Subgroups — #26199536

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254731	C₂	1	2	yes	maximal
2	#20253899	C₂	1	1	yes	maximal, characteristic
3	#20252319	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	#20242769	C₂	1	2	yes	maximal, complemented
8	#20242768	C₂	1	2	yes	maximal, complemented
9	#334×D₄₇	C₂	1	1	yes	maximal, characteristic
10	#20242077	C₂	1	2	yes	maximal
11	C₄₇×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689678	C₂²	1	2	yes	—
13	#15689677	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	#15689419	C₂²	1	2	yes	—
19	#15689418	C₂²	1	2	yes	—
20	#15689416	—	2	2	no	—
21	#15689363	—	2	2	no	—
22	#15689361	—	2	2	no	—
23	#15689331	—	2	2	no	—
24	#15689330	C₂²	1	2	yes	—
25	#15689253	C₂²	1	1	yes	characteristic
26	#15689251	—	2	1	no	—
27	Q₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	#15689247	—	2	1	no	—
29	#15689247	C₂²	1	1	yes	characteristic
30	#15689160	—	2	1	no	—
31	#15689160	—	2	1	no	—
32	#15689148	—	2	1	no	—
33	#15689145	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×D₄₇	C₂²	1	2	yes	—
35	#15689080	—	2	1	no	—
36	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
37	#15688874	C₂²	1	1	yes	characteristic
38	#15688873	C₂²	1	2	yes	—
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×#10625337	—	2	2	no	—
48	C₂×#10625324	—	2	1	no	—
49	C₂×C₈×D₄₇	—	2	2	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	#13094023	D₄	1	1	yes	characteristic
54	#13094022	D₄	1	1	yes	characteristic
55	#13094015	—	2	2	no	—
56	#13094014	—	2	1	no	—
57	#13094004	—	2	2	no	—
58	#13094004	D₄	1	1	yes	characteristic
59	#13094002	—	2	1	no	—
60	#13094000	—	2	2	no	—
61	#13094000	D₄	1	1	yes	characteristic
62	Q₈×Dic₄₇	—	2	2	no	—
63	C₄×Dic₉₄	—	2	1	no	—
64	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
65	#13093989	—	2	2	no	—
66	#13093982	—	2	2	no	—
67	#13093981	—	2	1	no	—
68	C₄²×D₄₇	C₂³	1	1	yes	characteristic
69	#13093958	—	2	1	no	—
70	#13093958	—	2	1	no	—
71	#13093958	—	2	2	no	—
72	#13093957	—	2	2	no	—
73	#13093955	—	2	2	no	—
74	#13093955	—	2	2	no	—
75	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
76	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
77	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
78	Dic₄₇⋊C₈	—	2	1	no	—
79	Dic₄₇⋊C₈	—	2	1	no	—
80	Dic₄₇⋊C₈	—	2	2	no	—
81	Dic₄₇⋊C₈	—	2	1	no	—
82	#13093949	—	2	2	no	—
83	#13093948	—	2	2	no	—
84	#13093948	—	2	2	no	—
85	#13093934	—	2	1	no	—
86	C₄×C₄₇⋊C₈	—	2	1	no	—
87	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₄₇	—	2	1	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	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
107	C₂×Dic₉₄	—	2	1	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
110	Q₁₆×C₄₇	—	8	2	no	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₇	—	2	1	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	2	2	no	—
118	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	—	2	1	no	—
120	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
121	#10625337	—	8	2	no	—
122	(C₂×D₄₇)⋊C₄	—	2	1	no	—
123	(C₂×D₄₇)⋊C₄	—	2	1	no	—
124	(C₂×D₄₇)⋊C₄	—	4	2	no	—
125	C₄⋊Dic₄₇	—	2	1	no	—
126	C₄⋊Dic₄₇	—	2	2	no	—
127	C₄⋊Dic₄₇	—	2	2	no	—
128	C₄⋊Dic₄₇	—	4	2	no	—
129	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
130	Dic₄₇⋊C₄	—	2	1	no	—
131	Dic₄₇⋊C₄	—	2	1	no	—
132	Dic₄₇⋊C₄	—	2	1	no	—
133	Dic₄₇⋊C₄	—	2	1	no	—
134	Dic₄₇⋊C₄	—	2	1	no	—
135	Dic₄₇⋊C₄	—	4	2	no	—
136	C₄×Dic₄₇	—	2	1	no	—
137	C₄×Dic₄₇	—	4	2	no	—
138	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₇	#41	1	1	yes	characteristic
141	C₂×C₄₇⋊C₈	—	2	1	no	—
142	C₂×C₄₇⋊C₈	—	2	2	no	—
143	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
145	#10625324	—	4	2	no	—
146	C₈×D₄₇	—	2	2	no	—
147	C₈×D₄₇	—	8	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	C₂×Q₁₆	1	2	yes	—
168	C₄×D₄₇	#137	1	2	yes	—
169	Dic₉₄	—	2	2	no	—
170	Dic₉₄	—	4	2	no	—
171	Dic₉₄	—	8	2	no	—
172	C₈×C₄₇	—	2	1	no	—
173	C₈×C₄₇	—	4	1	no	—
174	C₈×C₄₇	—	4	2	no	—
175	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
176	C₄₇⋊C₈	—	2	1	no	—
177	C₄₇⋊C₈	—	4	1	no	—
178	C₄₇⋊C₈	—	4	2	no	—
179	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
180	#19210	—	47	1	no	maximal, Sylow
181	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
182	C₂×D₄₇	—	2	1	no	—
183	C₂×D₄₇	—	2	1	no	—
184	C₂×D₄₇	#462	1	2	yes	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	—	4	1	no	—
188	C₄×C₄₇	—	8	2	no	—
189	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
190	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
191	C₄×C₄₇	#577	1	2	yes	—
192	Dic₄₇	—	2	1	no	—
193	Dic₄₇	—	2	1	no	—
194	Dic₄₇	—	4	1	no	—
195	Dic₄₇	—	4	1	no	—
196	Dic₄₇	—	8	2	no	—
197	Dic₄₇	#462	1	2	yes	—
198	#3249	—	47	2	no	—
199	#3160	—	47	1	no	—
200	#2960	—	47	1	no	—
201	#1696	—	47	1	no	—
202	#1696	—	47	1	no	—
203	#1694	—	47	1	no	—
204	#1694	D₄₇	1	1	yes	characteristic, complemented
205	#1672	—	47	2	no	—
206	#1672	—	47	2	no	—
207	C₂×#334	—	47	1	no	—
208	#1550	—	47	2	no	—
209	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
210	C₂×C₄₇	#3060	1	1	yes	characteristic
211	C₂×C₄₇	#3067	1	1	yes	characteristic
212	D₄₇	—	2	2	no	—
213	#533	—	47	1	no	—
214	#533	—	47	2	no	—
215	#500	—	47	1	no	—
216	#500	C₂×D₄₇	1	1	yes	characteristic
217	#499	—	94	1	no	—
218	#499	—	94	1	no	—
219	#498	—	47	2	no	—
220	#477	—	47	2	no	—
221	#477	—	47	2	no	—
222	#477	—	94	2	no	—
223	#467	—	94	2	no	—
224	#467	—	94	2	no	—
225	#462	—	47	2	no	—
226	#462	—	94	2	no	—
227	#462	C₂×D₄₇	1	2	yes	—
228	#446	—	47	1	no	—
229	#446	—	94	1	no	—
230	#446	C₂×D₄₇	1	1	yes	characteristic
231	C₈×Q₈	—	47	1	no	—
232	C₈×Q₈	—	47	1	no	—
233	C₈×Q₈	—	94	1	no	—
234	#427	—	94	1	no	—
235	#427	—	94	1	no	—
236	#424	—	94	1	no	—
237	#423	—	47	1	no	—
238	C₂×C₄⋊C₈	—	47	2	no	—
239	#405	—	94	1	no	—
240	C₂×C₄×C₈	—	47	1	no	—
241	#334	—	47	2	no	—
242	#334	—	47	1	no	—
243	#334	C₂×D₄₇	1	1	yes	characteristic
244	#333	—	47	2	no	—
245	#333	—	47	2	no	—
246	#332	—	47	2	no	—
247	#332	C₂×D₄₇	1	2	yes	—
248	C₄₇	#19210	1	1	yes	characteristic, complemented, Sylow
249	C₂×Q₁₆	—	2	2	no	—
250	C₂×Q₁₆	—	94	2	no	—
251	C₂×M₁₆	—	94	1	no	—
252	C₂²×C₈	—	47	1	no	—
253	C₂²×C₈	—	47	1	no	—
254	C₂²×C₈	—	94	2	no	—
255	C₄⋊Q₈	—	47	1	no	—
256	C₄⋊Q₈	—	47	1	no	—
257	C₄⋊Q₈	—	47	2	no	—
258	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
259	#125	—	94	1	no	—
260	#125	—	94	2	no	—
261	C₂²⋊Q₈	—	47	1	no	—
262	C₂²⋊Q₈	—	47	1	no	—
263	C₂²⋊Q₈	—	94	1	no	—
264	C₂²⋊Q₈	—	94	2	no	—
265	C₂²⋊Q₈	—	94	2	no	—
266	C₄×Q₈	—	47	1	no	—
267	C₄×Q₈	—	94	1	no	—
268	C₄×Q₈	—	94	2	no	—
269	C₄×Q₈	—	94	2	no	—
270	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
271	#117	—	94	1	no	—
272	#117	—	94	2	no	—
273	C₂×C₄²	—	47	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	94	2	no	—
277	#107	—	94	1	no	—
278	#107	—	94	1	no	—
279	#107	—	94	2	no	—
280	#106	—	94	2	no	—
281	#106	—	94	2	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	47	1	no	—
284	C₄⋊C₈	—	47	2	no	—
285	C₄⋊C₈	—	94	1	no	—
286	C₄⋊C₈	—	94	2	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	#97	—	2	1	no	—
297	#97	—	94	1	no	—
298	C₄×C₈	—	47	1	no	—
299	C₄×C₈	—	47	1	no	—
300	C₄×C₈	—	47	1	no	—
301	C₄×C₈	—	94	1	no	—
302	C₄×C₈	—	94	1	no	—
303	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	47	1	no	—
306	C₂×Q₈	—	94	1	no	—
307	C₂×Q₈	—	94	2	no	—
308	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
309	C₂²×C₄	—	47	1	no	—
310	C₂²×C₄	—	47	1	no	—
311	C₂²×C₄	—	47	1	no	—
312	Q₁₆	—	8	2	no	—
313	Q₁₆	—	376	2	no	—
314	M₁₆	—	188	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	47	1	no	—
318	C₂×C₈	—	47	4	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	94	2	no	—
321	C₂×C₈	—	94	1	no	—
322	C₂×C₈	—	94	2	no	—
323	C₂×C₈	—	376	2	no	—
324	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
325	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	47	1	no	—
330	C₄⋊C₄	—	94	1	no	—
331	C₄⋊C₄	—	94	1	no	—
332	C₄⋊C₄	—	94	1	no	—
333	C₄⋊C₄	—	94	1	no	—
334	C₄⋊C₄	—	94	1	no	—
335	C₄⋊C₄	—	94	1	no	—
336	C₄⋊C₄	—	94	2	no	—
337	C₄⋊C₄	—	94	2	no	—
338	C₄⋊C₄	—	188	2	no	—
339	C₄⋊C₄	—	188	2	no	—
340	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
341	C₂²⋊C₄	—	94	1	no	—
342	C₂²⋊C₄	—	94	1	no	—
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	—	188	2	no	—
350	C₄²	C₂³×D₄₇	1	1	yes	characteristic
351	C₂³	—	47	1	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	94	2	no	—
356	Q₈	—	376	2	no	—
357	Q₈	—	188	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	47	1	no	—
362	C₂×C₄	—	47	2	no	—
363	C₂×C₄	—	47	2	no	—
364	C₂×C₄	—	47	1	no	—
365	C₂×C₄	—	47	1	no	—
366	C₂×C₄	—	94	2	no	—
367	C₂×C₄	—	94	1	no	—
368	C₂×C₄	—	94	1	no	—
369	C₂×C₄	—	188	1	no	—
370	C₂×C₄	—	188	2	no	—
371	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
373	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	47	2	no	—
378	C₈	—	94	1	no	—
379	C₈	—	188	1	no	—
380	C₈	—	188	2	no	—
381	C₈	Q₁₆×D₄₇	1	2	yes	—
382	C₂²	—	47	2	no	—
383	C₂²	—	94	1	no	—
384	C₂²	—	94	1	no	—
385	C₂²	#121×D₄₇	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	8	2	no	—
390	C₄	—	47	2	no	—
391	C₄	—	94	1	no	—
392	C₄	—	94	1	no	—
393	C₄	—	376	2	no	—
394	C₄	—	188	1	no	—
395	C₄	—	188	1	no	—
396	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
397	C₄	#137×D₄₇	1	1	yes	characteristic
398	C₄	#15689933	1	2	yes	—
399	C₂	—	94	2	no	—
400	C₂	#462×D₄₇	1	1	yes	characteristic
401	C₂	#20253048	1	1	yes	characteristic
402	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	#19210
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. #20242769	C₂	semidirect
8. #20242768	C₂	semidirect
11. C₄₇×#1694	C₂	semidirect
148. C₈×D₄₇	Q₁₆	semidirect
204. #1694	D₄₇	semidirect
248. C₄₇	#19210	semidirect

Maximal subgroups

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

subgroup	index
#20254731	2
#20253899	2
#20252319	2
#20242917	2
#20242916	2
#20242903	2
#20242769	2
#20242768	2
#334×D₄₇	2
#20242077	2
C₄₇×#1694	2
#19210	47