Subgroups — #26370936

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253899	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245555	C₂	1	2	yes	maximal, complemented
4	#20245554	C₂	1	2	yes	maximal, complemented
5	#20245541	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245540	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245527	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245175	C₂	1	2	yes	maximal
9	#20245142	C₂	1	1	yes	maximal, characteristic
10	#20245106	C₂	1	2	yes	maximal
11	C₄₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	1	yes	characteristic
13	#15689678	—	2	1	no	—
14	C₄×Q₈×D₄₇	—	2	1	no	—
15	C₂×#13093963	C₂²	1	2	yes	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689524	—	2	1	no	—
18	#15689524	C₂²	1	2	yes	—
19	#15689519	—	2	1	no	—
20	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
22	#15689217	C₂²	1	1	yes	characteristic
23	C₄×#10625337	—	2	1	no	—
24	#15689215	—	2	1	no	—
25	#15689160	—	2	1	no	—
26	#15689160	C₂²	1	2	yes	—
27	#107×D₄₇	C₂²	1	1	yes	characteristic
28	#103×D₄₇	—	2	1	no	—
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	#15688950	C₂²	1	2	yes	—
31	#15688949	C₂²	1	2	yes	—
32	#15688948	C₂²	1	2	yes	—
33	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
34	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
35	#15688928	C₂²	1	2	yes	—
36	#15688927	C₂²	1	2	yes	—
37	#15688926	C₂²	1	2	yes	—
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#367	C₂²	1	2	yes	—
41	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₇×#358	C₂²	1	2	yes	—
43	C₂×Q₈×D₄₇	—	2	1	no	—
44	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
45	C₂×#10625337	—	2	1	no	—
46	Q₁₆×D₄₇	—	2	2	no	—
47	Q₁₆×D₄₇	Q₈	1	4	yes	complemented
48	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
49	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
50	#13094024	—	2	1	no	—
51	#13094015	—	2	2	no	—
52	#13094004	—	4	1	no	—
53	#13094000	—	4	1	no	—
54	Q₈×Dic₄₇	—	2	1	no	—
55	Q₈×Dic₄₇	—	4	1	no	—
56	C₄×Dic₉₄	—	2	1	no	—
57	C₄×Dic₉₄	—	4	1	no	—
58	#13093989	—	2	1	no	—
59	#13093989	—	4	1	no	—
60	#13093989	—	4	1	no	—
61	#13093982	—	2	2	no	—
62	C₄⋊C₄×D₄₇	—	2	1	no	—
63	C₄⋊C₄×D₄₇	—	4	1	no	—
64	C₄²×D₄₇	—	4	1	no	—
65	C₄²×D₄₇	D₄	1	1	yes	characteristic
66	#13093963	—	2	4	no	—
67	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
68	#13093958	—	2	1	no	—
69	#13093958	C₂³	1	1	yes	characteristic
70	#13093958	C₂³	1	2	yes	—
71	#13093957	—	2	1	no	—
72	#13093955	—	2	2	no	—
73	#13093955	—	2	2	no	—
74	#13093949	—	2	1	no	—
75	#13093948	—	2	1	no	—
76	#13093947	—	2	1	no	—
77	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₄₇	D₄	1	1	yes	characteristic
79	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₄₇	—	2	1	no	—
82	C₄×Q₈×C₄₇	—	2	1	no	—
83	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
84	#107×C₄₇	—	2	1	no	—
85	#107×C₄₇	C₂³	1	1	yes	characteristic
86	#107×C₄₇	C₂³	1	2	yes	—
87	#103×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₄₇	—	2	1	no	—
90	C₂×Q₈×C₄₇	—	4	1	no	—
91	Q₈×D₄₇	—	2	4	no	—
92	Q₈×D₄₇	—	4	2	no	—
93	C₂×C₄×D₄₇	—	4	1	no	—
94	C₂×C₄×D₄₇	—	4	1	no	—
95	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
96	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
97	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×Dic₉₄	—	2	1	no	—
99	C₂×Dic₉₄	—	4	1	no	—
100	C₂×Dic₉₄	—	4	1	no	—
101	Q₁₆×C₄₇	—	2	1	no	—
102	Q₁₆×C₄₇	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄⋊C₄×C₄₇	—	4	1	no	—
109	C₄²×C₄₇	—	4	1	no	—
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	#10625337	—	2	4	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	C₄⋊Dic₄₇	—	4	1	no	—
117	C₄⋊Dic₄₇	—	4	2	no	—
118	Dic₄₇⋊C₄	—	4	1	no	—
119	Dic₄₇⋊C₄	—	4	1	no	—
120	Dic₄₇⋊C₄	—	4	1	no	—
121	Dic₄₇⋊C₄	—	4	2	no	—
122	C₄×Dic₄₇	—	4	1	no	—
123	C₄×Dic₄₇	—	4	1	no	—
124	C₄×Dic₄₇	—	4	1	no	—
125	C₄×Dic₄₇	—	4	2	no	—
126	C₄×Dic₄₇	D₈	1	1	yes	characteristic
127	C₄×Dic₄₇	D₈	1	1	yes	characteristic
128	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Dic₁₈₈	—	2	1	no	—
132	Dic₁₈₈	C₂×Q₈	1	2	yes	—
133	C₈×D₄₇	—	2	2	no	—
134	C₈×D₄₇	C₂×Q₈	1	2	yes	—
135	C₈×D₄₇	#41	1	2	yes	—
136	C₁₆×C₄₇	—	2	2	no	—
137	C₄₇⋊C₁₆	—	2	2	no	—
138	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
139	Q₈×C₄₇	—	2	2	no	—
140	Q₈×C₄₇	—	4	1	no	—
141	Q₈×C₄₇	—	8	2	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	—	4	1	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₄₇	—	4	1	no	—
149	C₂×Dic₄₇	—	4	1	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
152	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	4	1	no	—
156	C₄×D₄₇	—	4	4	no	—
157	C₄×D₄₇	—	8	2	no	—
158	C₄×D₄₇	C₂²⋊Q₈	1	2	yes	—
159	C₄×D₄₇	#137	1	2	yes	—
160	Dic₉₄	—	2	2	no	—
161	Dic₉₄	—	4	1	no	—
162	Dic₉₄	—	4	4	no	—
163	Dic₉₄	—	8	2	no	—
164	C₈×C₄₇	—	2	1	no	—
165	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
166	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
167	C₄₇⋊C₈	—	2	1	no	—
168	C₄₇⋊C₈	C₂²⋊Q₈	1	2	yes	—
169	#33628	—	47	1	no	maximal, Sylow
170	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₄₇	—	2	1	no	—
172	C₂×D₄₇	—	2	1	no	—
173	C₂×D₄₇	#474	1	2	yes	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	4	2	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	—	8	1	no	—
178	C₄×C₄₇	#510	1	2	yes	—
179	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
181	Dic₄₇	—	2	1	no	—
182	Dic₄₇	—	2	1	no	—
183	Dic₄₇	—	4	2	no	—
184	Dic₄₇	—	8	2	no	—
185	Dic₄₇	—	8	1	no	—
186	Dic₄₇	#474	1	2	yes	—
187	#3160	—	47	1	no	—
188	C₂×C₄×Q₁₆	—	47	1	no	—
189	#2229	—	47	2	no	—
190	#2229	—	47	2	no	—
191	#2227	—	47	1	no	—
192	#2227	—	47	1	no	—
193	#2225	—	47	1	no	—
194	#2225	D₄₇	1	1	yes	characteristic, complemented
195	#2159	—	47	2	no	—
196	#2150	—	47	1	no	—
197	#2141	—	47	2	no	—
198	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
199	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
200	C₂×C₄₇	#3414	1	1	yes	characteristic
201	D₄₇	—	2	2	no	—
202	C₂²×Q₁₆	—	47	1	no	—
203	#533	—	94	1	no	—
204	C₂×C₄×Q₈	—	94	1	no	—
205	C₂×M₃₂	—	47	2	no	—
206	#500	—	47	1	no	—
207	#500	C₂×D₄₇	1	1	yes	characteristic
208	#499	—	47	2	no	—
209	#499	—	94	1	no	—
210	#498	—	94	1	no	—
211	C₄×Q₁₆	—	47	1	no	—
212	C₄×Q₁₆	—	47	1	no	—
213	C₄×Q₁₆	—	47	1	no	—
214	C₄×Q₁₆	—	94	1	no	—
215	C₄×Q₁₆	—	94	1	no	—
216	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
217	#427	—	47	2	no	—
218	#427	—	94	1	no	—
219	C₂×#107	—	47	1	no	—
220	C₂×#103	—	94	1	no	—
221	C₂×C₄×C₈	—	47	1	no	—
222	#367	—	47	2	no	—
223	#367	C₂×D₄₇	1	2	yes	—
224	#366	—	47	2	no	—
225	#366	—	47	2	no	—
226	C₄⋊C₁₆	—	47	2	no	—
227	C₄⋊C₁₆	—	47	1	no	—
228	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
229	#358	—	47	2	no	—
230	#358	—	47	2	no	—
231	#358	—	47	2	no	—
232	#358	C₂×D₄₇	1	2	yes	—
233	C₄₇	#33628	1	1	yes	characteristic, complemented, Sylow
234	C₂²×Q₈	—	94	1	no	—
235	C₂×Q₁₆	—	47	4	no	—
236	C₂×Q₁₆	—	47	1	no	—
237	C₂×Q₁₆	—	94	2	no	—
238	C₂×Q₁₆	—	94	1	no	—
239	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
240	C₂²×C₈	—	47	1	no	—
241	C₂²×C₈	—	47	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	94	1	no	—
244	#125	—	94	2	no	—
245	C₂²⋊Q₈	—	188	1	no	—
246	C₂²⋊Q₈	—	188	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	94	1	no	—
249	C₄×Q₈	—	94	1	no	—
250	C₄×Q₈	—	94	1	no	—
251	C₄×Q₈	—	188	1	no	—
252	C₄×Q₈	—	188	1	no	—
253	C₄×Q₈	—	188	1	no	—
254	C₄×Q₈	—	188	1	no	—
255	#117	—	94	2	no	—
256	C₂×C₄⋊C₄	—	94	1	no	—
257	C₂×C₄⋊C₄	—	188	1	no	—
258	C₂×C₄²	—	47	1	no	—
259	C₂×C₄²	—	188	1	no	—
260	M₃₂	—	94	4	no	—
261	C₂×C₁₆	—	47	2	no	—
262	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
263	#107	—	2	1	no	—
264	#107	—	47	2	no	—
265	#107	—	47	1	no	—
266	#107	—	94	1	no	—
267	#107	—	94	1	no	—
268	#107	C₂²×D₄₇	1	2	yes	—
269	#107	C₂²×D₄₇	1	1	yes	characteristic
270	#106	—	94	2	no	—
271	#106	—	94	2	no	—
272	#103	—	2	1	no	—
273	#103	—	94	1	no	—
274	#103	—	94	1	no	—
275	#103	—	94	1	no	—
276	C₄×C₈	—	47	1	no	—
277	C₄×C₈	—	47	1	no	—
278	C₄×C₈	—	47	1	no	—
279	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	94	4	no	—
283	C₂×Q₈	—	94	1	no	—
284	C₂×Q₈	—	188	1	no	—
285	C₂×Q₈	—	188	2	no	—
286	C₂×Q₈	—	188	1	no	—
287	C₂²×C₄	—	47	1	no	—
288	C₂²×C₄	—	47	1	no	—
289	C₂²×C₄	—	47	1	no	—
290	C₂²×C₄	—	188	1	no	—
291	C₂²×C₄	—	188	1	no	—
292	Q₁₆	—	2	1	no	—
293	Q₁₆	—	47	2	no	—
294	Q₁₆	—	94	4	no	—
295	Q₁₆	—	94	1	no	—
296	Q₁₆	Q₈×D₄₇	1	2	yes	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	2	no	—
299	C₂×C₈	—	47	2	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
303	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	94	1	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	188	1	no	—
311	C₄⋊C₄	—	188	1	no	—
312	C₄⋊C₄	—	188	2	no	—
313	C₄⋊C₄	—	188	1	no	—
314	C₄⋊C₄	—	188	2	no	—
315	C₄⋊C₄	—	188	1	no	—
316	C₄⋊C₄	—	188	1	no	—
317	C₂²⋊C₄	—	188	2	no	—
318	C₄²	—	4	1	no	—
319	C₄²	—	47	1	no	—
320	C₄²	—	47	1	no	—
321	C₄²	—	47	1	no	—
322	C₄²	—	188	1	no	—
323	C₄²	—	188	1	no	—
324	C₄²	—	188	2	no	—
325	C₄²	—	188	1	no	—
326	C₄²	D₄×D₄₇	1	1	yes	characteristic
327	C₁₆	—	2	2	no	—
328	C₁₆	—	94	2	no	—
329	C₂³	—	47	1	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	8	2	no	—
333	Q₈	—	94	2	no	—
334	Q₈	—	376	2	no	—
335	Q₈	—	188	4	no	—
336	Q₈	—	188	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	47	1	no	—
341	C₂×C₄	—	47	2	no	—
342	C₂×C₄	—	47	2	no	—
343	C₂×C₄	—	47	1	no	—
344	C₂×C₄	—	47	1	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	376	2	no	—
347	C₂×C₄	—	188	1	no	—
348	C₂×C₄	—	188	4	no	—
349	C₂×C₄	—	188	1	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	—	188	1	no	—
352	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
353	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	47	2	no	—
357	C₈	—	94	1	no	—
358	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
359	C₈	#41×D₄₇	1	1	yes	characteristic
360	C₂²	—	47	2	no	—
361	C₂²	—	94	1	no	—
362	C₂²	—	94	1	no	—
363	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	—	47	2	no	—
369	C₄	—	94	1	no	—
370	C₄	—	94	1	no	—
371	C₄	—	376	1	no	—
372	C₄	—	376	2	no	—
373	C₄	—	188	2	no	—
374	C₄	#15689562	1	2	yes	—
375	C₄	C₂²⋊Q₈×D₄₇	1	1	yes	characteristic
376	C₄	#137×D₄₇	1	1	yes	characteristic
377	C₂	—	94	2	no	—
378	C₂	#474×D₄₇	1	1	yes	characteristic
379	C₂	C₂×#15689562	1	1	yes	characteristic
380	C₂	#20256523	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2225	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₄₇	C₂×D₈
hypercenter	#2225	D₄₇
hypocenter	C₄₇	#33628
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
3. #20245555	C₂	semidirect
4. #20245554	C₂	semidirect
5. #20245541	C₂	semidirect
6. #20245540	C₂	semidirect
7. #20245527	C₂	semidirect
11. C₄₇×#2225	C₂	semidirect
47. Q₁₆×D₄₇	Q₈	semidirect
194. #2225	D₄₇	semidirect
233. C₄₇	#33628	semidirect

Maximal subgroups

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

subgroup	index
#20253899	2
C₄×Q₁₆×D₄₇	2
#20245555	2
#20245554	2
#20245541	2
#20245540	2
#20245527	2
#20245175	2
#20245142	2
#20245106	2
C₄₇×#2225	2
#33628	47