D₅₃	1	3	yes	—
277	Q₈	—	106	96	no	—
278	Q₈	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
279	D₄	—	2	96	no	—
280	D₄	—	106	96	no	—
281	C₂×C₄	—	2	12	no	—
282	C₂×C₄	—	2	96	no	—
283	C₂×C₄	—	53	4	no	—
284	C₂×C₄	—	53	12	no	—
285	C₂×C₄	—	53	12	no	—
286	C₂×C₄	—	106	36	no	—
287	C₂×C₄	—	106	36	no	—
288	C₂×C₄	—	106	12	no	—
289	C₂×C₄	—	106	96	no	—
290	C₂×C₄	C₂²×C₂²⋊D₅₃	1	72	yes	—
291	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
292	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
293	C₂²	—	2	12	no	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	12	no	—
296	C₂²	—	4	72	no	—
297	C₂²	—	53	8	no	—
298	C₂²	—	106	48	no	—
299	C₂²	—	106	4	no	—
300	C₂²	—	106	4	no	—
301	C₂²	#16811280	1	16	yes	direct product factor
302	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
303	C₂²	C₂×#41×D₅₃	1	6	yes	—
304	C₂²	C₂×#13133264	1	6	yes	—
305	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	106	12	no	—
308	C₄	—	106	12	no	—
309	C₄	—	106	4	no	—
310	C₄	C₂³×C₂²⋊D₅₃	1	24	yes	—
311	C₂	—	4	12	no	—
312	C₂	—	106	8	no	—
313	C₂	C₂×#16811280	1	12	yes	direct product factor
314	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
315	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
316	C₂	C₂²×#13133264	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×#548
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅₃	C₂	semidirect
2. C₂×#16811280	C₂	direct (N × H)
3. C₂²×#13133204	C₂	semidirect
4. C₂²×#13133196	C₂	semidirect
5. C₂²×#13133188	C₂	semidirect
6. C₂²×Q₈×Dic₅₃	C₂	semidirect
7. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
8. C₂³×#41×C₅₃	C₂	semidirect
11. C₂²×Q₈×D₅₃	C₂²	semidirect
14. #16811280	C₂²	direct (N × H)
15. C₂×#13133204	C₂²	semidirect
16. C₂×#13133196	C₂²	semidirect
17. C₂×#13133188	C₂²	semidirect
18. C₂×Q₈×Dic₅₃	C₂²	semidirect
19. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
27. C₂²×#41×C₅₃	C₂²	semidirect
35. C₂×Q₈×D₅₃	C₂³	semidirect
43. #13133204	C₂³	semidirect
44. #13133196	C₂³	semidirect
45. #13133188	C₂³	semidirect
46. Q₈×Dic₅₃	C₂³	semidirect
47. C₄×C₂²⋊D₅₃	C₂³	semidirect
56. C₂×#41×C₅₃	C₂³	semidirect
57. C₂²×Q₈×C₅₃	D₄	semidirect
72. C₂³×D₅₃	#41	semidirect
75. C₂×Q₈×C₅₃	C₂×D₄	semidirect
116. C₂²×D₅₃	C₂×#41	semidirect
118. Q₈×C₅₃	C₂²×D₄	semidirect
139. C₂²×C₅₃	#548	semidirect
147. C₂×D₅₃	C₂²×#41	semidirect
153. C₂³×#41	D₅₃	semidirect
162. C₂×C₅₃	C₂×#548	semidirect
168. C₂²×#41	C₂×D₅₃	semidirect
189. C₅₃	C₂²×#548	semidirect
193. C₂×#41	C₂²×D₅₃	semidirect
198. C₂²×Q₈	C₂²⋊D₅₃	semidirect
237. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
278. Q₈	C₂²×C₂²⋊D₅₃	semidirect
301. C₂²	#16811280	direct (N × H)
313. C₂	C₂×#16811280	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₅₃	2
C₂×#16811280	2
C₂²×#13133204	2
C₂²×#13133196	2
C₂²×#13133188	2
C₂²×Q₈×Dic₅₃	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂³×#41×C₅₃	2
C₂²×#548	53

Subgroups — C22×#16811280 #28291028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811280 #28291028