Subgroups — C₄₇×#18065 #26057763

Order 12032.

652 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2990	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1593	C₂	1	2	yes	maximal, complemented
4	C₄₇×#1582	C₂	1	4	yes	maximal, complemented
5	C₄₇×#1580	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1489	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₄₇	C₄	1	1	yes	characteristic
10	C₄×#41×C₄₇	C₂²	1	2	yes	—
11	C₄₇×#446	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₄₇	C₄	1	2	yes	—
15	C₄×SD₁₆×C₄₇	C₂²	1	2	yes	—
16	C₄×D₈×C₄₇	C₄	1	1	yes	characteristic
17	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#435	—	2	4	no	—
19	C₈×D₄×C₄₇	C₂²	1	4	yes	—
20	C₄₇×#432	—	2	2	no	—
21	C₄₇×#431	C₂²	1	1	yes	characteristic
22	C₄₇×#427	C₄	1	1	yes	characteristic
23	C₄₇×#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
25	C₄₇×#416	C₄	1	2	yes	—
26	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#334	C₂²	1	2	yes	—
29	C₄₇×#334	C₂²	1	2	yes	—
30	C₄₇×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₄₇×D₄⋊C₈	C₂²	1	4	yes	—
32	C₄₇×#322	C₂²	1	2	yes	—
33	C₄₇×#322	C₂²	1	2	yes	—
34	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
35	#135×C₄₇	—	2	4	no	—
36	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	1	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	4	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
46	C₄×D₄×C₄₇	—	2	2	no	—
47	C₄×D₄×C₄₇	—	2	2	no	—
48	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₄₇	C₂³	1	2	yes	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
55	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
56	#106×C₄₇	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₄₇	—	2	4	no	—
58	C₄⋊C₈×C₄₇	—	2	2	no	—
59	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
60	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
61	#103×C₄₇	C₂×C₄	1	2	yes	—
62	#103×C₄₇	C₂×C₄	1	2	yes	—
63	#102×C₄₇	C₂×C₄	1	2	yes	—
64	#102×C₄₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₄₇	—	2	4	no	—
66	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
67	#97×C₄₇	—	2	4	no	—
68	#97×C₄₇	—	2	1	no	—
69	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	2	yes	—
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	2	yes	—
75	#41×C₄₇	—	2	4	no	—
76	#41×C₄₇	—	4	2	no	—
77	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	Q₁₆×C₄₇	—	2	2	no	—
87	SD₁₆×C₄₇	—	2	4	no	—
88	D₈×C₄₇	M₁₆	1	4	yes	complemented
89	M₁₆×C₄₇	—	4	2	no	—
90	M₁₆×C₄₇	—	8	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	4	1	no	—
95	C₂×C₈×C₄₇	—	8	4	no	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
105	C₂²⋊C₄×C₄₇	—	2	2	no	—
106	C₂²⋊C₄×C₄₇	—	2	2	no	—
107	C₂²⋊C₄×C₄₇	—	2	2	no	—
108	C₄²×C₄₇	—	2	2	no	—
109	C₄²×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	—	2	2	no	—
111	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₄₇	—	2	2	no	—
118	Q₈×C₄₇	—	2	2	no	—
119	D₄×C₄₇	—	2	2	no	—
120	D₄×C₄₇	—	4	4	no	—
121	D₄×C₄₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₄₇	—	2	1	no	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	2	4	no	—
143	C₈×C₄₇	—	4	2	no	—
144	C₈×C₄₇	—	4	4	no	—
145	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
146	#18065	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	4	2	no	—
158	C₄×C₄₇	—	4	2	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	#444	1	2	yes	—
161	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₄₇	#568	1	1	yes	characteristic
163	C₄×C₄₇	#577	1	2	yes	—
164	#2990	C₂×C₄₇	1	2	yes	complemented
165	C₄×#135	C₂×C₄₇	1	1	yes	characteristic
166	#1593	C₂×C₄₇	1	2	yes	complemented
167	#1582	C₂×C₄₇	1	4	yes	complemented
168	#1580	C₂×C₄₇	1	2	yes	complemented
169	#1566	C₂×C₄₇	1	1	yes	characteristic, complemented
170	#1489	C₂×C₄₇	1	2	yes	complemented
171	#1451	C₂×C₄₇	1	1	yes	characteristic, complemented
172	C₂×C₄₇	—	4	1	no	—
173	C₂×C₄₇	—	4	4	no	—
174	C₂×C₄₇	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₄₇	#2952	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
177	C₂×#135	C₄×C₄₇	1	1	yes	characteristic
178	C₄×#41	C₂²×C₄₇	1	2	yes	—
179	#446	C₂²×C₄₇	1	4	yes	—
180	C₄×Q₁₆	C₄×C₄₇	1	1	yes	characteristic
181	C₄×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
182	C₄×SD₁₆	C₄×C₄₇	1	2	yes	—
183	C₄×SD₁₆	C₂²×C₄₇	1	2	yes	—
184	C₄×D₈	C₄×C₄₇	1	1	yes	characteristic
185	C₄×D₈	C₂²×C₄₇	1	1	yes	characteristic
186	#435	—	2	4	no	—
187	C₈×D₄	C₂²×C₄₇	1	4	yes	—
188	#432	—	2	2	no	—
189	#431	C₂²×C₄₇	1	1	yes	characteristic
190	#427	C₄×C₄₇	1	1	yes	characteristic
191	#423	C₂²×C₄₇	1	2	yes	—
192	C₂×C₄⋊C₈	—	2	2	no	—
193	#416	C₄×C₄₇	1	2	yes	—
194	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
196	#334	C₂²×C₄₇	1	2	yes	—
197	#334	C₂²×C₄₇	1	2	yes	—
198	Q₈⋊C₈	C₂²×C₄₇	1	4	yes	—
199	D₄⋊C₈	C₂²×C₄₇	1	4	yes	—
200	#322	C₂²×C₄₇	1	2	yes	—
201	#322	C₂²×C₄₇	1	2	yes	—
202	C₄₇	#18065	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×#41	C₂×C₄×C₄₇	1	2	yes	—
204	#135	—	2	4	no	—
205	C₂×Q₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
206	C₂×SD₁₆	C₂×C₄×C₄₇	1	2	yes	—
207	C₂×D₈	C₂×C₄×C₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂²×C₈	—	2	4	no	—
211	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
213	C₄×Q₈	C₂×C₄×C₄₇	1	2	yes	—
214	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂×C₄×C₄₇	1	2	yes	—
218	C₄×D₄	C₂³×C₄₇	1	2	yes	—
219	#117	—	2	2	no	—
220	#117	C₂×C₄×C₄₇	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
223	#107	C₂×C₄×C₄₇	1	1	yes	characteristic
224	#107	C₂×C₄×C₄₇	1	1	yes	characteristic
225	#106	C₂×C₄×C₄₇	1	2	yes	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	D₄×C₄₇	1	2	yes	—
229	C₄⋊C₈	C₂³×C₄₇	1	4	yes	—
230	#103	C₂×C₄×C₄₇	1	2	yes	—
231	#103	C₂×C₄×C₄₇	1	2	yes	—
232	#102	C₂×C₄×C₄₇	1	2	yes	—
233	#102	C₂×C₄×C₄₇	1	2	yes	—
234	C₂²⋊C₈	—	2	4	no	—
235	C₂²⋊C₈	D₄×C₄₇	1	2	yes	—
236	#97	—	2	4	no	—
237	#97	—	2	1	no	—
238	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
239	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₄₇	1	2	yes	—
241	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₄₇	1	2	yes	—
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	C₂²×C₄×C₄₇	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂²×C₄×C₄₇	1	2	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
254	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	SD₁₆	—	2	4	no	—
257	D₈	M₁₆×C₄₇	1	4	yes	complemented
258	M₁₆	—	4	2	no	—
259	M₁₆	—	8	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	8	4	no	—
265	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
273	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
282	C₄²	#41×C₄₇	1	1	yes	characteristic
283	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	C₄×D₄×C₄₇	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	4	no	—
290	D₄	C₂×M₁₆×C₄₇	1	4	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×M₁₆×C₄₇	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	C₄₇×#444	1	2	yes	—
329	C₄	C₂²×M₁₆×C₄₇	1	1	yes	characteristic
330	C₄	C₄₇×#568	1	1	yes	characteristic
331	C₄	C₄₇×#577	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	4	4	no	—
334	C₂	D₄×M₁₆×C₄₇	1	1	yes	characteristic
335	C₂	C₄₇×#2952	1	1	yes	characteristic
336	C₂	C₂×C₄₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₄²	C₂⁴×C₄₇
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#135×C₄₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₄₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#2990	C₂	semidirect
3. C₄₇×#1593	C₂	semidirect
4. C₄₇×#1582	C₂	semidirect
5. C₄₇×#1580	C₂	semidirect
6. C₄₇×#1566	C₂	semidirect
7. C₄₇×#1489	C₂	semidirect
8. C₄₇×#1451	C₂	semidirect
88. D₈×C₄₇	M₁₆	semidirect
146. #18065	C₄₇	direct (N × H)
164. #2990	C₂×C₄₇	semidirect
166. #1593	C₂×C₄₇	semidirect
167. #1582	C₂×C₄₇	semidirect
168. #1580	C₂×C₄₇	semidirect
169. #1566	C₂×C₄₇	semidirect
170. #1489	C₂×C₄₇	semidirect
171. #1451	C₂×C₄₇	semidirect
202. C₄₇	#18065	direct (N × H)
257. D₈	M₁₆×C₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₇×#2990	2
C₄×#135×C₄₇	2
C₄₇×#1593	2
C₄₇×#1582	2
C₄₇×#1580	2
C₄₇×#1566	2
C₄₇×#1489	2
C₄₇×#1451	2
#18065	47

Subgroups — C47×#18065 #26057763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18065 #26057763