Subgroups — #26116058

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
2	#20244194	C₂	1	1	yes	maximal, characteristic
3	#20244193	C₂	1	1	yes	maximal, characteristic
4	#20244191	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243544	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094021	C₂²	1	1	yes	characteristic
9	#15689471	—	2	2	no	—
10	#15689471	C₄	1	4	yes	complemented
11	C₂×#13093955	—	2	2	no	—
12	C₂×#13093948	—	2	2	no	—
13	C₂×#13093946	—	2	2	no	—
14	C₂×#13093944	—	2	2	no	—
15	C₂×#13093944	—	2	2	no	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15688993	C₂²	1	1	yes	characteristic
19	#15688844	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#384	C₂²	1	1	yes	characteristic
23	C₄₇×#340	—	2	2	no	—
24	C₂²×D₁₈₈	—	2	1	no	—
25	C₂²×D₁₈₈	—	2	1	no	—
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	#13094021	—	2	1	no	—
30	#13094021	—	2	1	no	—
31	#13094021	—	2	1	no	—
32	#13094021	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	2	1	no	—
35	C₂×C₄⋊Dic₄₇	—	2	1	no	—
36	#13093955	—	4	4	no	—
37	#13093948	—	4	4	no	—
38	#13093946	—	4	4	no	—
39	#13093944	—	2	8	no	—
40	#13093944	—	4	4	no	—
41	C₄×C₄₇⋊C₈	—	2	1	no	—
42	C₄×C₄₇⋊C₈	—	2	1	no	—
43	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
45	C₂²×Q₈×C₄₇	—	2	1	no	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₄⋊Q₈×C₄₇	—	2	1	no	—
48	C₄⋊Q₈×C₄₇	—	2	1	no	—
49	C₄⋊Q₈×C₄₇	—	2	1	no	—
50	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₄₇	—	2	1	no	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₂³×D₄₇	—	4	1	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	—	4	1	no	—
62	C₂×Q₈×C₄₇	—	4	2	no	—
63	C₂×Q₈×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	1	no	—
65	C₂²×C₄×C₄₇	—	4	1	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	4	1	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	4	2	no	—
74	C₂×D₁₈₈	—	4	2	no	—
75	C₂×D₁₈₈	—	4	2	no	—
76	C₂×D₁₈₈	—	4	2	no	—
77	C₂×D₁₈₈	—	4	2	no	—
78	C₂×D₁₈₈	—	4	2	no	—
79	C₂×D₁₈₈	—	4	1	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×D₁₈₈	—	4	1	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	4	1	no	—
88	C₄⋊C₄×C₄₇	—	4	1	no	—
89	C₄⋊C₄×C₄₇	—	4	2	no	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	—	4	2	no	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	#41	1	1	yes	characteristic
95	C₄²×C₄₇	#41	1	1	yes	characteristic
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	4	2	no	—
98	C₄⋊Dic₄₇	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
105	C₂³×C₄₇	#127	1	1	yes	characteristic
106	C₂²×D₄₇	—	4	2	no	—
107	C₂²×D₄₇	—	8	1	no	—
108	C₂²×D₄₇	—	8	1	no	—
109	C₂²×D₄₇	—	8	1	no	—
110	C₂²×D₄₇	—	8	1	no	—
111	C₂²×D₄₇	—	8	1	no	—
112	C₂²×D₄₇	—	8	1	no	—
113	Q₈×C₄₇	—	4	4	no	—
114	Q₈×C₄₇	—	8	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	8	1	no	—
117	C₂×C₄×C₄₇	—	8	1	no	—
118	C₂×C₄×C₄₇	—	8	1	no	—
119	C₂×C₄×C₄₇	—	8	1	no	—
120	C₂×C₄×C₄₇	—	8	1	no	—
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
128	C₂×C₄×C₄₇	#135	1	2	yes	—
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×Dic₄₇	—	8	1	no	—
134	C₂×Dic₄₇	—	8	1	no	—
135	C₂×Dic₄₇	—	8	1	no	—
136	D₁₈₈	—	4	4	no	—
137	D₁₈₈	—	4	4	no	—
138	D₁₈₈	—	4	4	no	—
139	D₁₈₈	—	8	2	no	—
140	D₁₈₈	—	8	2	no	—
141	D₁₈₈	—	8	2	no	—
142	C₈×C₄₇	—	2	4	no	—
143	C₄₇⋊C₈	—	2	4	no	—
144	C₄₇⋊C₈	—	4	2	no	—
145	#9827	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	#390	1	1	yes	characteristic
147	C₂²×C₄₇	#492	1	1	yes	characteristic
148	C₂²×C₄₇	#495	1	1	yes	characteristic
149	C₂²×C₄₇	#496	1	1	yes	characteristic
150	C₂²×C₄₇	#496	1	1	yes	characteristic
151	C₂²×C₄₇	#496	1	1	yes	characteristic
152	C₂²×C₄₇	#497	1	1	yes	characteristic
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	—	8	2	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₂×D₄₇	—	16	1	no	—
157	C₂×D₄₇	—	16	1	no	—
158	C₂×D₄₇	—	16	1	no	—
159	C₂×D₄₇	—	16	1	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	—	16	1	no	—
162	C₄×C₄₇	#417	1	2	yes	—
163	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
164	C₄×C₄₇	#442	1	2	yes	—
165	C₄×C₄₇	#465	1	2	yes	—
166	C₄×C₄₇	#469	1	2	yes	—
167	C₄×C₄₇	#489	1	2	yes	—
168	Dic₄₇	—	16	1	no	—
169	C₂×#486	—	47	1	no	—
170	#1990	D₄₇	1	1	yes	characteristic, complemented
171	#1957	—	47	1	no	—
172	#1957	—	47	1	no	—
173	#1956	—	47	1	no	—
174	#1844	—	47	1	no	—
175	C₂×#322	—	47	1	no	—
176	C₂×C₄₇	#1714	1	1	yes	characteristic
177	C₂×C₄₇	#1715	1	1	yes	characteristic
178	C₂×C₄₇	#1717	1	1	yes	characteristic
179	C₂×C₄₇	#1720	1	1	yes	characteristic
180	C₂×C₄₇	#1959	1	1	yes	characteristic
181	C₂×C₄₇	#1968	1	1	yes	characteristic
182	C₂×C₄₇	#1972	1	1	yes	characteristic
183	D₄₇	—	16	2	no	—
184	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
185	C₂×#127	—	47	1	no	—
186	#486	—	47	4	no	—
187	#486	—	94	2	no	—
188	C₂×#106	—	94	2	no	—
189	C₂×#103	—	94	2	no	—
190	C₂×#102	—	94	2	no	—
191	C₂×#102	—	94	2	no	—
192	C₂×#102	—	94	2	no	—
193	C₂×C₄×C₈	—	47	1	no	—
194	C₂×C₄×C₈	—	47	1	no	—
195	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
196	#384	—	47	1	no	—
197	#384	C₂×D₄₇	1	1	yes	characteristic
198	#340	—	2	2	no	—
199	#322	—	94	2	no	—
200	C₄₇	#9827	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×D₄	—	94	1	no	—
203	C₂²×D₄	—	94	1	no	—
204	C₂²×D₄	—	94	1	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	—	47	2	no	—
207	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
212	#127	—	47	2	no	—
213	#127	—	94	1	no	—
214	#127	—	94	1	no	—
215	#127	—	94	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	94	1	no	—
222	C₂×C₄⋊C₄	—	94	1	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	#106	—	188	4	no	—
226	#103	—	188	4	no	—
227	#102	—	94	8	no	—
228	#102	—	188	4	no	—
229	#102	—	188	4	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	47	2	no	—
233	C₄×C₈	—	47	2	no	—
234	C₄×C₈	—	94	1	no	—
235	C₄×C₈	—	94	1	no	—
236	C₂⁴	—	188	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×D₄	—	94	2	no	—
242	C₂×D₄	—	94	2	no	—
243	C₂×D₄	—	94	2	no	—
244	C₂×D₄	—	188	1	no	—
245	C₂×D₄	—	188	1	no	—
246	C₂×D₄	—	188	1	no	—
247	C₂×D₄	—	188	2	no	—
248	C₂×D₄	—	188	2	no	—
249	C₂×D₄	—	188	2	no	—
250	C₂×D₄	—	188	2	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂×D₄	—	188	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	188	1	no	—
256	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	47	4	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	2	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	#10625336	1	4	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	188	2	no	—
276	C₄⋊C₄	—	188	2	no	—
277	C₄⋊C₄	—	188	2	no	—
278	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
279	C₄²	#10625350	1	1	yes	characteristic
280	C₄²	D₄×D₄₇	1	1	yes	characteristic
281	C₄²	#10625352	1	1	yes	characteristic
282	C₂³	—	376	1	no	—
283	C₂³	—	376	1	no	—
284	C₂³	—	376	1	no	—
285	C₂³	—	376	1	no	—
286	C₂³	—	376	1	no	—
287	C₂³	—	376	1	no	—
288	C₂³	—	188	2	no	—
289	C₂³	#13094020	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	376	2	no	—
293	D₄	—	376	2	no	—
294	D₄	—	376	2	no	—
295	D₄	—	188	4	no	—
296	D₄	—	188	4	no	—
297	D₄	—	188	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	376	1	no	—
305	C₂×C₄	—	376	1	no	—
306	C₂×C₄	—	376	1	no	—
307	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₂×C₄	#13093996	1	1	yes	characteristic
311	C₂×C₄	#13093997	1	1	yes	characteristic
312	C₂×C₄	#13094010	1	1	yes	characteristic
313	C₂×C₄	C₂×#10625336	1	2	yes	—
314	C₂×C₄	#13094044	1	2	yes	—
315	C₂×C₄	#13094048	1	2	yes	—
316	C₂×C₄	#13094051	1	2	yes	—
317	C₂×C₄	#13094051	1	2	yes	—
318	C₂×C₄	#13094056	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	—	94	4	no	—
321	C₈	—	188	2	no	—
322	C₂²	—	376	2	no	—
323	C₂²	—	376	2	no	—
324	C₂²	—	376	2	no	—
325	C₂²	—	752	1	no	—
326	C₂²	—	752	1	no	—
327	C₂²	—	752	1	no	—
328	C₂²	—	752	1	no	—
329	C₂²	#15689023	1	1	yes	characteristic
330	C₂²	#15689497	1	1	yes	characteristic
331	C₂²	#15689505	1	1	yes	characteristic
332	C₂²	#15689509	1	1	yes	characteristic
333	C₂²	#15689509	1	1	yes	characteristic
334	C₂²	#15689511	1	1	yes	characteristic
335	C₂²	#15689516	1	1	yes	characteristic
336	C₄	—	8	2	no	—
337	C₄	—	16	1	no	—
338	C₄	—	752	1	no	—
339	C₄	#15689123	1	2	yes	—
340	C₄	C₄×#10625336	1	2	yes	—
341	C₄	#15689232	1	2	yes	—
342	C₄	#15689354	1	2	yes	—
343	C₄	#15689375	1	2	yes	—
344	C₄	#15689484	1	2	yes	—
345	C₂	—	752	2	no	—
346	C₂	#20243011	1	1	yes	characteristic
347	C₂	#20243017	1	1	yes	characteristic
348	C₂	#20243024	1	1	yes	characteristic
349	C₂	#20243045	1	1	yes	characteristic
350	C₂	#20244201	1	1	yes	characteristic
351	C₂	#20244245	1	1	yes	characteristic
352	C₂	#20244261	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1990	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1990	D₄₇
hypocenter	C₄₇	#9827
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
4. #20244191	C₂	semidirect
5. #20243544	C₂	semidirect
6. C₂×#15688844	C₂	semidirect
7. C₄₇×#1990	C₂	semidirect
10. #15689471	C₄	semidirect
170. #1990	D₄₇	semidirect
200. C₄₇	#9827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689471	2
#20244194	2
#20244193	2
#20244191	2
#20243544	2
C₂×#15688844	2
C₄₇×#1990	2
#9827	47