Subgroups — #26199715

Order 12032.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254569	C₂	1	2	yes	maximal
2	#20253974	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252363	C₂	1	1	yes	maximal, characteristic
4	#20242917	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242912	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242883	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242878	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242847	C₂	1	2	yes	maximal, complemented
9	#20242840	C₂	1	2	yes	maximal, complemented
10	#20242096	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689680	C₂²	1	2	yes	—
13	#15689657	C₂²	1	1	yes	characteristic
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689525	—	2	2	no	—
16	#15689503	C₂²	1	1	yes	characteristic
17	#15689478	—	2	2	no	—
18	#15689474	C₂²	1	2	yes	—
19	#15689465	—	2	2	no	—
20	#15689448	—	2	2	no	—
21	#15689419	C₂²	1	2	yes	—
22	#15689414	C₂²	1	2	yes	—
23	#15689409	C₂²	1	2	yes	—
24	#15689404	C₂²	1	2	yes	—
25	#15689364	—	2	1	no	—
26	#15689364	—	2	1	no	—
27	#15689253	C₂²	1	1	yes	characteristic
28	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₈₈	—	2	1	no	—
30	C₄×Dic₁₈₈	—	2	1	no	—
31	#15689202	C₂²	1	1	yes	characteristic
32	#15689194	—	2	1	no	—
33	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
34	C₈×C₂²⋊D₄₇	—	2	1	no	—
35	#15689182	—	2	1	no	—
36	#15689180	C₂²	1	1	yes	characteristic
37	#15689077	—	2	1	no	—
38	#15688871	C₂²	1	2	yes	—
39	#15688871	C₂²	1	2	yes	—
40	#15688868	C₂²	1	2	yes	—
41	#15688868	C₂²	1	2	yes	—
42	C₄₇×#431	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
44	C₄₇×#335	C₂²	1	2	yes	—
45	C₄₇×#334	C₂²	1	1	yes	characteristic
46	C₄₇×#334	C₂²	1	1	yes	characteristic
47	C₂×Dic₁₈₈	—	2	1	no	—
48	C₂×Dic₁₈₈	—	2	1	no	—
49	C₂×#10625324	—	2	1	no	—
50	C₂×C₈×D₄₇	—	2	1	no	—
51	#13094024	C₂³	1	2	yes	—
52	#13094024	C₂³	1	2	yes	—
53	#13094018	—	2	1	no	—
54	#13094016	—	2	2	no	—
55	#13094016	D₄	1	1	yes	characteristic
56	#13094012	D₄	1	1	yes	characteristic
57	#13094007	—	2	1	no	—
58	#13094003	—	2	2	no	—
59	#13094003	—	2	2	no	—
60	#13094003	D₄	1	1	yes	characteristic
61	#13093998	D₄	1	1	yes	characteristic
62	C₄×Dic₉₄	—	2	2	no	—
63	C₄×Dic₉₄	—	2	2	no	—
64	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
65	C₄×D₁₈₈	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₄₇	—	2	1	no	—
67	#13093983	—	2	2	no	—
68	#13093981	—	2	1	no	—
69	#13093958	—	2	1	no	—
70	#13093958	—	2	1	no	—
71	#13093958	—	2	2	no	—
72	#13093958	—	2	2	no	—
73	#13093956	—	2	2	no	—
74	#13093956	—	2	2	no	—
75	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
77	Dic₄₇⋊C₈	—	2	1	no	—
78	Dic₄₇⋊C₈	—	2	1	no	—
79	#13093949	—	2	2	no	—
80	#13093949	—	2	2	no	—
81	#13093949	—	2	2	no	—
82	#13093949	—	2	2	no	—
83	#13093946	—	2	2	no	—
84	#13093946	—	2	2	no	—
85	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
86	(C₂×D₄₇)⋊C₈	—	2	1	no	—
87	(C₂×D₄₇)⋊C₈	—	2	1	no	—
88	#13093935	—	2	1	no	—
89	#13093934	—	2	1	no	—
90	C₈×Dic₄₇	—	2	1	no	—
91	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
92	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
95	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
96	C₂²⋊C₈×C₄₇	—	2	2	no	—
97	C₄×C₈×C₄₇	—	2	2	no	—
98	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
99	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
100	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
105	C₂×C₂²⋊D₄₇	—	2	1	no	—
106	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₇	—	2	1	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	2	2	no	—
110	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₂×C₈×C₄₇	—	4	1	no	—
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₇	#41	1	1	yes	characteristic
122	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₄₇	—	2	1	no	—
124	C₂²⋊Dic₄₇	—	4	2	no	—
125	(C₂×D₄₇)⋊C₄	—	2	1	no	—
126	(C₂×D₄₇)⋊C₄	—	2	1	no	—
127	(C₂×D₄₇)⋊C₄	—	2	1	no	—
128	(C₂×D₄₇)⋊C₄	—	2	1	no	—
129	C₄⋊Dic₄₇	—	2	2	no	—
130	C₄⋊Dic₄₇	—	2	2	no	—
131	C₄⋊Dic₄₇	—	2	2	no	—
132	C₄⋊Dic₄₇	—	2	2	no	—
133	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
134	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
135	Dic₄₇⋊C₄	—	2	1	no	—
136	Dic₄₇⋊C₄	—	2	1	no	—
137	Dic₄₇⋊C₄	—	2	1	no	—
138	Dic₄₇⋊C₄	—	2	1	no	—
139	Dic₄₇⋊C₄	—	4	2	no	—
140	Dic₄₇⋊C₄	—	4	2	no	—
141	C₄×Dic₄₇	—	2	1	no	—
142	C₄×Dic₄₇	—	4	2	no	—
143	C₂×C₄₇⋊C₈	—	2	1	no	—
144	C₂×C₄₇⋊C₈	—	2	1	no	—
145	Dic₁₈₈	—	4	2	no	—
146	Dic₁₈₈	—	4	2	no	—
147	#10625324	—	4	2	no	—
148	C₈×D₄₇	—	4	2	no	—
149	C₂³×C₄₇	#121	1	1	yes	characteristic
150	C₂²×D₄₇	—	2	1	no	—
151	C₂×C₄×C₄₇	—	2	1	no	—
152	C₂×C₄×C₄₇	—	2	1	no	—
153	C₂×C₄×C₄₇	—	2	2	no	—
154	C₂×C₄×C₄₇	—	4	1	no	—
155	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₄₇	—	4	2	no	—
162	C₂×Dic₄₇	—	2	1	no	—
163	C₂×Dic₄₇	—	2	1	no	—
164	C₂×Dic₄₇	—	2	1	no	—
165	C₂×Dic₄₇	—	4	2	no	—
166	C₂×Dic₄₇	—	4	2	no	—
167	D₁₈₈	—	2	2	no	—
168	C₄×D₄₇	—	4	1	no	—
169	C₄×D₄₇	—	4	1	no	—
170	Dic₉₄	—	2	2	no	—
171	Dic₉₄	—	4	2	no	—
172	Dic₉₄	—	4	2	no	—
173	Dic₉₄	—	8	2	no	—
174	C₈×C₄₇	—	2	1	no	—
175	C₈×C₄₇	—	2	1	no	—
176	C₈×C₄₇	—	2	1	no	—
177	C₈×C₄₇	—	4	2	no	—
178	C₈×C₄₇	—	4	2	no	—
179	C₈×C₄₇	#135	1	2	yes	—
180	C₄₇⋊C₈	—	4	1	no	—
181	C₄₇⋊C₈	—	4	1	no	—
182	#19226	—	47	1	no	maximal, Sylow
183	C₂²×C₄₇	—	2	1	no	—
184	C₂²×C₄₇	—	2	1	no	—
185	C₂²×C₄₇	—	2	1	no	—
186	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
187	C₂×D₄₇	—	4	1	no	—
188	C₂×D₄₇	—	4	1	no	—
189	C₂×D₄₇	—	4	1	no	—
190	C₄×C₄₇	—	2	1	no	—
191	C₄×C₄₇	—	2	1	no	—
192	C₄×C₄₇	—	2	1	no	—
193	C₄×C₄₇	—	2	1	no	—
194	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
195	C₄×C₄₇	#575	1	1	yes	characteristic
196	C₄×C₄₇	#578	1	2	yes	—
197	Dic₄₇	—	4	1	no	—
198	Dic₄₇	—	4	1	no	—
199	Dic₄₇	—	4	1	no	—
200	Dic₄₇	—	8	2	no	—
201	Dic₄₇	—	8	2	no	—
202	#3233	—	47	2	no	—
203	#3170	—	47	1	no	—
204	#2966	—	47	1	no	—
205	#1696	—	47	1	no	—
206	#1695	—	47	1	no	—
207	#1690	—	47	1	no	—
208	#1689	—	47	1	no	—
209	#1684	—	47	2	no	—
210	#1683	—	47	2	no	—
211	#1564	D₄₇	1	1	yes	characteristic, complemented
212	#1554	—	47	2	no	—
213	C₂×C₄₇	—	4	1	no	—
214	C₂×C₄₇	#3035	1	1	yes	characteristic
215	C₂×C₄₇	#3059	1	1	yes	characteristic
216	C₂×C₄₇	#3067	1	1	yes	characteristic
217	D₄₇	—	8	1	no	—
218	#533	—	47	2	no	—
219	#529	—	47	1	no	—
220	#500	—	47	1	no	—
221	#499	—	94	2	no	—
222	#494	—	47	1	no	—
223	#488	—	94	2	no	—
224	#487	—	47	2	no	—
225	#485	—	94	2	no	—
226	#482	—	94	2	no	—
227	#477	—	47	2	no	—
228	#476	—	47	2	no	—
229	#475	—	47	2	no	—
230	#474	—	47	2	no	—
231	#467	—	94	1	no	—
232	#467	—	94	1	no	—
233	#446	—	47	1	no	—
234	C₈×Q₈	—	47	1	no	—
235	C₄×Q₁₆	—	94	1	no	—
236	C₄×Q₁₆	—	94	1	no	—
237	#436	—	47	1	no	—
238	#435	—	94	1	no	—
239	C₈×D₄	—	47	1	no	—
240	C₈×D₄	—	94	1	no	—
241	#433	—	94	1	no	—
242	#432	—	47	1	no	—
243	#431	C₂×D₄₇	1	2	yes	—
244	#405	—	94	1	no	—
245	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
246	#335	C₂×D₄₇	1	2	yes	—
247	#334	C₂×D₄₇	1	1	yes	characteristic
248	#334	C₂×D₄₇	1	1	yes	characteristic
249	#333	—	47	2	no	—
250	#333	—	47	2	no	—
251	#332	—	47	2	no	—
252	#332	—	47	2	no	—
253	C₄₇	#19226	1	1	yes	characteristic, complemented, Sylow
254	C₂×Q₁₆	—	94	1	no	—
255	C₂×Q₁₆	—	94	1	no	—
256	C₂×M₁₆	—	94	1	no	—
257	C₂²×C₈	—	94	1	no	—
258	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
259	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
260	C₄⋊Q₈	—	47	2	no	—
261	C₄⋊Q₈	—	47	2	no	—
262	#126	—	94	1	no	—
263	#125	—	47	1	no	—
264	#125	—	94	2	no	—
265	#124	—	47	1	no	—
266	#123	—	94	1	no	—
267	C₂²⋊Q₈	—	47	1	no	—
268	C₂²⋊Q₈	—	94	2	no	—
269	C₂²⋊Q₈	—	94	2	no	—
270	#121	—	47	1	no	—
271	C₄×Q₈	—	47	1	no	—
272	C₄×Q₈	—	94	2	no	—
273	C₄×Q₈	—	94	2	no	—
274	C₄×D₄	—	47	1	no	—
275	C₄×D₄	—	94	1	no	—
276	#117	—	94	1	no	—
277	#117	—	94	2	no	—
278	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
279	#107	—	94	2	no	—
280	#107	—	94	2	no	—
281	#107	—	94	1	no	—
282	#107	—	94	1	no	—
283	#106	—	94	2	no	—
284	#106	—	94	2	no	—
285	C₄⋊C₈	—	47	1	no	—
286	C₄⋊C₈	—	47	1	no	—
287	C₄⋊C₈	—	94	1	no	—
288	C₄⋊C₈	—	94	1	no	—
289	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
290	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
291	#103	—	94	2	no	—
292	#103	—	94	2	no	—
293	#103	—	94	2	no	—
294	#103	—	94	2	no	—
295	#102	—	94	2	no	—
296	#102	—	94	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	94	1	no	—
299	C₂²⋊C₈	—	94	1	no	—
300	C₂²⋊C₈	—	94	1	no	—
301	#97	—	94	1	no	—
302	#97	—	94	1	no	—
303	C₄×C₈	—	2	2	no	—
304	C₄×C₈	—	94	1	no	—
305	C₄×C₈	D₁₈₈	1	1	yes	characteristic
306	C₄×C₈	D₁₈₈	1	1	yes	characteristic
307	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
308	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
309	C₂×Q₈	—	47	1	no	—
310	C₂×Q₈	—	94	2	no	—
311	C₂×Q₈	—	94	2	no	—
312	C₂×D₄	—	47	1	no	—
313	C₂×D₄	—	94	1	no	—
314	C₂²×C₄	—	94	1	no	—
315	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
316	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
317	C₂²×C₄	#10625354	1	1	yes	characteristic
318	Q₁₆	—	188	2	no	—
319	Q₁₆	—	188	2	no	—
320	M₁₆	—	188	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	94	1	no	—
326	C₂×C₈	—	94	1	no	—
327	C₂×C₈	—	188	2	no	—
328	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
329	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
330	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
331	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
332	C₄⋊C₄	—	47	1	no	—
333	C₄⋊C₄	—	47	1	no	—
334	C₄⋊C₄	—	94	1	no	—
335	C₄⋊C₄	—	94	1	no	—
336	C₄⋊C₄	—	94	1	no	—
337	C₄⋊C₄	—	94	1	no	—
338	C₄⋊C₄	—	94	2	no	—
339	C₄⋊C₄	—	94	2	no	—
340	C₄⋊C₄	—	94	2	no	—
341	C₄⋊C₄	—	94	2	no	—
342	C₄⋊C₄	—	188	2	no	—
343	C₄⋊C₄	—	188	2	no	—
344	C₂²⋊C₄	—	94	1	no	—
345	C₂²⋊C₄	—	94	1	no	—
346	C₂²⋊C₄	—	94	1	no	—
347	C₂²⋊C₄	—	94	1	no	—
348	C₂²⋊C₄	—	94	1	no	—
349	C₂²⋊C₄	—	188	2	no	—
350	C₄²	—	94	1	no	—
351	C₄²	—	188	2	no	—
352	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
353	C₄²	D₄×D₄₇	1	1	yes	characteristic
354	C₄²	#10625354	1	1	yes	characteristic
355	C₄²	C₂³×D₄₇	1	1	yes	characteristic
356	C₂³	—	94	1	no	—
357	C₂³	#13093999	1	1	yes	characteristic
358	Q₈	—	94	2	no	—
359	Q₈	—	376	2	no	—
360	Q₈	—	188	2	no	—
361	Q₈	—	188	2	no	—
362	D₄	—	94	2	no	—
363	D₄	—	188	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	94	1	no	—
369	C₂×C₄	—	94	1	no	—
370	C₂×C₄	—	94	1	no	—
371	C₂×C₄	—	188	2	no	—
372	C₂×C₄	—	188	2	no	—
373	C₂×C₄	—	188	1	no	—
374	C₂×C₄	—	188	1	no	—
375	C₂×C₄	#13093999	1	1	yes	characteristic
376	C₂×C₄	#13093999	1	1	yes	characteristic
377	C₂×C₄	#13093999	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
380	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	188	1	no	—
387	C₈	—	188	1	no	—
388	C₈	#13094046	1	2	yes	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	188	1	no	—
393	C₂²	—	188	1	no	—
394	C₂²	—	188	1	no	—
395	C₂²	C₂×#13093999	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	376	2	no	—
401	C₄	—	376	2	no	—
402	C₄	—	188	1	no	—
403	C₄	—	188	1	no	—
404	C₄	—	188	1	no	—
405	C₄	C₂×#13094046	1	1	yes	characteristic
406	C₄	#15689923	1	1	yes	characteristic
407	C₄	#15689937	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	376	1	no	—
410	C₂	#20252845	1	1	yes	characteristic
411	C₂	#20253044	1	1	yes	characteristic
412	C₂	#20253109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	#13093998	D₄
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#19226
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #20253974	C₂	semidirect
4. #20242917	C₂	semidirect
5. #20242912	C₂	semidirect
6. #20242883	C₂	semidirect
7. #20242878	C₂	semidirect
8. #20242847	C₂	semidirect
9. #20242840	C₂	semidirect
10. #20242096	C₂	semidirect
11. C₄₇×#1564	C₂	semidirect
211. #1564	D₄₇	semidirect
253. C₄₇	#19226	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20254569	2
#20253974	2
#20252363	2
#20242917	2
#20242912	2
#20242883	2
#20242878	2
#20242847	2
#20242840	2
#20242096	2
C₄₇×#1564	2
#19226	47