Subgroups — C₂×#16811420 #20289459

Order 6784.

1484 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₅₃	C₂	1	2	yes	maximal, complemented
2	#16811420	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13133249	C₂	1	2	yes	maximal, complemented
4	C₂×#13133246	C₂	1	2	yes	maximal, complemented
5	C₂×#13133244	C₂	1	2	yes	maximal, complemented
6	C₂×#13133241	C₂	1	2	yes	maximal, complemented
7	C₂×#13133239	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133237	C₂	1	2	yes	maximal, complemented
9	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#11719949	C₂²	1	2	yes	—
12	C₂×#11719945	—	2	2	no	—
13	C₂×#11719945	C₂²	1	2	yes	complemented
14	C₂×#11719947	—	2	2	no	—
15	C₂×#11719947	C₂²	1	2	yes	—
16	#41×D₅₃	—	2	8	no	—
17	#41×D₅₃	C₂²	1	16	yes	complemented
18	C₂×Q₈×D₅₃	C₂²	1	2	yes	complemented
19	C₂×D₄×D₅₃	—	2	2	no	—
20	C₂×D₄×D₅₃	C₂²	1	2	yes	complemented
21	C₂²×C₄×D₅₃	—	2	2	no	—
22	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	#13133249	C₂²	1	16	yes	complemented
24	#13133246	C₂²	1	16	yes	complemented
25	#13133244	C₂²	1	16	yes	complemented
26	#13133241	C₂²	1	16	yes	complemented
27	#13133239	C₂²	1	8	yes	complemented
28	#13133237	C₂²	1	16	yes	complemented
29	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
30	C₂×#11719925	C₂²	1	2	yes	complemented
31	C₂×#11719913	C₂²	1	2	yes	complemented
32	C₂×#11719923	C₂²	1	2	yes	complemented
33	C₂×#11719924	C₂²	1	2	yes	complemented
34	C₂×D₄₂₄	C₂²	1	1	yes	characteristic, complemented
35	C₂×#11719922	C₂²	1	2	yes	complemented
36	C₂×#11719912	C₂²	1	2	yes	complemented
37	C₂×#11719912	C₂²	1	2	yes	complemented
38	C₂×#11719912	C₂²	1	8	yes	complemented
39	C₂²×#88322	C₂²	1	1	yes	characteristic, complemented
40	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
41	#135×C₅₃	C₂²	1	8	yes	complemented
42	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
44	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂³×D₅₃	—	2	2	no	—
47	C₂³×D₅₃	D₄	1	1	yes	characteristic
48	#41×C₅₃	—	2	4	no	—
49	#41×C₅₃	C₂³	1	8	yes	complemented
50	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₅₃	—	2	2	no	—
56	C₂×C₂²⋊D₅₃	—	2	2	no	—
57	C₂×C₂²⋊D₅₃	—	2	2	no	—
58	C₂²×Dic₅₃	—	2	2	no	—
59	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
60	#11719949	—	2	4	no	—
61	#11719949	C₂³	1	8	yes	—
62	Q₈×D₅₃	—	2	4	no	—
63	Q₈×D₅₃	C₂³	1	8	yes	complemented
64	#11719947	—	2	16	no	—
65	#11719947	—	2	4	no	—
66	#11719947	C₂³	1	8	yes	—
67	D₄×D₅₃	—	2	16	no	—
68	D₄×D₅₃	—	2	4	no	—
69	D₄×D₅₃	C₂³	1	8	yes	complemented
70	#11719945	—	2	4	no	—
71	#11719945	—	2	16	no	—
72	#11719945	C₂³	1	8	yes	complemented
73	C₂×D₂₁₂	—	2	2	no	—
74	C₂×D₂₁₂	C₂³	1	2	yes	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	16	no	—
79	C₂×C₄×D₅₃	D₄	1	2	yes	—
80	C₂×C₄×D₅₃	D₄	1	4	yes	—
81	C₂×C₄×D₅₃	C₂³	1	2	yes	—
82	C₂×C₄×D₅₃	C₂³	1	4	yes	—
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	C₂³	1	2	yes	—
85	Q₁₆×C₅₃	C₂³	1	4	yes	complemented
86	SD₁₆×C₅₃	C₂³	1	8	yes	complemented
87	D₈×C₅₃	C₂³	1	4	yes	complemented
88	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂³	1	4	yes	complemented
91	#11719925	C₂³	1	8	yes	complemented
92	#11719924	C₂³	1	8	yes	complemented
93	#11719923	C₂³	1	8	yes	complemented
94	#11719922	C₂³	1	8	yes	complemented
95	C₂×#88322	C₂³	1	2	yes	—
96	C₂×#88322	C₂³	1	4	yes	complemented
97	Dic₂₁₂	C₂³	1	4	yes	complemented
98	D₄₂₄	C₂³	1	4	yes	complemented
99	#11719913	C₂³	1	8	yes	complemented
100	#11719912	C₂³	1	8	yes	complemented
101	#11719912	C₂³	1	8	yes	complemented
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₅₃	—	2	2	no	—
105	C₂²×D₅₃	—	2	8	no	—
106	C₂²×D₅₃	—	4	2	no	—
107	C₂²×D₅₃	—	4	8	no	—
108	C₂²×D₅₃	C₂×D₄	1	2	yes	—
109	C₂²×D₅₃	C₂×D₄	1	4	yes	—
110	Q₈×C₅₃	—	2	2	no	—
111	Q₈×C₅₃	C₂⁴	1	4	yes	—
112	D₄×C₅₃	—	2	8	no	—
113	D₄×C₅₃	—	2	2	no	—
114	D₄×C₅₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	8	no	—
117	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
121	C₂²⋊D₅₃	—	2	8	no	—
122	C₂²⋊D₅₃	—	2	8	no	—
123	C₂²⋊D₅₃	—	2	8	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	2	8	no	—
126	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
127	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
128	D₂₁₂	—	2	8	no	—
129	D₂₁₂	—	2	2	no	—
130	D₂₁₂	C₂⁴	1	4	yes	—
131	C₄×D₅₃	—	2	8	no	—
132	C₄×D₅₃	—	2	8	no	—
133	C₄×D₅₃	—	2	8	no	—
134	C₄×D₅₃	C₂×D₄	1	8	yes	—
135	C₄×D₅₃	C₂×D₄	1	4	yes	—
136	C₄×D₅₃	C₂⁴	1	4	yes	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	2	8	no	—
139	Dic₁₀₆	C₂⁴	1	4	yes	—
140	C₈×C₅₃	C₂⁴	1	2	yes	—
141	C₈×C₅₃	C₂⁴	1	2	yes	—
142	#88322	C₂⁴	1	4	yes	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	—	4	4	no	—
146	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
147	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₅₃	—	2	4	no	—
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	—	4	4	no	—
151	C₂×D₅₃	—	4	16	no	—
152	C₂×D₅₃	—	4	4	no	—
153	C₂×D₅₃	C₂²×D₄	1	4	yes	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₅₃	C₂²×D₄	1	2	yes	—
157	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
158	Dic₅₃	—	2	4	no	—
159	Dic₅₃	C₂²×D₄	1	4	yes	—
160	C₂×#575	—	53	1	no	maximal, Sylow
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	—	4	4	no	—
163	C₂×C₅₃	#575	1	2	yes	complemented
164	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
165	D₅₃	—	2	4	no	—
166	D₅₃	—	4	4	no	—
167	C₂²×#41	—	53	2	no	—
168	#575	—	53	16	no	—
169	C₂×#137	—	53	2	no	—
170	C₂×#137	—	53	2	no	—
171	C₂×AGL(1,ℤ/8)	—	53	2	no	—
172	C₂×AGL(1,ℤ/8)	—	53	2	no	—
173	C₂×#135	—	53	2	no	—
174	C₂×#135	—	53	1	no	—
175	C₂×#135	D₅₃	1	1	yes	characteristic, complemented
176	C₂²×M₁₆	—	53	1	no	—
177	C₅₃	C₂×#575	1	1	yes	characteristic, complemented, Sylow
178	C₂×#41	—	53	16	no	—
179	C₂×#41	—	53	2	no	—
180	C₂×#41	—	53	2	no	—
181	C₂×#41	—	53	2	no	—
182	C₂×#41	—	106	2	no	—
183	C₂×#41	—	106	2	no	—
184	C₂×#41	—	106	8	no	—
185	C₂×#41	C₂×D₅₃	1	2	yes	complemented
186	C₂²×Q₈	—	53	2	no	—
187	C₂²×D₄	—	53	2	no	—
188	C₂²×D₄	—	106	2	no	—
189	C₂³×C₄	—	53	1	no	—
190	C₂³×C₄	—	106	2	no	—
191	#137	—	53	16	no	—
192	#137	—	53	16	no	—
193	AGL(1,ℤ/8)	—	53	16	no	—
194	AGL(1,ℤ/8)	—	53	16	no	—
195	#135	—	53	16	no	—
196	#135	—	53	8	no	—
197	#135	C₂×D₅₃	1	8	yes	complemented
198	C₂×Q₁₆	—	53	2	no	—
199	C₂×Q₁₆	—	53	1	no	—
200	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
201	C₂×SD₁₆	—	53	2	no	—
202	C₂×SD₁₆	—	53	2	no	—
203	C₂×SD₁₆	—	53	2	no	—
204	C₂×SD₁₆	C₂×D₅₃	1	2	yes	complemented
205	C₂×D₈	—	53	2	no	—
206	C₂×D₈	—	53	1	no	—
207	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic, complemented
208	C₂×M₁₆	—	53	2	no	—
209	C₂×M₁₆	—	53	2	no	—
210	C₂×M₁₆	—	53	8	no	—
211	C₂²×C₈	—	53	1	no	—
212	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
213	C₂⁴	—	53	1	no	—
214	C₂⁴	—	106	2	no	—
215	#41	—	2	4	no	—
216	#41	—	53	8	no	—
217	#41	—	53	8	no	—
218	#41	—	53	8	no	—
219	#41	—	106	16	no	—
220	#41	—	106	16	no	—
221	#41	—	106	4	no	—
222	#41	—	106	4	no	—
223	#41	—	106	4	no	—
224	#41	C₂²×D₅₃	1	8	yes	complemented
225	C₂×Q₈	—	53	8	no	—
226	C₂×Q₈	—	53	2	no	—
227	C₂×Q₈	—	106	2	no	—
228	C₂×Q₈	—	106	4	no	—
229	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	53	8	no	—
232	C₂×D₄	—	53	2	no	—
233	C₂×D₄	—	106	2	no	—
234	C₂×D₄	—	106	4	no	—
235	C₂×D₄	—	106	2	no	—
236	C₂×D₄	—	106	16	no	—
237	C₂×D₄	—	106	2	no	—
238	C₂×D₄	—	106	2	no	—
239	C₂×D₄	C₂²×D₅₃	1	2	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	53	1	no	—
242	C₂²×C₄	—	53	4	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	4	no	—
246	C₂²×C₄	—	106	2	no	—
247	C₂²×C₄	—	106	16	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
252	Q₁₆	—	53	8	no	—
253	Q₁₆	—	53	4	no	—
254	Q₁₆	C₂²×D₅₃	1	4	yes	complemented
255	SD₁₆	—	53	8	no	—
256	SD₁₆	—	53	8	no	—
257	SD₁₆	—	53	8	no	—
258	SD₁₆	C₂²×D₅₃	1	8	yes	complemented
259	D₈	—	53	8	no	—
260	D₈	—	53	4	no	—
261	D₈	C₂²×D₅₃	1	4	yes	complemented
262	M₁₆	—	53	8	no	—
263	M₁₆	—	53	8	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	4	no	—
266	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×D₅₃	1	4	yes	complemented
269	C₂³	—	2	2	no	—
270	C₂³	—	53	2	no	—
271	C₂³	—	53	4	no	—
272	C₂³	—	106	8	no	—
273	C₂³	—	106	2	no	—
274	C₂³	—	212	2	no	—
275	C₂³	—	212	8	no	—
276	C₂³	D₄×D₅₃	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	53	4	no	—
279	Q₈	—	106	8	no	—
280	Q₈	—	106	2	no	—
281	Q₈	C₂³×D₅₃	1	4	yes	—
282	D₄	—	2	8	no	—
283	D₄	—	2	2	no	—
284	D₄	—	53	4	no	—
285	D₄	—	106	8	no	—
286	D₄	—	106	8	no	—
287	D₄	—	106	8	no	—
288	D₄	—	106	8	no	—
289	D₄	—	106	2	no	—
290	D₄	C₂³×D₅₃	1	4	yes	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	53	2	no	—
294	C₂×C₄	—	53	4	no	—
295	C₂×C₄	—	53	8	no	—
296	C₂×C₄	—	53	4	no	—
297	C₂×C₄	—	53	4	no	—
298	C₂×C₄	—	106	8	no	—
299	C₂×C₄	—	106	8	no	—
300	C₂×C₄	—	106	8	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	8	no	—
303	C₂×C₄	D₄×D₅₃	1	2	yes	—
304	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₅₃	1	2	yes	—
307	C₈	—	53	4	no	—
308	C₈	C₂³×D₅₃	1	2	yes	—
309	C₈	C₂³×D₅₃	1	2	yes	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	53	4	no	—
314	C₂²	—	106	4	no	—
315	C₂²	—	106	4	no	—
316	C₂²	—	212	4	no	—
317	C₂²	—	212	16	no	—
318	C₂²	—	212	4	no	—
319	C₂²	C₂×D₄×D₅₃	1	2	yes	—
320	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	53	4	no	—
323	C₄	—	106	4	no	—
324	C₄	C₂×D₄×D₅₃	1	2	yes	—
325	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
326	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	4	4	no	—
329	C₂	—	106	4	no	—
330	C₂	—	212	4	no	—
331	C₂	#16811420	1	2	yes	direct product factor
332	C₂	C₂²×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂⁵
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×#135	D₅₃
hypocenter	C₅₃	C₂×#575
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₅₃	C₂	semidirect
2. #16811420	C₂	direct (N × H)
3. C₂×#13133249	C₂	semidirect
4. C₂×#13133246	C₂	semidirect
5. C₂×#13133244	C₂	semidirect
6. C₂×#13133241	C₂	semidirect
7. C₂×#13133239	C₂	semidirect
8. C₂×#13133237	C₂	semidirect
9. C₂²×#11719912	C₂	semidirect
10. C₂×#135×C₅₃	C₂	semidirect
13. C₂×#11719945	C₂²	semidirect
17. #41×D₅₃	C₂²	semidirect
18. C₂×Q₈×D₅₃	C₂²	semidirect
20. C₂×D₄×D₅₃	C₂²	semidirect
23. #13133249	C₂²	semidirect
24. #13133246	C₂²	semidirect
25. #13133244	C₂²	semidirect
26. #13133241	C₂²	semidirect
27. #13133239	C₂²	semidirect
28. #13133237	C₂²	semidirect
29. C₂×Dic₂₁₂	C₂²	semidirect
30. C₂×#11719925	C₂²	semidirect
31. C₂×#11719913	C₂²	semidirect
32. C₂×#11719923	C₂²	semidirect
33. C₂×#11719924	C₂²	semidirect
34. C₂×D₄₂₄	C₂²	semidirect
35. C₂×#11719922	C₂²	semidirect
36. C₂×#11719912	C₂²	semidirect
37. C₂×#11719912	C₂²	semidirect
38. C₂×#11719912	C₂²	semidirect
39. C₂²×#88322	C₂²	semidirect
40. C₂×#41×C₅₃	C₂²	semidirect
41. #135×C₅₃	C₂²	semidirect
42. C₂×Q₁₆×C₅₃	C₂²	semidirect
43. C₂×SD₁₆×C₅₃	C₂²	semidirect
44. C₂×D₈×C₅₃	C₂²	semidirect
45. C₂²×C₈×C₅₃	C₂²	semidirect
49. #41×C₅₃	C₂³	semidirect
63. Q₈×D₅₃	C₂³	semidirect
69. D₄×D₅₃	C₂³	semidirect
72. #11719945	C₂³	semidirect
85. Q₁₆×C₅₃	C₂³	semidirect
86. SD₁₆×C₅₃	C₂³	semidirect
87. D₈×C₅₃	C₂³	semidirect
90. C₂×C₈×C₅₃	C₂³	semidirect
91. #11719925	C₂³	semidirect
92. #11719924	C₂³	semidirect
93. #11719923	C₂³	semidirect
94. #11719922	C₂³	semidirect
96. C₂×#88322	C₂³	semidirect
97. Dic₂₁₂	C₂³	semidirect
98. D₄₂₄	C₂³	semidirect
99. #11719913	C₂³	semidirect
100. #11719912	C₂³	semidirect
101. #11719912	C₂³	semidirect
163. C₂×C₅₃	#575	semidirect
175. C₂×#135	D₅₃	semidirect
177. C₅₃	C₂×#575	semidirect
185. C₂×#41	C₂×D₅₃	semidirect
197. #135	C₂×D₅₃	semidirect
200. C₂×Q₁₆	C₂×D₅₃	semidirect
204. C₂×SD₁₆	C₂×D₅₃	semidirect
207. C₂×D₈	C₂×D₅₃	semidirect
212. C₂²×C₈	C₂×D₅₃	semidirect
224. #41	C₂²×D₅₃	semidirect
254. Q₁₆	C₂²×D₅₃	semidirect
258. SD₁₆	C₂²×D₅₃	semidirect
261. D₈	C₂²×D₅₃	semidirect
268. C₂×C₈	C₂²×D₅₃	semidirect
331. C₂	#16811420	direct (N × H)

Subgroups — C2×#16811420 #20289459

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811420 #20289459