Subgroups — #20288564

Order 6784.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811355	C₂	1	4	yes	maximal, complemented
2	#16811347	C₂	1	2	yes	maximal, complemented
3	#16811338	C₂	1	2	yes	maximal, complemented
4	#16811319	C₂	1	4	yes	maximal, complemented
5	#16811293	C₂	1	2	yes	maximal, complemented
6	#16811277	C₂	1	2	yes	maximal, complemented
7	#16811210	C₂	1	4	yes	maximal, complemented
8	#16811187	C₂	1	2	yes	maximal, complemented
9	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811133	C₂	1	4	yes	maximal, complemented
11	#16811106	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#11719947	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	1	yes	characteristic, complemented
19	#13133213	C₂²	1	4	yes	—
20	#13133212	C₂²	1	8	yes	complemented
21	#13133209	C₂²	1	2	yes	complemented
22	#13133206	C₂²	1	4	yes	—
23	#13133203	C₂²	1	8	yes	complemented
24	#13133201	C₂²	1	8	yes	complemented
25	#13133199	C₂²	1	8	yes	complemented
26	#13133197	C₂²	1	4	yes	complemented
27	#13133196	C₂²	1	2	yes	—
28	#13133196	C₂²	1	2	yes	—
29	#13133191	C₂²	1	4	yes	—
30	#13133191	C₂²	1	4	yes	—
31	#13133189	C₂²	1	8	yes	complemented
32	#13133188	C₂²	1	4	yes	complemented
33	#13133185	C₂²	1	8	yes	—
34	Q₈×Dic₅₃	C₂²	1	4	yes	complemented
35	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
36	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
37	C₄×D₂₁₂	C₂²	1	2	yes	—
38	C₄×D₂₁₂	C₂²	1	2	yes	—
39	C₄×D₂₁₂	C₂²	1	2	yes	—
40	C₄×D₂₁₂	C₂²	1	2	yes	—
41	D₄×Dic₅₃	C₂²	1	8	yes	—
42	D₄×Dic₅₃	C₂²	1	4	yes	complemented
43	#13133177	C₂²	1	8	yes	complemented
44	#13133176	C₂²	1	2	yes	—
45	#13133176	C₂²	1	4	yes	complemented
46	#13133175	C₂²	1	4	yes	complemented
47	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
49	C₂×#11719921	C₂²	1	2	yes	—
50	C₂×#41×C₅₃	C₂²	1	2	yes	—
51	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
53	#124×C₅₃	C₂²	1	2	yes	complemented
54	#121×C₅₃	C₂²	1	2	yes	—
55	#121×C₅₃	C₂²	1	2	yes	—
56	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
57	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
58	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
59	C₂³×D₅₃	C₂³	1	2	yes	—
60	#41×C₅₃	—	2	8	no	—
61	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
62	C₂×D₄×C₅₃	C₂³	1	2	yes	—
63	C₂×D₄×C₅₃	C₂³	1	4	yes	—
64	C₂×D₄×C₅₃	C₂³	1	2	yes	—
65	C₂×D₄×C₅₃	C₂³	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
71	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
72	C₂²×Dic₅₃	C₂³	1	2	yes	—
73	C₂²×Dic₅₃	C₂³	1	8	yes	—
74	#11719947	—	4	8	no	—
75	C₂×D₂₁₂	—	2	4	no	—
76	C₂×D₂₁₂	C₂³	1	2	yes	—
77	C₂×D₂₁₂	C₂³	1	2	yes	—
78	C₂×C₄×D₅₃	—	4	4	no	—
79	C₂×C₄×D₅₃	C₂³	1	4	yes	—
80	C₂×C₄×D₅₃	C₂³	1	4	yes	—
81	C₂×Dic₁₀₆	C₂³	1	2	yes	—
82	C₂×Dic₁₀₆	C₂³	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
85	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
92	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
93	#11719921	C₂³	1	4	yes	—
94	#11719921	C₂³	1	4	yes	—
95	#11719921	C₂³	1	8	yes	—
96	#11719921	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	8	yes	—
101	C₄⋊Dic₅₃	C₂³	1	2	yes	—
102	C₄⋊Dic₅₃	C₂³	1	2	yes	—
103	C₄⋊Dic₅₃	C₂³	1	2	yes	—
104	#11719919	C₂³	1	4	yes	—
105	#11719919	C₂³	1	4	yes	—
106	C₄×Dic₅₃	C₂³	1	8	yes	—
107	C₄×Dic₅₃	C₂³	1	4	yes	—
108	C₂³×C₅₃	C₂⁴	1	4	yes	—
109	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂²×D₅₃	—	2	4	no	—
111	C₂²×D₅₃	—	4	2	no	—
112	C₂²×D₅₃	—	4	2	no	—
113	C₂²×D₅₃	C₂⁴	1	4	yes	—
114	Q₈×C₅₃	—	2	4	no	—
115	D₄×C₅₃	—	2	8	no	—
116	D₄×C₅₃	—	2	4	no	—
117	D₄×C₅₃	—	2	4	no	—
118	D₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	2	8	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	#41	1	4	yes	complemented
123	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
127	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	4	8	no	—
131	C₂²⋊D₅₃	—	4	8	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	8	no	—
134	C₂×Dic₅₃	C₂⁴	1	4	yes	—
135	C₂×Dic₅₃	C₂⁴	1	4	yes	—
136	C₂×Dic₅₃	C₂⁴	1	4	yes	—
137	D₂₁₂	—	2	4	no	—
138	D₂₁₂	—	2	4	no	—
139	C₄×D₅₃	—	2	8	no	—
140	C₄×D₅₃	—	4	4	no	—
141	Dic₁₀₆	—	2	4	no	—
142	Dic₁₀₆	—	4	2	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	4	no	—
146	C₂²×C₅₃	—	2	4	no	—
147	C₂²×C₅₃	—	2	4	no	—
148	C₂²×C₅₃	C₂×#41	1	2	yes	—
149	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
150	C₂×D₅₃	—	2	4	no	—
151	C₂×D₅₃	—	2	4	no	—
152	C₂×D₅₃	—	2	4	no	—
153	C₂×D₅₃	—	4	8	no	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	C₂×#41	1	2	yes	—
159	C₄×C₅₃	C₂×#41	1	2	yes	—
160	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
161	Dic₅₃	—	2	4	no	—
162	Dic₅₃	—	2	4	no	—
163	Dic₅₃	—	2	4	no	—
164	#3506	—	53	1	no	maximal, Sylow
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	—	4	4	no	—
167	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
168	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₅₃	#585	1	1	yes	characteristic
170	D₅₃	—	4	4	no	—
171	#559	—	53	4	no	—
172	#556	—	53	2	no	—
173	#555	—	53	2	no	—
174	#553	—	53	4	no	—
175	#550	—	53	2	no	—
176	#547	—	53	2	no	—
177	#538	—	53	4	no	—
178	#535	—	53	2	no	—
179	#533	—	53	1	no	—
180	#532	D₅₃	1	1	yes	characteristic, complemented
181	#525	—	53	4	no	—
182	#520	—	53	2	no	—
183	C₂×C₄×D₄	—	53	1	no	—
184	C₅₃	#3506	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	53	4	no	—
186	C₂×#41	C₂×D₅₃	1	2	yes	—
187	C₂²×D₄	—	53	1	no	—
188	C₂³×C₄	—	53	2	no	—
189	C₄⋊Q₈	—	53	1	no	—
190	C₄⋊Q₈	—	53	1	no	—
191	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
192	#127	—	53	4	no	—
193	#127	C₂×D₅₃	1	1	yes	characteristic, complemented
194	#126	—	53	8	no	—
195	#125	—	53	2	no	—
196	#124	—	53	4	no	—
197	#124	—	53	8	no	—
198	#124	C₂×D₅₃	1	2	yes	complemented
199	#123	—	53	8	no	—
200	#123	—	53	8	no	—
201	C₂²⋊Q₈	—	53	2	no	—
202	C₂²⋊Q₈	—	53	2	no	—
203	C₂²⋊Q₈	—	53	4	no	—
204	#121	—	53	4	no	—
205	#121	—	53	8	no	—
206	#121	—	53	4	no	—
207	#121	—	53	4	no	—
208	#121	C₂×D₅₃	1	2	yes	—
209	#121	C₂×D₅₃	1	2	yes	—
210	#120	—	53	8	no	—
211	C₄×Q₈	—	53	4	no	—
212	C₄×Q₈	—	53	2	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×D₄	—	53	8	no	—
215	C₄×D₄	—	53	4	no	—
216	C₄×D₄	—	53	8	no	—
217	C₄×D₄	—	53	2	no	—
218	C₄×D₄	—	53	2	no	—
219	C₄×D₄	—	53	2	no	—
220	C₄×D₄	—	53	2	no	—
221	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
222	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
223	#117	—	53	2	no	—
224	#117	—	53	4	no	—
225	#117	—	53	4	no	—
226	C₂×C₄⋊C₄	—	53	1	no	—
227	C₂×C₄⋊C₄	—	53	4	no	—
228	C₂×C₂²⋊C₄	—	53	2	no	—
229	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
230	C₂⁴	—	53	2	no	—
231	#41	—	2	8	no	—
232	#41	—	212	8	no	—
233	C₂×Q₈	—	53	2	no	—
234	C₂×Q₈	—	53	2	no	—
235	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
236	C₂×D₄	—	53	8	no	—
237	C₂×D₄	—	53	8	no	—
238	C₂×D₄	—	53	2	no	—
239	C₂×D₄	—	53	2	no	—
240	C₂×D₄	—	106	4	no	—
241	C₂×D₄	C₂²×D₅₃	1	4	yes	—
242	C₂×D₄	C₂²×D₅₃	1	2	yes	—
243	C₂×D₄	C₂²×D₅₃	1	2	yes	—
244	C₂×D₄	C₂²×D₅₃	1	2	yes	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	8	no	—
247	C₂²×C₄	—	53	4	no	—
248	C₂²×C₄	—	53	4	no	—
249	C₂²×C₄	—	212	4	no	—
250	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
252	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
254	C₄⋊C₄	—	53	2	no	—
255	C₄⋊C₄	—	53	2	no	—
256	C₄⋊C₄	—	53	2	no	—
257	C₄⋊C₄	—	53	8	no	—
258	C₄⋊C₄	—	53	2	no	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	4	no	—
261	C₄⋊C₄	—	53	4	no	—
262	C₄⋊C₄	—	53	2	no	—
263	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
264	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
265	C₂²⋊C₄	—	53	4	no	—
266	C₂²⋊C₄	—	53	8	no	—
267	C₂²⋊C₄	—	53	4	no	—
268	C₂²⋊C₄	—	53	4	no	—
269	C₂²⋊C₄	—	53	8	no	—
270	C₂²⋊C₄	—	53	8	no	—
271	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
272	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
273	C₄²	—	53	8	no	—
274	C₄²	—	53	4	no	—
275	C₄²	C₂²×D₅₃	1	1	yes	characteristic
276	C₄²	C₂²×D₅₃	1	1	yes	characteristic
277	C₄²	C₂²×D₅₃	1	1	yes	characteristic
278	C₄²	C₂²×D₅₃	1	1	yes	characteristic
279	C₂³	—	53	4	no	—
280	C₂³	—	106	4	no	—
281	C₂³	—	212	2	no	—
282	C₂³	—	212	2	no	—
283	C₂³	C₂³×D₅₃	1	4	yes	—
284	C₂³	C₂³×D₅₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	106	4	no	—
287	Q₈	—	212	2	no	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	106	4	no	—
293	D₄	—	106	4	no	—
294	D₄	—	212	8	no	—
295	D₄	—	212	8	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	53	4	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	106	8	no	—
303	C₂×C₄	—	212	4	no	—
304	C₂×C₄	—	212	2	no	—
305	C₂×C₄	—	212	8	no	—
306	C₂×C₄	#11719947	1	4	yes	complemented
307	C₂×C₄	C₂³×D₅₃	1	4	yes	—
308	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	106	4	no	—
320	C₂²	—	106	4	no	—
321	C₂²	—	106	4	no	—
322	C₂²	—	212	8	no	—
323	C₂²	C₂×#11719947	1	2	yes	—
324	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	106	4	no	—
330	C₄	—	106	4	no	—
331	C₄	—	106	4	no	—
332	C₄	C₂×#11719947	1	2	yes	—
333	C₄	C₂×#11719947	1	2	yes	—
334	C₄	#13133263	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	212	4	no	—
338	C₂	C₂²×#11719947	1	1	yes	characteristic
339	C₂	C₂×#13133263	1	1	yes	characteristic
340	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3506
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #16811355	C₂	semidirect
2. #16811347	C₂	semidirect
3. #16811338	C₂	semidirect
4. #16811319	C₂	semidirect
5. #16811293	C₂	semidirect
6. #16811277	C₂	semidirect
7. #16811210	C₂	semidirect
8. #16811187	C₂	semidirect
9. #16811180	C₂	semidirect
10. #16811133	C₂	semidirect
11. #16811106	C₂	semidirect
12. C₂×C₄×D₂₁₂	C₂	semidirect
13. C₅₃×#532	C₂	semidirect
14. C₂×#11719947	C₂²	semidirect
17. #13133217	C₂²	semidirect
18. #13133217	C₂²	semidirect
20. #13133212	C₂²	semidirect
21. #13133209	C₂²	semidirect
23. #13133203	C₂²	semidirect
24. #13133201	C₂²	semidirect
25. #13133199	C₂²	semidirect
26. #13133197	C₂²	semidirect
31. #13133189	C₂²	semidirect
32. #13133188	C₂²	semidirect
34. Q₈×Dic₅₃	C₂²	semidirect
35. C₄×Dic₁₀₆	C₂²	semidirect
36. C₄×Dic₁₀₆	C₂²	semidirect
42. D₄×Dic₅₃	C₂²	semidirect
43. #13133177	C₂²	semidirect
45. #13133176	C₂²	semidirect
46. #13133175	C₂²	semidirect
48. C₂×C₄⋊Dic₅₃	C₂²	semidirect
51. C₄⋊Q₈×C₅₃	C₂²	semidirect
52. #127×C₅₃	C₂²	semidirect
53. #124×C₅₃	C₂²	semidirect
56. C₄×D₄×C₅₃	C₂²	semidirect
57. C₄×D₄×C₅₃	C₂²	semidirect
122. C₂×C₄×C₅₃	#41	semidirect
180. #532	D₅₃	semidirect
184. C₅₃	#3506	semidirect
191. C₄⋊Q₈	C₂×D₅₃	semidirect
193. #127	C₂×D₅₃	semidirect
198. #124	C₂×D₅₃	semidirect
221. C₄×D₄	C₂×D₅₃	semidirect
222. C₄×D₄	C₂×D₅₃	semidirect
306. C₂×C₄	#11719947	semidirect