Subgroups — C₄₇×#29773 #26069471

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#467	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2855	C₂	1	2	yes	maximal
4	C₄₇×#2086	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2066	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2056	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1892	C₂	1	2	yes	maximal
8	C₄₇×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#467	—	2	4	no	—
15	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×#103×C₄₇	C₂²	1	2	yes	—
21	C₂×#103×C₄₇	C₂²	1	4	yes	—
22	C₄₇×#391	C₂²	1	4	yes	—
23	C₄₇×#389	C₂²	1	4	yes	—
24	C₄₇×#387	—	2	4	no	—
25	C₄₇×#385	—	2	2	no	—
26	C₄₇×#381	—	2	2	no	—
27	C₄₇×#380	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₄₇×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₄₇	—	2	2	no	—
35	C₂×Q₁₆×C₄₇	—	4	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	4	2	no	—
38	C₂²×C₈×C₄₇	—	4	2	no	—
39	C₂²×C₈×C₄₇	D₄	1	4	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₄₇	—	2	4	no	—
44	C₂²⋊Q₈×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	#107×C₄₇	—	2	2	no	—
56	#106×C₄₇	—	4	2	no	—
57	C₄⋊C₈×C₄₇	—	2	4	no	—
58	C₄⋊C₈×C₄₇	—	2	4	no	—
59	#103×C₄₇	—	2	4	no	—
60	#103×C₄₇	—	4	4	no	—
61	#95×C₄₇	—	2	2	no	—
62	#95×C₄₇	—	2	4	no	—
63	#95×C₄₇	—	2	4	no	—
64	#95×C₄₇	—	2	4	no	—
65	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×Q₈×C₄₇	—	2	2	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₄₇	—	4	4	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	4	no	—
90	C₂×C₈×C₄₇	—	4	4	no	—
91	C₂×C₈×C₄₇	—	4	4	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	4	4	no	—
99	C₄⋊C₄×C₄₇	—	4	8	no	—
100	C₂²⋊C₄×C₄₇	—	4	4	no	—
101	C₂²⋊C₄×C₄₇	—	4	4	no	—
102	C₄²×C₄₇	—	2	4	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	#121	1	2	yes	—
108	C₂³×C₄₇	#137	1	4	yes	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₄₇	—	4	2	no	—
111	Q₈×C₄₇	—	4	2	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	#121	1	2	yes	—
131	C₂×C₄×C₄₇	#135	1	4	yes	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	4	2	no	—
139	#29773	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	8	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	#470	1	4	yes	—
144	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
145	C₂²×C₄₇	#537	1	2	yes	—
146	C₂²×C₄₇	C₂×#135	1	2	yes	—
147	C₂²×C₄₇	C₂×#137	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	—	8	2	no	—
152	C₄×C₄₇	—	8	4	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	#534	1	1	yes	characteristic
155	C₄×C₄₇	#536	1	1	yes	characteristic
156	C₄×C₄₇	#539	1	2	yes	—
157	C₂×#467	C₂×C₄₇	1	1	yes	characteristic
158	C₂²×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic
159	#2855	C₂×C₄₇	1	2	yes	—
160	#2086	C₂×C₄₇	1	2	yes	complemented
161	#2066	C₂×C₄₇	1	4	yes	complemented
162	#2056	C₂×C₄₇	1	2	yes	complemented
163	#1892	C₂×C₄₇	1	2	yes	—
164	#1807	C₂×C₄₇	1	1	yes	characteristic
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	#2618	1	1	yes	characteristic
167	C₂×C₄₇	#3050	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#470	1	1	yes	characteristic
169	C₂×C₄₇	#3248	1	4	yes	—
170	C₂²×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
173	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
174	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
175	#467	—	2	4	no	—
176	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
177	C₂×#106	C₂²×C₄₇	1	2	yes	—
178	C₂×C₄⋊C₈	—	2	4	no	—
179	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
180	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
181	C₂×#103	C₂²×C₄₇	1	2	yes	—
182	C₂×#103	C₂²×C₄₇	1	4	yes	—
183	#391	C₂²×C₄₇	1	4	yes	—
184	#389	C₂²×C₄₇	1	4	yes	—
185	#387	—	2	4	no	—
186	#385	—	2	2	no	—
187	#381	—	2	2	no	—
188	#380	C₂²×C₄₇	1	2	yes	—
189	#340	C₂²×C₄₇	1	2	yes	—
190	#340	C₂²×C₄₇	1	4	yes	—
191	C₄₇	#29773	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
193	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
194	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×Q₁₆	—	4	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	—	4	2	no	—
201	C₂²×C₈	D₄×C₄₇	1	4	yes	—
202	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
205	C₂²⋊Q₈	—	2	4	no	—
206	C₂²⋊Q₈	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₄₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
217	#107	—	2	2	no	—
218	#106	—	4	2	no	—
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	4	4	no	—
223	#95	—	2	2	no	—
224	#95	—	2	4	no	—
225	#95	—	2	4	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
245	C₂²×C₄	#41×C₄₇	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
247	Q₁₆	—	4	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	#121×C₄₇	1	2	yes	—
270	C₂³	#137×C₄₇	1	4	yes	—
271	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	#121×C₄₇	1	2	yes	—
293	C₂×C₄	#135×C₄₇	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
296	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
297	C₂×C₄	2₋¹⁺⁴×C₄₇	1	2	yes	—
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	8	no	—
304	C₂²	C₄₇×#470	1	4	yes	—
305	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
306	C₂²	C₄₇×#537	1	2	yes	—
307	C₂²	C₂×#135×C₄₇	1	2	yes	—
308	C₂²	C₂×#137×C₄₇	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	4	no	—
315	C₄	C₄₇×#534	1	1	yes	characteristic
316	C₄	C₄₇×#536	1	1	yes	characteristic
317	C₄	C₄₇×#539	1	2	yes	—
318	C₂	—	2	4	no	—
319	C₂	C₄₇×#2618	1	1	yes	characteristic
320	C₂	C₄₇×#3050	1	1	yes	characteristic
321	C₂	C₂×C₄₇×#470	1	1	yes	characteristic
322	C₂	C₄₇×#3248	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₇×#2086	C₂	semidirect
5. C₄₇×#2066	C₂	semidirect
6. C₄₇×#2056	C₂	semidirect
139. #29773	C₄₇	direct (N × H)
160. #2086	C₂×C₄₇	semidirect
161. #2066	C₂×C₄₇	semidirect
162. #2056	C₂×C₄₇	semidirect
191. C₄₇	#29773	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×#467	2
C₂²×C₄⋊C₈×C₄₇	2
C₄₇×#2855	2
C₄₇×#2086	2
C₄₇×#2066	2
C₄₇×#2056	2
C₄₇×#1892	2
C₄₇×#1807	2
#29773	47

Subgroups — C47×#29773 #26069471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29773 #26069471