Subgroups — C₄₇×#11212 #26050910

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1740	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1733	C₂	1	2	yes	maximal
5	C₂×C₄₇×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₂×#117×C₄₇	C₂²	1	2	yes	—
8	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#424	—	2	4	no	—
12	C₄₇×#423	—	2	2	no	—
13	C₄₇×#384	C₄	1	2	yes	complemented
14	C₄₇×#382	C₄	1	2	yes	complemented
15	C₄₇×#380	—	2	2	no	—
16	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
17	C₄₇×#342	—	2	2	no	—
18	C₄₇×#339	—	2	4	no	—
19	C₄₇×#339	—	2	4	no	—
20	C₂³×C₄×C₄₇	—	2	1	no	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂×M₁₆×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	#117×C₄₇	—	2	2	no	—
30	#117×C₄₇	—	2	2	no	—
31	#117×C₄₇	—	2	2	no	—
32	#117×C₄₇	—	2	2	no	—
33	#117×C₄₇	—	2	2	no	—
34	#117×C₄₇	—	2	2	no	—
35	#117×C₄₇	—	2	2	no	—
36	#117×C₄₇	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	—	2	1	no	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
54	C₄⋊C₈×C₄₇	—	2	8	no	—
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	4	2	no	—
78	M₁₆×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	4	8	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₂²⋊C₄×C₄₇	—	2	4	no	—
88	C₂²⋊C₄×C₄₇	—	2	4	no	—
89	C₂²⋊C₄×C₄₇	—	4	2	no	—
90	C₂²⋊C₄×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	—	2	1	no	—
92	C₄²×C₄₇	—	2	1	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	4	no	—
98	C₄²×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	—	4	2	no	—
100	C₄²×C₄₇	—	4	1	no	—
101	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
102	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	#121	1	2	yes	—
106	C₂³×C₄₇	#121	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	8	4	no	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#121	1	2	yes	—
127	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#123	1	2	yes	—
132	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#125	1	2	yes	—
135	C₂×C₄×C₄₇	#125	1	2	yes	—
136	C₂×C₄×C₄₇	#126	1	2	yes	—
137	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₄₇	—	4	4	no	—
140	#11212	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	—	4	1	no	—
144	C₂²×C₄₇	—	4	1	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	#380	1	1	yes	characteristic
147	C₂²×C₄₇	#386	1	1	yes	characteristic
148	C₂²×C₄₇	#391	1	1	yes	characteristic
149	C₂²×C₄₇	#396	1	2	yes	—
150	C₂²×C₄₇	#398	1	2	yes	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	8	4	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	#387	1	2	yes	—
157	C₄×C₄₇	#389	1	2	yes	—
158	C₄×C₄₇	#394	1	1	yes	characteristic
159	C₄×C₄₇	#395	1	1	yes	characteristic
160	C₄×C₄₇	#398	1	1	yes	characteristic
161	C₄×C₄₇	#400	1	1	yes	characteristic
162	#2905	C₂×C₄₇	1	1	yes	characteristic
163	#2285	C₂×C₄₇	1	1	yes	characteristic
164	#1740	C₂×C₄₇	1	1	yes	characteristic
165	#1733	C₂×C₄₇	1	2	yes	—
166	C₂×#339	C₂×C₄₇	1	2	yes	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	2	1	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	#1441	1	1	yes	characteristic
172	C₂×C₄₇	#1884	1	2	yes	—
173	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
174	C₂×#117	C₂²×C₄₇	1	2	yes	—
175	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
176	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
177	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
178	#424	—	2	4	no	—
179	#423	—	2	2	no	—
180	#384	C₄×C₄₇	1	2	yes	complemented
181	#382	C₄×C₄₇	1	2	yes	complemented
182	#380	—	2	2	no	—
183	C₄×C₄⋊C₄	—	2	2	no	—
184	#342	—	2	2	no	—
185	#339	—	2	4	no	—
186	#339	—	2	4	no	—
187	C₄₇	#11212	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	Q₈×C₄₇	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₄₇	1	2	yes	—
220	C₂×C₄²	D₄×C₄₇	1	2	yes	—
221	C₂×C₄²	Q₈×C₄₇	1	2	yes	—
222	C₄⋊C₈	—	2	8	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
239	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
240	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
241	C₂²×C₄	#41×C₄₇	1	2	yes	—
242	C₂²×C₄	#41×C₄₇	1	2	yes	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	8	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₄⋊C₄×C₄₇	1	2	yes	—
270	C₄²	C₄⋊C₄×C₄₇	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
273	C₂³	#121×C₄₇	1	2	yes	—
274	C₂³	#121×C₄₇	1	1	yes	characteristic
275	C₂³	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
276	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
290	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
291	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
292	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
293	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
294	C₂×C₄	#121×C₄₇	1	2	yes	—
295	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
297	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
298	C₂×C₄	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
299	C₂×C₄	#123×C₄₇	1	2	yes	—
300	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
301	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
302	C₂×C₄	#125×C₄₇	1	2	yes	—
303	C₂×C₄	#125×C₄₇	1	2	yes	—
304	C₂×C₄	#126×C₄₇	1	2	yes	—
305	C₂×C₄	C₄⋊Q₈×C₄₇	1	2	yes	—
306	C₂×C₄	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₄₇×#380	1	1	yes	characteristic
314	C₂²	C₄₇×#386	1	1	yes	characteristic
315	C₂²	C₄₇×#391	1	1	yes	characteristic
316	C₂²	C₄₇×#396	1	2	yes	—
317	C₂²	C₄₇×#398	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	4	no	—
322	C₄	—	8	2	no	—
323	C₄	C₄₇×#387	1	2	yes	—
324	C₄	C₄₇×#389	1	2	yes	—
325	C₄	C₄₇×#394	1	1	yes	characteristic
326	C₄	C₄₇×#395	1	1	yes	characteristic
327	C₄	C₄₇×#398	1	1	yes	characteristic
328	C₄	C₄₇×#400	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	C₄₇×#1441	1	1	yes	characteristic
334	C₂	C₄₇×#1884	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₇×#384	C₄	semidirect
14. C₄₇×#382	C₄	semidirect
140. #11212	C₄₇	direct (N × H)
180. #384	C₄×C₄₇	semidirect
181. #382	C₄×C₄₇	semidirect
187. C₄₇	#11212	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2905	2
C₄₇×#2285	2
C₄₇×#1740	2
C₄₇×#1733	2
C₂×C₄₇×#339	2
#11212	47

Subgroups — C47×#11212 #26050910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11212 #26050910