Subgroups — C₂×#20274104 #27366349

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810828	C₂	1	2	yes	maximal
3	#20274104	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810368	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719923	—	2	2	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	complemented
11	#16810828	—	2	4	no	—
12	#16810828	C₂²	1	8	yes	—
13	C₂×#13133151	—	2	1	no	—
14	C₂×#13133151	—	2	1	no	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×#13133140	—	2	2	no	—
17	#16810376	C₂²	1	4	yes	complemented
18	#16810368	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₅₃	—	2	2	no	—
22	C₈×D₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂×#11719923	—	2	8	no	—
28	C₂×#11719923	—	4	8	no	—
29	C₂²×#88322	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	#13133151	—	2	4	no	—
37	#13133151	—	2	4	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	C₂³	1	4	yes	—
40	#13133140	—	2	8	no	—
41	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	—	4	4	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×D₄×C₅₃	—	2	1	no	—
49	C₄×D₄×C₅₃	—	2	1	no	—
50	C₄×D₄×C₅₃	D₄	1	2	yes	—
51	C₄×D₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	1	no	—
56	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
57	C₂²⋊C₈×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂⁴×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	D₈	1	4	yes	—
64	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	4	2	no	—
74	C₂×Dic₁₀₆	—	2	4	no	—
75	C₂×Dic₁₀₆	—	4	4	no	—
76	C₂×Dic₁₀₆	—	4	4	no	—
77	C₂×Dic₁₀₆	—	4	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	8	no	—
82	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂²⋊C₄×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719923	—	4	16	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	4	4	no	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	—	2	4	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	#121	1	1	yes	characteristic
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	C₂×D₈	1	4	yes	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	#135	1	2	yes	—
120	C₂×C₄×C₅₃	#137	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₅₃	—	4	4	no	—
125	C₂×Dic₅₃	—	8	4	no	—
126	Dic₁₀₆	—	4	4	no	—
127	Dic₁₀₆	—	4	4	no	—
128	Dic₁₀₆	—	8	4	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	C₂×SD₁₆	1	4	yes	complemented
132	#88322	—	4	4	no	—
133	C₂×#1673	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	4	8	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	#461	1	2	yes	—
139	C₂²×C₅₃	#466	1	2	yes	—
140	C₂²×C₅₃	#473	1	2	yes	—
141	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	#461	1	2	yes	—
146	C₄×C₅₃	#466	1	2	yes	—
147	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
150	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
151	Dic₅₃	—	8	4	no	—
152	C₂×#500	—	53	1	no	—
153	C₂×#461	—	53	2	no	—
154	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
155	#1673	—	53	8	no	—
156	C₂×#335	—	53	1	no	—
157	C₂×#332	—	53	2	no	—
158	C₂×C₅₃	—	4	4	no	—
159	C₂×C₅₃	#1673	1	4	yes	complemented
160	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
163	C₂²×SD₁₆	—	106	2	no	—
164	C₂³×C₈	—	2	1	no	—
165	C₂×C₄⋊Q₈	—	53	2	no	—
166	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
167	#500	—	53	4	no	—
168	#500	—	106	2	no	—
169	#461	—	53	8	no	—
170	#461	—	106	4	no	—
171	C₈×D₄	—	2	2	no	—
172	C₈×D₄	C₂×D₅₃	1	4	yes	—
173	C₂×#107	—	106	1	no	—
174	C₂×#107	—	106	1	no	—
175	C₂×C₄⋊C₈	—	53	2	no	—
176	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
177	C₂×#103	—	106	2	no	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
180	#335	—	53	4	no	—
181	#332	—	53	8	no	—
182	C₅₃	C₂×#1673	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	106	2	no	—
184	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂×SD₁₆	—	106	8	no	—
187	C₂×SD₁₆	—	212	8	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	4	4	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	53	4	no	—
195	C₄⋊Q₈	—	106	2	no	—
196	C₄⋊Q₈	—	106	2	no	—
197	C₄⋊Q₈	—	106	2	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₂₁₂	1	2	yes	—
201	C₄×D₄	C₂²×D₅₃	1	2	yes	—
202	C₂×C₄⋊C₄	—	106	2	no	—
203	C₂×C₄⋊C₄	—	106	2	no	—
204	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	#107	—	106	4	no	—
208	#107	—	106	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	53	4	no	—
211	C₄⋊C₈	—	106	2	no	—
212	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
213	#103	—	106	8	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
216	C₄×C₈	C₂²×D₅₃	1	2	yes	—
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	106	4	no	—
219	C₂×Q₈	—	212	4	no	—
220	C₂×Q₈	—	212	4	no	—
221	C₂×Q₈	—	212	2	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	D₄₂₄	1	4	yes	—
225	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	212	2	no	—
232	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂²×C₄	#11719945	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	SD₁₆	—	212	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	—	212	4	no	—
242	C₂×C₈	#11719923	1	4	yes	complemented
243	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	106	4	no	—
247	C₄⋊C₄	—	106	4	no	—
248	C₄⋊C₄	—	212	2	no	—
249	C₄⋊C₄	—	212	2	no	—
250	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
254	C₄²	#11719945	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₄²	C₂³×D₅₃	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#13133191	1	1	yes	characteristic
262	Q₈	—	424	4	no	—
263	Q₈	—	212	4	no	—
264	Q₈	—	212	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂×D₄₂₄	1	4	yes	—
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	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	424	4	no	—
275	C₂×C₄	—	212	4	no	—
276	C₂×C₄	#13133191	1	1	yes	characteristic
277	C₂×C₄	#13133191	1	1	yes	characteristic
278	C₂×C₄	#13133191	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
280	C₂×C₄	C₂×#11719923	1	2	yes	—
281	C₂×C₄	#13133239	1	2	yes	—
282	C₂×C₄	#13133247	1	2	yes	—
283	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
285	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	212	4	no	—
289	C₈	C₂×#11719923	1	4	yes	complemented
290	C₂²	—	2	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	8	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#16810828	1	2	yes	—
295	C₂²	#16810859	1	2	yes	—
296	C₂²	#16810898	1	2	yes	—
297	C₂²	C₂×#13133191	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	424	4	no	—
302	C₄	#16810828	1	2	yes	—
303	C₄	#16810859	1	2	yes	—
304	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
305	C₄	C₂²×#11719923	1	1	yes	characteristic
306	C₄	C₂×#13133239	1	1	yes	characteristic
307	C₄	C₂×#13133247	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	#20274104	1	4	yes	direct product factor
310	C₂	C₂×#16810828	1	1	yes	characteristic
311	C₂	C₂×#16810859	1	1	yes	characteristic
312	C₂	C₂×#16810898	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	C₂×#1673
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16811028	C₂	semidirect
3. #20274104	C₂	direct (N × H)
4. C₂×#16810376	C₂	semidirect
5. C₂×#16810368	C₂	semidirect
10. #16811028	C₂²	semidirect
17. #16810376	C₂²	semidirect
18. #16810368	C₂²	semidirect
82. C₂×C₈×C₅₃	SD₁₆	semidirect
131. C₈×C₅₃	C₂×SD₁₆	semidirect
159. C₂×C₅₃	#1673	semidirect
182. C₅₃	C₂×#1673	semidirect
242. C₂×C₈	#11719923	semidirect
289. C₈	C₂×#11719923	semidirect
309. C₂	#20274104	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×#16811028	2
C₂×#16810828	2
#20274104	2
C₂×#16810376	2
C₂×#16810368	2
C₂×C₈×D₄×C₅₃	2
C₂×#1673	53

Subgroups — C2×#20274104 #27366349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274104 #27366349