Subgroups — #26225076

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689522	C₂	1	1	yes	maximal, characteristic
2	#125×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245943	C₂	1	1	yes	maximal, characteristic
4	#20244353	C₂	1	1	yes	maximal, characteristic
5	#20244130	C₂	1	2	yes	maximal
6	#20244125	C₂	1	2	yes	maximal
7	#20243550	C₂	1	1	yes	maximal, characteristic
8	#20243284	C₂	1	2	yes	maximal
9	#20243284	C₂	1	2	yes	maximal
10	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1991	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094016	C₂²	1	1	yes	characteristic
13	#15689522	C₄	1	8	yes	complemented
14	C₂×#13093957	C₂²	1	2	yes	—
15	C₂×#13093957	C₂²	1	2	yes	—
16	C₂×#13093955	C₂²	1	2	yes	—
17	C₂×#13093955	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	1	yes	characteristic
19	C₂×#13093934	C₂²	1	1	yes	characteristic
20	C₂×#13093934	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15689022	C₂²	1	2	yes	—
24	#15689019	C₂²	1	2	yes	—
25	#15689009	—	2	2	no	—
26	#15689005	—	2	2	no	—
27	#15688989	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×Dic₄₇	—	2	2	no	—
29	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
30	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
31	#15688974	—	2	2	no	—
32	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
33	#15688886	C₂²	1	2	yes	—
34	#15688886	C₂²	1	2	yes	—
35	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#382	C₂²	1	1	yes	characteristic
38	C₄₇×#340	C₂²	1	2	yes	—
39	C₄₇×#340	C₂²	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	#13094016	—	2	2	no	—
43	#13094016	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	—	2	2	no	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	—	2	2	no	—
48	C₂×Dic₄₇⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	—	2	2	no	—
51	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
52	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	#13093957	—	2	4	no	—
56	#13093957	C₂×C₄	1	8	yes	—
57	#13093955	—	2	4	no	—
58	#13093955	C₂×C₄	1	8	yes	—
59	#13093934	—	2	2	no	—
60	#13093934	—	2	2	no	—
61	#13093934	Q₈	1	4	yes	—
62	#13093934	Q₈	1	4	yes	—
63	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
64	C₄₇⋊#95	—	2	2	no	—
65	C₄₇⋊#95	—	2	2	no	—
66	C₄₇⋊#95	—	2	2	no	—
67	C₄₇⋊#95	—	2	2	no	—
68	C₄₇⋊#95	—	2	2	no	—
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	#125×C₄₇	—	2	2	no	—
72	#125×C₄₇	C₂×C₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₇	—	2	2	no	—
78	#95×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	8	no	—
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₇	#41	1	2	yes	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	4	4	no	—
105	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
106	Dic₄₇⋊C₄	—	2	8	no	—
107	Dic₄₇⋊C₄	—	4	4	no	—
108	C₄×Dic₄₇	—	2	1	no	—
109	C₄×Dic₄₇	—	2	1	no	—
110	C₄×Dic₄₇	—	4	2	no	—
111	C₄×Dic₄₇	—	4	2	no	—
112	C₄×Dic₄₇	—	4	4	no	—
113	C₄×Dic₄₇	—	4	4	no	—
114	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
115	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
116	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₇⋊C₈	#41	1	2	yes	—
121	C₂×C₄₇⋊C₈	#41	1	2	yes	—
122	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
150	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
154	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
155	#21728	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄₇	#532	1	1	yes	characteristic
159	C₂²×C₄₇	#575	1	2	yes	—
160	C₂²×C₄₇	#575	1	2	yes	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	—	8	2	no	—
164	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₄₇	#533	1	1	yes	characteristic
167	C₄×C₄₇	#533	1	1	yes	characteristic
168	Dic₄₇	—	2	2	no	—
169	Dic₄₇	—	2	2	no	—
170	Dic₄₇	—	4	2	no	—
171	Dic₄₇	—	4	4	no	—
172	Dic₄₇	—	8	2	no	—
173	C₂×#499	—	47	1	no	—
174	C₄×#125	—	47	1	no	—
175	C₄×#125	—	47	1	no	—
176	#1991	—	47	1	no	—
177	#1991	D₄₇	1	1	yes	characteristic
178	#1944	—	47	2	no	—
179	#1943	—	47	2	no	—
180	#1846	—	47	1	no	—
181	#1776	—	47	2	no	—
182	#1776	—	47	2	no	—
183	C₄×#97	—	47	1	no	—
184	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
185	C₂×C₄₇	#2943	1	2	yes	—
186	C₂×C₄₇	#3166	1	1	yes	characteristic
187	C₂×C₄₇	#3166	1	1	yes	characteristic
188	C₂×C₄₇	#3183	1	2	yes	—
189	C₂×#125	—	47	1	no	—
190	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
191	#499	—	47	8	no	—
192	C₂×#107	—	47	2	no	—
193	C₂×#107	—	47	2	no	—
194	C₂×#106	—	47	2	no	—
195	C₂×#106	—	47	2	no	—
196	C₂×#97	—	47	1	no	—
197	C₂×#97	—	47	1	no	—
198	C₂×#97	—	47	1	no	—
199	C₂×#97	—	47	1	no	—
200	C₂×C₄×C₈	—	47	1	no	—
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#389	—	47	2	no	—
203	#389	—	47	2	no	—
204	#387	—	94	2	no	—
205	#387	—	94	2	no	—
206	#382	—	47	1	no	—
207	#382	C₂×D₄₇	1	1	yes	characteristic
208	C₄×C₄⋊C₄	—	47	2	no	—
209	C₄×C₄⋊C₄	—	47	2	no	—
210	C₄×C₄⋊C₄	—	94	2	no	—
211	C₄×C₄⋊C₄	—	94	2	no	—
212	C₄³	—	47	1	no	—
213	#340	—	47	2	no	—
214	#340	—	47	2	no	—
215	#340	C₂×D₄₇	1	2	yes	—
216	#340	C₂×D₄₇	1	2	yes	—
217	C₄₇	#21728	1	1	yes	characteristic, complemented, Sylow
218	C₂²×C₈	—	47	1	no	—
219	C₂²×C₈	—	47	1	no	—
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
222	#125	—	2	2	no	—
223	#125	—	47	4	no	—
224	#125	—	94	2	no	—
225	#125	C₄×D₄₇	1	4	yes	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	47	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₄⋊C₄	—	94	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
234	C₂×C₄²	—	47	1	no	—
235	C₂×C₄²	—	47	1	no	—
236	C₂×C₄²	—	47	2	no	—
237	C₂×C₄²	—	47	2	no	—
238	C₂×C₄²	—	94	2	no	—
239	C₂×C₄²	—	94	2	no	—
240	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	#107	—	47	8	no	—
242	#107	—	94	4	no	—
243	#106	—	47	8	no	—
244	#106	—	94	4	no	—
245	#97	—	47	4	no	—
246	#97	—	47	4	no	—
247	#97	—	94	2	no	—
248	#97	—	94	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	47	4	no	—
251	#95	—	2	2	no	—
252	#95	—	94	2	no	—
253	#95	—	94	2	no	—
254	#95	—	94	2	no	—
255	#95	—	94	2	no	—
256	#95	—	94	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	47	1	no	—
260	C₂²×C₄	—	47	1	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	D₄×D₄₇	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	47	2	no	—
271	C₂×C₈	—	47	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	—	47	2	no	—
274	C₂×C₈	—	47	2	no	—
275	C₂×C₈	—	47	2	no	—
276	C₂×C₈	Q₈×D₄₇	1	2	yes	—
277	C₂×C₈	Q₈×D₄₇	1	2	yes	—
278	C₄⋊C₄	—	2	8	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	47	4	no	—
283	C₄⋊C₄	—	94	8	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₄⋊C₄	—	94	2	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₄⋊C₄	—	188	4	no	—
288	C₄⋊C₄	—	188	4	no	—
289	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
290	C₄²	—	47	2	no	—
291	C₄²	—	47	2	no	—
292	C₄²	—	94	1	no	—
293	C₄²	—	94	1	no	—
294	C₄²	—	188	2	no	—
295	C₄²	—	188	2	no	—
296	C₄²	—	188	4	no	—
297	C₄²	—	188	4	no	—
298	C₄²	C₂×C₄×D₄₇	1	2	yes	—
299	C₄²	#10625352	1	2	yes	—
300	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
323	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
326	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
327	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	47	4	no	—
332	C₈	—	47	4	no	—
333	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
334	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
335	C₂²	#15689672	1	1	yes	characteristic
336	C₂²	#15689920	1	2	yes	—
337	C₂²	#15689920	1	2	yes	—
338	C₄	—	2	4	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	94	2	no	—
342	C₄	—	94	2	no	—
343	C₄	—	376	2	no	—
344	C₄	—	188	4	no	—
345	C₄	—	188	2	no	—
346	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
347	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
348	C₄	#15689675	1	1	yes	characteristic
349	C₄	#15689675	1	1	yes	characteristic
350	C₂	#20245957	1	1	yes	characteristic
351	C₂	#20252184	1	2	yes	—
352	C₂	#20253936	1	1	yes	characteristic
353	C₂	#20253936	1	1	yes	characteristic
354	C₂	#20254116	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1991	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1991	D₄₇
hypocenter	C₄₇	#21728
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1991

Complements

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

N	H ≅ G/N	type
13. #15689522	C₄	semidirect
217. C₄₇	#21728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689522	2
#125×Dic₄₇	2
#20245943	2
#20244353	2
#20244130	2
#20244125	2
#20243550	2
#20243284	2
C₄×#13093934	2
C₄₇×#1991	2
#21728	47