Subgroups — C₄₇×#10803 #26050501

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#344	C₂	1	2	yes	maximal
3	C₄₇×#1742	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1740	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1729	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₇	C₂²	1	2	yes	—
10	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#405	—	2	4	no	—
12	C₂×#97×C₄₇	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	complemented
14	C₄₇×#344	—	2	4	no	—
15	C₄₇×#343	C₄	1	4	yes	complemented
16	C₄₇×#342	C₄	1	4	yes	—
17	C₄₇×#336	—	2	4	no	—
18	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₄₇	—	2	2	no	—
20	C₂×M₁₆×C₄₇	—	2	2	no	—
21	C₂×M₁₆×C₄₇	—	2	2	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂×M₁₆×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
28	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	1	no	—
31	C₂²×C₈×C₄₇	—	2	1	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₇	D₄	1	4	yes	—
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	2	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	#97×C₄₇	—	2	4	no	—
50	C₄×C₈×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	4	2	no	—
69	M₁₆×C₄₇	—	4	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₂²⋊C₄×C₄₇	—	2	4	no	—
88	C₂²⋊C₄×C₄₇	—	2	4	no	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	4	2	no	—
93	C₄²×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₄₇	#117	1	2	yes	—
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	8	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#124	1	2	yes	—
119	C₂×C₄×C₄₇	#125	1	2	yes	—
120	C₂×C₄×C₄₇	#126	1	2	yes	—
121	C₂×C₄×C₄₇	#126	1	2	yes	—
122	C₈×C₄₇	—	2	4	no	—
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	—	4	4	no	—
126	#10803	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	—	4	1	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	#376	1	1	yes	characteristic
133	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	#387	1	2	yes	—
136	C₂²×C₄₇	#388	1	2	yes	—
137	C₄×C₄₇	—	4	4	no	—
138	C₄×C₄₇	—	8	4	no	—
139	C₄×C₄₇	#380	1	1	yes	characteristic
140	C₄×C₄₇	#381	1	1	yes	characteristic
141	C₄×C₄₇	#382	1	1	yes	characteristic
142	C₄×C₄₇	#383	1	1	yes	characteristic
143	C₄×C₄₇	#387	1	2	yes	—
144	C₄×C₄₇	#388	1	2	yes	—
145	C₂×#405	C₂×C₄₇	1	1	yes	characteristic
146	C₂×#344	C₂×C₄₇	1	2	yes	—
147	#1742	C₂×C₄₇	1	1	yes	characteristic
148	#1740	C₂×C₄₇	1	1	yes	characteristic
149	#1729	C₂×C₄₇	1	2	yes	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	#1434	1	1	yes	characteristic
155	C₂×C₄₇	#1757	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
159	C₂×#117	C₂²×C₄₇	1	2	yes	—
160	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
161	#405	—	2	4	no	—
162	C₂×#97	C₄×C₄₇	1	2	yes	complemented
163	C₂×C₄×C₈	C₄×C₄₇	1	2	yes	complemented
164	#344	—	2	4	no	—
165	#343	C₄×C₄₇	1	4	yes	complemented
166	#342	C₄×C₄₇	1	4	yes	—
167	#336	—	2	4	no	—
168	C₄₇	#10803	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
178	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
179	C₂×M₁₆	Q₈×C₄₇	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
187	C₂²×C₈	D₄×C₄₇	1	4	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₄²×C₄₇	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
210	C₂²×C₄	#41×C₄₇	1	2	yes	—
211	C₂²×C₄	#41×C₄₇	1	2	yes	—
212	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
213	C₂²×C₄	#41×C₄₇	1	2	yes	—
214	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
233	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
234	C₂×C₈	C₄⋊C₄×C₄₇	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄²×C₄₇	1	1	yes	characteristic
247	C₂³	#117×C₄₇	1	2	yes	—
248	C₂³	C₄×D₄×C₄₇	1	2	yes	—
249	C₂³	C₄×D₄×C₄₇	1	1	yes	characteristic
250	C₂³	C₄×Q₈×C₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
257	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
258	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
259	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
260	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
261	C₂×C₄	#117×C₄₇	1	2	yes	—
262	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
263	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
264	C₂×C₄	C₄×Q₈×C₄₇	1	1	yes	characteristic
265	C₂×C₄	#121×C₄₇	1	2	yes	—
266	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
267	C₂×C₄	#123×C₄₇	1	2	yes	—
268	C₂×C₄	#123×C₄₇	1	2	yes	—
269	C₂×C₄	#124×C₄₇	1	2	yes	—
270	C₂×C₄	#125×C₄₇	1	2	yes	—
271	C₂×C₄	#126×C₄₇	1	2	yes	—
272	C₂×C₄	#126×C₄₇	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	C₄₇×#376	1	1	yes	characteristic
283	C₂²	C₄×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
284	C₂²	C₄×C₄⋊C₄×C₄₇	1	1	yes	characteristic
285	C₂²	C₄₇×#387	1	2	yes	—
286	C₂²	C₄₇×#388	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	C₄₇×#380	1	1	yes	characteristic
290	C₄	C₄₇×#381	1	1	yes	characteristic
291	C₄	C₄₇×#382	1	1	yes	characteristic
292	C₄	C₄₇×#383	1	1	yes	characteristic
293	C₄	C₄₇×#387	1	2	yes	—
294	C₄	C₄₇×#388	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	C₄₇×#1434	1	1	yes	characteristic
300	C₂	C₄₇×#1757	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂³×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#97×C₄₇	C₄	semidirect
13. C₂×C₄×C₈×C₄₇	C₄	semidirect
15. C₄₇×#343	C₄	semidirect
126. #10803	C₄₇	direct (N × H)
162. C₂×#97	C₄×C₄₇	semidirect
163. C₂×C₄×C₈	C₄×C₄₇	semidirect
165. #343	C₄×C₄₇	semidirect
168. C₄₇	#10803	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×#405	2
C₂×C₄₇×#344	2
C₄₇×#1742	2
C₄₇×#1740	2
C₄₇×#1729	2
#10803	47

Subgroups — C47×#10803 #26050501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10803 #26050501