Subgroups — #20288527

Order 6784.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
2	#16811351	C₂	1	2	yes	maximal, complemented
3	#16811333	C₂	1	2	yes	maximal, complemented
4	#16811331	C₂	1	4	yes	maximal, complemented
5	Q₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
6	#16811286	C₂	1	4	yes	maximal, complemented
7	#16811253	C₂	1	4	yes	maximal, complemented
8	#16811251	C₂	1	2	yes	maximal, complemented
9	#16811178	C₂	1	2	yes	maximal, complemented
10	C₂×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
12	#16811103	C₂	1	2	yes	maximal
13	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
14	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#11719945	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
17	C₂×Q₈×D₅₃	C₂²	1	4	yes	—
18	#13133217	C₂²	1	2	yes	—
19	#13133217	C₂²	1	4	yes	—
20	#13133217	C₂²	1	2	yes	—
21	#13133217	C₂²	1	2	yes	—
22	#13133217	C₂²	1	2	yes	—
23	#13133216	C₂²	1	8	yes	—
24	#13133216	C₂²	1	4	yes	—
25	#13133216	C₂²	1	4	yes	—
26	#13133215	C₂²	1	4	yes	—
27	#13133209	C₂²	1	4	yes	—
28	#13133208	C₂²	1	4	yes	—
29	#13133207	C₂²	1	8	yes	—
30	#13133197	C₂²	1	4	yes	—
31	#13133197	C₂²	1	8	yes	—
32	#13133195	C₂²	1	8	yes	—
33	#13133194	C₂²	1	8	yes	—
34	#13133193	C₂²	1	4	yes	—
35	Q₈×Dic₅₃	C₂²	1	4	yes	—
36	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₁₀₆	C₂²	1	4	yes	—
39	#13133182	C₂²	1	4	yes	—
40	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
41	C₄×D₂₁₂	C₂²	1	4	yes	—
42	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
44	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
45	#13133177	C₂²	1	8	yes	—
46	#13133176	C₂²	1	1	yes	characteristic
47	#13133175	C₂²	1	4	yes	—
48	#13133174	C₂²	1	2	yes	—
49	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
50	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
51	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
52	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
53	C₄²×D₅₃	C₂²	1	2	yes	—
54	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
56	#125×C₅₃	C₂²	1	2	yes	—
57	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
58	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
59	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
60	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
61	#117×C₅₃	C₂²	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
64	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
69	C₂²×Dic₅₃	C₂³	1	4	yes	—
70	Q₈×D₅₃	—	4	8	no	—
71	#11719945	Q₈	1	8	yes	complemented
72	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂³	1	2	yes	—
74	C₂×C₄×D₅₃	C₂³	1	2	yes	—
75	C₂×C₄×D₅₃	C₂³	1	8	yes	—
76	C₂×C₄×D₅₃	C₂³	1	2	yes	—
77	C₂×Dic₁₀₆	—	4	4	no	—
78	C₂×Dic₁₀₆	C₂³	1	4	yes	—
79	C₂×Dic₁₀₆	C₂³	1	8	yes	—
80	C₂×Dic₁₀₆	C₂³	1	4	yes	—
81	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
84	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
87	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
88	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₄²×C₅₃	C₂³	1	4	yes	—
94	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
95	#11719921	C₂³	1	2	yes	—
96	#11719921	C₂³	1	2	yes	—
97	#11719921	C₂³	1	8	yes	—
98	C₄⋊Dic₅₃	C₂³	1	4	yes	—
99	C₄⋊Dic₅₃	C₂³	1	4	yes	—
100	C₄⋊Dic₅₃	C₂³	1	4	yes	—
101	C₄⋊Dic₅₃	C₂³	1	4	yes	—
102	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₅₃	C₂³	1	8	yes	—
105	#11719919	C₂³	1	8	yes	—
106	#11719919	C₂³	1	8	yes	—
107	#11719919	C₂³	1	2	yes	—
108	#11719919	C₂³	1	2	yes	—
109	#11719919	C₂³	1	8	yes	—
110	C₄×Dic₅₃	C₂³	1	2	yes	—
111	C₄×Dic₅₃	C₂³	1	2	yes	—
112	C₄×Dic₅₃	C₂³	1	2	yes	—
113	C₄×Dic₅₃	C₂³	1	8	yes	—
114	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂²×D₅₃	C₂⁴	1	2	yes	—
116	Q₈×C₅₃	—	2	4	no	—
117	Q₈×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	complemented
121	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
123	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
125	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₅₃	C₂×Q₈	1	8	yes	complemented
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	C₂⁴	1	2	yes	—
132	C₂×Dic₅₃	C₂⁴	1	8	yes	—
133	C₂×Dic₅₃	C₂⁴	1	2	yes	—
134	C₂×Dic₅₃	C₂⁴	1	2	yes	—
135	D₂₁₂	C₂×Q₈	1	4	yes	complemented
136	C₄×D₅₃	—	2	4	no	—
137	C₄×D₅₃	—	2	4	no	—
138	C₄×D₅₃	—	4	8	no	—
139	C₄×D₅₃	C₂×Q₈	1	8	yes	complemented
140	Dic₁₀₆	—	2	8	no	—
141	Dic₁₀₆	—	4	4	no	—
142	Dic₁₀₆	—	4	8	no	—
143	Dic₁₀₆	C₂×Q₈	1	4	yes	complemented
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
147	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
148	C₂×D₅₃	—	2	2	no	—
149	C₂×D₅₃	—	2	2	no	—
150	C₂×D₅₃	C₂²×Q₈	1	4	yes	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	4	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
156	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
157	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
158	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
159	Dic₅₃	—	2	8	no	—
160	Dic₅₃	—	2	2	no	—
161	Dic₅₃	—	2	2	no	—
162	Dic₅₃	C₂²×Q₈	1	4	yes	—
163	#3503	—	53	1	no	maximal, Sylow
164	C₂×C₅₃	—	2	2	no	—
165	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
166	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
168	D₅₃	—	2	4	no	—
169	Q₈²	—	53	2	no	—
170	#557	—	53	2	no	—
171	#554	—	53	4	no	—
172	#554	—	53	2	no	—
173	D₄×Q₈	—	53	4	no	—
174	D₄×Q₈	—	53	2	no	—
175	#544	—	53	4	no	—
176	#544	—	53	2	no	—
177	#533	—	53	2	no	—
178	#533	D₅₃	1	1	yes	characteristic, complemented
179	C₂×C₄⋊Q₈	—	53	2	no	—
180	C₂×C₄⋊Q₈	—	53	1	no	—
181	#520	—	53	2	no	—
182	C₄×#41	—	53	1	no	—
183	C₅₃	#3503	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	53	1	no	—
185	C₂²×Q₈	—	53	4	no	—
186	C₂²×Q₈	—	53	2	no	—
187	C₄⋊Q₈	—	53	4	no	—
188	C₄⋊Q₈	—	53	4	no	—
189	C₄⋊Q₈	—	53	4	no	—
190	C₄⋊Q₈	—	53	8	no	—
191	C₄⋊Q₈	—	53	2	no	—
192	C₄⋊Q₈	—	53	2	no	—
193	C₄⋊Q₈	—	53	2	no	—
194	C₄⋊Q₈	—	53	2	no	—
195	C₄⋊Q₈	—	53	4	no	—
196	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
198	#125	—	53	4	no	—
199	#125	—	53	8	no	—
200	#125	—	53	4	no	—
201	#125	C₂×D₅₃	1	2	yes	—
202	C₂²⋊Q₈	—	53	4	no	—
203	C₂²⋊Q₈	—	53	4	no	—
204	C₂²⋊Q₈	—	53	8	no	—
205	C₂²⋊Q₈	—	53	8	no	—
206	C₂²⋊Q₈	—	53	8	no	—
207	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
208	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
209	C₄×Q₈	—	53	4	no	—
210	C₄×Q₈	—	53	4	no	—
211	C₄×Q₈	—	53	1	no	—
212	C₄×Q₈	—	53	1	no	—
213	C₄×Q₈	—	53	4	no	—
214	C₄×Q₈	C₂×D₅₃	1	2	yes	—
215	C₄×Q₈	C₂×D₅₃	1	2	yes	—
216	C₄×D₄	—	53	2	no	—
217	C₄×D₄	—	53	2	no	—
218	C₄×D₄	—	53	8	no	—
219	C₄×D₄	—	53	1	no	—
220	C₄×D₄	—	53	1	no	—
221	C₄×D₄	—	53	4	no	—
222	#117	—	53	1	no	—
223	#117	—	53	4	no	—
224	#117	—	53	2	no	—
225	#117	C₂×D₅₃	1	2	yes	—
226	C₂×C₄⋊C₄	—	53	2	no	—
227	C₂×C₄⋊C₄	—	53	2	no	—
228	C₂×C₄⋊C₄	—	53	4	no	—
229	C₂×C₄⋊C₄	—	53	4	no	—
230	C₂×C₄²	—	53	2	no	—
231	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
232	#41	—	53	8	no	—
233	C₂×Q₈	—	53	1	no	—
234	C₂×Q₈	—	53	4	no	—
235	C₂×Q₈	—	53	8	no	—
236	C₂×Q₈	—	53	4	no	—
237	C₂×Q₈	—	212	4	no	—
238	C₂×Q₈	—	212	8	no	—
239	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
240	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
241	C₂×D₄	—	53	2	no	—
242	C₂×D₄	—	53	1	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	4	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	8	no	—
248	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
251	C₄⋊C₄	—	53	4	no	—
252	C₄⋊C₄	—	53	4	no	—
253	C₄⋊C₄	—	53	8	no	—
254	C₄⋊C₄	—	53	4	no	—
255	C₄⋊C₄	—	53	8	no	—
256	C₄⋊C₄	—	53	4	no	—
257	C₄⋊C₄	—	53	2	no	—
258	C₄⋊C₄	—	53	1	no	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	1	no	—
261	C₄⋊C₄	—	53	8	no	—
262	C₄⋊C₄	—	53	8	no	—
263	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
264	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
265	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
266	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
267	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
268	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
269	C₂²⋊C₄	—	53	2	no	—
270	C₂²⋊C₄	—	53	2	no	—
271	C₂²⋊C₄	—	53	2	no	—
272	C₂²⋊C₄	—	53	8	no	—
273	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
274	C₄²	—	53	2	no	—
275	C₄²	—	53	2	no	—
276	C₄²	—	53	2	no	—
277	C₄²	—	53	8	no	—
278	C₄²	C₂²×D₅₃	1	1	yes	characteristic
279	C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₄²	C₂²×D₅₃	1	1	yes	characteristic
281	C₄²	C₂²×D₅₃	1	1	yes	characteristic
282	C₄²	C₂²×D₅₃	1	4	yes	—
283	C₂³	—	53	2	no	—
284	C₂³	C₂³×D₅₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	53	4	no	—
288	Q₈	—	106	8	no	—
289	Q₈	—	212	4	no	—
290	Q₈	—	212	8	no	—
291	D₄	—	53	8	no	—
292	D₄	—	53	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	8	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	8	no	—
300	C₂×C₄	—	106	4	no	—
301	C₂×C₄	—	106	4	no	—
302	C₂×C₄	—	212	8	no	—
303	C₂×C₄	—	212	4	no	—
304	C₂×C₄	Q₈×D₅₃	1	4	yes	complemented
305	C₂×C₄	C₂³×D₅₃	1	4	yes	—
306	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₅₃	1	4	yes	—
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	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	53	4	no	—
316	C₂²	—	106	2	no	—
317	C₂²	—	106	2	no	—
318	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
319	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	53	4	no	—
325	C₄	—	106	2	no	—
326	C₄	—	106	8	no	—
327	C₄	—	106	2	no	—
328	C₄	C₂×Q₈×D₅₃	1	2	yes	—
329	C₄	C₂×Q₈×D₅₃	1	2	yes	—
330	C₄	#13133263	1	2	yes	—
331	C₄	#13133264	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	106	4	no	—
334	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
335	C₂	C₂×#13133263	1	1	yes	characteristic
336	C₂	C₂×#13133264	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	#3503
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. Q₈×Dic₁₀₆	C₂	semidirect
2. #16811351	C₂	semidirect
3. #16811333	C₂	semidirect
4. #16811331	C₂	semidirect
5. Q₈×D₂₁₂	C₂	semidirect
6. #16811286	C₂	semidirect
7. #16811253	C₂	semidirect
8. #16811251	C₂	semidirect
9. #16811178	C₂	semidirect
10. C₂×#13133217	C₂	semidirect
11. C₄⋊Q₈×D₅₃	C₂	semidirect
14. C₅₃×#533	C₂	semidirect
71. #11719945	Q₈	semidirect
120. C₂×C₄×C₅₃	C₂×Q₈	semidirect
129. C₂²⋊D₅₃	C₂×Q₈	semidirect
135. D₂₁₂	C₂×Q₈	semidirect
139. C₄×D₅₃	C₂×Q₈	semidirect
143. Dic₁₀₆	C₂×Q₈	semidirect
178. #533	D₅₃	semidirect
183. C₅₃	#3503	semidirect
304. C₂×C₄	Q₈×D₅₃	semidirect