Subgroups — C₂×#20289545 #28300166

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88324	C₂	1	2	yes	maximal
2	#20289545	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88324	—	2	2	no	—
5	C₂×C₄⋊#88324	—	2	4	no	—
6	C₂×C₄⋊#88324	—	2	2	no	—
7	C₂×C₄⋊#88324	C₄	1	8	yes	—
8	C₂×C₄⋊#88324	C₂²	1	4	yes	—
9	C₂³×#88322	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	—	2	4	no	—
12	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	8	no	—
16	C₂²×#88324	—	2	4	no	—
17	C₂²×#88324	—	4	4	no	—
18	C₂²×#88324	—	4	4	no	—
19	C₄⋊#88324	—	2	8	no	—
20	C₄⋊#88324	—	2	4	no	—
21	C₄⋊#88324	C₂×C₄	1	8	yes	—
22	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂³×Dic₅₃	Q₈	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	C₈	1	4	yes	complemented
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
31	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
34	C₈×Dic₅₃	—	2	4	no	—
35	C₈×Dic₅₃	—	2	4	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₈	1	4	yes	complemented
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	2	2	no	—
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂²×Dic₅₃	C₂²⋊C₄	1	4	yes	—
53	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₄⋊C₄	1	4	yes	—
55	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	D₈	1	2	yes	—
57	C₂²×Dic₅₃	SD₁₆	1	4	yes	—
58	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
59	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×#88324	—	2	8	no	—
62	C₂×#88324	—	4	16	no	—
63	C₂×#88324	—	4	4	no	—
64	C₂×#88324	—	4	4	no	—
65	C₂×C₈×C₅₃	—	2	8	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
69	C₄×Dic₅₃	C₄²	1	2	yes	—
70	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₅₃	C₂²⋊C₄	1	4	yes	—
72	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
75	C₂×#88322	—	2	8	no	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	C₂×C₈	1	4	yes	complemented
79	C₂³×C₅₃	#95	1	2	yes	—
80	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
81	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
82	C₂³×C₅₃	#106	1	2	yes	—
83	C₂³×C₅₃	#107	1	2	yes	—
84	C₂³×C₅₃	#108	1	2	yes	—
85	C₂³×C₅₃	#108	1	2	yes	—
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#95	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
89	C₂×C₄×C₅₃	#97	1	4	yes	—
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	8	no	—
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	#95	1	2	yes	—
106	C₂×Dic₅₃	#95	1	2	yes	—
107	C₂×Dic₅₃	#102	1	8	yes	—
108	C₂×Dic₅₃	#103	1	8	yes	—
109	C₂×Dic₅₃	#106	1	2	yes	—
110	C₂×Dic₅₃	#107	1	2	yes	—
111	C₂×Dic₅₃	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
114	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
115	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
116	#88324	—	4	8	no	—
117	C₈×C₅₃	—	2	4	no	—
118	#88322	—	2	4	no	—
119	C₂×#1277	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	#334	1	4	yes	—
121	C₂²×C₅₃	#334	1	4	yes	—
122	C₂²×C₅₃	#335	1	4	yes	—
123	C₂²×C₅₃	#335	1	4	yes	—
124	C₂²×C₅₃	#336	1	4	yes	—
125	C₂²×C₅₃	#340	1	4	yes	—
126	C₂²×C₅₃	#341	1	4	yes	—
127	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
134	C₄×C₅₃	#336	1	4	yes	—
135	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	Dic₅₃	—	2	2	no	—
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	#340	1	4	yes	—
142	Dic₅₃	C₂×#102	1	2	yes	—
143	Dic₅₃	C₂×#103	1	2	yes	—
144	C₂²×C₄⋊C₈	—	53	2	no	—
145	C₂²×C₄×C₈	—	53	1	no	—
146	#1277	—	53	4	no	—
147	C₂×C₅₃	#1277	1	8	yes	complemented
148	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
155	C₂³×C₈	—	53	1	no	—
156	C₂³×C₈	—	106	2	no	—
157	C₂³×C₈	#6157	1	1	yes	characteristic
158	C₂²×C₄²	—	53	1	no	—
159	C₂×C₄⋊C₈	—	53	8	no	—
160	C₂×C₄⋊C₈	—	53	4	no	—
161	C₂×C₄⋊C₈	—	106	2	no	—
162	C₂×C₄⋊C₈	—	106	4	no	—
163	C₂×C₄×C₈	—	53	2	no	—
164	C₂×C₄×C₈	—	53	2	no	—
165	C₂×C₄×C₈	—	106	4	no	—
166	C₅₃	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	53	1	no	—
168	C₂³×C₄	—	53	1	no	—
169	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	53	2	no	—
173	C₂²×C₈	—	53	4	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₂²×C₈	—	106	8	no	—
177	C₂²×C₈	—	106	4	no	—
178	C₂²×C₈	—	212	4	no	—
179	C₂²×C₈	—	212	4	no	—
180	C₂²×C₈	#88324	1	4	yes	complemented
181	C₂²×C₈	C₂×#6157	1	2	yes	—
182	C₂×C₄²	—	53	4	no	—
183	C₂×C₄²	—	53	1	no	—
184	C₂×C₄²	—	53	1	no	—
185	C₂×C₄²	—	53	4	no	—
186	C₂×C₄²	—	53	1	no	—
187	C₂×C₄²	—	53	1	no	—
188	C₄⋊C₈	—	53	8	no	—
189	C₄⋊C₈	—	106	8	no	—
190	C₄⋊C₈	—	106	4	no	—
191	C₄×C₈	—	106	4	no	—
192	C₄×C₈	—	106	4	no	—
193	C₂⁴	C₄⋊#6157	1	1	yes	characteristic
194	C₂²×C₄	—	53	4	no	—
195	C₂²×C₄	—	53	4	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	53	4	no	—
201	C₂²×C₄	—	53	1	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	106	2	no	—
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
206	C₂²×C₄	#11719936	1	2	yes	—
207	C₂²×C₄	C₄×#6157	1	2	yes	—
208	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
209	C₂²×C₄	C₂×#88324	1	2	yes	—
210	C₂²×C₄	#11719940	1	2	yes	—
211	C₂²×C₄	C₂²⋊#6157	1	2	yes	—
212	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
213	C₂×C₈	—	2	8	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	53	4	no	—
217	C₂×C₈	—	106	8	no	—
218	C₂×C₈	—	106	2	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	—	106	8	no	—
221	C₂×C₈	—	212	4	no	—
222	C₂×C₈	—	212	4	no	—
223	C₂×C₈	—	212	16	no	—
224	C₂×C₈	C₂×#88324	1	4	yes	complemented
225	C₄²	—	53	4	no	—
226	C₄²	—	53	4	no	—
227	C₄²	—	53	2	no	—
228	C₄²	—	53	2	no	—
229	C₄²	—	53	2	no	—
230	C₄²	—	53	2	no	—
231	C₂³	(C₂×#6157)⋊C₄	1	2	yes	—
232	C₂³	C₄⋊#88324	1	2	yes	—
233	C₂³	#13133284	1	2	yes	—
234	C₂³	#13133285	1	2	yes	—
235	C₂³	#13133286	1	2	yes	—
236	C₂³	#13133287	1	2	yes	—
237	C₂³	#13133288	1	2	yes	—
238	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	53	2	no	—
244	C₂×C₄	—	53	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	53	8	no	—
247	C₂×C₄	—	53	8	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	106	1	no	—
251	C₂×C₄	—	106	1	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	8	no	—
255	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
256	C₂×C₄	C₄×#88324	1	4	yes	—
257	C₂×C₄	#13133272	1	4	yes	—
258	C₂×C₄	C₂²⋊#88324	1	4	yes	—
259	C₂×C₄	#13133274	1	4	yes	—
260	C₂×C₄	C₄⋊#88324	1	2	yes	—
261	C₂×C₄	#13133284	1	2	yes	—
262	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
264	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
265	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	106	4	no	—
270	C₈	—	212	8	no	—
271	C₂²	#16811484	1	4	yes	—
272	C₂²	#16811485	1	4	yes	—
273	C₂²	#16811486	1	4	yes	—
274	C₂²	#16811487	1	4	yes	—
275	C₂²	#16811489	1	4	yes	—
276	C₂²	#16811497	1	4	yes	—
277	C₂²	#16811499	1	4	yes	—
278	C₂²	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
279	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
280	C₂²	C₂×#13133284	1	1	yes	characteristic
281	C₂²	C₂×#13133285	1	1	yes	characteristic
282	C₂²	C₂×#13133286	1	1	yes	characteristic
283	C₂²	C₂×#13133287	1	1	yes	characteristic
284	C₂²	C₂×#13133288	1	1	yes	characteristic
285	C₄	—	53	2	no	—
286	C₄	—	53	4	no	—
287	C₄	—	53	2	no	—
288	C₄	—	106	2	no	—
289	C₄	—	106	2	no	—
290	C₄	#16811489	1	4	yes	—
291	C₄	C₂×C₄×#88324	1	1	yes	characteristic
292	C₄	C₂×#13133272	1	1	yes	characteristic
293	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
294	C₄	C₂×#13133274	1	1	yes	characteristic
295	C₂	#20289545	1	8	yes	direct product factor
296	C₂	C₂×#16811484	1	1	yes	characteristic
297	C₂	C₂×#16811485	1	1	yes	characteristic
298	C₂	C₂×#16811486	1	1	yes	characteristic
299	C₂	C₂×#16811487	1	1	yes	characteristic
300	C₂	C₂×#16811489	1	1	yes	characteristic
301	C₂	C₂×#16811497	1	1	yes	characteristic
302	C₂	C₂×#16811499	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂×#1277
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #20289545	C₂	direct (N × H)
26. C₂²×#88322	C₈	semidirect
39. C₂²×C₈×C₅₃	C₈	semidirect
68. C₂×C₈×C₅₃	C₂×C₈	semidirect
78. C₂×#88322	C₂×C₈	semidirect
147. C₂×C₅₃	#1277	semidirect
166. C₅₃	C₂×#1277	semidirect
180. C₂²×C₈	#88324	semidirect
224. C₂×C₈	C₂×#88324	semidirect
295. C₂	#20289545	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×C₄⋊#88324	2
#20289545	2
C₂²×C₈×Dic₅₃	2
C₂×#1277	53

Subgroups — C2×#20289545 #28300166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289545 #28300166