Subgroups — C₄₇×#19001 #26058699

Order 12032.

960 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#562	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3213	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1662	C₂	1	4	yes	maximal, complemented
5	C₄₇×#1656	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1534	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1527	C₂	1	2	yes	maximal, complemented
8	C₄₇×#562	—	2	2	no	—
9	C₄₇×#562	—	2	4	no	—
10	C₄₇×#562	C₂²	1	4	yes	—
11	C₄₇×#532	C₂²	1	1	yes	characteristic
12	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₄₇	C₂²	1	2	yes	—
14	C₂×#123×C₄₇	—	2	2	no	—
15	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
16	C₂×#121×C₄₇	—	2	2	no	—
17	C₄₇×#496	C₂²	1	4	yes	complemented
18	C₄₇×#491	C₂²	1	2	yes	complemented
19	C₄₇×#490	C₂²	1	2	yes	complemented
20	C₄₇×#489	C₂²	1	4	yes	complemented
21	C₄₇×#488	—	2	2	no	—
22	C₄₇×#463	—	2	4	no	—
23	C₄₇×#448	—	2	2	no	—
24	C₄₇×#447	—	2	2	no	—
25	C₄₇×#432	C₂²	1	2	yes	—
26	C₄₇×#417	—	2	2	no	—
27	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#330	C₂²	1	4	yes	complemented
29	C₄₇×#329	C₂²	1	8	yes	complemented
30	C₂×#41×C₄₇	—	2	1	no	—
31	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
32	C₂²×D₄×C₄₇	—	2	2	no	—
33	C₂²×D₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	—	2	2	no	—
35	C₂×SD₁₆×C₄₇	—	2	4	no	—
36	C₂×D₈×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	1	no	—
38	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
39	#127×C₄₇	C₂³	1	1	yes	characteristic
40	#125×C₄₇	—	2	2	no	—
41	#125×C₄₇	D₄	1	2	yes	—
42	#125×C₄₇	C₂³	1	2	yes	—
43	#124×C₄₇	—	2	2	no	—
44	#124×C₄₇	—	2	2	no	—
45	#124×C₄₇	—	2	1	no	—
46	#124×C₄₇	D₄	1	4	yes	—
47	#124×C₄₇	C₂³	1	4	yes	—
48	#123×C₄₇	—	2	8	no	—
49	#123×C₄₇	—	2	8	no	—
50	C₂²⋊Q₈×C₄₇	—	2	8	no	—
51	C₂²⋊Q₈×C₄₇	—	2	8	no	—
52	#121×C₄₇	—	2	8	no	—
53	#121×C₄₇	—	2	8	no	—
54	#121×C₄₇	—	2	2	no	—
55	C₄×D₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₇	—	2	2	no	—
64	#103×C₄₇	—	2	4	no	—
65	#103×C₄₇	—	2	4	no	—
66	#102×C₄₇	—	2	4	no	—
67	#102×C₄₇	—	2	4	no	—
68	C₂²⋊C₈×C₄₇	—	2	2	no	—
69	#97×C₄₇	C₂³	1	4	yes	—
70	#97×C₄₇	C₂³	1	2	yes	—
71	C₄×C₈×C₄₇	—	2	1	no	—
72	C₂⁴×C₄₇	—	2	2	no	—
73	#41×C₄₇	—	4	4	no	—
74	C₂×Q₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	—	2	1	no	—
76	C₂×Q₈×C₄₇	—	8	2	no	—
77	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₄₇	—	2	8	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	2	1	no	—
82	C₂×D₄×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	8	2	no	—
84	C₂×D₄×C₄₇	—	8	4	no	—
85	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	—	2	8	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	—	8	4	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
93	SD₁₆×C₄₇	—	8	4	no	—
94	D₈×C₄₇	—	8	4	no	—
95	M₁₆×C₄₇	—	4	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	2	8	no	—
99	C₄⋊C₄×C₄₇	—	2	4	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	2	4	no	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₄₇	—	2	8	no	—
105	C₂²⋊C₄×C₄₇	—	2	4	no	—
106	C₂²⋊C₄×C₄₇	—	2	4	no	—
107	C₂²⋊C₄×C₄₇	—	2	4	no	—
108	C₂²⋊C₄×C₄₇	—	2	4	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₄²×C₄₇	C₂×D₄	1	2	yes	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	C₂³×C₄₇	—	2	4	no	—
114	C₂³×C₄₇	—	4	4	no	—
115	C₂³×C₄₇	—	4	4	no	—
116	C₂³×C₄₇	—	4	4	no	—
117	C₂³×C₄₇	—	4	1	no	—
118	C₂³×C₄₇	#120	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	4	no	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	8	2	no	—
122	D₄×C₄₇	—	4	4	no	—
123	D₄×C₄₇	—	4	4	no	—
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	8	2	no	—
126	D₄×C₄₇	—	8	8	no	—
127	D₄×C₄₇	—	8	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	8	2	no	—
136	C₂×C₄×C₄₇	—	8	8	no	—
137	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#120	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
141	C₈×C₄₇	—	4	2	no	—
142	C₈×C₄₇	—	4	4	no	—
143	#19001	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	—	8	8	no	—
149	C₂²×C₄₇	—	8	1	no	—
150	C₂²×C₄₇	—	8	2	no	—
151	C₂²×C₄₇	#454	1	2	yes	—
152	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	—	4	4	no	—
159	C₄×C₄₇	—	8	1	no	—
160	C₂×#562	C₂×C₄₇	1	1	yes	characteristic, complemented
161	#3213	C₂×C₄₇	1	2	yes	complemented
162	#3142	C₂×C₄₇	1	1	yes	characteristic, complemented
163	#1662	C₂×C₄₇	1	4	yes	complemented
164	#1656	C₂×C₄₇	1	4	yes	complemented
165	#1534	C₂×C₄₇	1	1	yes	characteristic, complemented
166	#1527	C₂×C₄₇	1	2	yes	complemented
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	8	4	no	—
169	C₂×C₄₇	—	16	1	no	—
170	C₂×C₄₇	C₂×#454	1	1	yes	characteristic
171	C₂×C₄₇	#3019	1	2	yes	—
172	#562	—	2	2	no	—
173	#562	—	2	4	no	—
174	#562	C₂²×C₄₇	1	4	yes	—
175	#532	C₂²×C₄₇	1	1	yes	characteristic
176	C₂×#125	C₂²×C₄₇	1	1	yes	characteristic
177	C₂×#124	C₂²×C₄₇	1	2	yes	—
178	C₂×#123	—	2	2	no	—
179	C₂×C₂²⋊Q₈	—	2	2	no	—
180	C₂×#121	—	2	2	no	—
181	#496	C₂²×C₄₇	1	4	yes	complemented
182	#491	C₂²×C₄₇	1	2	yes	complemented
183	#490	C₂²×C₄₇	1	2	yes	complemented
184	#489	C₂²×C₄₇	1	4	yes	complemented
185	#488	—	2	2	no	—
186	#463	—	2	4	no	—
187	#448	—	2	2	no	—
188	#447	—	2	2	no	—
189	#432	C₂²×C₄₇	1	2	yes	—
190	#417	—	2	2	no	—
191	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
192	#330	C₂²×C₄₇	1	4	yes	complemented
193	#329	C₂²×C₄₇	1	8	yes	complemented
194	C₄₇	#19001	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×#41	—	2	1	no	—
196	C₂²×Q₈	D₄×C₄₇	1	2	yes	—
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	D₄×C₄₇	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×D₈	—	2	4	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
204	#127	C₂³×C₄₇	1	1	yes	characteristic
205	#125	—	2	2	no	—
206	#125	D₄×C₄₇	1	2	yes	—
207	#125	C₂³×C₄₇	1	2	yes	—
208	#124	—	2	2	no	—
209	#124	—	2	2	no	—
210	#124	—	2	1	no	—
211	#124	D₄×C₄₇	1	4	yes	—
212	#124	C₂³×C₄₇	1	4	yes	—
213	#123	—	2	8	no	—
214	#123	—	2	8	no	—
215	C₂²⋊Q₈	—	2	8	no	—
216	C₂²⋊Q₈	—	2	8	no	—
217	#121	—	2	8	no	—
218	#121	—	2	8	no	—
219	#121	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	D₄×C₄₇	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	#103	—	2	4	no	—
230	#103	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	C₂³×C₄₇	1	2	yes	—
235	#97	C₂³×C₄₇	1	4	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₂⁴	—	2	2	no	—
238	#41	—	4	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	8	2	no	—
242	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
243	C₂×D₄	—	2	8	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	8	2	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	8	4	no	—
256	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
258	SD₁₆	—	8	4	no	—
259	D₈	—	8	4	no	—
260	M₁₆	—	4	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	C₂×D₄×C₄₇	1	4	yes	—
269	C₂²⋊C₄	—	2	8	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₄₇	1	2	yes	—
277	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	1	no	—
283	C₂³	#120×C₄₇	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	2	no	—
291	D₄	—	8	8	no	—
292	D₄	—	8	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	8	no	—
302	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
303	C₂×C₄	#120×C₄₇	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	8	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₄₇×#454	1	2	yes	—
316	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	8	4	no	—
326	C₂	—	16	1	no	—
327	C₂	C₂×C₄₇×#454	1	1	yes	characteristic
328	C₂	C₄₇×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#562	C₂	semidirect
2. C₄₇×#3213	C₂	semidirect
3. C₄₇×#3142	C₂	semidirect
4. C₄₇×#1662	C₂	semidirect
5. C₄₇×#1656	C₂	semidirect
6. C₄₇×#1534	C₂	semidirect
7. C₄₇×#1527	C₂	semidirect
17. C₄₇×#496	C₂²	semidirect
18. C₄₇×#491	C₂²	semidirect
19. C₄₇×#490	C₂²	semidirect
20. C₄₇×#489	C₂²	semidirect
28. C₄₇×#330	C₂²	semidirect
29. C₄₇×#329	C₂²	semidirect
143. #19001	C₄₇	direct (N × H)
160. C₂×#562	C₂×C₄₇	semidirect
161. #3213	C₂×C₄₇	semidirect
162. #3142	C₂×C₄₇	semidirect
163. #1662	C₂×C₄₇	semidirect
164. #1656	C₂×C₄₇	semidirect
165. #1534	C₂×C₄₇	semidirect
166. #1527	C₂×C₄₇	semidirect
181. #496	C₂²×C₄₇	semidirect
182. #491	C₂²×C₄₇	semidirect
183. #490	C₂²×C₄₇	semidirect
184. #489	C₂²×C₄₇	semidirect
192. #330	C₂²×C₄₇	semidirect
193. #329	C₂²×C₄₇	semidirect
194. C₄₇	#19001	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#562	2
C₄₇×#3213	2
C₄₇×#3142	2
C₄₇×#1662	2
C₄₇×#1656	2
C₄₇×#1534	2
C₄₇×#1527	2
#19001	47

Subgroups — C47×#19001 #26058699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19001 #26058699