Subgroups — C₄₇×#20017 #26059715

Order 12032.

600 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3394	C₂	1	2	yes	maximal, complemented
2	C₄₇×#3340	C₂	1	2	yes	maximal, complemented
3	C₄₇×#1705	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1672	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1665	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1652	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1630	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1624	C₂	1	1	yes	maximal, characteristic
9	C₈×SD₁₆×C₄₇	C₂	1	2	yes	maximal, complemented
10	C₄₇×#559	C₂²	1	1	yes	characteristic
11	C₄₇×#556	C₂²	1	2	yes	—
12	C₄₇×#555	C₂²	1	1	yes	characteristic
13	C₄₇×#498	C₂²	1	2	yes	—
14	C₄₇×#498	C₂²	1	1	yes	characteristic
15	C₄₇×#495	—	2	2	no	—
16	C₄₇×#492	C₂²	1	2	yes	—
17	C₄₇×#492	C₂²	1	1	yes	characteristic
18	C₄₇×#485	—	2	2	no	—
19	C₄₇×#482	—	2	2	no	—
20	C₄₇×#477	C₂²	1	2	yes	—
21	C₄₇×#474	C₂²	1	2	yes	—
22	C₄₇×#465	—	2	2	no	—
23	C₄₇×#464	—	2	2	no	—
24	C₄₇×#463	—	2	2	no	—
25	C₄₇×#462	C₂²	1	2	yes	—
26	C₄₇×#459	C₂²	1	2	yes	—
27	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄×SD₁₆×C₄₇	—	2	2	no	—
29	C₄×SD₁₆×C₄₇	C₂²	1	2	yes	—
30	C₈×D₄×C₄₇	C₂²	1	2	yes	—
31	C₄₇×#427	—	2	2	no	—
32	C₄₇×#334	C₂²	1	2	yes	—
33	C₄₇×#333	C₂²	1	1	yes	characteristic
34	C₄₇×#332	C₂²	1	2	yes	—
35	C₄₇×#331	C₂²	1	1	yes	characteristic
36	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₄₇	—	2	2	no	—
40	C₂×SD₁₆×C₄₇	—	2	2	no	—
41	C₂×SD₁₆×C₄₇	—	2	2	no	—
42	C₂×SD₁₆×C₄₇	—	4	1	no	—
43	C₂×SD₁₆×C₄₇	D₄	1	2	yes	—
44	C₂×D₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	#127×C₄₇	C₂³	1	1	yes	characteristic
49	#126×C₄₇	—	4	2	no	—
50	#124×C₄₇	—	2	2	no	—
51	#124×C₄₇	—	2	2	no	—
52	#124×C₄₇	—	4	1	no	—
53	#123×C₄₇	—	4	1	no	—
54	C₂²⋊Q₈×C₄₇	—	2	2	no	—
55	#121×C₄₇	—	2	2	no	—
56	C₄×Q₈×C₄₇	—	2	2	no	—
57	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
58	C₄×D₄×C₄₇	—	2	2	no	—
59	C₄×D₄×C₄₇	C₂³	1	2	yes	—
60	#117×C₄₇	—	2	2	no	—
61	#117×C₄₇	—	2	2	no	—
62	#107×C₄₇	—	2	2	no	—
63	#107×C₄₇	—	2	2	no	—
64	#106×C₄₇	—	2	2	no	—
65	#106×C₄₇	—	2	2	no	—
66	#106×C₄₇	—	4	1	no	—
67	#106×C₄₇	D₄	1	2	yes	—
68	C₄⋊C₈×C₄₇	—	2	2	no	—
69	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
70	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
71	#103×C₄₇	—	2	2	no	—
72	#103×C₄₇	—	2	2	no	—
73	#103×C₄₇	D₄	1	2	yes	—
74	#102×C₄₇	—	2	2	no	—
75	#102×C₄₇	—	2	2	no	—
76	#102×C₄₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₄₇	—	2	2	no	—
78	C₄×C₈×C₄₇	—	2	2	no	—
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	1	no	—
83	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	1	no	—
86	C₂×D₄×C₄₇	—	4	1	no	—
87	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	Q₁₆×C₄₇	—	8	2	no	—
91	SD₁₆×C₄₇	—	2	4	no	—
92	SD₁₆×C₄₇	—	4	4	no	—
93	SD₁₆×C₄₇	—	8	2	no	—
94	SD₁₆×C₄₇	—	8	2	no	—
95	D₈×C₄₇	—	8	2	no	—
96	C₂×C₈×C₄₇	—	2	1	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	4	4	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	4	2	no	—
107	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	1	no	—
112	C₂²⋊C₄×C₄₇	—	4	2	no	—
113	C₂²⋊C₄×C₄₇	—	4	2	no	—
114	C₂²⋊C₄×C₄₇	—	4	2	no	—
115	C₄²×C₄₇	—	2	2	no	—
116	C₄²×C₄₇	—	4	1	no	—
117	C₄²×C₄₇	—	4	2	no	—
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	C₂³×C₄₇	—	2	2	no	—
120	C₂³×C₄₇	—	4	1	no	—
121	Q₈×C₄₇	—	2	2	no	—
122	Q₈×C₄₇	—	2	2	no	—
123	Q₈×C₄₇	—	4	2	no	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	8	1	no	—
126	D₄×C₄₇	—	2	2	no	—
127	D₄×C₄₇	—	2	2	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	4	2	no	—
130	D₄×C₄₇	—	8	1	no	—
131	D₄×C₄₇	—	16	1	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	—	16	1	no	—
140	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	—	4	2	no	—
145	C₈×C₄₇	—	4	2	no	—
146	C₈×C₄₇	#135	1	4	yes	—
147	#20017	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₇	—	4	2	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	2	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	D₄²	1	1	yes	characteristic
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	4	2	no	—
156	C₄×C₄₇	—	4	2	no	—
157	C₄×C₄₇	—	4	2	no	—
158	C₄×C₄₇	—	8	2	no	—
159	C₄×C₄₇	—	8	1	no	—
160	C₄×C₄₇	C₂×#135	1	2	yes	—
161	C₄×C₄₇	#577	1	2	yes	—
162	#3394	C₂×C₄₇	1	2	yes	complemented
163	#3340	C₂×C₄₇	1	2	yes	complemented
164	#1705	C₂×C₄₇	1	1	yes	characteristic, complemented
165	#1672	C₂×C₄₇	1	2	yes	complemented
166	#1665	C₂×C₄₇	1	2	yes	complemented
167	#1652	C₂×C₄₇	1	2	yes	complemented
168	#1630	C₂×C₄₇	1	1	yes	characteristic, complemented
169	#1624	C₂×C₄₇	1	1	yes	characteristic
170	C₈×SD₁₆	C₂×C₄₇	1	2	yes	complemented
171	C₂×C₄₇	—	8	2	no	—
172	C₂×C₄₇	—	16	1	no	—
173	C₂×C₄₇	#3284	1	2	yes	—
174	C₂×C₄₇	#3291	1	1	yes	characteristic
175	#559	C₂²×C₄₇	1	1	yes	characteristic
176	#556	C₂²×C₄₇	1	2	yes	—
177	#555	C₂²×C₄₇	1	1	yes	characteristic
178	#498	C₂²×C₄₇	1	2	yes	—
179	#498	C₂²×C₄₇	1	1	yes	characteristic
180	#495	—	2	2	no	—
181	#492	C₂²×C₄₇	1	2	yes	—
182	#492	C₂²×C₄₇	1	1	yes	characteristic
183	#485	—	2	2	no	—
184	#482	—	2	2	no	—
185	#477	C₂²×C₄₇	1	2	yes	—
186	#474	C₂²×C₄₇	1	2	yes	—
187	#465	—	2	2	no	—
188	#464	—	2	2	no	—
189	#463	—	2	2	no	—
190	#462	C₂²×C₄₇	1	2	yes	—
191	#459	C₂²×C₄₇	1	2	yes	—
192	C₈×Q₈	C₂²×C₄₇	1	2	yes	—
193	C₄×SD₁₆	—	2	2	no	—
194	C₄×SD₁₆	C₂²×C₄₇	1	2	yes	—
195	C₈×D₄	C₂²×C₄₇	1	2	yes	—
196	#427	—	2	2	no	—
197	#334	C₂²×C₄₇	1	2	yes	—
198	#333	C₂²×C₄₇	1	1	yes	characteristic
199	#332	C₂²×C₄₇	1	2	yes	—
200	#331	C₂²×C₄₇	1	1	yes	characteristic
201	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	—
202	D₄⋊C₈	C₂²×C₄₇	1	2	yes	—
203	C₈²	C₂²×C₄₇	1	1	yes	characteristic
204	C₄₇	#20017	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×SD₁₆	—	4	1	no	—
209	C₂×SD₁₆	D₄×C₄₇	1	2	yes	—
210	C₂×D₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
213	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
214	#127	C₂³×C₄₇	1	1	yes	characteristic
215	#126	—	4	2	no	—
216	#124	—	2	2	no	—
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	1	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	C₂³×C₄₇	1	2	yes	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#106	—	2	2	no	—
231	#106	—	2	2	no	—
232	#106	—	4	1	no	—
233	#106	D₄×C₄₇	1	2	yes	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
236	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	D₄×C₄₇	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	D₄×C₄₇	1	2	yes	—
243	C₂²⋊C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₄₇	1	2	yes	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	Q₁₆	—	8	2	no	—
257	SD₁₆	—	2	4	no	—
258	SD₁₆	—	4	4	no	—
259	SD₁₆	—	8	2	no	—
260	SD₁₆	—	8	2	no	—
261	D₈	—	8	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
274	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	—	4	1	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	16	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
307	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	#135×C₄₇	1	4	yes	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	D₄²×C₄₇	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	C₂×#135×C₄₇	1	2	yes	—
326	C₄	C₄₇×#577	1	2	yes	—
327	C₂	—	8	2	no	—
328	C₂	—	16	1	no	—
329	C₂	C₄₇×#3284	1	2	yes	—
330	C₂	C₄₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#1624	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3394	C₂	semidirect
2. C₄₇×#3340	C₂	semidirect
3. C₄₇×#1705	C₂	semidirect
4. C₄₇×#1672	C₂	semidirect
5. C₄₇×#1665	C₂	semidirect
6. C₄₇×#1652	C₂	semidirect
7. C₄₇×#1630	C₂	semidirect
9. C₈×SD₁₆×C₄₇	C₂	semidirect
147. #20017	C₄₇	direct (N × H)
162. #3394	C₂×C₄₇	semidirect
163. #3340	C₂×C₄₇	semidirect
164. #1705	C₂×C₄₇	semidirect
165. #1672	C₂×C₄₇	semidirect
166. #1665	C₂×C₄₇	semidirect
167. #1652	C₂×C₄₇	semidirect
168. #1630	C₂×C₄₇	semidirect
170. C₈×SD₁₆	C₂×C₄₇	semidirect
204. C₄₇	#20017	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×#3394	2
C₄₇×#3340	2
C₄₇×#1705	2
C₄₇×#1672	2
C₄₇×#1665	2
C₄₇×#1652	2
C₄₇×#1630	2
C₄₇×#1624	2
C₈×SD₁₆×C₄₇	2
#20017	47

Subgroups — C47×#20017 #26059715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#20017 #26059715