Subgroups — C₂×#20276869 #27538704

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#20276869	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810448	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810423	C₂	1	1	yes	maximal, characteristic
7	C₂×#16810423	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁₂	—	2	1	no	—
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₄×D₄₂₄	—	2	2	no	—
14	C₄×D₄₂₄	C₂²	1	4	yes	—
15	C₂×#13133151	C₂²	1	1	yes	characteristic
16	C₂×#13133139	—	2	1	no	—
17	#16810450	C₂²	1	4	yes	complemented
18	#16810448	C₂²	1	4	yes	complemented
19	C₄⋊#11719908	C₂²	1	4	yes	complemented
20	#16810423	C₂²	1	4	yes	—
21	#16810423	C₂²	1	4	yes	—
22	C₂×#125×C₅₃	—	2	1	no	—
23	C₅₃×#499	—	2	2	no	—
24	C₅₃×#499	C₂²	1	4	yes	complemented
25	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₃	—	2	1	no	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×D₂₁₂	—	2	1	no	—
30	C₂²×C₄×D₅₃	—	4	1	no	—
31	C₂×D₄₂₄	—	2	1	no	—
32	C₂×D₄₂₄	—	2	2	no	—
33	C₂×D₄₂₄	Q₈	1	4	yes	—
34	C₂×D₄₂₄	C₂³	1	2	yes	—
35	C₄×D₂₁₂	—	2	4	no	—
36	C₄×D₂₁₂	—	4	2	no	—
37	C₂×C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×#11719921	—	4	1	no	—
39	C₂×#11719908	—	2	2	no	—
40	C₂×#11719908	—	2	2	no	—
41	C₂×#11719908	C₂³	1	2	yes	—
42	C₂×#11719908	C₂³	1	2	yes	—
43	#13133151	—	2	1	no	—
44	#13133151	C₂³	1	2	yes	—
45	#13133139	—	2	4	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#125×C₅₃	—	2	4	no	—
49	#125×C₅₃	—	4	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	#107×C₅₃	—	2	1	no	—
56	#107×C₅₃	—	2	1	no	—
57	#107×C₅₃	C₂³	1	2	yes	—
58	#107×C₅₃	C₂³	1	2	yes	—
59	#106×C₅₃	—	2	4	no	—
60	C₄×C₈×C₅₃	D₄	1	2	yes	—
61	C₄×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂³×D₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	4	1	no	—
69	C₂×D₂₁₂	—	2	4	no	—
70	C₂×D₂₁₂	—	2	2	no	—
71	C₂×D₂₁₂	—	4	1	no	—
72	C₂×D₂₁₂	—	4	2	no	—
73	C₂×C₄×D₅₃	—	4	4	no	—
74	C₂×C₄×D₅₃	—	8	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₃	#41	1	2	yes	—
80	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	4	1	no	—
84	C₄⋊C₄×C₅₃	—	4	1	no	—
85	C₄⋊C₄×C₅₃	—	4	4	no	—
86	C₄⋊C₄×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	D₈	1	2	yes	—
88	C₄²×C₅₃	C₂×D₄	1	2	yes	—
89	#11719921	—	4	4	no	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	4	1	no	—
92	D₄₂₄	—	2	4	no	—
93	D₄₂₄	—	2	2	no	—
94	D₄₂₄	C₂×Q₈	1	4	yes	—
95	#11719908	—	2	2	no	—
96	#11719908	—	2	2	no	—
97	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₅₃	—	4	2	no	—
99	C₂²×D₅₃	—	4	4	no	—
100	C₂²×D₅₃	—	8	2	no	—
101	C₂²×D₅₃	—	8	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	8	2	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#137	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₅₃	—	4	2	no	—
115	C₂×Dic₅₃	—	8	2	no	—
116	D₂₁₂	—	2	4	no	—
117	D₂₁₂	—	4	2	no	—
118	D₂₁₂	—	4	4	no	—
119	C₄×D₅₃	—	8	8	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×#2228	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	#474	1	2	yes	—
126	C₂²×C₅₃	#508	1	2	yes	—
127	C₂²×C₅₃	#509	1	2	yes	—
128	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₅₃	—	4	4	no	—
130	C₂×D₅₃	—	8	2	no	—
131	C₂×D₅₃	—	8	2	no	—
132	C₂×D₅₃	—	8	8	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	8	2	no	—
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	#474	1	2	yes	—
138	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
140	Dic₅₃	—	8	2	no	—
141	C₂×#499	D₅₃	1	1	yes	characteristic, complemented
142	C₂×C₄×D₈	—	53	1	no	—
143	#2228	—	53	8	no	—
144	C₂×#367	—	53	1	no	—
145	C₂×#366	—	53	1	no	—
146	C₂×C₄⋊C₁₆	—	53	1	no	—
147	C₂×#357	—	53	1	no	—
148	C₂×#357	—	53	1	no	—
149	C₂×C₅₃	#2228	1	4	yes	complemented
150	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
153	D₅₃	—	8	4	no	—
154	C₂²×D₈	—	53	1	no	—
155	C₂×#125	—	2	1	no	—
156	C₂×C₄×D₄	—	106	1	no	—
157	C₂²×C₁₆	—	53	1	no	—
158	C₂²×C₁₆	—	53	1	no	—
159	#499	—	2	2	no	—
160	#499	C₂×D₅₃	1	4	yes	complemented
161	C₄×D₈	—	53	4	no	—
162	C₄×D₈	—	106	2	no	—
163	C₂×#107	—	53	1	no	—
164	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
165	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
166	C₂×#106	—	2	1	no	—
167	C₂×#102	—	106	1	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	#367	—	53	4	no	—
170	#366	—	53	4	no	—
171	C₄⋊C₁₆	—	53	4	no	—
172	#357	—	53	4	no	—
173	#357	—	53	4	no	—
174	C₅₃	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₄	—	106	1	no	—
176	C₂³×C₄	—	212	1	no	—
177	C₂×D₈	—	53	4	no	—
178	C₂×D₈	—	53	2	no	—
179	C₂×D₈	—	106	2	no	—
180	C₂×D₈	—	106	1	no	—
181	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	106	4	no	—
186	C₄×D₄	—	212	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	106	1	no	—
192	C₂×C₂²⋊C₄	—	212	1	no	—
193	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₁₆	—	53	2	no	—
195	C₂×C₁₆	—	53	2	no	—
196	C₂×C₁₆	—	106	2	no	—
197	C₂×C₁₆	—	106	2	no	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	53	2	no	—
201	#107	—	106	1	no	—
202	#107	C₂²×D₅₃	1	2	yes	—
203	#107	C₂²×D₅₃	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	106	4	no	—
206	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
207	C₄×C₈	C₂²×D₅₃	1	2	yes	—
208	C₂⁴	—	212	1	no	—
209	C₂×D₄	—	106	4	no	—
210	C₂×D₄	—	106	2	no	—
211	C₂×D₄	—	212	1	no	—
212	C₂×D₄	—	212	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	424	4	no	—
216	C₂²×C₄	—	212	1	no	—
217	C₂²×C₄	—	212	4	no	—
218	C₂²×C₄	#11719922	1	1	yes	characteristic
219	C₂²×C₄	#11719922	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
221	D₈	—	53	4	no	—
222	D₈	—	106	4	no	—
223	D₈	—	106	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	Q₈×D₅₃	1	2	yes	—
226	C₂×C₈	#11719949	1	2	yes	—
227	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	106	2	no	—
237	C₄⋊C₄	—	212	1	no	—
238	C₂²⋊C₄	—	212	4	no	—
239	C₄²	#11719922	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
241	C₁₆	—	106	2	no	—
242	C₁₆	—	106	2	no	—
243	C₂³	—	424	2	no	—
244	C₂³	—	424	2	no	—
245	C₂³	—	212	2	no	—
246	C₂³	—	212	4	no	—
247	C₂³	C₂×#11719922	1	1	yes	characteristic
248	D₄	—	106	4	no	—
249	D₄	—	212	4	no	—
250	D₄	—	212	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	424	8	no	—
258	C₂×C₄	—	424	2	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	#13133193	1	2	yes	—
261	C₂×C₄	C₂×#11719922	1	2	yes	—
262	C₂×C₄	C₂×#11719922	1	2	yes	—
263	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
264	C₂×C₄	#13133248	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#13133193	1	2	yes	—
268	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
269	C₈	C₂×#11719949	1	1	yes	characteristic
270	C₂²	—	424	2	no	—
271	C₂²	—	424	2	no	—
272	C₂²	—	424	8	no	—
273	C₂²	—	212	4	no	—
274	C₂²	#16810900	1	2	yes	—
275	C₂²	#16811053	1	2	yes	—
276	C₂²	#16811058	1	2	yes	—
277	C₂²	C₂²×#11719922	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	424	2	no	—
283	C₄	#16810900	1	2	yes	—
284	C₄	C₂×#13133193	1	1	yes	characteristic
285	C₄	C₂×#13133248	1	1	yes	characteristic
286	C₂	—	424	4	no	—
287	C₂	#20276869	1	4	yes	direct product factor
288	C₂	C₂×#16810900	1	1	yes	characteristic
289	C₂	C₂×#16811053	1	1	yes	characteristic
290	C₂	C₂×#16811058	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	C₂×#2228
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #20276869	C₂	direct (N × H)
3. C₂×#16810450	C₂	semidirect
4. C₂×#16810448	C₂	semidirect
5. C₂×C₄⋊#11719908	C₂	semidirect
8. C₂×C₅₃×#499	C₂	semidirect
17. #16810450	C₂²	semidirect
18. #16810448	C₂²	semidirect
19. C₄⋊#11719908	C₂²	semidirect
24. C₅₃×#499	C₂²	semidirect
141. C₂×#499	D₅₃	semidirect
149. C₂×C₅₃	#2228	semidirect
160. #499	C₂×D₅₃	semidirect
174. C₅₃	C₂×#2228	semidirect
287. C₂	#20276869	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×C₄×D₄₂₄	2
#20276869	2
C₂×#16810450	2
C₂×#16810448	2
C₂×C₄⋊#11719908	2
C₂×#16810423	2
C₂×C₅₃×#499	2
C₂×#2228	53

Subgroups — C2×#20276869 #27538704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276869 #27538704