Subgroups — C₄₇×#11372 #26051070

Order 12032.

798 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	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#340	C₂	1	2	yes	maximal
3	C₂×C₄₇×#340	C₂	1	2	yes	maximal
4	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₄₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#382	—	2	2	no	—
12	C₄₇×#382	C₄	1	4	yes	complemented
13	C₂×#95×C₄₇	—	2	2	no	—
14	C₂×#95×C₄₇	—	2	2	no	—
15	C₄₇×#340	—	2	4	no	—
16	C₄₇×#340	—	2	4	no	—
17	C₄₇×#336	—	2	2	no	—
18	C₂³×C₄×C₄₇	—	2	1	no	—
19	C₂³×C₄×C₄₇	—	2	1	no	—
20	C₂³×C₄×C₄₇	—	2	2	no	—
21	C₂³×C₄×C₄₇	D₄	1	2	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	—	2	2	no	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	D₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	#95×C₄₇	—	2	8	no	—
49	#95×C₄₇	—	4	4	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	—	4	1	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	2	yes	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	4	2	no	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	—	4	2	no	—
92	C₄⋊C₄×C₄₇	—	4	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₄₇	#123	1	2	yes	—
99	C₂³×C₄₇	#124	1	1	yes	characteristic
100	C₂³×C₄₇	#127	1	1	yes	characteristic
101	C₂³×C₄₇	#135	1	4	yes	—
102	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
103	C₂³×C₄₇	#137	1	2	yes	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	8	no	—
115	C₂×C₄×C₄₇	—	8	4	no	—
116	C₂×C₄×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	—	8	2	no	—
118	C₂×C₄×C₄₇	—	8	2	no	—
119	C₂×C₄×C₄₇	—	8	2	no	—
120	C₂×C₄×C₄₇	—	8	2	no	—
121	C₂×C₄×C₄₇	—	8	1	no	—
122	C₂×C₄×C₄₇	—	8	1	no	—
123	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#121	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	#123	1	2	yes	—
131	C₂×C₄×C₄₇	#123	1	2	yes	—
132	C₂×C₄×C₄₇	#123	1	2	yes	—
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	#126	1	2	yes	—
138	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₄₇	—	4	4	no	—
140	#11372	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₇	#381	1	1	yes	characteristic
142	C₂²×C₄₇	#390	1	1	yes	characteristic
143	C₂²×C₄₇	#391	1	1	yes	characteristic
144	C₂²×C₄₇	#399	1	2	yes	—
145	C₂²×C₄₇	#400	1	2	yes	—
146	C₂²×C₄₇	#416	1	2	yes	—
147	C₂²×C₄₇	#417	1	2	yes	—
148	C₂²×C₄₇	#478	1	4	yes	—
149	C₂²×C₄₇	#479	1	4	yes	—
150	C₂²×C₄₇	#482	1	4	yes	—
151	C₂²×C₄₇	#485	1	4	yes	—
152	C₂²×C₄₇	#488	1	2	yes	—
153	C₂²×C₄₇	#489	1	2	yes	—
154	C₂²×C₄₇	#495	1	2	yes	—
155	C₂²×C₄₇	#496	1	1	yes	characteristic
156	C₂²×C₄₇	#497	1	1	yes	characteristic
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	—	8	4	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	#387	1	2	yes	—
162	C₄×C₄₇	#387	1	2	yes	—
163	C₄×C₄₇	#397	1	2	yes	—
164	C₄×C₄₇	#398	1	2	yes	—
165	C₂×#382	C₂×C₄₇	1	1	yes	characteristic
166	C₂×#340	C₂×C₄₇	1	2	yes	—
167	C₂×#340	C₂×C₄₇	1	2	yes	—
168	C₂×#336	C₂×C₄₇	1	1	yes	characteristic
169	C₄⋊#95	C₂×C₄₇	1	1	yes	characteristic
170	C₂×C₄₇	#1445	1	1	yes	characteristic
171	C₂×C₄₇	#1825	1	2	yes	—
172	C₂×C₄₇	#1958	1	1	yes	characteristic
173	C₂×C₄₇	#1959	1	1	yes	characteristic
174	C₂×C₄₇	#1989	1	2	yes	—
175	C₂×C₄₇	#2092	1	2	yes	—
176	C₂×C₄₇	#2097	1	2	yes	—
177	C₂×C₄₇	#2101	1	4	yes	—
178	C₂³×C₈	C₂²×C₄₇	1	2	yes	—
179	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
182	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
183	#382	—	2	2	no	—
184	#382	C₄×C₄₇	1	4	yes	complemented
185	C₂×#95	—	2	2	no	—
186	C₂×#95	—	2	2	no	—
187	#340	—	2	4	no	—
188	#340	—	2	4	no	—
189	#336	—	2	2	no	—
190	C₄₇	#11372	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	—	2	1	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	D₄×C₄₇	1	2	yes	—
195	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	D₄×C₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
216	C₂×C₄⋊C₄	Q₈×C₄₇	1	4	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
220	C₂×C₄²	D₄×C₄₇	1	2	yes	—
221	#95	—	2	8	no	—
222	#95	—	4	4	no	—
223	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
243	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
247	C₂²×C₄	#41×C₄₇	1	2	yes	—
248	C₂²×C₄	#41×C₄₇	1	2	yes	—
249	C₂²×C₄	#41×C₄₇	1	2	yes	—
250	C₂²×C₄	#41×C₄₇	1	2	yes	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	4	no	—
269	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
270	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
271	C₂³	#123×C₄₇	1	2	yes	—
272	C₂³	#124×C₄₇	1	1	yes	characteristic
273	C₂³	#127×C₄₇	1	1	yes	characteristic
274	C₂³	#135×C₄₇	1	4	yes	—
275	C₂³	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
276	C₂³	#137×C₄₇	1	2	yes	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
297	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
299	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
300	C₂×C₄	#121×C₄₇	1	2	yes	—
301	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
302	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
303	C₂×C₄	#123×C₄₇	1	2	yes	—
304	C₂×C₄	#123×C₄₇	1	2	yes	—
305	C₂×C₄	#123×C₄₇	1	2	yes	—
306	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
307	C₂×C₄	#125×C₄₇	1	2	yes	—
308	C₂×C₄	#125×C₄₇	1	2	yes	—
309	C₂×C₄	#126×C₄₇	1	2	yes	—
310	C₂×C₄	#126×C₄₇	1	2	yes	—
311	C₂×C₄	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₂²	C₄₇×#381	1	1	yes	characteristic
314	C₂²	C₄₇×#390	1	1	yes	characteristic
315	C₂²	C₄₇×#391	1	1	yes	characteristic
316	C₂²	C₄₇×#399	1	2	yes	—
317	C₂²	C₄₇×#400	1	2	yes	—
318	C₂²	C₄₇×#416	1	2	yes	—
319	C₂²	C₄₇×#417	1	2	yes	—
320	C₂²	C₄₇×#478	1	4	yes	—
321	C₂²	C₄₇×#479	1	4	yes	—
322	C₂²	C₄₇×#482	1	4	yes	—
323	C₂²	C₄₇×#485	1	4	yes	—
324	C₂²	C₄₇×#488	1	2	yes	—
325	C₂²	C₄₇×#489	1	2	yes	—
326	C₂²	C₄₇×#495	1	2	yes	—
327	C₂²	C₄₇×#496	1	1	yes	characteristic
328	C₂²	C₄₇×#497	1	1	yes	characteristic
329	C₄	—	4	4	no	—
330	C₄	—	8	4	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	C₄₇×#387	1	2	yes	—
334	C₄	C₄₇×#387	1	2	yes	—
335	C₄	C₄₇×#397	1	2	yes	—
336	C₄	C₄₇×#398	1	2	yes	—
337	C₂	C₄₇×#1445	1	1	yes	characteristic
338	C₂	C₄₇×#1825	1	2	yes	—
339	C₂	C₄₇×#1958	1	1	yes	characteristic
340	C₂	C₄₇×#1959	1	1	yes	characteristic
341	C₂	C₄₇×#1989	1	2	yes	—
342	C₂	C₄₇×#2092	1	2	yes	—
343	C₂	C₄₇×#2097	1	2	yes	—
344	C₂	C₄₇×#2101	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₄₇×#382	C₄	semidirect
140. #11372	C₄₇	direct (N × H)
184. #382	C₄×C₄₇	semidirect
190. C₄₇	#11372	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×#382	2
C₂×C₄₇×#340	2
C₂×C₄₇×#336	2
C₄₇×C₄⋊#95	2
#11372	47

Subgroups — C47×#11372 #26051070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11372 #26051070