Subgroups — #20253861

Order 6016.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094053	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094048	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094043	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689693	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689686	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689517	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689514	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689511	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689506	C₂	1	1	yes	maximal, characteristic, complemented
11	#15689505	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689495	C₂	1	1	yes	maximal, characteristic, complemented
13	#15689494	C₂	1	1	yes	maximal, characteristic, complemented
14	#15689080	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₇×#496	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10625352	C₂²	1	1	yes	characteristic
17	C₂×#10625352	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₄₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₄₇	C₂²	1	1	yes	characteristic
20	#13094053	—	2	4	no	—
21	#13094048	—	2	4	no	—
22	#13094043	—	2	4	no	—
23	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×#10625337	C₂²	1	1	yes	characteristic, complemented
26	C₂×#10625325	C₂²	1	1	yes	characteristic
27	C₂×#10625325	C₂²	1	1	yes	characteristic
28	C₂×#10625335	C₂²	1	1	yes	characteristic
29	C₂×#10625335	C₂²	1	1	yes	characteristic
30	C₂×#10625335	C₂²	1	1	yes	characteristic, complemented
31	C₂×#10625335	C₂²	1	1	yes	characteristic, complemented
32	C₂×#10625335	C₂²	1	1	yes	characteristic, complemented
33	C₂×#10625336	C₂²	1	1	yes	characteristic
34	SD₁₆×D₄₇	—	2	4	no	—
35	C₂×#10625334	C₂²	1	1	yes	characteristic
36	C₂×#10625324	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
38	#13094024	C₂²	1	1	yes	characteristic, complemented
39	#13094022	C₂²	1	1	yes	characteristic, complemented
40	#13094020	C₂²	1	1	yes	characteristic
41	#13094012	C₂²	1	1	yes	characteristic
42	#13094011	C₂²	1	1	yes	characteristic
43	#13094011	C₂²	1	1	yes	characteristic, complemented
44	#13094006	—	2	2	no	—
45	#13094001	—	2	2	no	—
46	#13093996	—	2	2	no	—
47	#13093992	—	2	2	no	—
48	#13093981	C₂²	1	1	yes	characteristic
49	#13093934	C₂²	1	1	yes	characteristic, complemented
50	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
54	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
56	#127×C₄₇	C₂²	1	1	yes	characteristic
57	#124×C₄₇	C₂²	1	1	yes	characteristic, complemented
58	#97×C₄₇	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₄₇	—	2	1	no	—
60	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₇	—	2	1	no	—
66	C₂×C₂²⋊D₄₇	—	2	1	no	—
67	C₂×C₂²⋊D₄₇	—	2	1	no	—
68	C₂×C₂²⋊D₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	1	no	—
70	C₂²×Dic₄₇	—	2	1	no	—
71	Q₈×D₄₇	—	2	2	no	—
72	Q₈×D₄₇	—	2	2	no	—
73	#10625352	—	2	2	no	—
74	#10625352	—	2	2	no	—
75	#10625352	—	2	2	no	—
76	#10625352	—	2	2	no	—
77	D₄×D₄₇	—	2	2	no	—
78	D₄×D₄₇	—	2	2	no	—
79	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₄₇	—	2	1	no	—
81	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₉₄	—	2	1	no	—
83	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
84	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
85	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
86	SD₁₆×C₄₇	—	2	2	no	—
87	SD₁₆×C₄₇	—	2	2	no	—
88	D₈×C₄₇	—	2	2	no	—
89	D₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₄₇	—	2	2	no	—
95	C₂²⋊Dic₄₇	—	2	2	no	—
96	#10625337	—	2	2	no	—
97	#10625336	—	2	2	no	—
98	#10625335	—	2	2	no	—
99	#10625335	—	2	2	no	—
100	#10625335	—	2	2	no	—
101	#10625335	—	2	2	no	—
102	#10625335	—	2	2	no	—
103	#10625334	—	2	2	no	—
104	(C₂×D₄₇)⋊C₄	—	2	2	no	—
105	(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	Dic₄₇⋊C₄	—	2	2	no	—
108	Dic₄₇⋊C₄	D₄	1	2	yes	—
109	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
110	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
112	Dic₁₈₈	—	2	2	no	—
113	Dic₁₈₈	—	2	2	no	—
114	#10625325	—	2	2	no	—
115	#10625325	—	2	2	no	—
116	#10625324	D₄	1	4	yes	complemented
117	C₈×D₄₇	D₄	1	4	yes	complemented
118	C₂³×C₄₇	—	2	1	no	—
119	C₂³×C₄₇	—	2	1	no	—
120	C₂³×C₄₇	—	2	1	no	—
121	C₂²×D₄₇	—	2	1	no	—
122	C₂²×D₄₇	—	4	2	no	—
123	C₂²×D₄₇	—	8	2	no	—
124	C₂²×D₄₇	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₄₇	—	2	1	no	—
126	Q₈×C₄₇	—	2	1	no	—
127	D₄×C₄₇	—	2	1	no	—
128	D₄×C₄₇	—	2	1	no	—
129	D₄×C₄₇	—	2	1	no	—
130	D₄×C₄₇	—	2	1	no	—
131	D₄×C₄₇	—	2	1	no	—
132	D₄×C₄₇	—	2	1	no	—
133	D₄×C₄₇	—	8	2	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
137	C₂²⋊D₄₇	—	4	2	no	—
138	C₂²⋊D₄₇	—	4	2	no	—
139	C₂²⋊D₄₇	—	4	2	no	—
140	C₂²⋊D₄₇	—	8	2	no	—
141	C₂×Dic₄₇	—	2	1	no	—
142	C₂×Dic₄₇	—	2	1	no	—
143	C₂×Dic₄₇	—	2	1	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
147	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
148	D₁₈₈	—	2	1	no	—
149	D₁₈₈	—	2	1	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₄×D₄₇	C₂×D₄	1	2	yes	—
152	C₄×D₄₇	C₂×D₄	1	2	yes	—
153	Dic₉₄	—	2	1	no	—
154	Dic₉₄	—	2	1	no	—
155	Dic₉₄	—	2	1	no	—
156	Dic₉₄	—	2	1	no	—
157	Dic₉₄	—	2	1	no	—
158	Dic₉₄	—	2	1	no	—
159	Dic₉₄	—	4	2	no	—
160	C₈×C₄₇	C₂×D₄	1	2	yes	—
161	C₈×C₄₇	C₂×D₄	1	2	yes	—
162	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
163	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
164	C₂²×C₄₇	—	4	1	no	—
165	C₂²×C₄₇	—	4	1	no	—
166	C₂²×C₄₇	—	4	1	no	—
167	C₂²×C₄₇	—	4	2	no	—
168	C₂²×C₄₇	—	4	2	no	—
169	C₂²×C₄₇	—	4	2	no	—
170	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×D₄₇	—	2	2	no	—
172	C₂×D₄₇	—	4	1	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₂×D₄₇	—	8	4	no	—
175	C₂×D₄₇	#127	1	2	yes	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	—	4	1	no	—
178	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
179	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
180	Dic₄₇	—	2	2	no	—
181	Dic₄₇	—	4	1	no	—
182	Dic₄₇	—	4	1	no	—
183	Dic₄₇	—	4	1	no	—
184	Dic₄₇	#127	1	2	yes	—
185	#3154	—	47	1	no	maximal, Sylow
186	C₂×C₄₇	—	8	1	no	—
187	C₂×C₄₇	—	8	1	no	—
188	C₂×C₄₇	—	8	1	no	—
189	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
190	C₂×C₄₇	#577	1	2	yes	—
191	D₄₇	—	2	2	no	—
192	D₄₇	—	8	1	no	—
193	C₂×#137	—	47	1	no	—
194	C₂×AGL(1,ℤ/8)	—	47	1	no	—
195	C₂×#135	—	47	1	no	—
196	C₂²×SD₁₆	—	47	1	no	—
197	#536	—	47	1	no	—
198	#535	—	47	1	no	—
199	#497	—	47	1	no	—
200	#497	—	47	1	no	—
201	#496	—	47	1	no	—
202	#496	D₄₇	1	1	yes	characteristic, complemented
203	#495	—	47	1	no	—
204	#495	—	47	1	no	—
205	#492	—	47	1	no	—
206	#492	—	47	1	no	—
207	#405	—	47	1	no	—
208	C₄₇	#3154	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	47	1	no	—
210	C₂×#41	—	47	1	no	—
211	C₂²×Q₈	—	47	1	no	—
212	C₂²×D₄	—	47	1	no	—
213	#137	—	94	4	no	—
214	AGL(1,ℤ/8)	—	94	4	no	—
215	#135	—	94	4	no	—
216	C₂×Q₁₆	—	47	1	no	—
217	C₂×Q₁₆	—	47	1	no	—
218	C₂×Q₁₆	—	47	1	no	—
219	C₂×SD₁₆	—	47	1	no	—
220	C₂×SD₁₆	—	47	1	no	—
221	C₂×SD₁₆	—	47	1	no	—
222	C₂×SD₁₆	—	47	1	no	—
223	C₂×SD₁₆	—	47	1	no	—
224	C₂×SD₁₆	—	47	1	no	—
225	C₂×SD₁₆	—	47	1	no	—
226	C₂×SD₁₆	—	47	1	no	—
227	C₂×SD₁₆	—	94	4	no	—
228	C₂×SD₁₆	C₂×D₄₇	1	1	yes	characteristic
229	C₂×SD₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
230	C₂×D₈	—	47	1	no	—
231	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
232	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic, complemented
233	C₂×M₁₆	—	47	1	no	—
234	C₂²×C₈	—	47	1	no	—
235	C₄⋊Q₈	—	47	1	no	—
236	C₄⋊Q₈	—	47	1	no	—
237	#127	—	47	1	no	—
238	#127	C₂×D₄₇	1	1	yes	characteristic
239	#124	—	47	1	no	—
240	#124	—	47	1	no	—
241	#124	—	47	1	no	—
242	#124	C₂×D₄₇	1	1	yes	characteristic, complemented
243	#123	—	94	2	no	—
244	C₂²⋊Q₈	—	94	2	no	—
245	#121	—	94	2	no	—
246	#120	—	94	2	no	—
247	#117	—	47	1	no	—
248	#97	—	47	1	no	—
249	#97	C₂×D₄₇	1	1	yes	characteristic, complemented
250	C₄×C₈	—	47	1	no	—
251	C₄×C₈	—	47	1	no	—
252	C₂⁴	—	94	1	no	—
253	#41	—	94	2	no	—
254	#41	—	94	2	no	—
255	#41	—	94	2	no	—
256	#41	—	94	2	no	—
257	C₂×Q₈	—	47	1	no	—
258	C₂×Q₈	—	47	1	no	—
259	C₂×Q₈	—	47	1	no	—
260	C₂×Q₈	—	94	1	no	—
261	C₂×Q₈	—	94	2	no	—
262	C₂×Q₈	—	94	2	no	—
263	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	47	1	no	—
266	C₂×D₄	—	94	2	no	—
267	C₂×D₄	—	94	2	no	—
268	C₂×D₄	—	94	2	no	—
269	C₂×D₄	—	94	1	no	—
270	C₂×D₄	—	94	1	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
273	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	—	47	1	no	—
276	C₂²×C₄	—	94	1	no	—
277	C₂²×C₄	—	94	1	no	—
278	C₂²×C₄	—	94	1	no	—
279	Q₁₆	—	94	2	no	—
280	Q₁₆	—	94	2	no	—
281	Q₁₆	—	94	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	94	2	no	—
285	SD₁₆	—	94	2	no	—
286	SD₁₆	—	94	2	no	—
287	SD₁₆	—	94	2	no	—
288	SD₁₆	—	94	2	no	—
289	SD₁₆	—	94	2	no	—
290	SD₁₆	—	94	2	no	—
291	SD₁₆	—	94	2	no	—
292	D₈	—	2	2	no	—
293	D₈	—	2	2	no	—
294	D₈	—	94	2	no	—
295	M₁₆	—	47	4	no	—
296	C₂×C₈	—	47	1	no	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	4	no	—
299	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
301	C₄⋊C₄	—	47	2	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	47	2	no	—
306	C₂²⋊C₄	—	94	2	no	—
307	C₂²⋊C₄	—	94	2	no	—
308	C₂²⋊C₄	—	94	2	no	—
309	C₄²	—	47	1	no	—
310	C₄²	C₂²×D₄₇	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	2	1	no	—
314	C₂³	—	47	1	no	—
315	C₂³	—	94	1	no	—
316	C₂³	—	376	2	no	—
317	C₂³	—	188	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	94	1	no	—
321	Q₈	—	94	1	no	—
322	Q₈	—	94	1	no	—
323	Q₈	—	94	1	no	—
324	Q₈	—	94	1	no	—
325	Q₈	—	94	1	no	—
326	Q₈	—	188	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	2	1	no	—
332	D₄	—	2	1	no	—
333	D₄	—	8	2	no	—
334	D₄	—	94	1	no	—
335	D₄	—	94	1	no	—
336	D₄	—	376	2	no	—
337	D₄	—	188	2	no	—
338	D₄	—	188	2	no	—
339	D₄	—	188	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	47	1	no	—
342	C₂×C₄	—	47	2	no	—
343	C₂×C₄	—	47	2	no	—
344	C₂×C₄	—	47	2	no	—
345	C₂×C₄	—	94	1	no	—
346	C₂×C₄	—	94	1	no	—
347	C₂×C₄	—	94	1	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	D₄×D₄₇	1	2	yes	—
352	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
353	C₈	—	47	2	no	—
354	C₈	—	47	2	no	—
355	C₈	D₄×D₄₇	1	2	yes	—
356	C₈	D₄×D₄₇	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	47	2	no	—
364	C₂²	—	94	2	no	—
365	C₂²	—	376	4	no	—
366	C₂²	—	188	1	no	—
367	C₂²	—	188	2	no	—
368	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	47	2	no	—
372	C₄	—	94	2	no	—
373	C₄	—	188	1	no	—
374	C₄	—	188	1	no	—
375	C₄	—	188	1	no	—
376	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
377	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	8	1	no	—
381	C₂	—	94	2	no	—
382	C₂	—	376	1	no	—
383	C₂	#127×D₄₇	1	1	yes	characteristic
384	C₂	#15689933	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#496	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#496	D₄₇
hypocenter	C₄₇	#3154
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. C₂×#13094053	C₂	semidirect
2. C₂×#13094048	C₂	semidirect
3. C₂×#13094043	C₂	semidirect
4. C₂×SD₁₆×D₄₇	C₂	semidirect
5. #15689693	C₂	semidirect
6. #15689686	C₂	semidirect
7. #15689517	C₂	semidirect
8. #15689514	C₂	semidirect
9. #15689511	C₂	semidirect
10. #15689506	C₂	semidirect
11. #15689505	C₂	semidirect
12. #15689495	C₂	semidirect
13. #15689494	C₂	semidirect
14. #15689080	C₂	semidirect
15. C₄₇×#496	C₂	semidirect
23. C₂×Dic₁₈₈	C₂²	semidirect
24. C₂×Dic₁₈₈	C₂²	semidirect
25. C₂×#10625337	C₂²	semidirect
30. C₂×#10625335	C₂²	semidirect
31. C₂×#10625335	C₂²	semidirect
32. C₂×#10625335	C₂²	semidirect
38. #13094024	C₂²	semidirect
39. #13094022	C₂²	semidirect
43. #13094011	C₂²	semidirect
49. #13093934	C₂²	semidirect
50. C₄×C₄₇⋊C₈	C₂²	semidirect
51. C₈×Dic₄₇	C₂²	semidirect
53. C₂×SD₁₆×C₄₇	C₂²	semidirect
55. C₂×D₈×C₄₇	C₂²	semidirect
57. #124×C₄₇	C₂²	semidirect
58. #97×C₄₇	C₂²	semidirect
116. #10625324	D₄	semidirect
117. C₈×D₄₇	D₄	semidirect
202. #496	D₄₇	semidirect
208. C₄₇	#3154	semidirect
229. C₂×SD₁₆	C₂×D₄₇	semidirect
232. C₂×D₈	C₂×D₄₇	semidirect
242. #124	C₂×D₄₇	semidirect
249. #97	C₂×D₄₇	semidirect