Subgroups — C₄₇×#20673 #26060371

Order 12032.

1292 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#430	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₇×#413	C₂	1	2	yes	maximal
4	C₄₇×#1813	C₂	1	8	yes	maximal, complemented
5	C₂×C₄₇×#338	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#568	—	2	8	no	—
8	C₂×#131×C₄₇	C₄	1	4	yes	—
9	C₂×#131×C₄₇	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₄₇	C₄	1	4	yes	—
11	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#430	—	2	8	no	—
15	C₂×#108×C₄₇	C₂²	1	8	yes	complemented
16	C₄₇×#413	—	2	4	no	—
17	C₄₇×#413	C₂²	1	8	yes	—
18	C₂×#101×C₄₇	—	2	2	no	—
19	C₂×#100×C₄₇	—	2	2	no	—
20	C₄₇×#338	C₂²	1	8	yes	complemented
21	C₂×#41×C₄₇	—	2	4	no	—
22	C₂×#41×C₄₇	—	2	4	no	—
23	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×#41×C₄₇	D₄	1	2	yes	—
25	C₂×#41×C₄₇	Q₈	1	2	yes	—
26	C₂×#41×C₄₇	C₂³	1	2	yes	—
27	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	#131×C₄₇	—	2	16	no	—
33	#131×C₄₇	—	2	16	no	—
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	16	no	—
36	C₂×M₁₆×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
46	C₂×M₁₆×C₄₇	C₂³	1	8	yes	—
47	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	#108×C₄₇	—	2	16	no	—
53	#101×C₄₇	—	2	8	no	—
54	#100×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	#41×C₄₇	—	2	4	no	—
58	#41×C₄₇	—	2	4	no	—
59	#41×C₄₇	—	2	8	no	—
60	#41×C₄₇	—	2	8	no	—
61	#41×C₄₇	—	2	4	no	—
62	#41×C₄₇	—	2	4	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₄₇	—	2	8	no	—
67	C₂×D₄×C₄₇	—	2	4	no	—
68	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	#41	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₄₇	—	2	4	no	—
86	M₁₆×C₄₇	—	2	4	no	—
87	M₁₆×C₄₇	—	2	16	no	—
88	M₁₆×C₄₇	—	2	8	no	—
89	M₁₆×C₄₇	—	2	8	no	—
90	M₁₆×C₄₇	—	4	4	no	—
91	M₁₆×C₄₇	—	4	4	no	—
92	M₁₆×C₄₇	C₂²⋊C₄	1	8	yes	—
93	M₁₆×C₄₇	C₂²⋊C₄	1	8	yes	—
94	C₂×C₈×C₄₇	—	2	4	no	—
95	C₂×C₈×C₄₇	—	2	16	no	—
96	C₂×C₈×C₄₇	—	2	8	no	—
97	C₂×C₈×C₄₇	—	4	4	no	—
98	C₂×C₈×C₄₇	—	4	4	no	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
102	C₂³×C₄₇	—	2	4	no	—
103	C₂³×C₄₇	—	8	1	no	—
104	C₂³×C₄₇	—	8	2	no	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
109	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	4	no	—
111	Q₈×C₄₇	—	2	4	no	—
112	D₄×C₄₇	—	2	8	no	—
113	D₄×C₄₇	—	2	8	no	—
114	D₄×C₄₇	—	2	4	no	—
115	D₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#117	1	2	yes	—
126	C₂×C₄×C₄₇	#117	1	2	yes	—
127	C₂×C₄×C₄₇	#121	1	4	yes	—
128	C₂×C₄×C₄₇	#121	1	4	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₈×C₄₇	—	2	4	no	—
135	C₈×C₄₇	—	4	8	no	—
136	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
137	#20673	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	—	4	1	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	—	8	4	no	—
145	C₂²×C₄₇	#380	1	2	yes	—
146	C₂²×C₄₇	#380	1	2	yes	—
147	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	#380	1	2	yes	—
151	C₄×C₄₇	#380	1	2	yes	—
152	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#121	1	2	yes	—
155	C₂×#568	C₂×C₄₇	1	1	yes	characteristic
156	C₂×#430	C₂×C₄₇	1	2	yes	complemented
157	C₂×#413	C₂×C₄₇	1	2	yes	—
158	#1813	C₂×C₄₇	1	8	yes	complemented
159	C₂×#338	C₂×C₄₇	1	2	yes	complemented
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	4	4	no	—
165	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
166	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
167	#568	—	2	8	no	—
168	C₂×#131	C₄×C₄₇	1	4	yes	—
169	C₂×#131	C₂²×C₄₇	1	4	yes	—
170	C₂²×M₁₆	C₄×C₄₇	1	4	yes	—
171	C₂²×M₁₆	C₂²×C₄₇	1	4	yes	—
172	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
173	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
174	#430	—	2	8	no	—
175	C₂×#108	C₂²×C₄₇	1	8	yes	complemented
176	#413	—	2	4	no	—
177	#413	C₂²×C₄₇	1	8	yes	—
178	C₂×#101	—	2	2	no	—
179	C₂×#100	—	2	2	no	—
180	#338	C₂²×C₄₇	1	8	yes	complemented
181	C₄₇	#20673	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#41	—	2	4	no	—
183	C₂×#41	—	2	4	no	—
184	C₂×#41	C₂×C₄×C₄₇	1	2	yes	—
185	C₂×#41	D₄×C₄₇	1	2	yes	—
186	C₂×#41	Q₈×C₄₇	1	2	yes	—
187	C₂×#41	C₂³×C₄₇	1	2	yes	—
188	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
189	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
190	C₂²×D₄	Q₈×C₄₇	1	2	yes	—
191	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
192	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
193	#131	—	2	16	no	—
194	#131	—	2	16	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	16	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
203	C₂×M₁₆	C₂×C₄×C₄₇	1	8	yes	—
204	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
205	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
206	C₂×M₁₆	D₄×C₄₇	1	8	yes	—
207	C₂×M₁₆	C₂³×C₄₇	1	8	yes	—
208	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	C₂×C₄×C₄₇	1	8	yes	—
211	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
212	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
213	#108	—	2	16	no	—
214	#101	—	2	8	no	—
215	#100	—	2	8	no	—
216	C₂⁴	C₄⋊C₄×C₄₇	1	2	yes	—
217	C₂⁴	C₂×Q₈×C₄₇	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	#41	—	2	4	no	—
220	#41	—	2	8	no	—
221	#41	—	2	8	no	—
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	C₄⋊C₄×C₄₇	1	2	yes	—
226	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	C₄⋊C₄×C₄₇	1	2	yes	—
230	C₂×D₄	C₄⋊C₄×C₄₇	1	4	yes	—
231	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
232	C₂×D₄	C₂×Q₈×C₄₇	1	4	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
236	C₂²×C₄	C₄⋊C₄×C₄₇	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
244	C₂²×C₄	#41×C₄₇	1	4	yes	—
245	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
246	M₁₆	—	2	4	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	16	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	2	8	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	4	no	—
253	M₁₆	C₂²⋊C₄×C₄₇	1	8	yes	—
254	M₁₆	C₂²⋊C₄×C₄₇	1	8	yes	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	16	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	—	4	4	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
261	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
262	C₂×C₈	C₂×D₄×C₄₇	1	8	yes	—
263	C₂³	—	2	4	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	2	no	—
266	C₂³	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
267	C₂³	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
268	C₂³	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
269	C₂³	C₂²⋊Q₈×C₄₇	1	4	yes	—
270	C₂³	C₂²×Q₈×C₄₇	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	4	no	—
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
283	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
284	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
286	C₂×C₄	#117×C₄₇	1	2	yes	—
287	C₂×C₄	#117×C₄₇	1	2	yes	—
288	C₂×C₄	#121×C₄₇	1	4	yes	—
289	C₂×C₄	#121×C₄₇	1	4	yes	—
290	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
291	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	4	8	no	—
297	C₈	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	8	4	no	—
305	C₂²	C₄₇×#380	1	2	yes	—
306	C₂²	C₄₇×#380	1	2	yes	—
307	C₂²	C₂²×C₄⋊C₄×C₄₇	1	1	yes	characteristic
308	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	C₄₇×#380	1	2	yes	—
311	C₄	C₄₇×#380	1	2	yes	—
312	C₄	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
313	C₄	C₂×#117×C₄₇	1	1	yes	characteristic
314	C₄	C₂×#121×C₄₇	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	2	no	—
319	C₂	—	4	4	no	—
320	C₂	C₂×C₄₇×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₇	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
socle	C₂×C₄₇	C₂×#380
prime radical	C₂²×#41×C₄₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₇ ◁ C₂²×M₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#430	C₂	semidirect
4. C₄₇×#1813	C₂	semidirect
5. C₂×C₄₇×#338	C₂	semidirect
15. C₂×#108×C₄₇	C₂²	semidirect
20. C₄₇×#338	C₂²	semidirect
137. #20673	C₄₇	direct (N × H)
156. C₂×#430	C₂×C₄₇	semidirect
158. #1813	C₂×C₄₇	semidirect
159. C₂×#338	C₂×C₄₇	semidirect
175. C₂×#108	C₂²×C₄₇	semidirect
180. #338	C₂²×C₄₇	semidirect
181. C₄₇	#20673	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×#568	2
C₂×C₄₇×#430	2
C₂×C₄₇×#413	2
C₄₇×#1813	2
C₂×C₄₇×#338	2
#20673	47

Subgroups — C47×#20673 #26060371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#20673 #26060371