Subgroups — C₂³×#13133190 #28290728

Order 13568.

19998 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133190	C₂	1	56	yes	maximal, direct product factor
6	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
10	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
11	C₂³×D₂₁₂	—	2	16	no	—
12	C₂³×D₂₁₂	C₂²	1	28	yes	complemented
13	C₂³×C₂²⋊D₅₃	—	2	16	no	—
14	C₂³×C₂²⋊D₅₃	—	2	16	no	—
15	C₂³×C₂²⋊D₅₃	C₂²	1	28	yes	complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	28	yes	complemented
17	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₄×D₅₃	—	2	16	no	—
20	C₂³×C₄×D₅₃	C₂²	1	28	yes	complemented
21	C₂×#13133190	C₂²	1	448	yes	direct product factor
22	C₂²×#11719919	C₂²	1	28	yes	complemented
23	C₂²×#11719921	C₂²	1	28	yes	complemented
24	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
25	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	28	yes	complemented
28	C₂⁴×D₅₃	—	2	8	no	—
29	C₂⁴×D₅₃	—	2	8	no	—
30	C₂⁴×D₅₃	—	2	8	no	—
31	C₂⁴×D₅₃	—	2	8	no	—
32	C₂⁴×D₅₃	—	2	8	no	—
33	C₂⁴×D₅₃	D₄	1	16	yes	complemented
34	C₂⁴×D₅₃	C₂³	1	14	yes	—
35	C₂⁴×D₅₃	C₂³	1	14	yes	—
36	C₂²×D₂₁₂	—	2	224	no	—
37	C₂²×D₂₁₂	C₂³	1	112	yes	complemented
38	C₂²×C₂²⋊D₅₃	—	2	224	no	—
39	C₂²×C₂²⋊D₅₃	—	2	224	no	—
40	C₂²×C₂²⋊D₅₃	C₂³	1	112	yes	complemented
41	C₂²×C₂²⋊D₅₃	C₂³	1	112	yes	complemented
42	C₂³×Dic₅₃	—	2	8	no	—
43	C₂³×Dic₅₃	D₄	1	16	yes	complemented
44	C₂³×Dic₅₃	C₂³	1	14	yes	—
45	C₂³×Dic₅₃	C₂³	1	14	yes	—
46	C₂²×C₄×D₅₃	—	2	224	no	—
47	C₂²×C₄×D₅₃	C₂³	1	112	yes	complemented
48	#13133190	C₂³	1	512	yes	direct product factor
49	C₂×#11719919	C₂³	1	112	yes	complemented
50	C₂×#11719921	C₂³	1	112	yes	complemented
51	C₂⁵×C₅₃	—	2	8	no	—
52	C₂⁵×C₅₃	—	2	8	no	—
53	C₂⁵×C₅₃	—	2	8	no	—
54	C₂⁵×C₅₃	C₂³	1	14	yes	—
55	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	—	2	8	no	—
57	C₂³×C₄×C₅₃	—	2	8	no	—
58	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂³	1	112	yes	complemented
61	C₂⁴×C₅₃	—	2	56	no	—
62	C₂⁴×C₅₃	—	2	56	no	—
63	C₂⁴×C₅₃	—	2	56	no	—
64	C₂⁴×C₅₃	—	4	64	no	—
65	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
66	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
67	C₂⁴×C₅₃	#41	1	8	yes	—
68	C₂⁴×C₅₃	C₂⁴	1	28	yes	—
69	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
70	C₂³×D₅₃	—	2	128	no	—
71	C₂³×D₅₃	—	2	56	no	—
72	C₂³×D₅₃	—	2	56	no	—
73	C₂³×D₅₃	—	2	56	no	—
74	C₂³×D₅₃	—	2	56	no	—
75	C₂³×D₅₃	—	2	56	no	—
76	C₂³×D₅₃	—	4	64	no	—
77	C₂³×D₅₃	C₂×D₄	1	112	yes	complemented
78	C₂³×D₅₃	C₂⁴	1	28	yes	—
79	C₂³×D₅₃	C₂⁴	1	28	yes	—
80	C₂²×C₄×C₅₃	—	2	56	no	—
81	C₂²×C₄×C₅₃	—	2	56	no	—
82	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
84	C₂×C₂²⋊D₅₃	—	2	448	no	—
85	C₂×C₂²⋊D₅₃	—	2	448	no	—
86	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
87	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
88	C₂²×Dic₅₃	—	2	56	no	—
89	C₂²×Dic₅₃	C₂×D₄	1	112	yes	complemented
90	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
91	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
92	C₂×D₂₁₂	—	2	448	no	—
93	C₂×D₂₁₂	C₂⁴	1	64	yes	complemented
94	C₂×C₄×D₅₃	—	2	448	no	—
95	C₂×C₄×D₅₃	C₂⁴	1	64	yes	complemented
96	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
97	#11719921	C₂⁴	1	64	yes	complemented
98	#11719919	C₂⁴	1	64	yes	complemented
99	C₂³×C₅₃	—	2	56	no	—
100	C₂³×C₅₃	—	2	56	no	—
101	C₂³×C₅₃	—	2	56	no	—
102	C₂³×C₅₃	—	4	224	no	—
103	C₂³×C₅₃	#121	1	64	yes	complemented
104	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
105	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
106	C₂³×C₅₃	C₂×#41	1	28	yes	—
107	C₂³×C₅₃	C₂⁵	1	8	yes	—
108	C₂³×C₅₃	C₂⁵	1	7	yes	—
109	C₂²×D₅₃	—	2	448	no	—
110	C₂²×D₅₃	—	2	56	no	—
111	C₂²×D₅₃	—	2	56	no	—
112	C₂²×D₅₃	—	2	56	no	—
113	C₂²×D₅₃	—	2	56	no	—
114	C₂²×D₅₃	—	2	56	no	—
115	C₂²×D₅₃	—	4	224	no	—
116	C₂²×D₅₃	C₂²×D₄	1	112	yes	complemented
117	C₂²×D₅₃	C₂⁵	1	8	yes	—
118	C₂²×D₅₃	C₂⁵	1	8	yes	—
119	C₂×C₄×C₅₃	—	2	56	no	—
120	C₂×C₄×C₅₃	—	2	56	no	—
121	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
122	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
123	C₂²⋊D₅₃	—	2	128	no	—
124	C₂²⋊D₅₃	—	2	128	no	—
125	C₂×Dic₅₃	—	2	56	no	—
126	C₂×Dic₅₃	C₂²×D₄	1	112	yes	complemented
127	C₂×Dic₅₃	C₂⁵	1	8	yes	—
128	C₂×Dic₅₃	C₂⁵	1	8	yes	—
129	D₂₁₂	—	2	128	no	—
130	C₄×D₅₃	—	2	128	no	—
131	C₂³×#121	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	8	no	—
133	C₂²×C₅₃	—	2	8	no	—
134	C₂²×C₅₃	—	2	8	no	—
135	C₂²×C₅₃	—	4	112	no	—
136	C₂²×C₅₃	C₂×#121	1	112	yes	complemented
137	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
138	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
139	C₂²×C₅₃	C₂²×#41	1	14	yes	—
140	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
141	C₂×D₅₃	—	2	8	no	—
142	C₂×D₅₃	—	2	8	no	—
143	C₂×D₅₃	—	2	224	no	—
144	C₂×D₅₃	—	2	8	no	—
145	C₂×D₅₃	—	2	8	no	—
146	C₂×D₅₃	—	2	8	no	—
147	C₂×D₅₃	—	4	112	no	—
148	C₂×D₅₃	C₂³×D₄	1	16	yes	complemented
149	C₄×C₅₃	—	2	8	no	—
150	C₄×C₅₃	—	2	8	no	—
151	Dic₅₃	—	2	8	no	—
152	Dic₅₃	C₂³×D₄	1	16	yes	complemented
153	C₂⁴×D₄	—	53	1	no	—
154	C₂⁴×D₄	—	53	1	no	—
155	C₂⁴×D₄	—	53	1	no	—
156	C₂⁵×C₄	—	53	1	no	—
157	C₂²×#121	—	53	56	no	—
158	C₂³×C₄⋊C₄	—	53	1	no	—
159	C₂³×C₂²⋊C₄	—	53	1	no	—
160	C₂³×C₂²⋊C₄	D₅₃	1	1	yes	characteristic, complemented
161	C₂×C₅₃	—	4	8	no	—
162	C₂×C₅₃	C₂²×#121	1	28	yes	complemented
163	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
164	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
165	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
166	D₅₃	—	2	16	no	—
167	D₅₃	—	4	8	no	—
168	C₂⁶	—	53	1	no	—
169	C₂⁶	—	53	1	no	—
170	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
171	C₂³×D₄	—	53	28	no	—
172	C₂³×D₄	—	53	28	no	—
173	C₂³×D₄	—	53	28	no	—
174	C₂³×D₄	—	106	16	no	—
175	C₂³×D₄	—	106	16	no	—
176	C₂³×D₄	—	106	16	no	—
177	C₂⁴×C₄	—	53	1	no	—
178	C₂⁴×C₄	—	53	1	no	—
179	C₂⁴×C₄	—	53	28	no	—
180	C₂⁴×C₄	—	106	16	no	—
181	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
182	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#121	—	53	448	no	—
184	C₂²×C₄⋊C₄	—	53	28	no	—
185	C₂²×C₂²⋊C₄	—	53	28	no	—
186	C₂²×C₂²⋊C₄	C₂×D₅₃	1	28	yes	complemented
187	C₅₃	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	—	2	8	no	—
189	C₂⁵	—	2	8	no	—
190	C₂⁵	—	2	8	no	—
191	C₂⁵	—	53	16	no	—
192	C₂⁵	—	53	14	no	—
193	C₂⁵	—	53	14	no	—
194	C₂⁵	—	106	8	no	—
195	C₂⁵	—	106	8	no	—
196	C₂⁵	—	106	8	no	—
197	C₂⁵	—	106	8	no	—
198	C₂⁵	—	106	8	no	—
199	C₂⁵	C₂²×D₅₃	1	14	yes	—
200	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
201	C₂²×D₄	—	53	112	no	—
202	C₂²×D₄	—	53	112	no	—
203	C₂²×D₄	—	53	112	no	—
204	C₂²×D₄	—	106	224	no	—
205	C₂²×D₄	—	106	224	no	—
206	C₂²×D₄	—	106	224	no	—
207	C₂³×C₄	—	2	8	no	—
208	C₂³×C₄	—	2	8	no	—
209	C₂³×C₄	—	53	16	no	—
210	C₂³×C₄	—	53	14	no	—
211	C₂³×C₄	—	53	14	no	—
212	C₂³×C₄	—	53	112	no	—
213	C₂³×C₄	—	106	224	no	—
214	C₂³×C₄	—	106	8	no	—
215	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
216	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
217	#121	—	53	512	no	—
218	C₂×C₄⋊C₄	—	53	112	no	—
219	C₂×C₂²⋊C₄	—	53	112	no	—
220	C₂×C₂²⋊C₄	C₂²×D₅₃	1	112	yes	complemented
221	C₂⁴	—	2	56	no	—
222	C₂⁴	—	2	56	no	—
223	C₂⁴	—	2	56	no	—
224	C₂⁴	—	4	64	no	—
225	C₂⁴	—	53	112	no	—
226	C₂⁴	—	53	28	no	—
227	C₂⁴	—	53	28	no	—
228	C₂⁴	—	106	128	no	—
229	C₂⁴	—	106	56	no	—
230	C₂⁴	—	106	56	no	—
231	C₂⁴	—	106	56	no	—
232	C₂⁴	—	106	56	no	—
233	C₂⁴	—	106	56	no	—
234	C₂⁴	—	212	64	no	—
235	C₂⁴	#11719945	1	8	yes	—
236	C₂⁴	D₄×D₅₃	1	8	yes	—
237	C₂⁴	D₄×D₅₃	1	8	yes	—
238	C₂⁴	C₂³×D₅₃	1	28	yes	—
239	C₂⁴	C₂³×D₅₃	1	7	yes	—
240	C₂×D₄	—	53	64	no	—
241	C₂×D₄	—	53	64	no	—
242	C₂×D₄	—	53	64	no	—
243	C₂×D₄	—	106	448	no	—
244	C₂×D₄	—	106	448	no	—
245	C₂×D₄	—	106	448	no	—
246	C₂²×C₄	—	2	56	no	—
247	C₂²×C₄	—	2	56	no	—
248	C₂²×C₄	—	53	112	no	—
249	C₂²×C₄	—	53	28	no	—
250	C₂²×C₄	—	53	28	no	—
251	C₂²×C₄	—	53	64	no	—
252	C₂²×C₄	—	106	448	no	—
253	C₂²×C₄	—	106	56	no	—
254	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
255	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
256	C₄⋊C₄	—	53	64	no	—
257	C₂²⋊C₄	—	53	64	no	—
258	C₂²⋊C₄	C₂³×D₅₃	1	64	yes	complemented
259	C₂³	—	2	56	no	—
260	C₂³	—	2	56	no	—
261	C₂³	—	2	56	no	—
262	C₂³	—	4	224	no	—
263	C₂³	—	53	112	no	—
264	C₂³	—	53	8	no	—
265	C₂³	—	53	8	no	—
266	C₂³	—	106	448	no	—
267	C₂³	—	106	56	no	—
268	C₂³	—	106	56	no	—
269	C₂³	—	106	56	no	—
270	C₂³	—	106	56	no	—
271	C₂³	—	106	56	no	—
272	C₂³	—	212	224	no	—
273	C₂³	#13133190	1	64	yes	direct product factor
274	C₂³	C₂×D₄×D₅₃	1	28	yes	—
275	C₂³	C₂×D₄×D₅₃	1	28	yes	—
276	C₂³	C₂×#11719945	1	28	yes	—
277	C₂³	C₂⁴×D₅₃	1	8	yes	—
278	C₂³	C₂⁴×D₅₃	1	7	yes	—
279	D₄	—	106	128	no	—
280	D₄	—	106	128	no	—
281	D₄	—	106	128	no	—
282	C₂×C₄	—	2	56	no	—
283	C₂×C₄	—	2	56	no	—
284	C₂×C₄	—	53	112	no	—
285	C₂×C₄	—	53	8	no	—
286	C₂×C₄	—	53	8	no	—
287	C₂×C₄	—	106	128	no	—
288	C₂×C₄	—	106	56	no	—
289	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
290	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	4	112	no	—
295	C₂²	—	53	16	no	—
296	C₂²	—	106	224	no	—
297	C₂²	—	106	8	no	—
298	C₂²	—	106	8	no	—
299	C₂²	—	106	8	no	—
300	C₂²	—	106	8	no	—
301	C₂²	—	106	8	no	—
302	C₂²	—	212	112	no	—
303	C₂²	C₂×#13133190	1	112	yes	direct product factor
304	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
305	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
306	C₂²	C₂²×#11719945	1	14	yes	—
307	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
308	C₄	—	2	8	no	—
309	C₄	—	2	8	no	—
310	C₄	—	53	16	no	—
311	C₄	—	106	8	no	—
312	C₂	—	4	8	no	—
313	C₂	—	106	16	no	—
314	C₂	—	212	8	no	—
315	C₂	C₂²×#13133190	1	28	yes	direct product factor
316	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
317	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
318	C₂	C₂³×#11719945	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₂²⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#121
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₁₂	C₂	semidirect
2. C₂⁴×C₂²⋊D₅₃	C₂	semidirect
3. C₂⁴×C₂²⋊D₅₃	C₂	semidirect
4. C₂⁴×C₄×D₅₃	C₂	semidirect
5. C₂²×#13133190	C₂	direct (N × H)
6. C₂³×#11719919	C₂	semidirect
7. C₂³×#11719921	C₂	semidirect
8. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
15. C₂³×C₂²⋊D₅₃	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
20. C₂³×C₄×D₅₃	C₂²	semidirect
21. C₂×#13133190	C₂²	direct (N × H)
22. C₂²×#11719919	C₂²	semidirect
23. C₂²×#11719921	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
33. C₂⁴×D₅₃	D₄	semidirect
37. C₂²×D₂₁₂	C₂³	semidirect
40. C₂²×C₂²⋊D₅₃	C₂³	semidirect
41. C₂²×C₂²⋊D₅₃	C₂³	semidirect
43. C₂³×Dic₅₃	D₄	semidirect
47. C₂²×C₄×D₅₃	C₂³	semidirect
48. #13133190	C₂³	direct (N × H)
49. C₂×#11719919	C₂³	semidirect
50. C₂×#11719921	C₂³	semidirect
60. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
77. C₂³×D₅₃	C₂×D₄	semidirect
86. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
87. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
89. C₂²×Dic₅₃	C₂×D₄	semidirect
93. C₂×D₂₁₂	C₂⁴	semidirect
95. C₂×C₄×D₅₃	C₂⁴	semidirect
96. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
97. #11719921	C₂⁴	semidirect
98. #11719919	C₂⁴	semidirect
103. C₂³×C₅₃	#121	semidirect
116. C₂²×D₅₃	C₂²×D₄	semidirect
126. C₂×Dic₅₃	C₂²×D₄	semidirect
136. C₂²×C₅₃	C₂×#121	semidirect
148. C₂×D₅₃	C₂³×D₄	semidirect
152. Dic₅₃	C₂³×D₄	semidirect
160. C₂³×C₂²⋊C₄	D₅₃	semidirect
162. C₂×C₅₃	C₂²×#121	semidirect
186. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
187. C₅₃	C₂³×#121	semidirect
220. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
258. C₂²⋊C₄	C₂³×D₅₃	semidirect
273. C₂³	#13133190	direct (N × H)
303. C₂²	C₂×#13133190	direct (N × H)
315. C₂	C₂²×#13133190	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂⁴×C₂²⋊D₅₃	2
C₂⁴×C₄×D₅₃	2
C₂²×#13133190	2
C₂³×#11719919	2
C₂³×#11719921	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂³×#121	53

Subgroups — C23×#13133190 #28290728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133190 #28290728