Subgroups — C₄₇×#19965 #26059663

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3400	C₂	1	2	yes	maximal, complemented
2	C₄₇×#3365	C₂	1	2	yes	maximal, complemented
3	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1686	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1682	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1648	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1620	C₂	1	2	yes	maximal
8	C₄₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#559	C₂²	1	2	yes	—
11	C₄₇×#557	C₂²	1	2	yes	—
12	C₄₇×#500	C₂²	1	1	yes	characteristic
13	C₄₇×#497	—	2	2	no	—
14	C₄₇×#493	C₂²	1	1	yes	characteristic
15	C₄₇×#492	C₂²	1	2	yes	—
16	C₄₇×#488	—	2	2	no	—
17	C₄₇×#486	C₂²	1	2	yes	—
18	C₄₇×#478	—	2	2	no	—
19	C₄₇×#475	C₂²	1	2	yes	—
20	C₄₇×#464	—	2	2	no	—
21	C₄₇×#464	—	2	2	no	—
22	C₄₇×#462	C₂²	1	2	yes	—
23	C₄₇×#461	—	2	2	no	—
24	C₄₇×#460	C₂²	1	2	yes	—
25	C₄₇×#460	C₂²	1	2	yes	—
26	C₄₇×#446	C₂²	1	2	yes	—
27	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#440	—	2	2	no	—
29	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₄₇	—	2	2	no	—
32	C₄₇×#335	C₂²	1	1	yes	characteristic
33	C₄₇×#335	C₂²	1	1	yes	characteristic
34	C₄₇×#332	C₂²	1	2	yes	—
35	C₄₇×#331	C₂²	1	2	yes	—
36	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₄₇	—	2	2	no	—
40	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₄₇	—	2	2	no	—
43	C₂×SD₁₆×C₄₇	—	2	2	no	—
44	C₂×SD₁₆×C₄₇	—	2	2	no	—
45	C₂×SD₁₆×C₄₇	—	2	2	no	—
46	C₂×D₈×C₄₇	—	4	1	no	—
47	C₄⋊Q₈×C₄₇	—	2	2	no	—
48	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
49	#127×C₄₇	C₂³	1	2	yes	—
50	#125×C₄₇	—	2	2	no	—
51	#125×C₄₇	—	4	2	no	—
52	#124×C₄₇	—	2	2	no	—
53	#124×C₄₇	—	2	2	no	—
54	#123×C₄₇	—	4	2	no	—
55	C₄×Q₈×C₄₇	—	2	2	no	—
56	C₄×Q₈×C₄₇	—	2	2	no	—
57	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
58	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
59	C₄×D₄×C₄₇	—	2	2	no	—
60	C₄×D₄×C₄₇	—	2	2	no	—
61	#107×C₄₇	—	4	1	no	—
62	#107×C₄₇	D₄	1	1	yes	characteristic
63	#107×C₄₇	D₄	1	1	yes	characteristic
64	#106×C₄₇	—	2	2	no	—
65	C₄⋊C₈×C₄₇	—	2	2	no	—
66	C₄⋊C₈×C₄₇	—	2	2	no	—
67	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
69	#103×C₄₇	—	2	2	no	—
70	#103×C₄₇	—	2	2	no	—
71	#103×C₄₇	D₄	1	2	yes	—
72	#103×C₄₇	D₄	1	2	yes	—
73	#102×C₄₇	—	2	2	no	—
74	#102×C₄₇	—	2	2	no	—
75	#102×C₄₇	—	2	2	no	—
76	#102×C₄₇	—	2	2	no	—
77	#97×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	1	yes	characteristic
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	—	4	2	no	—
89	Q₁₆×C₄₇	—	2	2	no	—
90	Q₁₆×C₄₇	—	4	1	no	—
91	Q₁₆×C₄₇	—	8	2	no	—
92	SD₁₆×C₄₇	—	4	4	no	—
93	SD₁₆×C₄₇	—	4	4	no	—
94	SD₁₆×C₄₇	—	8	2	no	—
95	SD₁₆×C₄₇	—	8	2	no	—
96	D₈×C₄₇	—	8	2	no	—
97	C₂×C₈×C₄₇	—	2	1	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	4	2	no	—
109	C₄⋊C₄×C₄₇	—	4	2	no	—
110	C₄⋊C₄×C₄₇	—	4	2	no	—
111	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₄₇	—	4	2	no	—
114	C₂²⋊C₄×C₄₇	—	4	2	no	—
115	C₂²⋊C₄×C₄₇	—	4	2	no	—
116	C₄²×C₄₇	—	2	2	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	—	4	2	no	—
119	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	2	2	no	—
122	Q₈×C₄₇	—	2	2	no	—
123	Q₈×C₄₇	—	2	2	no	—
124	Q₈×C₄₇	—	2	2	no	—
125	Q₈×C₄₇	—	4	2	no	—
126	Q₈×C₄₇	—	4	2	no	—
127	D₄×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	4	2	no	—
130	D₄×C₄₇	—	4	2	no	—
131	D₄×C₄₇	—	8	2	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	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	16	2	no	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₈×C₄₇	—	2	2	no	—
142	C₈×C₄₇	—	2	1	no	—
143	C₈×C₄₇	—	2	1	no	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	—	4	2	no	—
146	C₈×C₄₇	—	4	2	no	—
147	C₈×C₄₇	#135	1	2	yes	—
148	#19965	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₄₇	—	8	2	no	—
150	C₂²×C₄₇	—	8	2	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	D₄²	1	1	yes	characteristic
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	4	2	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	C₂×#135	1	1	yes	characteristic
160	C₄×C₄₇	#575	1	1	yes	characteristic
161	C₄×C₄₇	#576	1	1	yes	characteristic
162	C₄×C₄₇	#578	1	1	yes	characteristic
163	#3400	C₂×C₄₇	1	2	yes	complemented
164	#3365	C₂×C₄₇	1	2	yes	complemented
165	#1706	C₂×C₄₇	1	1	yes	characteristic
166	#1686	C₂×C₄₇	1	2	yes	complemented
167	#1682	C₂×C₄₇	1	2	yes	complemented
168	#1648	C₂×C₄₇	1	2	yes	complemented
169	#1620	C₂×C₄₇	1	2	yes	—
170	#1591	C₂×C₄₇	1	1	yes	characteristic, complemented
171	C₈×Q₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
172	C₂×C₄₇	—	16	2	no	—
173	C₂×C₄₇	#3277	1	1	yes	characteristic
174	C₂×C₄₇	#3287	1	1	yes	characteristic
175	C₂×C₄₇	#3291	1	1	yes	characteristic
176	#559	C₂²×C₄₇	1	2	yes	—
177	#557	C₂²×C₄₇	1	2	yes	—
178	#500	C₂²×C₄₇	1	1	yes	characteristic
179	#497	—	2	2	no	—
180	#493	C₂²×C₄₇	1	1	yes	characteristic
181	#492	C₂²×C₄₇	1	2	yes	—
182	#488	—	2	2	no	—
183	#486	C₂²×C₄₇	1	2	yes	—
184	#478	—	2	2	no	—
185	#475	C₂²×C₄₇	1	2	yes	—
186	#464	—	2	2	no	—
187	#464	—	2	2	no	—
188	#462	C₂²×C₄₇	1	2	yes	—
189	#461	—	2	2	no	—
190	#460	C₂²×C₄₇	1	2	yes	—
191	#460	C₂²×C₄₇	1	2	yes	—
192	#446	C₂²×C₄₇	1	2	yes	—
193	C₈×Q₈	C₂²×C₄₇	1	2	yes	—
194	#440	—	2	2	no	—
195	C₄×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
196	C₄×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
197	C₄×SD₁₆	—	2	2	no	—
198	#335	C₂²×C₄₇	1	1	yes	characteristic
199	#335	C₂²×C₄₇	1	1	yes	characteristic
200	#332	C₂²×C₄₇	1	2	yes	—
201	#331	C₂²×C₄₇	1	2	yes	—
202	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	—
203	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	—
204	C₈²	C₂²×C₄₇	1	1	yes	characteristic
205	C₄₇	#19965	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	D₄×C₄₇	1	1	yes	characteristic
208	C₂×Q₁₆	D₄×C₄₇	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	4	1	no	—
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
216	#127	C₂³×C₄₇	1	2	yes	—
217	#125	—	2	2	no	—
218	#125	—	4	2	no	—
219	#124	—	2	2	no	—
220	#124	—	2	2	no	—
221	#123	—	4	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
225	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	#107	—	4	1	no	—
229	#107	D₄×C₄₇	1	1	yes	characteristic
230	#107	D₄×C₄₇	1	1	yes	characteristic
231	#106	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
235	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	D₄×C₄₇	1	2	yes	—
239	#103	D₄×C₄₇	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
251	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	4	1	no	—
258	Q₁₆	—	8	2	no	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	4	4	no	—
261	SD₁₆	—	8	2	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	8	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	2	no	—
286	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
287	C₂³	—	4	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
307	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	#135×C₄₇	1	2	yes	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	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	—	4	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
326	C₄	C₄₇×#575	1	1	yes	characteristic
327	C₄	C₄₇×#576	1	1	yes	characteristic
328	C₄	C₄₇×#578	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	C₄₇×#3277	1	1	yes	characteristic
331	C₂	C₄₇×#3287	1	1	yes	characteristic
332	C₂	C₄₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#493	C₂²
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×#3400	C₂	semidirect
2. C₄₇×#3365	C₂	semidirect
4. C₄₇×#1686	C₂	semidirect
5. C₄₇×#1682	C₂	semidirect
6. C₄₇×#1648	C₂	semidirect
8. C₄₇×#1591	C₂	semidirect
9. C₈×Q₁₆×C₄₇	C₂	semidirect
148. #19965	C₄₇	direct (N × H)
163. #3400	C₂×C₄₇	semidirect
164. #3365	C₂×C₄₇	semidirect
166. #1686	C₂×C₄₇	semidirect
167. #1682	C₂×C₄₇	semidirect
168. #1648	C₂×C₄₇	semidirect
170. #1591	C₂×C₄₇	semidirect
171. C₈×Q₁₆	C₂×C₄₇	semidirect
205. C₄₇	#19965	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×#3400	2
C₄₇×#3365	2
C₄₇×#1706	2
C₄₇×#1686	2
C₄₇×#1682	2
C₄₇×#1648	2
C₄₇×#1620	2
C₄₇×#1591	2
C₈×Q₁₆×C₄₇	2
#19965	47

Subgroups — C47×#19965 #26059663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19965 #26059663