Subgroups — #26373230

Order 12032.

556 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	#20255820	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255146	C₂	1	1	yes	maximal, characteristic
3	#20245611	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245578	C₂	1	2	yes	maximal, complemented
5	#20245531	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245481	C₂	1	2	yes	maximal, complemented
7	#20245325	C₂	1	2	yes	maximal, complemented
8	C₄×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245207	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245092	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	C₂²	1	2	yes	—
13	#15689846	—	2	1	no	—
14	#15689793	—	2	1	no	—
15	C₂×#13093970	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
17	#15689528	C₂²	1	1	yes	characteristic
18	#15689463	—	2	2	no	—
19	#15689408	—	2	1	no	—
20	#15689361	C₂²	1	2	yes	—
21	#15689356	C₂²	1	2	yes	—
22	#15689320	—	2	1	no	—
23	#15689265	—	2	2	no	—
24	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
25	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
26	#15689160	C₂²	1	2	yes	—
27	#15688951	C₂²	1	2	yes	—
28	#15688948	C₂²	1	1	yes	characteristic
29	#15688928	C₂²	1	2	yes	—
30	#15688927	C₂²	1	1	yes	characteristic
31	#15688926	C₂²	1	1	yes	characteristic
32	#15688924	C₂²	1	2	yes	—
33	#15688924	C₂²	1	2	yes	—
34	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#493	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#357	C₂²	1	2	yes	—
41	C₂×#10625352	—	2	2	no	—
42	#13094043	—	2	4	no	—
43	#13094043	—	4	2	no	—
44	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
45	C₂×#10625335	—	2	2	no	—
46	C₂×C₈×D₄₇	C₂³	1	2	yes	—
47	#13094024	—	2	1	no	—
48	#13094020	—	4	1	no	—
49	#13094019	—	4	2	no	—
50	#13094012	—	4	1	no	—
51	#13094000	—	2	2	no	—
52	#13093996	—	4	2	no	—
53	#13093995	—	2	2	no	—
54	C₄×Dic₉₄	—	2	1	no	—
55	C₄×Dic₉₄	—	4	1	no	—
56	C₄×D₁₈₈	D₄	1	1	yes	characteristic
57	D₄×Dic₄₇	—	4	1	no	—
58	#13093982	—	2	2	no	—
59	#13093970	—	2	2	no	—
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₁₆×D₄₇	—	4	2	no	—
63	#13093958	—	2	1	no	—
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093958	C₂³	1	2	yes	—
66	#13093955	—	2	2	no	—
67	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
68	#13093949	—	2	1	no	—
69	#13093948	—	2	2	no	—
70	#13093945	—	2	2	no	—
71	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
72	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×D₈×C₄₇	—	2	1	no	—
74	C₂×D₈×C₄₇	C₂³	1	2	yes	—
75	#127×C₄₇	—	2	1	no	—
76	C₄×Q₈×C₄₇	—	2	1	no	—
77	SD₃₂×C₄₇	—	4	2	no	—
78	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#103×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	2	1	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂×D₄×C₄₇	—	4	1	no	—
86	C₂×C₂²⋊D₄₇	—	4	2	no	—
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	C₂²×Dic₄₇	—	4	2	no	—
89	#10625352	—	4	4	no	—
90	#10625352	—	8	2	no	—
91	C₂×D₁₈₈	D₈	1	1	yes	characteristic
92	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
93	C₂×Dic₉₄	—	2	1	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	Q₁₆×C₄₇	—	2	1	no	—
96	Q₁₆×C₄₇	—	2	1	no	—
97	D₈×C₄₇	—	2	2	no	—
98	D₈×C₄₇	—	2	2	no	—
99	D₈×C₄₇	—	8	1	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₇	—	2	1	no	—
103	C₄⋊C₄×C₄₇	—	4	1	no	—
104	C₄²×C₄₇	—	4	1	no	—
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₄₇	—	4	2	no	—
107	#10625335	—	4	4	no	—
108	(C₂×D₄₇)⋊C₄	—	4	2	no	—
109	(C₂×D₄₇)⋊C₄	—	4	2	no	—
110	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
111	C₄⋊Dic₄₇	—	2	1	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
115	Dic₄₇⋊C₄	—	4	1	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	C₄×Dic₄₇	—	4	1	no	—
118	C₄×Dic₄₇	—	4	2	no	—
119	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
120	Dic₁₈₈	—	2	1	no	—
121	Dic₁₈₈	#41	1	2	yes	—
122	C₈×D₄₇	—	2	2	no	—
123	C₈×D₄₇	—	2	2	no	—
124	C₁₆×C₄₇	—	2	2	no	—
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₂³×C₄₇	—	4	2	no	—
128	C₂²×D₄₇	C₂×D₈	1	2	yes	—
129	Q₈×C₄₇	—	4	1	no	—
130	Q₈×C₄₇	—	4	1	no	—
131	D₄×C₄₇	—	4	2	no	—
132	D₄×C₄₇	—	4	2	no	—
133	D₄×C₄₇	—	8	2	no	—
134	D₄×C₄₇	—	16	1	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₄₇	—	8	4	no	—
141	C₂²⋊D₄₇	—	16	2	no	—
142	C₂×Dic₄₇	—	4	1	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	8	4	no	—
146	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
147	D₁₈₈	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	2	2	no	—
150	Dic₉₄	—	2	2	no	—
151	Dic₉₄	—	4	1	no	—
152	Dic₉₄	—	8	2	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₄₇⋊C₈	—	2	2	no	—
157	#33871	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₇	—	2	2	no	—
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	8	1	no	—
167	C₄×C₄₇	—	8	1	no	—
168	C₄×C₄₇	#508	1	2	yes	—
169	C₄×C₄₇	#546	1	1	yes	characteristic
170	C₄×C₄₇	#577	1	1	yes	characteristic
171	Dic₄₇	—	2	2	no	—
172	Dic₄₇	—	4	2	no	—
173	Dic₄₇	—	8	2	no	—
174	Dic₄₇	—	8	1	no	—
175	#3342	—	47	1	no	—
176	#3287	—	47	1	no	—
177	#2240	—	47	1	no	—
178	#2234	—	47	2	no	—
179	#2225	—	47	1	no	—
180	#2217	—	47	2	no	—
181	#2207	D₄₇	1	1	yes	characteristic, complemented
182	#2189	—	47	2	no	—
183	C₄×Q₃₂	—	47	1	no	—
184	D₄×C₁₆	—	47	1	no	—
185	#2138	—	47	2	no	—
186	C₂×C₄₇	—	16	2	no	—
187	C₂×C₄₇	#3294	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
189	C₂×C₄₇	#3416	1	1	yes	characteristic
190	D₄₇	—	4	2	no	—
191	C₂×#135	—	47	2	no	—
192	#556	—	94	1	no	—
193	#550	—	94	1	no	—
194	C₂×Q₃₂	—	47	1	no	—
195	C₂×SD₃₂	C₂×D₄₇	1	2	yes	—
196	C₂²×C₁₆	—	47	2	no	—
197	#500	—	47	1	no	—
198	#493	C₂×D₄₇	1	1	yes	characteristic
199	#485	—	94	2	no	—
200	#475	—	94	1	no	—
201	#467	—	47	2	no	—
202	#466	—	47	2	no	—
203	#460	—	94	1	no	—
204	#449	—	94	2	no	—
205	C₄×Q₁₆	—	47	1	no	—
206	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
207	C₈×D₄	—	47	1	no	—
208	#427	—	47	2	no	—
209	#367	—	47	2	no	—
210	#366	—	47	1	no	—
211	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
212	#358	—	47	1	no	—
213	#358	—	47	2	no	—
214	#358	—	47	1	no	—
215	#357	—	47	2	no	—
216	#357	—	47	2	no	—
217	#357	C₂×D₄₇	1	2	yes	—
218	C₂²⋊C₁₆	—	47	2	no	—
219	C₄×C₁₆	—	47	1	no	—
220	C₄₇	#33871	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	94	2	no	—
222	#135	—	94	4	no	—
223	#135	—	188	2	no	—
224	C₂×Q₁₆	—	47	1	no	—
225	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	94	2	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×D₈	C₂²×D₄₇	1	2	yes	—
229	C₂²×C₈	—	47	2	no	—
230	C₄⋊Q₈	—	94	1	no	—
231	#127	—	2	1	no	—
232	#127	—	188	1	no	—
233	#126	—	188	2	no	—
234	#124	—	188	1	no	—
235	C₂²⋊Q₈	—	94	2	no	—
236	#121	—	94	2	no	—
237	#121	—	188	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	94	1	no	—
240	C₄×Q₈	—	188	1	no	—
241	C₄×D₄	—	47	1	no	—
242	C₄×D₄	—	188	1	no	—
243	#117	—	94	2	no	—
244	Q₃₂	—	94	2	no	—
245	SD₃₂	—	4	2	no	—
246	C₂×C₁₆	—	47	1	no	—
247	C₂×C₁₆	—	47	1	no	—
248	C₂×C₁₆	—	188	2	no	—
249	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
250	#107	—	47	2	no	—
251	#107	—	47	1	no	—
252	#107	—	94	1	no	—
253	#107	C₂²×D₄₇	1	1	yes	characteristic
254	#106	—	94	2	no	—
255	C₄⋊C₈	—	47	1	no	—
256	#103	—	2	1	no	—
257	#103	—	94	2	no	—
258	#103	—	94	1	no	—
259	#102	—	94	2	no	—
260	C₂²⋊C₈	—	47	2	no	—
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	#41	—	376	2	no	—
263	#41	—	188	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	94	1	no	—
266	C₂×Q₈	—	188	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	47	1	no	—
271	C₂×D₄	—	188	2	no	—
272	C₂×D₄	—	188	2	no	—
273	C₂²×C₄	—	47	2	no	—
274	C₂²×C₄	—	188	2	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	47	2	no	—
278	Q₁₆	—	94	1	no	—
279	SD₁₆	—	188	4	no	—
280	D₈	—	2	2	no	—
281	D₈	—	2	2	no	—
282	D₈	—	8	1	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	47	1	no	—
291	C₄⋊C₄	—	94	1	no	—
292	C₄⋊C₄	—	94	2	no	—
293	C₄⋊C₄	—	188	2	no	—
294	C₄⋊C₄	—	188	1	no	—
295	C₄⋊C₄	—	188	1	no	—
296	C₂²⋊C₄	—	47	2	no	—
297	C₂²⋊C₄	—	188	2	no	—
298	C₂²⋊C₄	—	188	2	no	—
299	C₂²⋊C₄	—	188	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	188	2	no	—
302	C₄²	—	188	1	no	—
303	C₄²	D₄×D₄₇	1	1	yes	characteristic
304	C₁₆	—	2	2	no	—
305	C₁₆	—	94	1	no	—
306	C₁₆	—	94	1	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	47	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	94	2	no	—
312	Q₈	—	376	2	no	—
313	Q₈	—	188	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	8	2	no	—
317	D₄	—	16	1	no	—
318	D₄	—	94	2	no	—
319	D₄	—	376	4	no	—
320	D₄	—	752	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	376	4	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	—	188	1	no	—
330	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
331	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	94	2	no	—
335	C₈	#41×D₄₇	1	1	yes	characteristic
336	C₈	#13094069	1	1	yes	characteristic
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	94	2	no	—
341	C₂²	—	94	2	no	—
342	C₂²	—	94	2	no	—
343	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	94	2	no	—
348	C₄	—	376	1	no	—
349	C₄	—	376	2	no	—
350	C₄	—	188	2	no	—
351	C₄	#15689554	1	2	yes	—
352	C₄	#15689770	1	1	yes	characteristic
353	C₄	#15689933	1	1	yes	characteristic
354	C₂	—	16	2	no	—
355	C₂	—	188	2	no	—
356	C₂	#20255204	1	1	yes	characteristic
357	C₂	C₂×#15689554	1	1	yes	characteristic
358	C₂	#20256534	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×#2207	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255146	C₂
hypercenter	#2207	D₄₇
hypocenter	C₄₇	#33871
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
1. #20255820	C₂	semidirect
3. #20245611	C₂	semidirect
4. #20245578	C₂	semidirect
5. #20245531	C₂	semidirect
6. #20245481	C₂	semidirect
7. #20245325	C₂	semidirect
8. C₄×#13093970	C₂	semidirect
9. #20245207	C₂	semidirect
10. #20245092	C₂	semidirect
11. C₄₇×#2207	C₂	semidirect
181. #2207	D₄₇	semidirect
220. C₄₇	#33871	semidirect

Maximal subgroups

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

subgroup	index
#20255820	2
#20255146	2
#20245611	2
#20245578	2
#20245531	2
#20245481	2
#20245325	2
C₄×#13093970	2
#20245207	2
#20245092	2
C₄₇×#2207	2
#33871	47