Subgroups — C₂×#20277329 #27542701

Order 13568.

5182 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	C₂³×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
2	#20277329	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16810498	C₂	1	8	yes	maximal, complemented
4	C₂×#16810483	C₂	1	4	yes	maximal, complemented
5	C₂³×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₃	C₂²	1	4	yes	—
7	C₂²×C₄⋊Dic₅₃	C₂²	1	8	yes	complemented
8	C₂²×C₂²⋊Dic₅₃	—	2	16	no	—
9	C₂²×C₂²⋊Dic₅₃	C₂²	1	16	yes	—
10	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
12	#16810498	C₂²	1	64	yes	complemented
13	#16810483	C₂²	1	32	yes	complemented
14	C₂×#13133124	C₂²	1	8	yes	complemented
15	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₅₃	C₄	1	16	yes	complemented
18	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
20	C₂³×Dic₅₃	—	2	16	no	—
21	C₂³×Dic₅₃	—	4	16	no	—
22	C₂³×Dic₅₃	C₂³	1	8	yes	—
23	C₂³×Dic₅₃	C₂³	1	16	yes	—
24	C₂×C₄⋊Dic₅₃	—	4	16	no	—
25	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	complemented
26	C₂×C₂²⋊Dic₅₃	—	2	16	no	—
27	C₂×C₂²⋊Dic₅₃	—	2	16	no	—
28	C₂×C₂²⋊Dic₅₃	—	2	64	no	—
29	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
30	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
31	#13133124	C₂³	1	32	yes	complemented
32	C₂⁵×C₅₃	—	4	8	no	—
33	C₂⁵×C₅₃	C₂×C₄	1	8	yes	—
34	C₂⁵×C₅₃	D₄	1	8	yes	—
35	C₂⁵×C₅₃	Q₈	1	8	yes	—
36	C₂⁵×C₅₃	C₂³	1	2	yes	—
37	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
38	C₂⁵×C₅₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₅₃	—	4	8	no	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
41	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	32	yes	complemented
44	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	64	yes	complemented
45	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
46	C₂⁴×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	—	2	16	no	—
48	C₂⁴×C₅₃	—	4	32	no	—
49	C₂⁴×C₅₃	—	4	32	no	—
50	C₂⁴×C₅₃	—	4	32	no	—
51	C₂⁴×C₅₃	—	4	16	no	—
52	C₂⁴×C₅₃	C₄⋊C₄	1	64	yes	complemented
53	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	—
56	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
57	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
58	C₂⁴×C₅₃	C₂×D₄	1	16	yes	—
59	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
60	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
61	C₂⁴×C₅₃	C₂×Q₈	1	16	yes	—
62	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
63	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
64	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	16	no	—
66	C₂²×C₄×C₅₃	—	4	32	no	—
67	C₂²×C₄×C₅₃	—	4	16	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	complemented
70	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
71	C₂²×Dic₅₃	—	2	32	no	—
72	C₂²×Dic₅₃	—	2	16	no	—
73	C₂²×Dic₅₃	—	2	32	no	—
74	C₂²×Dic₅₃	—	4	32	no	—
75	C₂²×Dic₅₃	—	4	64	no	—
76	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
77	C₂²⋊C₄×C₅₃	C₂²×C₄	1	64	yes	complemented
78	C₂²⋊Dic₅₃	—	2	32	no	—
79	C₂²⋊Dic₅₃	—	2	32	no	—
80	C₄⋊Dic₅₃	—	4	32	no	—
81	C₂³×C₅₃	—	2	16	no	—
82	C₂³×C₅₃	—	2	32	no	—
83	C₂³×C₅₃	—	2	32	no	—
84	C₂³×C₅₃	—	2	16	no	—
85	C₂³×C₅₃	—	2	16	no	—
86	C₂³×C₅₃	—	2	32	no	—
87	C₂³×C₅₃	—	2	16	no	—
88	C₂³×C₅₃	—	2	8	no	—
89	C₂³×C₅₃	—	4	128	no	—
90	C₂³×C₅₃	—	4	32	no	—
91	C₂³×C₅₃	—	4	32	no	—
92	C₂³×C₅₃	—	4	32	no	—
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	32	yes	—
94	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	64	yes	complemented
96	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
97	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
98	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
99	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
100	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
101	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
102	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
103	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
104	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
105	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
106	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	32	no	—
108	C₂×C₄×C₅₃	—	2	16	no	—
109	C₂×C₄×C₅₃	—	4	32	no	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
111	C₂×Dic₅₃	—	2	16	no	—
112	C₂×Dic₅₃	—	2	32	no	—
113	C₂×Dic₅₃	—	4	64	no	—
114	C₂×Dic₅₃	—	4	32	no	—
115	C₂×#2315	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	32	no	—
117	C₂²×C₅₃	—	2	64	no	—
118	C₂²×C₅₃	—	2	8	no	—
119	C₂²×C₅₃	—	2	16	no	—
120	C₂²×C₅₃	—	2	16	no	—
121	C₂²×C₅₃	—	2	8	no	—
122	C₂²×C₅₃	—	2	16	no	—
123	C₂²×C₅₃	—	2	8	no	—
124	C₂²×C₅₃	—	4	64	no	—
125	C₂²×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
126	C₂²×C₅₃	C₂²×C₄⋊C₄	1	16	yes	—
127	C₂²×C₅₃	C₂²×C₄⋊C₄	1	4	yes	—
128	C₂²×C₅₃	#518	1	8	yes	—
129	C₂²×C₅₃	#534	1	8	yes	—
130	C₂²×C₅₃	#543	1	8	yes	—
131	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
133	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
135	C₄×C₅₃	—	2	16	no	—
136	Dic₅₃	—	4	16	no	—
137	C₂³×C₂²⋊C₄	—	53	2	no	—
138	C₂³×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
139	#2315	—	53	16	no	—
140	C₂×#385	—	53	8	no	—
141	C₂×#380	—	53	4	no	—
142	C₂×C₅₃	—	2	8	no	—
143	C₂×C₅₃	—	2	16	no	—
144	C₂×C₅₃	#2315	1	8	yes	complemented
145	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#518	1	2	yes	—
147	C₂×C₅₃	C₂×#534	1	2	yes	—
148	C₂×C₅₃	C₂×#543	1	2	yes	—
149	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
150	C₂⁴×C₄	—	53	4	no	—
151	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
152	C₂²×C₄⋊C₄	—	53	8	no	—
153	C₂²×C₂²⋊C₄	—	53	16	no	—
154	C₂²×C₂²⋊C₄	—	53	4	no	—
155	C₂²×C₂²⋊C₄	—	53	4	no	—
156	C₂²×C₂²⋊C₄	—	106	16	no	—
157	C₂²×C₂²⋊C₄	Dic₅₃	1	16	yes	complemented
158	C₂²×C₂²⋊C₄	C₂×D₅₃	1	8	yes	—
159	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
160	#385	—	53	64	no	—
161	#380	—	53	32	no	—
162	C₂×#95	—	53	8	no	—
163	C₅₃	C₂×#2315	1	1	yes	characteristic, complemented, Sylow
164	C₂⁵	—	4	8	no	—
165	C₂⁵	Dic₁₀₆	1	8	yes	—
166	C₂⁵	D₂₁₂	1	8	yes	—
167	C₂⁵	C₂×Dic₅₃	1	8	yes	—
168	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
169	C₂⁵	C₂²×D₅₃	1	2	yes	—
170	C₂⁵	C₂²×D₅₃	1	4	yes	—
171	C₂³×C₄	—	4	8	no	—
172	C₂³×C₄	—	53	8	no	—
173	C₂³×C₄	—	53	16	no	—
174	C₂³×C₄	—	106	16	no	—
175	C₂³×C₄	—	212	16	no	—
176	C₂³×C₄	C₂×Dic₅₃	1	16	yes	complemented
177	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
178	C₂³×C₄	C₂²×D₅₃	1	8	yes	—
179	C₂×C₄⋊C₄	—	53	32	no	—
180	C₂×C₄⋊C₄	—	212	16	no	—
181	C₂×C₂²⋊C₄	—	53	16	no	—
182	C₂×C₂²⋊C₄	—	53	16	no	—
183	C₂×C₂²⋊C₄	—	106	16	no	—
184	C₂×C₂²⋊C₄	—	106	16	no	—
185	C₂×C₂²⋊C₄	—	106	64	no	—
186	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	32	yes	complemented
187	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	64	yes	complemented
188	C₂×C₂²⋊C₄	C₂²×D₅₃	1	16	yes	—
189	#95	—	53	32	no	—
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	—	4	32	no	—
193	C₂⁴	—	4	32	no	—
194	C₂⁴	—	4	32	no	—
195	C₂⁴	—	4	16	no	—
196	C₂⁴	C₄⋊Dic₅₃	1	64	yes	complemented
197	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
198	C₂⁴	C₂×Dic₁₀₆	1	8	yes	—
199	C₂⁴	C₂×Dic₁₀₆	1	16	yes	—
200	C₂⁴	C₂×D₂₁₂	1	4	yes	—
201	C₂⁴	C₂×D₂₁₂	1	8	yes	—
202	C₂⁴	C₂×D₂₁₂	1	16	yes	—
203	C₂⁴	C₂²×Dic₅₃	1	8	yes	—
204	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
205	C₂⁴	C₂²×Dic₅₃	1	16	yes	—
206	C₂⁴	C₂³×D₅₃	1	2	yes	—
207	C₂⁴	C₂³×D₅₃	1	4	yes	—
208	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	4	32	no	—
211	C₂²×C₄	—	4	16	no	—
212	C₂²×C₄	—	53	16	no	—
213	C₂²×C₄	—	106	32	no	—
214	C₂²×C₄	—	106	16	no	—
215	C₂²×C₄	—	106	32	no	—
216	C₂²×C₄	—	212	32	no	—
217	C₂²×C₄	—	212	64	no	—
218	C₂²×C₄	C₂²×Dic₅₃	1	16	yes	—
219	C₂²×C₄	C₂²×Dic₅₃	1	32	yes	complemented
220	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
221	C₄⋊C₄	—	212	32	no	—
222	C₂²⋊C₄	—	106	32	no	—
223	C₂²⋊C₄	—	106	32	no	—
224	C₂²⋊C₄	C₂²×Dic₅₃	1	64	yes	complemented
225	C₂³	—	2	16	no	—
226	C₂³	—	2	32	no	—
227	C₂³	—	2	32	no	—
228	C₂³	—	2	16	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	2	16	no	—
231	C₂³	—	2	16	no	—
232	C₂³	—	2	32	no	—
233	C₂³	—	4	32	no	—
234	C₂³	—	4	32	no	—
235	C₂³	—	4	32	no	—
236	C₂³	—	4	128	no	—
237	C₂³	C₂×C₄⋊Dic₅₃	1	16	yes	—
238	C₂³	C₂×C₄⋊Dic₅₃	1	32	yes	—
239	C₂³	C₂×C₄⋊Dic₅₃	1	64	yes	complemented
240	C₂³	C₂³×Dic₅₃	1	8	yes	—
241	C₂³	C₂³×Dic₅₃	1	4	yes	—
242	C₂³	C₂³×Dic₅₃	1	2	yes	—
243	C₂³	C₂²×D₂₁₂	1	4	yes	—
244	C₂³	C₂²×D₂₁₂	1	8	yes	—
245	C₂³	C₂²×D₂₁₂	1	2	yes	—
246	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
247	C₂³	C₂²×Dic₁₀₆	1	8	yes	—
248	C₂³	C₂²×Dic₁₀₆	1	2	yes	—
249	C₂³	#13133262	1	8	yes	—
250	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
251	C₂×C₄	—	2	32	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	4	32	no	—
254	C₂×C₄	—	106	16	no	—
255	C₂×C₄	—	106	32	no	—
256	C₂×C₄	—	212	64	no	—
257	C₂×C₄	—	212	32	no	—
258	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
259	C₂²	—	2	16	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	2	32	no	—
262	C₂²	—	2	64	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	16	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	16	no	—
267	C₂²	—	4	64	no	—
268	C₂²	C₂²×C₄⋊Dic₅₃	1	16	yes	—
269	C₂²	C₂²×C₄⋊Dic₅₃	1	8	yes	—
270	C₂²	C₂²×C₄⋊Dic₅₃	1	4	yes	—
271	C₂²	#16811098	1	8	yes	—
272	C₂²	#16811185	1	8	yes	—
273	C₂²	#16811249	1	8	yes	—
274	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
275	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
276	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
277	C₂²	C₂×#13133262	1	4	yes	—
278	C₄	—	2	16	no	—
279	C₄	—	212	16	no	—
280	C₂	—	2	16	no	—
281	C₂	—	2	8	no	—
282	C₂	#20277329	1	8	yes	direct product factor
283	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
284	C₂	C₂×#16811098	1	2	yes	—
285	C₂	C₂×#16811185	1	2	yes	—
286	C₂	C₂×#16811249	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₂²⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁶×C₅₃	C₂²
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂×#2315
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20277329	C₂	direct (N × H)
3. C₂×#16810498	C₂	semidirect
4. C₂×#16810483	C₂	semidirect
7. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
12. #16810498	C₂²	semidirect
13. #16810483	C₂²	semidirect
14. C₂×#13133124	C₂²	semidirect
17. C₂²×C₂²⋊C₄×C₅₃	C₄	semidirect
25. C₂×C₄⋊Dic₅₃	C₂³	semidirect
31. #13133124	C₂³	semidirect
40. C₂³×C₄×C₅₃	C₂×C₄	semidirect
43. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
44. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
52. C₂⁴×C₅₃	C₄⋊C₄	semidirect
69. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
77. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
95. C₂³×C₅₃	C₂×C₄⋊C₄	semidirect
144. C₂×C₅₃	#2315	semidirect
157. C₂²×C₂²⋊C₄	Dic₅₃	semidirect
163. C₅₃	C₂×#2315	semidirect
176. C₂³×C₄	C₂×Dic₅₃	semidirect
186. C₂×C₂²⋊C₄	C₂×Dic₅₃	semidirect
187. C₂×C₂²⋊C₄	C₂×Dic₅₃	semidirect
196. C₂⁴	C₄⋊Dic₅₃	semidirect
219. C₂²×C₄	C₂²×Dic₅₃	semidirect
224. C₂²⋊C₄	C₂²×Dic₅₃	semidirect
239. C₂³	C₂×C₄⋊Dic₅₃	semidirect
282. C₂	#20277329	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×C₂²⋊Dic₅₃	2
#20277329	2
C₂×#16810498	2
C₂×#16810483	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂×#2315	53

Subgroups — C2×#20277329 #27542701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20277329 #27542701