Subgroups — C₂×#20274077 #27366324

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810900	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20274077	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810369	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	—	2	1	no	—
8	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
9	#16810900	—	2	4	no	—
10	#16810900	C₂²	1	8	yes	—
11	C₈×D₂₁₂	—	2	2	no	—
12	C₈×D₂₁₂	C₂²	1	4	yes	—
13	C₂×#13133150	—	2	2	no	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133137	—	2	2	no	—
17	C₂×#13133129	—	2	1	no	—
18	#16810376	C₂²	1	4	yes	complemented
19	#16810369	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₅₃	—	2	1	no	—
21	C₂²×Q₁₆×C₅₃	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#494	—	2	2	no	—
24	C₅₃×#494	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	—	2	1	no	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂×C₈×D₅₃	—	2	4	no	—
31	C₂×C₈×D₅₃	—	4	4	no	—
32	C₄×D₂₁₂	—	2	1	no	—
33	C₄×D₂₁₂	—	2	1	no	—
34	C₄×D₂₁₂	D₄	1	2	yes	—
35	C₄×D₂₁₂	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
37	C₂×#11719921	—	2	1	no	—
38	#13133150	—	2	8	no	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	—	2	1	no	—
41	C₄⋊#88322	C₂³	1	2	yes	—
42	C₄⋊#88322	C₂³	1	4	yes	—
43	#13133137	—	2	8	no	—
44	#13133129	—	2	4	no	—
45	C₂²×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	—	2	4	no	—
47	C₂×Q₁₆×C₅₃	—	2	4	no	—
48	C₂×Q₁₆×C₅₃	—	4	4	no	—
49	C₂×Q₁₆×C₅₃	—	4	4	no	—
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	—	2	2	no	—
53	C₄⋊Q₈×C₅₃	—	2	2	no	—
54	C₄⋊Q₈×C₅₃	—	2	2	no	—
55	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂³×D₅₃	—	2	1	no	—
62	C₂×Q₈×C₅₃	—	2	4	no	—
63	C₂×Q₈×C₅₃	—	4	2	no	—
64	C₂×Q₈×C₅₃	—	4	4	no	—
65	C₂×Q₈×C₅₃	—	4	4	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	2	1	no	—
71	C₂×D₂₁₂	—	2	1	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×D₂₁₂	Q₁₆	1	4	yes	complemented
74	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	—	4	2	no	—
78	C₂×C₄×D₅₃	—	4	2	no	—
79	Q₁₆×C₅₃	—	4	8	no	—
80	Q₁₆×C₅₃	—	4	8	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	D₈	1	4	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₄⋊C₄×C₅₃	—	4	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	#11719921	—	2	2	no	—
94	#11719921	—	2	2	no	—
95	C₄⋊Dic₅₃	—	2	1	no	—
96	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂×#88322	—	4	4	no	—
101	C₈×D₅₃	—	4	8	no	—
102	C₂³×C₅₃	#121	1	1	yes	characteristic
103	C₂²×D₅₃	—	2	2	no	—
104	C₂²×D₅₃	—	2	4	no	—
105	C₂²×D₅₃	—	4	2	no	—
106	C₂²×D₅₃	—	4	2	no	—
107	Q₈×C₅₃	—	4	4	no	—
108	Q₈×C₅₃	—	4	4	no	—
109	Q₈×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	8	4	no	—
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₅₃	#135	1	2	yes	—
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	4	2	no	—
125	D₂₁₂	—	2	2	no	—
126	D₂₁₂	—	2	4	no	—
127	D₂₁₂	C₂×Q₁₆	1	4	yes	complemented
128	C₄×D₅₃	—	4	2	no	—
129	C₄×D₅₃	—	4	2	no	—
130	C₄×D₅₃	—	4	4	no	—
131	C₈×C₅₃	—	2	2	no	—
132	C₈×C₅₃	C₂×D₈	1	4	yes	—
133	#88322	—	4	2	no	—
134	#88322	—	4	4	no	—
135	C₂×#1669	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	#459	1	2	yes	—
137	C₂²×C₅₃	#467	1	2	yes	—
138	C₂²×C₅₃	#473	1	2	yes	—
139	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
140	C₂×D₅₃	—	2	4	no	—
141	C₂×D₅₃	—	4	8	no	—
142	C₂×D₅₃	—	4	2	no	—
143	C₂×D₅₃	—	4	2	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	8	4	no	—
147	C₄×C₅₃	#459	1	2	yes	—
148	C₄×C₅₃	#467	1	2	yes	—
149	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
151	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
152	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	Dic₅₃	—	4	2	no	—
154	C₂×#494	D₅₃	1	1	yes	characteristic, complemented
155	C₂×#474	—	53	2	no	—
156	C₂×C₈×D₄	—	53	1	no	—
157	#1669	—	53	8	no	—
158	C₂×#335	—	53	1	no	—
159	C₂×#332	—	53	2	no	—
160	C₂×C₅₃	#1669	1	4	yes	complemented
161	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
163	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
164	D₅₃	—	4	4	no	—
165	C₂²×Q₁₆	—	2	1	no	—
166	C₂²×Q₁₆	—	2	1	no	—
167	C₂³×C₈	—	106	1	no	—
168	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
169	C₂×C₄×D₄	—	53	1	no	—
170	#494	—	2	2	no	—
171	#494	C₂×D₅₃	1	4	yes	complemented
172	#474	—	53	8	no	—
173	#474	—	106	4	no	—
174	C₈×D₄	—	53	4	no	—
175	C₈×D₄	—	106	2	no	—
176	C₂×#107	—	106	2	no	—
177	C₂×C₄⋊C₈	—	53	1	no	—
178	C₂×C₄⋊C₈	—	53	2	no	—
179	C₂×#102	—	106	2	no	—
180	C₂×C₂²⋊C₈	—	106	1	no	—
181	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
182	#335	—	53	4	no	—
183	#332	—	53	8	no	—
184	C₅₃	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×D₄	—	53	1	no	—
187	C₂³×C₄	—	106	1	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	4	4	no	—
191	C₂×Q₁₆	—	4	4	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	—	106	4	no	—
195	C₂²×C₈	—	212	4	no	—
196	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
202	C₄×D₄	—	53	2	no	—
203	C₄×D₄	—	53	2	no	—
204	C₄×D₄	—	106	1	no	—
205	C₄×D₄	—	106	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	53	1	no	—
209	C₂×C₂²⋊C₄	—	106	1	no	—
210	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
211	#107	—	106	8	no	—
212	C₄⋊C₈	—	53	2	no	—
213	C₄⋊C₈	—	53	4	no	—
214	C₄⋊C₈	—	106	2	no	—
215	C₄⋊C₈	—	106	1	no	—
216	#102	—	106	8	no	—
217	C₂²⋊C₈	—	106	4	no	—
218	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
219	C₄×C₈	C₂²×D₅₃	1	2	yes	—
220	C₂⁴	—	106	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×D₄	—	53	4	no	—
226	C₂×D₄	—	53	2	no	—
227	C₂×D₄	—	106	1	no	—
228	C₂×D₄	—	106	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	106	1	no	—
231	C₂²×C₄	—	106	2	no	—
232	C₂²×C₄	—	106	2	no	—
233	C₂²×C₄	—	212	2	no	—
234	C₂²×C₄	—	212	2	no	—
235	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₄	#11719947	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	106	2	no	—
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	4	no	—
245	C₂×C₈	—	212	8	no	—
246	C₂×C₈	#11719922	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	53	2	no	—
254	C₄⋊C₄	—	106	1	no	—
255	C₂²⋊C₄	—	106	2	no	—
256	C₂²⋊C₄	—	106	2	no	—
257	C₄²	D₄×D₅₃	1	1	yes	characteristic
258	C₄²	#11719947	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₄²	C₂³×D₅₃	1	1	yes	characteristic
261	C₂³	—	106	2	no	—
262	C₂³	—	106	4	no	—
263	C₂³	—	212	2	no	—
264	C₂³	—	212	2	no	—
265	C₂³	#13133188	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	53	4	no	—
270	D₄	—	106	4	no	—
271	D₄	—	106	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	212	2	no	—
277	C₂×C₄	—	212	4	no	—
278	C₂×C₄	—	212	2	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	#13133188	1	1	yes	characteristic
281	C₂×C₄	#13133188	1	1	yes	characteristic
282	C₂×C₄	#13133188	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719922	1	2	yes	—
284	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
285	C₂×C₄	#13133240	1	2	yes	—
286	C₂×C₄	#13133243	1	2	yes	—
287	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	212	2	no	—
292	C₈	—	212	4	no	—
293	C₈	C₂×#11719922	1	4	yes	—
294	C₂²	—	106	4	no	—
295	C₂²	—	212	2	no	—
296	C₂²	—	212	8	no	—
297	C₂²	—	212	2	no	—
298	C₂²	#16810815	1	2	yes	—
299	C₂²	#16810861	1	2	yes	—
300	C₂²	#16810895	1	2	yes	—
301	C₂²	C₂×#13133188	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	4	no	—
305	C₄	—	212	2	no	—
306	C₄	#16810815	1	2	yes	—
307	C₄	#16810861	1	2	yes	—
308	C₄	C₂²×#11719922	1	1	yes	characteristic
309	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
310	C₄	C₂×#13133240	1	1	yes	characteristic
311	C₄	C₂×#13133243	1	1	yes	characteristic
312	C₂	—	212	4	no	—
313	C₂	#20274077	1	4	yes	direct product factor
314	C₂	C₂×#16810815	1	1	yes	characteristic
315	C₂	C₂×#16810861	1	1	yes	characteristic
316	C₂	C₂×#16810895	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₂₁₂	D₄
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	C₂×#1669
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20274077	C₂	direct (N × H)
4. C₂×#16810376	C₂	semidirect
5. C₂×#16810369	C₂	semidirect
6. C₂×C₅₃×#494	C₂	semidirect
18. #16810376	C₂²	semidirect
19. #16810369	C₂²	semidirect
24. C₅₃×#494	C₂²	semidirect
73. C₂×D₂₁₂	Q₁₆	semidirect
127. D₂₁₂	C₂×Q₁₆	semidirect
154. C₂×#494	D₅₃	semidirect
160. C₂×C₅₃	#1669	semidirect
171. #494	C₂×D₅₃	semidirect
184. C₅₃	C₂×#1669	semidirect
313. C₂	#20274077	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×#16810900	2
C₂×C₈×D₂₁₂	2
#20274077	2
C₂×#16810376	2
C₂×#16810369	2
C₂×C₅₃×#494	2
C₂×#1669	53

Subgroups — C2×#20274077 #27366324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274077 #27366324