Subgroups — C₂×#20274201 #27366596

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 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×#16810702	C₂	1	1	yes	maximal, characteristic
3	#20274201	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810374	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×#11719912	—	2	1	no	—
8	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
9	#16810900	—	2	4	no	—
10	#16810900	C₂²	1	8	yes	—
11	#16810702	—	2	2	no	—
12	#16810702	C₂²	1	4	yes	—
13	C₂×#13133150	—	2	2	no	—
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×#13133137	—	2	2	no	—
17	C₂×#13133129	—	2	1	no	—
18	#16810374	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	C₂×#11719912	—	2	4	no	—
29	C₂×#11719912	—	4	4	no	—
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	—	2	2	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	4	no	—
72	C₂×D₂₁₂	—	2	1	no	—
73	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	—	4	2	no	—
77	C₂×C₄×D₅₃	—	4	2	no	—
78	Q₁₆×C₅₃	—	4	8	no	—
79	Q₁₆×C₅₃	—	4	8	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	D₈	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
92	#11719921	—	2	2	no	—
93	#11719921	—	2	2	no	—
94	C₄⋊Dic₅₃	—	2	1	no	—
95	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	4	no	—
100	#11719912	—	4	8	no	—
101	C₂³×C₅₃	#121	1	1	yes	characteristic
102	C₂²×D₅₃	—	2	2	no	—
103	C₂²×D₅₃	—	4	2	no	—
104	C₂²×D₅₃	—	4	2	no	—
105	C₂²×D₅₃	—	4	2	no	—
106	Q₈×C₅₃	—	4	4	no	—
107	Q₈×C₅₃	—	4	4	no	—
108	Q₈×C₅₃	—	8	4	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₅₃	#137	1	4	yes	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	—	4	2	no	—
123	D₂₁₂	—	2	4	no	—
124	D₂₁₂	—	4	2	no	—
125	C₄×D₅₃	—	4	2	no	—
126	C₄×D₅₃	—	4	2	no	—
127	C₄×D₅₃	—	4	4	no	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	C₂×D₈	1	4	yes	—
130	#88322	—	4	2	no	—
131	#88322	—	4	4	no	—
132	C₂×#1689	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	#459	1	2	yes	—
134	C₂²×C₅₃	#470	1	2	yes	—
135	C₂²×C₅₃	#473	1	2	yes	—
136	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
137	C₂×D₅₃	—	4	2	no	—
138	C₂×D₅₃	—	4	2	no	—
139	C₂×D₅₃	—	4	2	no	—
140	C₂×D₅₃	—	8	4	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	8	4	no	—
144	C₄×C₅₃	#459	1	2	yes	—
145	C₄×C₅₃	#470	1	2	yes	—
146	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#137	1	2	yes	—
149	Dic₅₃	—	4	2	no	—
150	C₂×#494	D₅₃	1	1	yes	characteristic, complemented
151	C₂×#474	—	53	2	no	—
152	C₂×#436	—	53	1	no	—
153	#1689	—	53	8	no	—
154	C₂×#334	—	53	1	no	—
155	C₂×#332	—	53	2	no	—
156	C₂×C₅₃	#1689	1	4	yes	complemented
157	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#470	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
160	D₅₃	—	8	2	no	—
161	C₂²×Q₁₆	—	2	1	no	—
162	C₂²×Q₁₆	—	2	1	no	—
163	C₂²×M₁₆	—	106	1	no	—
164	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
165	C₂×C₄×D₄	—	53	1	no	—
166	#494	—	2	2	no	—
167	#494	C₂×D₅₃	1	4	yes	complemented
168	#474	—	53	8	no	—
169	#474	—	106	4	no	—
170	#436	—	53	4	no	—
171	#436	—	106	2	no	—
172	C₂×#107	—	106	2	no	—
173	C₂×C₄⋊C₈	—	53	1	no	—
174	C₂×C₄⋊C₈	—	53	2	no	—
175	C₂×#102	—	106	2	no	—
176	C₂×C₂²⋊C₈	—	106	1	no	—
177	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
178	#334	—	53	4	no	—
179	#332	—	53	8	no	—
180	C₅₃	C₂×#1689	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×D₄	—	53	1	no	—
183	C₂³×C₄	—	106	1	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	C₂×Q₁₆	—	4	4	no	—
188	C₂×M₁₆	—	106	4	no	—
189	C₂×M₁₆	—	212	4	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	—	106	1	no	—
192	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
198	C₄×D₄	—	53	2	no	—
199	C₄×D₄	—	53	2	no	—
200	C₄×D₄	—	106	1	no	—
201	C₄×D₄	—	106	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	53	1	no	—
205	C₂×C₂²⋊C₄	—	106	1	no	—
206	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	#107	—	106	8	no	—
208	C₄⋊C₈	—	53	2	no	—
209	C₄⋊C₈	—	53	4	no	—
210	C₄⋊C₈	—	106	2	no	—
211	C₄⋊C₈	—	106	1	no	—
212	#102	—	106	8	no	—
213	C₂²⋊C₈	—	106	4	no	—
214	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
215	C₄×C₈	C₂²×D₅₃	1	2	yes	—
216	C₂⁴	—	106	1	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×D₄	—	53	2	no	—
222	C₂×D₄	—	106	1	no	—
223	C₂×D₄	—	106	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	106	1	no	—
226	C₂²×C₄	—	106	2	no	—
227	C₂²×C₄	—	106	2	no	—
228	C₂²×C₄	—	212	2	no	—
229	C₂²×C₄	—	212	2	no	—
230	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
231	C₂²×C₄	#11719947	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	M₁₆	—	212	8	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	106	4	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	—	212	4	no	—
241	C₂×C₈	#11719922	1	4	yes	—
242	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	53	2	no	—
249	C₄⋊C₄	—	106	1	no	—
250	C₂²⋊C₄	—	106	2	no	—
251	C₂²⋊C₄	—	106	2	no	—
252	C₄²	D₄×D₅₃	1	1	yes	characteristic
253	C₄²	#11719947	1	1	yes	characteristic
254	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
255	C₄²	C₂³×D₅₃	1	1	yes	characteristic
256	C₂³	—	106	2	no	—
257	C₂³	—	212	2	no	—
258	C₂³	—	212	2	no	—
259	C₂³	—	212	2	no	—
260	C₂³	#13133188	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	106	4	no	—
265	D₄	—	212	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	—	212	4	no	—
272	C₂×C₄	—	212	2	no	—
273	C₂×C₄	—	212	2	no	—
274	C₂×C₄	#13133188	1	1	yes	characteristic
275	C₂×C₄	#13133188	1	1	yes	characteristic
276	C₂×C₄	#13133188	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719922	1	2	yes	—
278	C₂×C₄	#13133243	1	2	yes	—
279	C₂×C₄	#13133249	1	4	yes	—
280	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	212	2	no	—
285	C₈	—	212	4	no	—
286	C₈	C₂×#11719922	1	4	yes	—
287	C₂²	—	424	4	no	—
288	C₂²	—	212	2	no	—
289	C₂²	—	212	2	no	—
290	C₂²	—	212	2	no	—
291	C₂²	#16810815	1	2	yes	—
292	C₂²	#16810878	1	2	yes	—
293	C₂²	#16810895	1	2	yes	—
294	C₂²	C₂×#13133188	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	4	no	—
298	C₄	—	212	2	no	—
299	C₄	#16810815	1	2	yes	—
300	C₄	#16810878	1	2	yes	—
301	C₄	C₂²×#11719922	1	1	yes	characteristic
302	C₄	C₂×#13133243	1	1	yes	characteristic
303	C₄	C₂×#13133249	1	2	yes	—
304	C₂	—	424	2	no	—
305	C₂	#20274201	1	4	yes	direct product factor
306	C₂	C₂×#16810815	1	1	yes	characteristic
307	C₂	C₂×#16810878	1	1	yes	characteristic
308	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×#1689
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. #20274201	C₂	direct (N × H)
4. C₂×#16810374	C₂	semidirect
5. C₂×#16810369	C₂	semidirect
6. C₂×C₅₃×#494	C₂	semidirect
18. #16810374	C₂²	semidirect
19. #16810369	C₂²	semidirect
24. C₅₃×#494	C₂²	semidirect
150. C₂×#494	D₅₃	semidirect
156. C₂×C₅₃	#1689	semidirect
167. #494	C₂×D₅₃	semidirect
180. C₅₃	C₂×#1689	semidirect
305. C₂	#20274201	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×#16810702	2
#20274201	2
C₂×#16810374	2
C₂×#16810369	2
C₂×C₅₃×#494	2
C₂×#1689	53

Subgroups — C2×#20274201 #27366596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274201 #27366596