Subgroups — #26204291

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254517	C₂	1	2	yes	maximal, complemented
2	#20253968	C₂	1	1	yes	maximal, characteristic
3	#494×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253783	C₂	1	2	yes	maximal, complemented
5	#20243046	C₂	1	2	yes	maximal, complemented
6	#20243045	C₂	1	2	yes	maximal, complemented
7	#20242995	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242994	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242986	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	—	2	2	no	—
13	#15689674	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₄₇	C₂²	1	2	yes	—
15	#15689519	C₂²	1	2	yes	—
16	#15689503	C₂²	1	1	yes	characteristic
17	#15689502	C₂²	1	2	yes	—
18	#15689501	—	2	2	no	—
19	#15689500	C₂²	1	1	yes	characteristic
20	#15689484	—	2	4	no	—
21	#15689473	C₂²	1	2	yes	—
22	#15689471	C₂²	1	2	yes	—
23	#15689471	C₂²	1	2	yes	—
24	#15689471	C₂²	1	2	yes	—
25	#15689470	C₂²	1	2	yes	—
26	#15689470	C₂²	1	2	yes	—
27	#15689443	—	2	4	no	—
28	#15689412	—	2	4	no	—
29	#15689360	—	2	2	no	—
30	#15689205	—	2	2	no	—
31	#15689174	C₂²	1	1	yes	characteristic
32	#15689123	—	2	4	no	—
33	C₄×#10625324	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄₇×#498	C₂²	1	2	yes	—
40	C₄₇×#494	C₂²	1	1	yes	characteristic
41	C₄₇×#487	C₂²	1	2	yes	—
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625354	—	2	2	no	—
44	C₂×Q₈×D₄₇	—	2	2	no	—
45	C₂×Dic₁₈₈	—	2	2	no	—
46	C₂×#10625337	—	2	4	no	—
47	Q₁₆×D₄₇	—	4	8	no	—
48	C₂×D₃₇₆	—	2	2	no	—
49	C₂×#10625324	D₄	1	2	yes	—
50	C₂×C₈×D₄₇	D₄	1	2	yes	—
51	#13094024	C₂³	1	2	yes	—
52	#13094023	—	2	2	no	—
53	#13094023	—	2	2	no	—
54	#13094022	—	2	2	no	—
55	#13094021	C₂³	1	2	yes	—
56	#13094013	—	2	2	no	—
57	#13093999	—	2	2	no	—
58	#13093999	—	2	2	no	—
59	#13093986	—	2	2	no	—
60	#13093986	—	2	2	no	—
61	C₄⋊C₄×D₄₇	—	2	2	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	C₄²×D₄₇	C₂³	1	1	yes	characteristic
64	#13093955	—	2	4	no	—
65	Dic₄₇⋊C₈	D₄	1	2	yes	—
66	#13093948	—	2	4	no	—
67	#13093946	—	2	4	no	—
68	#13093946	—	2	4	no	—
69	#13093944	—	2	4	no	—
70	#13093944	—	2	4	no	—
71	#13093944	—	2	4	no	—
72	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
73	#13093934	D₄	1	2	yes	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	D₄	1	2	yes	—
78	C₂×Q₁₆×C₄₇	—	2	2	no	—
79	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
81	#107×C₄₇	—	2	2	no	—
82	#106×C₄₇	—	2	4	no	—
83	#103×C₄₇	—	2	4	no	—
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₇	—	2	2	no	—
88	C₂×Q₈×C₄₇	—	2	2	no	—
89	#10625354	—	4	8	no	—
90	Q₈×D₄₇	—	4	8	no	—
91	C₂×D₁₈₈	—	2	2	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₉₄	—	2	2	no	—
101	C₂×Dic₉₄	—	4	2	no	—
102	Q₁₆×C₄₇	—	4	4	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
111	#10625337	—	4	8	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	(C₂×D₄₇)⋊C₄	—	4	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
124	Dic₁₈₈	—	4	4	no	—
125	D₃₇₆	—	4	4	no	—
126	#10625324	—	2	4	no	—
127	C₈×D₄₇	—	2	4	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	C₂²×D₄₇	#127	1	1	yes	characteristic
130	Q₈×C₄₇	—	4	4	no	—
131	Q₈×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	#127	1	1	yes	characteristic
139	C₂×Dic₄₇	#127	1	1	yes	characteristic
140	C₂×Dic₄₇	#127	1	1	yes	characteristic
141	D₁₈₈	—	4	4	no	—
142	D₁₈₈	—	4	4	no	—
143	D₁₈₈	—	4	4	no	—
144	D₁₈₈	—	8	4	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	8	4	no	—
148	C₄×D₄₇	—	8	4	no	—
149	C₄×D₄₇	#135	1	4	yes	—
150	Dic₉₄	—	4	4	no	—
151	Dic₉₄	—	8	4	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
155	C₄₇⋊C₈	—	2	2	no	—
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	2	2	no	—
158	#19639	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
160	C₂×D₄₇	—	2	1	no	—
161	C₂×D₄₇	—	2	1	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₂×D₄₇	—	8	2	no	—
165	C₂×D₄₇	#495	1	2	yes	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	C₂×#135	1	2	yes	—
169	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
170	C₄×C₄₇	#578	1	2	yes	—
171	Dic₄₇	—	2	1	no	—
172	Dic₄₇	—	2	1	no	—
173	Dic₄₇	—	8	2	no	—
174	Dic₄₇	#495	1	2	yes	—
175	#3227	—	47	2	no	—
176	#3169	—	47	1	no	—
177	C₂×#494	—	47	1	no	—
178	#3141	—	47	2	no	—
179	#1720	—	47	2	no	—
180	#1720	—	47	2	no	—
181	#1709	—	47	1	no	—
182	#1709	—	47	1	no	—
183	#1707	—	47	1	no	—
184	#1707	D₄₇	1	1	yes	characteristic, complemented
185	C₈×M₁₆	—	47	1	no	—
186	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
187	C₂×C₄₇	#3151	1	1	yes	characteristic
188	C₂×C₄₇	#3156	1	1	yes	characteristic
189	D₄₇	—	2	2	no	—
190	D₄₇	—	16	2	no	—
191	C₂²×Q₁₆	—	94	2	no	—
192	#532	—	47	2	no	—
193	C₂×C₄⋊Q₈	—	47	2	no	—
194	#500	C₂×D₄₇	1	1	yes	characteristic
195	#498	—	47	2	no	—
196	#498	C₂×D₄₇	1	2	yes	—
197	#494	—	47	2	no	—
198	#494	—	47	1	no	—
199	#494	—	94	2	no	—
200	#494	C₂×D₄₇	1	1	yes	characteristic
201	#493	—	47	1	no	—
202	#489	—	94	4	no	—
203	#487	—	47	2	no	—
204	#487	C₂×D₄₇	1	2	yes	—
205	#486	—	47	2	no	—
206	#486	—	47	2	no	—
207	#486	—	47	2	no	—
208	#486	—	47	2	no	—
209	#486	—	47	2	no	—
210	#481	—	94	4	no	—
211	#476	—	94	4	no	—
212	#466	—	94	2	no	—
213	C₄×D₈	—	94	2	no	—
214	#431	—	47	1	no	—
215	#417	—	94	4	no	—
216	C₄×M₁₆	—	47	1	no	—
217	C₂×C₄×C₈	—	47	1	no	—
218	#322	—	47	1	no	—
219	#322	—	47	1	no	—
220	C₈²	—	47	1	no	—
221	C₈²	C₂×D₄₇	1	1	yes	characteristic
222	C₄₇	#19639	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	94	2	no	—
224	C₂²×Q₈	—	94	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	—	94	4	no	—
227	C₂×Q₁₆	—	94	2	no	—
228	C₂×Q₁₆	—	188	8	no	—
229	C₂×D₈	—	94	2	no	—
230	C₂×M₁₆	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₄⋊Q₈	—	47	2	no	—
233	C₄⋊Q₈	—	94	2	no	—
234	C₄⋊Q₈	—	94	2	no	—
235	C₄⋊Q₈	—	94	2	no	—
236	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
237	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
238	#127	—	47	2	no	—
239	#124	—	94	2	no	—
240	#121	—	94	2	no	—
241	#121	—	94	2	no	—
242	C₄×D₄	—	94	2	no	—
243	C₄×D₄	—	94	2	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	—	94	2	no	—
246	C₂×C₄²	—	47	1	no	—
247	#107	—	2	2	no	—
248	#106	—	2	4	no	—
249	#106	—	94	4	no	—
250	C₄⋊C₈	—	47	2	no	—
251	#103	—	2	4	no	—
252	#103	—	94	4	no	—
253	#102	—	94	4	no	—
254	#102	—	94	4	no	—
255	#102	—	94	4	no	—
256	#102	—	94	4	no	—
257	#102	—	94	4	no	—
258	C₂²⋊C₈	—	47	2	no	—
259	#97	—	47	2	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	47	1	no	—
262	C₄×C₈	—	47	1	no	—
263	C₄×C₈	—	47	1	no	—
264	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	#41	—	188	8	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	94	2	no	—
271	C₂×Q₈	—	188	2	no	—
272	C₂×Q₈	—	188	8	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	—	94	2	no	—
275	C₂×D₄	—	94	2	no	—
276	C₂×D₄	—	188	2	no	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	47	1	no	—
280	C₂²×C₄	—	188	2	no	—
281	C₂²×C₄	—	188	2	no	—
282	Q₁₆	—	4	4	no	—
283	Q₁₆	—	188	8	no	—
284	Q₁₆	—	188	4	no	—
285	D₈	—	188	4	no	—
286	M₁₆	—	94	4	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	94	4	no	—
291	C₂×C₈	D₄×D₄₇	1	2	yes	—
292	C₂×C₈	D₄×D₄₇	1	2	yes	—
293	C₂×C₈	D₄×D₄₇	1	2	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	188	2	no	—
301	C₄⋊C₄	—	188	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₂²⋊C₄	—	188	2	no	—
304	C₄²	—	47	1	no	—
305	C₄²	—	47	1	no	—
306	C₄²	—	47	1	no	—
307	C₄²	C₂³×D₄₇	1	1	yes	characteristic
308	C₂³	—	47	1	no	—
309	C₂³	—	188	2	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	376	4	no	—
313	Q₈	—	188	4	no	—
314	D₄	—	376	4	no	—
315	D₄	—	188	4	no	—
316	D₄	—	188	4	no	—
317	D₄	—	188	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	4	no	—
322	C₂×C₄	—	47	1	no	—
323	C₂×C₄	—	47	1	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	376	4	no	—
327	C₂×C₄	—	376	4	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	94	2	no	—
335	C₈	—	94	2	no	—
336	C₈	—	94	2	no	—
337	C₈	#13094051	1	4	yes	—
338	C₂²	—	47	2	no	—
339	C₂²	—	94	1	no	—
340	C₂²	—	94	1	no	—
341	C₂²	—	376	2	no	—
342	C₂²	—	376	2	no	—
343	C₂²	—	376	2	no	—
344	C₂²	#127×D₄₇	1	1	yes	characteristic
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	47	2	no	—
348	C₄	—	94	1	no	—
349	C₄	—	94	1	no	—
350	C₄	—	376	2	no	—
351	C₄	#135×D₄₇	1	2	yes	—
352	C₄	C₂×#13094051	1	2	yes	—
353	C₄	#15689936	1	2	yes	—
354	C₂	—	94	2	no	—
355	C₂	—	752	2	no	—
356	C₂	#495×D₄₇	1	1	yes	characteristic
357	C₂	#20253835	1	1	yes	characteristic
358	C₂	#20253876	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1707	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253968	C₂
hypercenter	#1707	D₄₇
hypocenter	C₄₇	#19639
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
1. #20254517	C₂	semidirect
3. #494×D₄₇	C₂	semidirect
4. #20253783	C₂	semidirect
5. #20243046	C₂	semidirect
6. #20243045	C₂	semidirect
7. #20242995	C₂	semidirect
8. #20242994	C₂	semidirect
9. #20242986	C₂	semidirect
10. C₈×#10625324	C₂	semidirect
11. C₄₇×#1707	C₂	semidirect
184. #1707	D₄₇	semidirect
222. C₄₇	#19639	semidirect

Maximal subgroups

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

subgroup	index
#20254517	2
#20253968	2
#494×D₄₇	2
#20253783	2
#20243046	2
#20243045	2
#20242995	2
#20242994	2
#20242986	2
C₈×#10625324	2
C₄₇×#1707	2
#19639	47