Subgroups — C₄₇×#32701 #26072399

Order 12032.

628 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2104	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2032	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2030	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2022	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1913	C₂	1	2	yes	maximal
6	C₄₇×#1899	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1890	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1828	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1820	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#543	C₂²	1	1	yes	characteristic
11	C₄₇×#534	C₂²	1	1	yes	characteristic
12	C₂×#123×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#518	C₂²	1	2	yes	—
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#99×C₄₇	C₂²	1	2	yes	—
16	C₂×#99×C₄₇	C₂²	1	2	yes	—
17	C₂×#99×C₄₇	C₂²	1	2	yes	—
18	C₂×#99×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#381	C₂²	1	1	yes	characteristic
20	C₄₇×#381	C₂²	1	2	yes	—
21	C₄₇×#381	C₂²	1	1	yes	characteristic
22	C₄₇×#342	C₂²	1	1	yes	characteristic
23	C₄₇×#342	C₂²	1	1	yes	characteristic
24	C₄₇×#342	C₂²	1	2	yes	—
25	C₄₇×#342	C₂²	1	2	yes	—
26	C₄₇×#342	C₂²	1	2	yes	—
27	C₄₇×#342	C₂²	1	2	yes	—
28	C₄₇×#342	C₂²	1	2	yes	—
29	C₄₇×#342	C₂²	1	2	yes	—
30	C₄₇×#342	C₂²	1	2	yes	—
31	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	#123×C₄₇	—	2	2	no	—
34	#123×C₄₇	—	2	2	no	—
35	#123×C₄₇	—	2	2	no	—
36	#123×C₄₇	—	4	2	no	—
37	C₂²⋊Q₈×C₄₇	—	4	1	no	—
38	C₂²⋊Q₈×C₄₇	—	4	2	no	—
39	#121×C₄₇	—	4	1	no	—
40	#120×C₄₇	—	2	2	no	—
41	C₄×D₄×C₄₇	—	4	2	no	—
42	C₄×D₄×C₄₇	—	4	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	—	4	1	no	—
45	#117×C₄₇	—	4	1	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	1	no	—
58	#99×C₄₇	—	4	2	no	—
59	#99×C₄₇	—	4	2	no	—
60	#99×C₄₇	—	4	2	no	—
61	#99×C₄₇	—	4	2	no	—
62	#95×C₄₇	—	4	1	no	—
63	#95×C₄₇	—	4	1	no	—
64	#95×C₄₇	—	4	2	no	—
65	C₂⁴×C₄₇	#41	1	2	yes	—
66	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂×Q₈×C₄₇	—	4	1	no	—
68	C₂×D₄×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	—	2	2	no	—
71	C₂×D₄×C₄₇	—	2	2	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	2	1	no	—
77	C₂²×C₄×C₄₇	—	2	1	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	#41	1	2	yes	—
89	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	—	4	2	no	—
92	C₄⋊C₄×C₄₇	—	4	1	no	—
93	C₄⋊C₄×C₄₇	—	4	2	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₄⋊C₄×C₄₇	—	4	3	no	—
96	C₂²⋊C₄×C₄₇	—	2	2	no	—
97	C₂²⋊C₄×C₄₇	—	2	2	no	—
98	C₂²⋊C₄×C₄₇	—	2	2	no	—
99	C₂²⋊C₄×C₄₇	—	2	2	no	—
100	C₂²⋊C₄×C₄₇	—	2	2	no	—
101	C₂²⋊C₄×C₄₇	—	2	2	no	—
102	C₂²⋊C₄×C₄₇	—	2	2	no	—
103	C₂²⋊C₄×C₄₇	—	2	2	no	—
104	C₂²⋊C₄×C₄₇	—	2	2	no	—
105	C₂²⋊C₄×C₄₇	—	2	2	no	—
106	C₂²⋊C₄×C₄₇	—	2	2	no	—
107	C₂²⋊C₄×C₄₇	—	2	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₄²×C₄₇	—	4	2	no	—
113	C₄²×C₄₇	—	4	2	no	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	8	1	no	—
118	C₂³×C₄₇	—	8	2	no	—
119	C₂³×C₄₇	C₂×#41	1	2	yes	—
120	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
124	Q₈×C₄₇	—	16	1	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	4	2	no	—
127	D₄×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	16	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	1	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	8	2	no	—
148	C₂×C₄×C₄₇	—	8	2	no	—
149	C₂×C₄×C₄₇	—	8	2	no	—
150	C₂×C₄×C₄₇	—	8	2	no	—
151	C₂×C₄×C₄₇	—	8	1	no	—
152	C₂×C₄×C₄₇	—	8	2	no	—
153	C₂×C₄×C₄₇	—	8	1	no	—
154	C₂×C₄×C₄₇	—	16	2	no	—
155	#32701	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₄₇	—	4	1	no	—
157	C₂²×C₄₇	—	4	2	no	—
158	C₂²×C₄₇	—	4	1	no	—
159	C₂²×C₄₇	—	4	1	no	—
160	C₂²×C₄₇	—	4	2	no	—
161	C₂²×C₄₇	—	4	2	no	—
162	C₂²×C₄₇	—	8	2	no	—
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	—	8	2	no	—
165	C₂²×C₄₇	#538	1	1	yes	characteristic
166	C₂²×C₄₇	#538	1	2	yes	—
167	C₂²×C₄₇	#551	1	2	yes	—
168	C₂²×C₄₇	#551	1	1	yes	characteristic
169	C₂²×C₄₇	#560	1	1	yes	characteristic
170	C₄×C₄₇	—	4	1	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	—	4	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	—	8	2	no	—
175	C₄×C₄₇	—	8	2	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	—	8	1	no	—
178	C₄×C₄₇	—	8	2	no	—
179	C₄×C₄₇	—	8	1	no	—
180	#2104	C₂×C₄₇	1	2	yes	complemented
181	#2032	C₂×C₄₇	1	2	yes	complemented
182	#2030	C₂×C₄₇	1	1	yes	characteristic, complemented
183	#2022	C₂×C₄₇	1	1	yes	characteristic
184	#1913	C₂×C₄₇	1	2	yes	—
185	#1899	C₂×C₄₇	1	2	yes	complemented
186	#1890	C₂×C₄₇	1	2	yes	complemented
187	#1828	C₂×C₄₇	1	2	yes	complemented
188	#1820	C₂×C₄₇	1	1	yes	characteristic, complemented
189	C₂×C₄₇	—	2	1	no	—
190	C₂×C₄₇	—	2	2	no	—
191	C₂×C₄₇	—	2	1	no	—
192	C₂×C₄₇	—	2	1	no	—
193	C₂×C₄₇	—	2	2	no	—
194	C₂×C₄₇	—	8	2	no	—
195	C₂×C₄₇	#2760	1	1	yes	characteristic
196	#543	C₂²×C₄₇	1	1	yes	characteristic
197	#534	C₂²×C₄₇	1	1	yes	characteristic
198	C₂×#123	C₂²×C₄₇	1	2	yes	—
199	#518	C₂²×C₄₇	1	2	yes	—
200	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
201	C₂×#99	C₂²×C₄₇	1	2	yes	—
202	C₂×#99	C₂²×C₄₇	1	2	yes	—
203	C₂×#99	C₂²×C₄₇	1	2	yes	—
204	C₂×#99	C₂²×C₄₇	1	2	yes	—
205	#381	C₂²×C₄₇	1	1	yes	characteristic
206	#381	C₂²×C₄₇	1	1	yes	characteristic
207	#381	C₂²×C₄₇	1	2	yes	—
208	#342	C₂²×C₄₇	1	2	yes	—
209	#342	C₂²×C₄₇	1	2	yes	—
210	#342	C₂²×C₄₇	1	1	yes	characteristic
211	#342	C₂²×C₄₇	1	2	yes	—
212	#342	C₂²×C₄₇	1	2	yes	—
213	#342	C₂²×C₄₇	1	1	yes	characteristic
214	#342	C₂²×C₄₇	1	2	yes	—
215	#342	C₂²×C₄₇	1	2	yes	—
216	#342	C₂²×C₄₇	1	2	yes	—
217	C₄₇	#32701	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
219	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
220	#123	—	2	2	no	—
221	#123	—	2	2	no	—
222	#123	—	2	2	no	—
223	#123	—	4	2	no	—
224	C₂²⋊Q₈	—	4	1	no	—
225	C₂²⋊Q₈	—	4	2	no	—
226	#121	—	4	1	no	—
227	#120	—	2	2	no	—
228	C₄×D₄	—	4	2	no	—
229	C₄×D₄	—	4	2	no	—
230	#117	—	2	2	no	—
231	#117	—	4	1	no	—
232	#117	—	4	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
238	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
239	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
240	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
241	C₂×C₂²⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
244	C₂×C₄²	—	2	1	no	—
245	#99	—	4	2	no	—
246	#99	—	4	2	no	—
247	#99	—	4	2	no	—
248	#99	—	4	2	no	—
249	#95	—	4	1	no	—
250	#95	—	4	1	no	—
251	#95	—	4	2	no	—
252	C₂⁴	#41×C₄₇	1	2	yes	—
253	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
254	C₂×Q₈	—	4	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
275	C₂²×C₄	#41×C₄₇	1	2	yes	—
276	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	3	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	8	1	no	—
305	C₂³	—	8	2	no	—
306	C₂³	C₂×#41×C₄₇	1	2	yes	—
307	C₂³	C₂×#41×C₄₇	1	1	yes	characteristic
308	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
309	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
310	C₂³	H(5,𝔽₂)×C₄₇	1	2	yes	—
311	Q₈	—	16	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	16	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	2	no	—
341	C₂×C₄	—	16	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	C₄₇×#538	1	1	yes	characteristic
352	C₂²	C₄₇×#538	1	2	yes	—
353	C₂²	C₄₇×#551	1	2	yes	—
354	C₂²	C₄₇×#551	1	1	yes	characteristic
355	C₂²	C₄₇×#560	1	1	yes	characteristic
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	2	no	—
366	C₂	—	2	1	no	—
367	C₂	—	2	2	no	—
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	2	2	no	—
371	C₂	—	8	2	no	—
372	C₂	C₄₇×#2760	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#2760
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₄₇
prime radical	C₄₇×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂⁴×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#2104	C₂	semidirect
2. C₄₇×#2032	C₂	semidirect
3. C₄₇×#2030	C₂	semidirect
6. C₄₇×#1899	C₂	semidirect
7. C₄₇×#1890	C₂	semidirect
8. C₄₇×#1828	C₂	semidirect
9. C₄₇×#1820	C₂	semidirect
155. #32701	C₄₇	direct (N × H)
180. #2104	C₂×C₄₇	semidirect
181. #2032	C₂×C₄₇	semidirect
182. #2030	C₂×C₄₇	semidirect
185. #1899	C₂×C₄₇	semidirect
186. #1890	C₂×C₄₇	semidirect
187. #1828	C₂×C₄₇	semidirect
188. #1820	C₂×C₄₇	semidirect
217. C₄₇	#32701	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×#2104	2
C₄₇×#2032	2
C₄₇×#2030	2
C₄₇×#2022	2
C₄₇×#1913	2
C₄₇×#1899	2
C₄₇×#1890	2
C₄₇×#1828	2
C₄₇×#1820	2
#32701	47

Subgroups — C47×#32701 #26072399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#32701 #26072399