Subgroups — #20288516

Order 6784.

1100 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811334	C₂	1	2	yes	maximal, complemented
2	#16811333	C₂	1	2	yes	maximal, complemented
3	#16811333	C₂	1	4	yes	maximal, complemented
4	#16811329	C₂	1	4	yes	maximal, complemented
5	Q₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
6	#16811288	C₂	1	2	yes	maximal, complemented
7	#16811251	C₂	1	2	yes	maximal, complemented
8	#16811248	C₂	1	4	yes	maximal, complemented
9	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊Q₈×D₅₃	C₂	1	4	yes	maximal, complemented
11	#16811097	C₂	1	2	yes	maximal
12	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₅₃	C₂²	1	4	yes	complemented
15	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
17	#13133217	C₂²	1	1	yes	characteristic, complemented
18	#13133217	C₂²	1	4	yes	complemented
19	#13133217	C₂²	1	1	yes	characteristic, complemented
20	#13133216	C₂²	1	4	yes	complemented
21	#13133209	C₂²	1	2	yes	complemented
22	#13133209	C₂²	1	4	yes	complemented
23	#13133208	C₂²	1	4	yes	complemented
24	#13133197	C₂²	1	8	yes	complemented
25	#13133197	C₂²	1	4	yes	complemented
26	#13133196	C₂²	1	2	yes	—
27	#13133196	C₂²	1	2	yes	—
28	#13133195	C₂²	1	8	yes	complemented
29	#13133195	C₂²	1	8	yes	complemented
30	#13133195	C₂²	1	8	yes	complemented
31	#13133194	C₂²	1	8	yes	—
32	#13133193	C₂²	1	4	yes	complemented
33	#13133193	C₂²	1	8	yes	complemented
34	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
35	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
36	C₄×D₂₁₂	C₂²	1	2	yes	—
37	C₄×D₂₁₂	C₂²	1	2	yes	—
38	C₄×D₂₁₂	C₂²	1	4	yes	—
39	C₄×D₂₁₂	C₂²	1	2	yes	—
40	C₄×D₂₁₂	C₂²	1	2	yes	—
41	C₄×D₂₁₂	C₂²	1	4	yes	—
42	#13133179	C₂²	1	4	yes	—
43	#13133179	C₂²	1	4	yes	—
44	#13133176	C₂²	1	2	yes	—
45	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
46	C₄⋊C₄×D₅₃	C₂²	1	8	yes	complemented
47	C₄⋊C₄×D₅₃	C₂²	1	4	yes	complemented
48	C₂×#11719921	C₂²	1	2	yes	—
49	C₂²⋊C₄×D₅₃	C₂²	1	8	yes	—
50	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	#125×C₅₃	C₂²	1	2	yes	complemented
53	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
55	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
56	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
57	#117×C₅₃	C₂²	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
59	C₂³×D₅₃	C₂³	1	2	yes	—
60	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
61	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂³	1	2	yes	—
66	Q₈×D₅₃	—	4	8	no	—
67	C₂×D₂₁₂	Q₈	1	8	yes	complemented
68	C₂×D₂₁₂	C₂³	1	2	yes	—
69	C₂×D₂₁₂	C₂³	1	2	yes	—
70	C₂×C₄×D₅₃	—	4	4	no	—
71	C₂×C₄×D₅₃	C₂³	1	4	yes	—
72	C₂×C₄×D₅₃	C₂³	1	8	yes	—
73	C₂×C₄×D₅₃	C₂³	1	8	yes	—
74	C₂×C₄×D₅₃	C₂³	1	4	yes	—
75	C₂×Dic₁₀₆	C₂³	1	2	yes	—
76	C₂×Dic₁₀₆	C₂³	1	2	yes	—
77	C₂×Dic₁₀₆	C₂³	1	8	yes	—
78	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
80	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
81	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₄²×C₅₃	C₂³	1	1	yes	characteristic
87	C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₄²×C₅₃	C₂³	1	4	yes	—
90	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
91	#11719921	C₂³	1	8	yes	—
92	#11719921	C₂³	1	8	yes	—
93	#11719921	C₂³	1	4	yes	—
94	#11719921	C₂³	1	4	yes	—
95	C₄⋊Dic₅₃	C₂³	1	2	yes	—
96	C₄⋊Dic₅₃	C₂³	1	8	yes	—
97	C₄⋊Dic₅₃	C₂³	1	2	yes	—
98	C₄⋊Dic₅₃	C₂³	1	2	yes	—
99	C₄⋊Dic₅₃	C₂³	1	2	yes	—
100	C₄⋊Dic₅₃	C₂³	1	2	yes	—
101	C₄⋊Dic₅₃	C₂³	1	2	yes	—
102	#11719919	C₂³	1	4	yes	—
103	#11719919	C₂³	1	8	yes	—
104	#11719919	C₂³	1	8	yes	—
105	#11719919	C₂³	1	4	yes	—
106	C₄×Dic₅₃	C₂³	1	4	yes	—
107	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²×D₅₃	—	4	2	no	—
109	C₂²×D₅₃	—	4	2	no	—
110	C₂²×D₅₃	C₂×Q₈	1	8	yes	complemented
111	C₂²×D₅₃	C₂⁴	1	4	yes	—
112	Q₈×C₅₃	—	2	4	no	—
113	Q₈×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	complemented
117	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
121	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	C₂⁴	1	4	yes	—
127	C₂×Dic₅₃	C₂⁴	1	4	yes	—
128	C₂×Dic₅₃	C₂⁴	1	4	yes	—
129	D₂₁₂	C₂×Q₈	1	8	yes	complemented
130	D₂₁₂	C₂×Q₈	1	8	yes	complemented
131	C₄×D₅₃	—	2	8	no	—
132	C₄×D₅₃	—	4	4	no	—
133	C₄×D₅₃	—	4	8	no	—
134	C₄×D₅₃	—	4	8	no	—
135	Dic₁₀₆	—	2	4	no	—
136	Dic₁₀₆	—	4	2	no	—
137	Dic₁₀₆	—	4	8	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
141	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
142	C₂×D₅₃	—	2	4	no	—
143	C₂×D₅₃	—	2	4	no	—
144	C₂×D₅₃	—	4	8	no	—
145	C₂×D₅₃	C₂²×Q₈	1	8	yes	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
152	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
153	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
154	Dic₅₃	—	2	4	no	—
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	2	4	no	—
157	#3501	—	53	1	no	maximal, Sylow
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
160	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₅₃	#585	1	1	yes	characteristic
162	D₅₃	—	2	8	no	—
163	#554	—	53	2	no	—
164	#554	—	53	4	no	—
165	#554	—	53	2	no	—
166	#554	—	53	4	no	—
167	D₄×Q₈	—	53	2	no	—
168	D₄×Q₈	—	53	2	no	—
169	#544	—	53	2	no	—
170	#543	—	53	4	no	—
171	#533	—	53	1	no	—
172	#533	D₅₃	1	1	yes	characteristic, complemented
173	C₂×C₂²⋊Q₈	—	53	4	no	—
174	#518	—	53	2	no	—
175	C₂×C₄×D₄	—	53	1	no	—
176	C₅₃	#3501	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	—	53	4	no	—
178	C₂²×D₄	—	53	1	no	—
179	C₂³×C₄	—	53	2	no	—
180	C₄⋊Q₈	—	53	4	no	—
181	C₄⋊Q₈	—	53	1	no	—
182	C₄⋊Q₈	—	53	1	no	—
183	C₄⋊Q₈	—	53	4	no	—
184	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
185	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
186	#125	—	53	4	no	—
187	#125	—	53	2	no	—
188	#125	—	53	4	no	—
189	#125	C₂×D₅₃	1	2	yes	complemented
190	C₂²⋊Q₈	—	53	4	no	—
191	C₂²⋊Q₈	—	53	8	no	—
192	C₂²⋊Q₈	—	53	2	no	—
193	C₂²⋊Q₈	—	53	2	no	—
194	C₂²⋊Q₈	—	53	4	no	—
195	C₂²⋊Q₈	—	53	8	no	—
196	C₂²⋊Q₈	—	53	8	no	—
197	C₂²⋊Q₈	—	53	8	no	—
198	C₂²⋊Q₈	—	53	8	no	—
199	C₂²⋊Q₈	—	53	8	no	—
200	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
201	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
202	C₄×Q₈	—	53	2	no	—
203	C₄×Q₈	—	53	2	no	—
204	C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
205	C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
206	C₄×D₄	—	53	4	no	—
207	C₄×D₄	—	53	4	no	—
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	—	53	2	no	—
210	C₄×D₄	—	53	2	no	—
211	C₄×D₄	—	53	2	no	—
212	C₄×D₄	—	53	4	no	—
213	C₄×D₄	—	53	4	no	—
214	#117	—	53	2	no	—
215	#117	C₂×D₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	—	53	1	no	—
217	C₂×C₄⋊C₄	—	53	8	no	—
218	C₂×C₄⋊C₄	—	53	4	no	—
219	C₂×C₂²⋊C₄	—	53	2	no	—
220	C₂×C₂²⋊C₄	—	53	8	no	—
221	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
222	C₂⁴	—	53	2	no	—
223	C₂×Q₈	—	53	2	no	—
224	C₂×Q₈	—	53	2	no	—
225	C₂×Q₈	—	53	8	no	—
226	C₂×Q₈	—	212	8	no	—
227	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
228	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
229	C₂×D₄	—	53	8	no	—
230	C₂×D₄	—	53	2	no	—
231	C₂×D₄	—	53	2	no	—
232	C₂²×C₄	—	53	2	no	—
233	C₂²×C₄	—	53	4	no	—
234	C₂²×C₄	—	53	8	no	—
235	C₂²×C₄	—	53	8	no	—
236	C₂²×C₄	—	53	4	no	—
237	C₂²×C₄	—	212	4	no	—
238	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
241	C₄⋊C₄	—	53	4	no	—
242	C₄⋊C₄	—	53	2	no	—
243	C₄⋊C₄	—	53	2	no	—
244	C₄⋊C₄	—	53	2	no	—
245	C₄⋊C₄	—	53	8	no	—
246	C₄⋊C₄	—	53	8	no	—
247	C₄⋊C₄	—	53	8	no	—
248	C₄⋊C₄	—	53	2	no	—
249	C₄⋊C₄	—	53	2	no	—
250	C₄⋊C₄	—	53	4	no	—
251	C₄⋊C₄	—	53	2	no	—
252	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
253	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
254	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
255	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
256	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
257	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
258	C₂²⋊C₄	—	53	4	no	—
259	C₂²⋊C₄	—	53	8	no	—
260	C₂²⋊C₄	—	53	8	no	—
261	C₂²⋊C₄	—	53	4	no	—
262	C₂²⋊C₄	—	53	4	no	—
263	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
264	C₄²	—	53	4	no	—
265	C₄²	C₂²×D₅₃	1	1	yes	characteristic
266	C₄²	C₂²×D₅₃	1	1	yes	characteristic
267	C₄²	C₂²×D₅₃	1	1	yes	characteristic
268	C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	C₄²	C₂²×D₅₃	1	4	yes	—
270	C₂³	—	53	4	no	—
271	C₂³	—	53	8	no	—
272	C₂³	—	212	2	no	—
273	C₂³	—	212	2	no	—
274	C₂³	C₂³×D₅₃	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	106	4	no	—
278	Q₈	—	212	2	no	—
279	Q₈	—	212	8	no	—
280	D₄	—	53	8	no	—
281	D₄	—	53	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	53	4	no	—
285	C₂×C₄	—	53	4	no	—
286	C₂×C₄	—	53	4	no	—
287	C₂×C₄	—	106	8	no	—
288	C₂×C₄	—	212	8	no	—
289	C₂×C₄	—	212	4	no	—
290	C₂×C₄	—	212	8	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	Q₈×D₅₃	1	4	yes	complemented
293	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₅₃	1	4	yes	—
297	C₂×C₄	C₂³×D₅₃	1	4	yes	—
298	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	53	8	no	—
304	C₂²	—	106	4	no	—
305	C₂²	—	106	4	no	—
306	C₂²	—	212	8	no	—
307	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
308	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	106	4	no	—
315	C₄	—	106	4	no	—
316	C₄	—	106	4	no	—
317	C₄	C₂×Q₈×D₅₃	1	2	yes	—
318	C₄	C₂×Q₈×D₅₃	1	2	yes	—
319	C₄	#13133263	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	106	8	no	—
322	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
323	C₂	C₂×#13133263	1	1	yes	characteristic
324	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16811334	C₂	semidirect
2. #16811333	C₂	semidirect
3. #16811333	C₂	semidirect
4. #16811329	C₂	semidirect
5. Q₈×D₂₁₂	C₂	semidirect
6. #16811288	C₂	semidirect
7. #16811251	C₂	semidirect
8. #16811248	C₂	semidirect
9. #16811180	C₂	semidirect
10. C₂²⋊Q₈×D₅₃	C₂	semidirect
13. C₅₃×#533	C₂	semidirect
14. C₂×Q₈×D₅₃	C₂²	semidirect
17. #13133217	C₂²	semidirect
18. #13133217	C₂²	semidirect
19. #13133217	C₂²	semidirect
20. #13133216	C₂²	semidirect
21. #13133209	C₂²	semidirect
22. #13133209	C₂²	semidirect
23. #13133208	C₂²	semidirect
24. #13133197	C₂²	semidirect
25. #13133197	C₂²	semidirect
28. #13133195	C₂²	semidirect
29. #13133195	C₂²	semidirect
30. #13133195	C₂²	semidirect
32. #13133193	C₂²	semidirect
33. #13133193	C₂²	semidirect
34. C₄×Dic₁₀₆	C₂²	semidirect
35. C₄×Dic₁₀₆	C₂²	semidirect
46. C₄⋊C₄×D₅₃	C₂²	semidirect
47. C₄⋊C₄×D₅₃	C₂²	semidirect
50. C₄⋊Q₈×C₅₃	C₂²	semidirect
51. C₄⋊Q₈×C₅₃	C₂²	semidirect
52. #125×C₅₃	C₂²	semidirect
55. C₄×Q₈×C₅₃	C₂²	semidirect
56. C₄×Q₈×C₅₃	C₂²	semidirect
67. C₂×D₂₁₂	Q₈	semidirect
110. C₂²×D₅₃	C₂×Q₈	semidirect
116. C₂×C₄×C₅₃	C₂×Q₈	semidirect
129. D₂₁₂	C₂×Q₈	semidirect
130. D₂₁₂	C₂×Q₈	semidirect
172. #533	D₅₃	semidirect
176. C₅₃	#3501	semidirect
184. C₄⋊Q₈	C₂×D₅₃	semidirect
185. C₄⋊Q₈	C₂×D₅₃	semidirect
189. #125	C₂×D₅₃	semidirect
204. C₄×Q₈	C₂×D₅₃	semidirect
205. C₄×Q₈	C₂×D₅₃	semidirect
292. C₂×C₄	Q₈×D₅₃	semidirect