Subgroups — #26370786

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253043	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245499	C₂	1	4	yes	maximal, complemented
5	#20245482	C₂	1	4	yes	maximal, complemented
6	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245107	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3058	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	#15689694	—	2	1	no	—
12	C₂×#13093968	—	2	4	no	—
13	C₂×#13093968	C₂²	1	2	yes	—
14	C₂×#13093968	C₂²	1	2	yes	—
15	#15689540	—	2	4	no	—
16	C₂×#13093964	C₂²	1	1	yes	characteristic
17	C₂×#13093964	C₂²	1	2	yes	complemented
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
20	#15689381	—	2	2	no	—
21	#15689364	C₂²	1	4	yes	complemented
22	#15689161	C₂²	1	1	yes	characteristic
23	#15689133	—	2	1	no	—
24	#15688950	C₂²	1	2	yes	complemented
25	#15688948	C₂²	1	2	yes	complemented
26	#15688926	C₂²	1	4	yes	complemented
27	#15688924	C₂²	1	4	yes	—
28	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#535	—	2	1	no	—
31	C₄₇×#469	—	2	2	no	—
32	C₄₇×#466	C₂²	1	4	yes	—
33	C₄₇×#427	C₂²	1	1	yes	characteristic
34	C₄₇×#418	—	2	1	no	—
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂×Dic₁₈₈	—	2	2	no	—
37	C₂×Dic₁₈₈	—	2	2	no	—
38	C₂×Dic₁₈₈	—	2	1	no	—
39	C₂×Dic₁₈₈	C₂³	1	2	yes	—
40	#13094022	—	4	1	no	—
41	#13094011	—	4	1	no	—
42	#13094005	—	4	2	no	—
43	#13094003	—	2	4	no	—
44	#13093983	—	2	1	no	—
45	#13093968	—	2	4	no	—
46	#13093968	—	2	4	no	—
47	#13093964	—	2	2	no	—
48	#13093964	—	2	4	no	—
49	C₂×C₄₇⋊C₁₆	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13093958	C₂³	1	2	yes	—
55	#13093956	—	2	1	no	—
56	#13093949	—	2	4	no	—
57	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂×D₈×C₄₇	—	2	2	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	2	no	—
62	C₂×D₈×C₄₇	C₂³	1	2	yes	—
63	#131×C₄₇	—	2	2	no	—
64	#131×C₄₇	—	2	2	no	—
65	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
66	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	#127×C₄₇	—	4	1	no	—
70	#124×C₄₇	—	4	1	no	—
71	#121×C₄₇	—	2	4	no	—
72	#120×C₄₇	—	4	2	no	—
73	#117×C₄₇	—	2	1	no	—
74	#107×C₄₇	C₂³	1	2	yes	—
75	#106×C₄₇	—	2	1	no	—
76	#102×C₄₇	—	2	4	no	—
77	C₂⁴×C₄₇	—	4	1	no	—
78	#41×C₄₇	—	2	2	no	—
79	#41×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	—	4	4	no	—
83	C₂×D₄×C₄₇	—	4	2	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂×D₄×C₄₇	—	4	2	no	—
86	C₂×D₄×C₄₇	D₈	1	2	yes	—
87	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₄₇	—	4	1	no	—
91	C₂×Dic₉₄	—	2	2	no	—
92	C₂×Dic₉₄	—	4	2	no	—
93	C₂×Dic₉₄	—	4	2	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	D₈×C₄₇	—	2	2	no	—
96	D₈×C₄₇	—	2	2	no	—
97	D₈×C₄₇	—	4	2	no	—
98	M₁₆×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	2	4	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₇	#41	1	2	yes	—
106	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	1	no	—
109	C₂²⋊C₄×C₄₇	—	4	4	no	—
110	C₄²×C₄₇	—	4	1	no	—
111	C₂²⋊Dic₄₇	—	4	1	no	—
112	C₂²⋊Dic₄₇	—	4	4	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	Dic₄₇⋊C₄	—	4	4	no	—
115	C₄×Dic₄₇	—	4	1	no	—
116	Dic₁₈₈	—	2	2	no	—
117	Dic₁₈₈	—	2	2	no	—
118	Dic₁₈₈	—	4	2	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	8	2	no	—
124	C₂³×C₄₇	—	16	2	no	—
125	C₂³×C₄₇	C₂×D₈	1	2	yes	—
126	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₄₇	—	2	2	no	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	2	2	no	—
130	D₄×C₄₇	—	4	2	no	—
131	D₄×C₄₇	—	4	2	no	—
132	D₄×C₄₇	—	8	2	no	—
133	D₄×C₄₇	—	16	4	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	#121	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#135	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	8	2	no	—
145	Dic₉₄	—	4	2	no	—
146	Dic₉₄	—	4	2	no	—
147	Dic₉₄	—	8	2	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	#121	1	2	yes	—
150	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
152	#33609	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	4	no	—
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	8	4	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	16	4	no	—
159	C₂²×C₄₇	#466	1	2	yes	—
160	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	#466	1	2	yes	—
164	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	2	no	—
168	C₂²×SD₃₂	—	47	1	no	—
169	C₂×#504	—	47	1	no	—
170	#3059	—	47	1	no	—
171	#3058	D₄₇	1	1	yes	characteristic
172	#2220	—	47	4	no	—
173	#2217	—	47	4	no	—
174	#2158	—	47	1	no	—
175	#2141	—	47	1	no	—
176	#2140	—	47	1	no	—
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	4	2	no	—
179	C₂×C₄₇	—	16	2	no	—
180	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
181	C₂×C₄₇	#3416	1	2	yes	—
182	C₂²×Q₁₆	—	47	1	no	—
183	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
185	#536	—	94	1	no	—
186	#535	—	2	1	no	—
187	C₂×SD₃₂	—	47	2	no	—
188	C₂×SD₃₂	—	47	2	no	—
189	C₂×SD₃₂	—	94	4	no	—
190	#504	—	94	4	no	—
191	C₂×M₃₂	—	47	1	no	—
192	C₂×M₃₂	—	47	2	no	—
193	C₂²×C₁₆	—	47	1	no	—
194	C₂²×C₁₆	—	47	2	no	—
195	#470	—	94	2	no	—
196	#469	—	2	2	no	—
197	#467	—	47	4	no	—
198	#466	C₂×D₄₇	1	4	yes	—
199	#427	—	47	1	no	—
200	#427	C₂×D₄₇	1	1	yes	characteristic
201	#419	—	94	1	no	—
202	#418	—	2	1	no	—
203	#367	—	47	2	no	—
204	#366	—	47	2	no	—
205	#358	—	47	4	no	—
206	#357	—	47	4	no	—
207	C₄₇	#33609	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂²×Q₈	—	94	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂×Q₁₆	—	47	2	no	—
212	C₂×Q₁₆	—	94	2	no	—
213	C₂×Q₁₆	—	94	2	no	—
214	C₂×Q₁₆	—	94	1	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂²×D₄₇	1	2	yes	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	188	1	no	—
226	#127	—	4	1	no	—
227	#124	—	4	1	no	—
228	#124	—	188	1	no	—
229	#123	—	188	2	no	—
230	C₂²⋊Q₈	—	94	4	no	—
231	#121	—	2	4	no	—
232	#120	—	4	2	no	—
233	#117	—	2	1	no	—
234	#117	—	94	1	no	—
235	SD₃₂	—	94	4	no	—
236	SD₃₂	—	94	4	no	—
237	M₃₂	—	94	4	no	—
238	M₃₂	—	94	2	no	—
239	C₂×C₁₆	—	47	4	no	—
240	C₂×C₁₆	—	47	1	no	—
241	C₂×C₁₆	—	47	1	no	—
242	C₂×C₁₆	—	47	2	no	—
243	C₂×C₁₆	—	94	4	no	—
244	#107	—	47	2	no	—
245	#107	C₂²×D₄₇	1	2	yes	—
246	#106	—	2	1	no	—
247	#106	—	94	1	no	—
248	#103	—	94	4	no	—
249	#102	—	2	4	no	—
250	C₂⁴	—	4	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	94	2	no	—
254	C₂×Q₈	—	188	2	no	—
255	C₂×Q₈	—	188	2	no	—
256	C₂×Q₈	—	188	1	no	—
257	C₂×Q₈	#10625334	1	1	yes	characteristic
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	#10625334	1	2	yes	—
264	C₂×D₄	#10625334	1	1	yes	characteristic
265	C₂²×C₄	—	188	1	no	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	Q₁₆	—	94	2	no	—
269	Q₁₆	—	94	2	no	—
270	Q₁₆	—	188	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	4	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	D₄×D₄₇	1	2	yes	—
279	C₂×C₈	#10625352	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
281	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₄⋊C₄	—	188	4	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	188	1	no	—
289	C₂²⋊C₄	—	188	4	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	188	1	no	—
292	C₁₆	—	47	2	no	—
293	C₁₆	—	47	2	no	—
294	C₁₆	—	94	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	16	2	no	—
298	C₂³	C₂×#10625334	1	2	yes	—
299	C₂³	C₂×#10625334	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	376	2	no	—
302	Q₈	—	188	2	no	—
303	Q₈	—	188	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	376	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	#13093996	1	2	yes	—
317	C₂×C₄	C₂×#10625334	1	2	yes	—
318	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
319	C₂×C₄	#13094044	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#13093996	1	2	yes	—
323	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
324	C₈	C₂×#10625352	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	8	4	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	16	4	no	—
331	C₂²	#15689357	1	2	yes	—
332	C₂²	C₂²×#10625334	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	376	2	no	—
336	C₄	—	376	2	no	—
337	C₄	#15689357	1	2	yes	—
338	C₄	C₂×#13093996	1	1	yes	characteristic
339	C₄	C₂×#13094044	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×#15689357	1	1	yes	characteristic
344	C₂	#20256535	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3058	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3058	D₄₇
hypocenter	C₄₇	#33609
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3058

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093968	C₂	semidirect
2. C₂×#15689540	C₂	semidirect
3. #20253043	C₂	semidirect
4. #20245499	C₂	semidirect
5. #20245482	C₂	semidirect
6. #20245170	C₂	semidirect
7. #20245107	C₂	semidirect
8. #20245104	C₂	semidirect
17. C₂×#13093964	C₂²	semidirect
19. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
21. #15689364	C₂²	semidirect
24. #15688950	C₂²	semidirect
25. #15688948	C₂²	semidirect
26. #15688926	C₂²	semidirect
50. C₂×C₄₇⋊C₁₆	D₄	semidirect
207. C₄₇	#33609	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093968	2
C₂×#15689540	2
#20253043	2
#20245499	2
#20245482	2
#20245170	2
#20245107	2
#20245104	2
C₄₇×#3058	2
#33609	47