Subgroups — #26198917

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 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	#20253426	C₂	1	2	yes	maximal
3	#446×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20242917	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242916	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242903	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242893	C₂	1	2	yes	maximal, complemented
8	#20242892	C₂	1	2	yes	maximal, complemented
9	#20242133	C₂	1	1	yes	maximal, characteristic
10	#20242044	C₂	1	2	yes	maximal
11	C₄₇×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689678	C₂²	1	2	yes	—
13	C₄×Q₈×D₄₇	C₂²	1	1	yes	characteristic
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689524	—	2	1	no	—
16	#15689524	—	2	1	no	—
17	#15689519	C₂²	1	2	yes	—
18	#15689457	—	2	2	no	—
19	#15689419	C₂²	1	2	yes	—
20	#15689418	C₂²	1	2	yes	—
21	#15689330	C₂²	1	2	yes	—
22	#15689277	—	2	2	no	—
23	#15689254	C₂²	1	1	yes	characteristic
24	#15689253	C₂²	1	1	yes	characteristic
25	#15689252	—	2	1	no	—
26	#15689251	—	2	1	no	—
27	#15689250	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₄₇	—	2	2	no	—
29	#15689215	—	2	2	no	—
30	#15689175	C₂²	1	2	yes	—
31	#15689160	—	2	1	no	—
32	#15689160	—	2	1	no	—
33	C₄⋊C₈×D₄₇	—	2	1	no	—
34	C₄⋊C₈×D₄₇	C₂²	1	1	yes	characteristic
35	#97×D₄₇	—	2	1	no	—
36	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
37	#15688875	C₂²	1	2	yes	—
38	#15688874	C₂²	1	1	yes	characteristic
39	#15688871	C₂²	1	2	yes	—
40	#15688871	C₂²	1	2	yes	—
41	#15688869	C₂²	1	2	yes	—
42	C₄₇×#500	C₂²	1	1	yes	characteristic
43	C₄₇×#462	C₂²	1	2	yes	—
44	C₄₇×#446	C₂²	1	1	yes	characteristic
45	C₄₇×#334	C₂²	1	1	yes	characteristic
46	C₄₇×#332	C₂²	1	2	yes	—
47	C₂×Q₈×D₄₇	D₄	1	1	yes	characteristic
48	C₂×#10625337	—	2	2	no	—
49	C₂×C₈×D₄₇	—	2	1	no	—
50	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
51	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
52	#13094024	C₂³	1	2	yes	—
53	#13094015	—	2	2	no	—
54	#13094004	—	2	2	no	—
55	#13094000	—	2	2	no	—
56	Q₈×Dic₄₇	—	2	2	no	—
57	Q₈×Dic₄₇	D₄	1	1	yes	characteristic
58	C₄×Dic₉₄	—	2	1	no	—
59	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
60	#13093989	—	2	1	no	—
61	#13093989	—	2	2	no	—
62	#13093989	D₄	1	1	yes	characteristic
63	#13093982	—	2	2	no	—
64	C₄⋊C₄×D₄₇	—	2	1	no	—
65	C₄⋊C₄×D₄₇	D₄	1	1	yes	characteristic
66	C₄²×D₄₇	—	2	1	no	—
67	C₄²×D₄₇	C₂³	1	1	yes	characteristic
68	#13093958	—	2	1	no	—
69	#13093958	—	2	1	no	—
70	#13093958	—	2	2	no	—
71	#13093957	—	2	2	no	—
72	#13093955	—	2	2	no	—
73	#13093955	—	2	2	no	—
74	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
75	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
77	Dic₄₇⋊C₈	—	2	1	no	—
78	Dic₄₇⋊C₈	—	2	1	no	—
79	Dic₄₇⋊C₈	—	2	1	no	—
80	#13093949	—	2	2	no	—
81	#13093948	—	2	2	no	—
82	#13093948	—	2	2	no	—
83	(C₂×D₄₇)⋊C₈	—	2	2	no	—
84	#13093935	—	2	1	no	—
85	#13093934	—	2	1	no	—
86	#13093934	—	2	1	no	—
87	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₄₇	—	2	2	no	—
89	C₈×Dic₄₇	D₄	1	1	yes	characteristic
90	C₈×Dic₄₇	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₄₇	—	2	2	no	—
92	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
93	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
94	#107×C₄₇	—	2	1	no	—
95	#107×C₄₇	—	2	1	no	—
96	C₄⋊C₈×C₄₇	—	2	1	no	—
97	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
98	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
99	#103×C₄₇	—	2	2	no	—
100	#97×C₄₇	—	2	1	no	—
101	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₄₇	—	2	2	no	—
103	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	Q₈×D₄₇	—	4	2	no	—
105	C₂×C₄×D₄₇	—	2	1	no	—
106	C₂×C₄×D₄₇	—	2	1	no	—
107	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
110	C₂×Dic₉₄	—	2	1	no	—
111	C₂×Dic₉₄	—	2	2	no	—
112	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₄₇	—	4	4	no	—
114	C₂×C₈×C₄₇	—	2	2	no	—
115	C₂×C₈×C₄₇	—	2	1	no	—
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₄₇	—	2	1	no	—
119	C₄⋊C₄×C₄₇	—	2	2	no	—
120	C₄⋊C₄×C₄₇	—	2	2	no	—
121	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₄₇	—	2	1	no	—
123	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
124	#10625337	—	4	4	no	—
125	(C₂×D₄₇)⋊C₄	—	4	2	no	—
126	C₄⋊Dic₄₇	—	2	1	no	—
127	C₄⋊Dic₄₇	—	2	2	no	—
128	C₄⋊Dic₄₇	—	2	2	no	—
129	C₄⋊Dic₄₇	—	4	2	no	—
130	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
131	Dic₄₇⋊C₄	—	2	1	no	—
132	Dic₄₇⋊C₄	—	2	1	no	—
133	Dic₄₇⋊C₄	—	2	1	no	—
134	Dic₄₇⋊C₄	—	4	2	no	—
135	C₄×Dic₄₇	—	2	1	no	—
136	C₄×Dic₄₇	—	2	1	no	—
137	C₄×Dic₄₇	—	2	1	no	—
138	C₄×Dic₄₇	—	4	2	no	—
139	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₄₇	#41	1	1	yes	characteristic
142	C₂×C₄₇⋊C₈	—	2	2	no	—
143	C₂×C₄₇⋊C₈	—	2	1	no	—
144	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
146	C₈×D₄₇	—	2	2	no	—
147	C₈×D₄₇	—	4	2	no	—
148	C₈×D₄₇	Q₁₆	1	4	yes	complemented
149	C₂²×D₄₇	#121	1	1	yes	characteristic
150	Q₈×C₄₇	—	2	2	no	—
151	Q₈×C₄₇	—	4	2	no	—
152	Q₈×C₄₇	—	4	2	no	—
153	C₂×C₄×C₄₇	—	2	1	no	—
154	C₂×C₄×C₄₇	—	2	1	no	—
155	C₂×C₄×C₄₇	—	4	2	no	—
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₄₇	—	2	1	no	—
160	C₂×Dic₄₇	—	2	1	no	—
161	C₂×Dic₄₇	—	4	2	no	—
162	C₂×Dic₄₇	#121	1	1	yes	characteristic
163	C₂×Dic₄₇	#121	1	1	yes	characteristic
164	C₂×Dic₄₇	#121	1	1	yes	characteristic
165	C₄×D₄₇	—	2	2	no	—
166	C₄×D₄₇	—	4	1	no	—
167	C₄×D₄₇	—	4	2	no	—
168	C₄×D₄₇	—	4	2	no	—
169	C₄×D₄₇	C₂×Q₁₆	1	2	yes	—
170	C₄×D₄₇	#137	1	2	yes	—
171	Dic₉₄	—	2	2	no	—
172	Dic₉₄	—	4	2	no	—
173	Dic₉₄	—	8	2	no	—
174	C₈×C₄₇	—	2	1	no	—
175	C₈×C₄₇	—	4	1	no	—
176	C₈×C₄₇	—	4	2	no	—
177	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
178	C₄₇⋊C₈	—	2	1	no	—
179	C₄₇⋊C₈	—	4	1	no	—
180	C₄₇⋊C₈	—	4	2	no	—
181	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
182	#19150	—	47	1	no	maximal, Sylow
183	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
184	C₂×D₄₇	—	2	1	no	—
185	C₂×D₄₇	—	2	1	no	—
186	C₂×D₄₇	#462	1	2	yes	—
187	C₄×C₄₇	—	2	1	no	—
188	C₄×C₄₇	—	4	1	no	—
189	C₄×C₄₇	—	4	1	no	—
190	C₄×C₄₇	—	8	2	no	—
191	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
192	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₇	#575	1	2	yes	—
194	Dic₄₇	—	2	1	no	—
195	Dic₄₇	—	2	1	no	—
196	Dic₄₇	—	4	1	no	—
197	Dic₄₇	—	4	1	no	—
198	Dic₄₇	—	8	2	no	—
199	Dic₄₇	#462	1	2	yes	—
200	#3160	—	47	1	no	—
201	#3104	—	47	2	no	—
202	C₂×#446	—	47	1	no	—
203	#1696	—	47	1	no	—
204	#1696	—	47	1	no	—
205	#1694	—	47	1	no	—
206	#1694	D₄₇	1	1	yes	characteristic, complemented
207	#1692	—	47	2	no	—
208	#1692	—	47	2	no	—
209	#1564	—	47	1	no	—
210	#1542	—	47	2	no	—
211	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
212	C₂×C₄₇	#3054	1	1	yes	characteristic
213	C₂×C₄₇	#3067	1	1	yes	characteristic
214	D₄₇	—	2	2	no	—
215	#533	—	47	2	no	—
216	C₂×C₄×Q₈	—	47	1	no	—
217	#500	—	47	1	no	—
218	#500	C₂×D₄₇	1	1	yes	characteristic
219	#499	—	94	1	no	—
220	#499	—	94	1	no	—
221	#498	—	47	2	no	—
222	#484	—	94	2	no	—
223	#477	—	47	2	no	—
224	#477	—	47	2	no	—
225	#462	—	47	2	no	—
226	#462	C₂×D₄₇	1	2	yes	—
227	#451	—	94	2	no	—
228	#446	—	47	1	no	—
229	#446	—	47	1	no	—
230	#446	—	47	1	no	—
231	#446	—	94	1	no	—
232	#446	—	94	1	no	—
233	#446	C₂×D₄₇	1	1	yes	characteristic
234	C₄×Q₁₆	—	94	2	no	—
235	C₄×Q₁₆	—	94	2	no	—
236	#431	—	47	2	no	—
237	#427	—	94	1	no	—
238	#427	—	94	1	no	—
239	C₂×C₄⋊C₈	—	47	1	no	—
240	C₂×C₄⋊C₈	—	94	1	no	—
241	C₂×#97	—	94	1	no	—
242	C₂×C₄×C₈	—	47	1	no	—
243	#335	—	47	2	no	—
244	#334	—	47	1	no	—
245	#334	C₂×D₄₇	1	1	yes	characteristic
246	#333	—	47	2	no	—
247	#333	—	47	2	no	—
248	#332	—	47	2	no	—
249	#332	C₂×D₄₇	1	2	yes	—
250	C₄₇	#19150	1	1	yes	characteristic, complemented, Sylow
251	C₂²×Q₈	—	47	1	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	94	2	no	—
254	C₂²×C₈	—	47	1	no	—
255	C₂²×C₈	—	47	1	no	—
256	C₂²×C₈	—	94	1	no	—
257	C₄⋊Q₈	—	47	2	no	—
258	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
259	#125	—	94	2	no	—
260	C₂²⋊Q₈	—	94	2	no	—
261	C₂²⋊Q₈	—	94	2	no	—
262	C₄×Q₈	—	47	1	no	—
263	C₄×Q₈	—	47	1	no	—
264	C₄×Q₈	—	47	1	no	—
265	C₄×Q₈	—	94	1	no	—
266	C₄×Q₈	—	94	1	no	—
267	C₄×Q₈	—	94	2	no	—
268	C₄×Q₈	—	94	2	no	—
269	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
270	#117	—	94	2	no	—
271	C₂×C₄⋊C₄	—	47	1	no	—
272	C₂×C₄⋊C₄	—	94	1	no	—
273	C₂×C₄²	—	47	1	no	—
274	C₂×C₄²	—	94	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	94	2	no	—
278	#107	—	94	1	no	—
279	#107	—	94	1	no	—
280	#107	—	94	2	no	—
281	#106	—	94	2	no	—
282	#106	—	94	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	47	1	no	—
285	C₄⋊C₈	—	47	2	no	—
286	C₄⋊C₈	—	94	1	no	—
287	C₄⋊C₈	—	94	1	no	—
288	C₄⋊C₈	—	94	1	no	—
289	C₄⋊C₈	—	94	1	no	—
290	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	94	2	no	—
294	#103	—	94	2	no	—
295	#103	—	94	2	no	—
296	C₂²⋊C₈	—	94	2	no	—
297	#97	—	2	1	no	—
298	#97	—	94	1	no	—
299	#97	—	94	1	no	—
300	#97	—	94	1	no	—
301	C₄×C₈	—	47	1	no	—
302	C₄×C₈	—	47	1	no	—
303	C₄×C₈	—	47	1	no	—
304	C₄×C₈	—	94	2	no	—
305	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	47	1	no	—
308	C₂×Q₈	—	94	1	no	—
309	C₂×Q₈	—	94	2	no	—
310	C₂×Q₈	—	188	2	no	—
311	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
312	C₂²×C₄	—	47	1	no	—
313	C₂²×C₄	—	47	1	no	—
314	C₂²×C₄	—	47	1	no	—
315	C₂²×C₄	—	94	1	no	—
316	C₂²×C₄	—	94	1	no	—
317	Q₁₆	—	4	4	no	—
318	Q₁₆	—	188	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	47	1	no	—
322	C₂×C₈	—	47	4	no	—
323	C₂×C₈	—	47	1	no	—
324	C₂×C₈	—	94	2	no	—
325	C₂×C₈	—	94	2	no	—
326	C₂×C₈	—	94	1	no	—
327	C₂×C₈	—	188	2	no	—
328	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
329	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	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	—	188	2	no	—
341	C₄⋊C₄	—	188	2	no	—
342	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
343	C₂²⋊C₄	—	188	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	47	1	no	—
346	C₄²	—	47	1	no	—
347	C₄²	—	47	1	no	—
348	C₄²	—	94	1	no	—
349	C₄²	—	94	1	no	—
350	C₄²	—	94	1	no	—
351	C₄²	—	188	2	no	—
352	C₄²	C₂³×D₄₇	1	1	yes	characteristic
353	C₂³	—	47	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	94	2	no	—
358	Q₈	—	376	2	no	—
359	Q₈	—	188	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	47	1	no	—
364	C₂×C₄	—	47	2	no	—
365	C₂×C₄	—	47	2	no	—
366	C₂×C₄	—	47	1	no	—
367	C₂×C₄	—	47	1	no	—
368	C₂×C₄	—	94	2	no	—
369	C₂×C₄	—	94	1	no	—
370	C₂×C₄	—	94	1	no	—
371	C₂×C₄	—	188	1	no	—
372	C₂×C₄	—	188	2	no	—
373	C₂×C₄	—	188	2	no	—
374	C₂×C₄	—	188	2	no	—
375	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
377	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₈	—	47	2	no	—
382	C₈	—	94	1	no	—
383	C₈	—	188	1	no	—
384	C₈	—	188	2	no	—
385	C₈	Q₁₆×D₄₇	1	2	yes	—
386	C₂²	—	47	2	no	—
387	C₂²	—	94	1	no	—
388	C₂²	—	94	1	no	—
389	C₂²	#121×D₄₇	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	47	2	no	—
395	C₄	—	94	1	no	—
396	C₄	—	94	1	no	—
397	C₄	—	376	2	no	—
398	C₄	—	188	1	no	—
399	C₄	—	188	1	no	—
400	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
401	C₄	#137×D₄₇	1	1	yes	characteristic
402	C₄	#15689919	1	2	yes	—
403	C₂	—	94	2	no	—
404	C₂	#462×D₄₇	1	1	yes	characteristic
405	C₂	#20252995	1	1	yes	characteristic
406	C₂	#20253110	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1694	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₄₇	#121
hypercenter	#1694	D₄₇
hypocenter	C₄₇	#19150
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1694

Complements

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

N	H ≅ G/N	type
4. #20242917	C₂	semidirect
5. #20242916	C₂	semidirect
6. #20242903	C₂	semidirect
7. #20242893	C₂	semidirect
8. #20242892	C₂	semidirect
11. C₄₇×#1694	C₂	semidirect
148. C₈×D₄₇	Q₁₆	semidirect
206. #1694	D₄₇	semidirect
250. C₄₇	#19150	semidirect

Maximal subgroups

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

subgroup	index
#20253899	2
#20253426	2
#446×D₄₇	2
#20242917	2
#20242916	2
#20242903	2
#20242893	2
#20242892	2
#20242133	2
#20242044	2
C₄₇×#1694	2
#19150	47