Subgroups — C₄₇×#8100 #26047798

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2280	C₂	1	2	yes	maximal
2	C₄₇×#2280	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1304	C₂	1	4	yes	maximal
4	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
5	C₂×#117×C₄₇	—	2	2	no	—
6	C₂×#117×C₄₇	C₄	1	2	yes	complemented
7	C₄₇×#381	—	2	2	no	—
8	C₄₇×#381	C₄	1	2	yes	complemented
9	C₄₇×#380	—	2	2	no	—
10	C₄₇×#380	—	2	2	no	—
11	C₄₇×#380	C₄	1	4	yes	complemented
12	C₄₇×#379	C₂²	1	4	yes	—
13	C₄₇×#379	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	complemented
17	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	—	4	4	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	D₄	1	2	yes	—
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	D₄	1	2	yes	—
27	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
29	#117×C₄₇	—	4	8	no	—
30	#117×C₄₇	—	4	4	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₄₇	—	2	4	no	—
39	C₂×C₄²×C₄₇	—	2	2	no	—
40	#95×C₄₇	—	2	4	no	—
41	#95×C₄₇	—	2	4	no	—
42	#95×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	4	2	no	—
44	C₂⁴×C₄₇	—	4	4	no	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₄⋊C₄×C₄₇	—	4	8	no	—
82	C₂²⋊C₄×C₄₇	—	4	8	no	—
83	C₂²⋊C₄×C₄₇	—	4	8	no	—
84	C₂²⋊C₄×C₄₇	—	4	8	no	—
85	C₂²⋊C₄×C₄₇	—	4	8	no	—
86	C₂²⋊C₄×C₄₇	—	4	4	no	—
87	C₄²×C₄₇	—	4	8	no	—
88	C₄²×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	16	no	—
102	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	8	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	—	4	8	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	8	no	—
129	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#95	1	2	yes	—
132	C₂×C₄×C₄₇	#95	1	2	yes	—
133	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#99	1	4	yes	—
136	#8100	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	4	8	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	—	4	8	no	—
151	C₂²×C₄₇	—	4	4	no	—
152	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#99	1	2	yes	—
154	C₂²×C₄₇	#410	1	4	yes	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	8	4	no	—
160	C₄×C₄₇	—	8	8	no	—
161	C₄×C₄₇	#342	1	2	yes	—
162	C₄×C₄₇	#342	1	2	yes	—
163	#2280	C₂×C₄₇	1	2	yes	—
164	#2280	C₂×C₄₇	1	1	yes	characteristic
165	#1304	C₂×C₄₇	1	4	yes	—
166	C₂×C₄₇	—	2	2	no	—
167	C₂×C₄₇	—	2	4	no	—
168	C₂×C₄₇	—	4	1	no	—
169	C₂×C₄₇	—	4	1	no	—
170	C₂×C₄₇	—	4	2	no	—
171	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
172	C₂×C₄₇	#1740	1	2	yes	—
173	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
174	C₂×#117	—	2	2	no	—
175	C₂×#117	C₄×C₄₇	1	2	yes	complemented
176	#381	—	2	2	no	—
177	#381	C₄×C₄₇	1	2	yes	complemented
178	#380	—	2	2	no	—
179	#380	—	2	2	no	—
180	#380	C₄×C₄₇	1	4	yes	complemented
181	#379	C₂²×C₄₇	1	4	yes	—
182	#379	C₂²×C₄₇	1	2	yes	—
183	C₄×C₂²⋊C₄	—	2	2	no	—
184	C₄×C₂²⋊C₄	—	2	2	no	—
185	C₄×C₂²⋊C₄	C₄×C₄₇	1	4	yes	complemented
186	C₄₇	#8100	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
188	C₂³×C₄	—	4	4	no	—
189	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
191	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
192	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
193	C₂³×C₄	D₄×C₄₇	1	2	yes	—
194	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₄₇	1	2	yes	—
197	C₂³×C₄	Q₈×C₄₇	1	1	yes	characteristic
198	C₂³×C₄	Q₈×C₄₇	1	1	yes	characteristic
199	#117	—	4	8	no	—
200	#117	—	4	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
207	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	—	2	4	no	—
212	#95	—	2	4	no	—
213	C₂⁴	—	4	2	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
216	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
217	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
218	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
219	C₂⁴	C₂×Q₈×C₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₄²×C₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₄²×C₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₄²×C₄₇	1	2	yes	—
235	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
236	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
237	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₄⋊C₄×C₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
244	C₂²×C₄	C₄⋊C₄×C₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
246	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
247	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
248	C₂²×C₄	C₄⋊C₄×C₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊C₄×C₄₇	1	1	yes	characteristic
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₂²⋊C₄	—	4	8	no	—
253	C₂²⋊C₄	—	4	8	no	—
254	C₂²⋊C₄	—	4	8	no	—
255	C₂²⋊C₄	—	4	8	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₄²	—	4	8	no	—
258	C₄²	—	4	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	16	no	—
272	C₂³	C₂×C₄²×C₄₇	1	1	yes	characteristic
273	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
274	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
275	C₂³	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	#95×C₄₇	1	1	yes	characteristic
300	C₂×C₄	#95×C₄₇	1	1	yes	characteristic
301	C₂×C₄	#95×C₄₇	1	2	yes	—
302	C₂×C₄	#95×C₄₇	1	2	yes	—
303	C₂×C₄	#95×C₄₇	1	1	yes	characteristic
304	C₂×C₄	#95×C₄₇	1	1	yes	characteristic
305	C₂×C₄	#99×C₄₇	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	8	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	4	no	—
321	C₂²	C₂×#95×C₄₇	1	1	yes	characteristic
322	C₂²	C₂×#99×C₄₇	1	2	yes	—
323	C₂²	C₄₇×#410	1	4	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	8	no	—
330	C₄	C₄₇×#342	1	2	yes	—
331	C₄	C₄₇×#342	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	2	4	no	—
334	C₂	—	4	1	no	—
335	C₂	—	4	1	no	—
336	C₂	—	4	2	no	—
337	C₂	C₂×C₄₇×#342	1	1	yes	characteristic
338	C₂	C₄₇×#1740	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	#95
commutator	C₂³	C₂×C₄²×C₄₇
Frattini, cosocle	C₂⁵	C₂³×C₄₇
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	C₂³
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂⁴×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×#117×C₄₇	C₄	semidirect
8. C₄₇×#381	C₄	semidirect
11. C₄₇×#380	C₄	semidirect
16. C₄×C₂²⋊C₄×C₄₇	C₄	semidirect
136. #8100	C₄₇	direct (N × H)
175. C₂×#117	C₄×C₄₇	semidirect
177. #381	C₄×C₄₇	semidirect
180. #380	C₄×C₄₇	semidirect
185. C₄×C₂²⋊C₄	C₄×C₄₇	semidirect
186. C₄₇	#8100	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×#2280	2
C₄₇×#1304	2
#8100	47

Subgroups — C47×#8100 #26047798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#8100 #26047798