Subgroups — C₄₇×#29638 #26069336

Order 12032.

1292 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#480	C₂	1	2	yes	maximal
3	C₄₇×#2854	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2057	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2023	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1823	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₄₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
12	C₂×#123×C₄₇	—	2	1	no	—
13	C₂×#121×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#480	—	2	8	no	—
18	C₂×#107×C₄₇	C₂²	1	4	yes	—
19	C₂×#102×C₄₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#390	C₂²	1	2	yes	—
22	C₄₇×#389	C₂²	1	2	yes	—
23	C₄₇×#388	—	2	2	no	—
24	C₄₇×#386	—	2	1	no	—
25	C₄₇×#380	C₂²	1	1	yes	characteristic
26	C₄₇×#340	C₂²	1	8	yes	—
27	C₄₇×#336	C₂²	1	4	yes	—
28	C₂²×Q₈×C₄₇	—	2	8	no	—
29	C₂²×Q₈×C₄₇	—	2	4	no	—
30	C₂²×Q₈×C₄₇	—	4	4	no	—
31	C₂²×Q₈×C₄₇	—	4	4	no	—
32	C₂²×D₄×C₄₇	—	2	1	no	—
33	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	—	2	2	no	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₄₇	—	2	16	no	—
40	C₄⋊Q₈×C₄₇	—	2	16	no	—
41	#123×C₄₇	—	4	4	no	—
42	#121×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×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	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	—	2	4	no	—
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	#107×C₄₇	—	2	8	no	—
60	#102×C₄₇	—	2	8	no	—
61	C₂²⋊C₈×C₄₇	—	2	4	no	—
62	#95×C₄₇	—	2	1	no	—
63	#95×C₄₇	—	2	2	no	—
64	C₂⁴×C₄₇	—	2	1	no	—
65	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₄₇	—	2	16	no	—
67	C₂×Q₈×C₄₇	—	4	8	no	—
68	C₂×Q₈×C₄₇	—	4	32	no	—
69	C₂×Q₈×C₄₇	—	4	8	no	—
70	C₂×D₄×C₄₇	—	2	2	no	—
71	C₂×D₄×C₄₇	—	2	4	no	—
72	C₂×D₄×C₄₇	—	4	2	no	—
73	C₂×D₄×C₄₇	—	4	2	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	4	no	—
80	C₂²×C₄×C₄₇	—	2	8	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	4	4	no	—
85	C₂²×C₄×C₄₇	—	4	4	no	—
86	C₂²×C₄×C₄₇	D₈	1	4	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₂×C₈×C₄₇	—	2	8	no	—
94	C₄⋊C₄×C₄₇	—	2	16	no	—
95	C₄⋊C₄×C₄₇	—	2	8	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	—	4	4	no	—
100	C₄⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₄₇	—	4	2	no	—
102	C₂²⋊C₄×C₄₇	—	4	2	no	—
103	C₂²⋊C₄×C₄₇	—	4	4	no	—
104	C₂²⋊C₄×C₄₇	—	4	4	no	—
105	C₄²×C₄₇	—	2	4	no	—
106	C₄²×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	4	1	no	—
110	C₂³×C₄₇	—	4	2	no	—
111	C₂³×C₄₇	—	4	1	no	—
112	C₂³×C₄₇	—	4	1	no	—
113	C₂³×C₄₇	—	4	2	no	—
114	C₂³×C₄₇	#120	1	2	yes	—
115	C₂³×C₄₇	C₂×D₈	1	2	yes	—
116	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₄₇	—	4	32	no	—
119	D₄×C₄₇	—	4	4	no	—
120	D₄×C₄₇	—	8	4	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	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	8	no	—
128	C₂×C₄×C₄₇	—	2	8	no	—
129	C₂×C₄×C₄₇	—	2	8	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	16	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	#120	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
141	C₂×C₄×C₄₇	#137	1	4	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
144	C₈×C₄₇	—	4	4	no	—
145	#29638	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	8	1	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	—	8	2	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	—	8	1	no	—
155	C₂²×C₄₇	#447	1	4	yes	—
156	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
157	C₂²×C₄₇	#536	1	2	yes	—
158	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#137	1	2	yes	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	—	4	8	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	—	8	1	no	—
167	C₄×C₄₇	#536	1	4	yes	—
168	C₂²×C₄⋊Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
169	C₂×#480	C₂×C₄₇	1	2	yes	—
170	#2854	C₂×C₄₇	1	1	yes	characteristic
171	#2057	C₂×C₄₇	1	4	yes	complemented
172	#2023	C₂×C₄₇	1	4	yes	complemented
173	#1823	C₂×C₄₇	1	2	yes	complemented
174	#1817	C₂×C₄₇	1	1	yes	characteristic, complemented
175	C₂×C₄₇	—	2	4	no	—
176	C₂×C₄₇	—	16	1	no	—
177	C₂×C₄₇	#2614	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#447	1	1	yes	characteristic
179	C₂×C₄₇	#3004	1	1	yes	characteristic
180	C₂×C₄₇	#3226	1	2	yes	—
181	C₂×C₄₇	#3238	1	2	yes	—
182	C₂³×Q₈	—	2	1	no	—
183	C₂×C₄⋊Q₈	—	2	8	no	—
184	C₂×C₄⋊Q₈	—	2	2	no	—
185	C₂×C₄⋊Q₈	C₂²×C₄₇	1	4	yes	—
186	C₂×#123	—	2	1	no	—
187	C₂×#121	C₂²×C₄₇	1	1	yes	characteristic
188	C₂²×C₄⋊C₄	—	2	1	no	—
189	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
190	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
191	#480	—	2	8	no	—
192	C₂×#107	C₂²×C₄₇	1	4	yes	—
193	C₂×#102	C₂²×C₄₇	1	4	yes	—
194	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
195	#390	C₂²×C₄₇	1	2	yes	—
196	#389	C₂²×C₄₇	1	2	yes	—
197	#388	—	2	2	no	—
198	#386	—	2	1	no	—
199	#380	C₂²×C₄₇	1	1	yes	characteristic
200	#340	C₂²×C₄₇	1	8	yes	—
201	#336	C₂²×C₄₇	1	4	yes	—
202	C₄₇	#29638	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²×Q₈	—	2	8	no	—
204	C₂²×Q₈	—	2	4	no	—
205	C₂²×Q₈	—	4	4	no	—
206	C₂²×Q₈	—	4	4	no	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
209	C₂³×C₄	—	2	2	no	—
210	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
211	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
212	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
214	C₄⋊Q₈	—	2	16	no	—
215	C₄⋊Q₈	—	2	16	no	—
216	#123	—	4	4	no	—
217	#121	—	2	4	no	—
218	C₂×C₄⋊C₄	—	2	8	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	D₄×C₄₇	1	8	yes	—
224	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	D₄×C₄₇	1	2	yes	—
233	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
234	#107	—	2	8	no	—
235	#102	—	2	8	no	—
236	C₂²⋊C₈	—	2	4	no	—
237	#95	—	2	1	no	—
238	#95	—	2	2	no	—
239	C₂⁴	—	2	1	no	—
240	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
241	C₂×Q₈	—	2	16	no	—
242	C₂×Q₈	—	4	8	no	—
243	C₂×Q₈	—	4	32	no	—
244	C₂×Q₈	—	4	8	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	D₈×C₄₇	1	4	yes	—
262	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
264	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
265	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	8	no	—
269	C₄⋊C₄	—	2	16	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	#120×C₄₇	1	2	yes	—
290	C₂³	C₂×D₈×C₄₇	1	2	yes	—
291	C₂³	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
292	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
293	Q₈	—	4	32	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	8	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	#120×C₄₇	1	2	yes	—
315	C₂×C₄	C₂×D₈×C₄₇	1	4	yes	—
316	C₂×C₄	#137×C₄₇	1	4	yes	—
317	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
318	C₂×C₄	2₋¹⁺⁴×C₄₇	1	4	yes	—
319	C₈	—	4	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₄₇×#447	1	4	yes	—
330	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
331	C₂²	C₄₇×#536	1	2	yes	—
332	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
333	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
334	C₂²	C₂×#137×C₄₇	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	4	8	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	1	no	—
341	C₄	C₄₇×#536	1	4	yes	—
342	C₂	—	2	4	no	—
343	C₂	—	16	1	no	—
344	C₂	C₄₇×#2614	1	1	yes	characteristic
345	C₂	C₂×C₄₇×#447	1	1	yes	characteristic
346	C₂	C₄₇×#3004	1	1	yes	characteristic
347	C₂	C₄₇×#3226	1	2	yes	—
348	C₂	C₄₇×#3238	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂×#121×C₄₇	C₂²
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
1. C₂²×C₄⋊Q₈×C₄₇	C₂	semidirect
4. C₄₇×#2057	C₂	semidirect
5. C₄₇×#2023	C₂	semidirect
6. C₄₇×#1823	C₂	semidirect
7. C₄₇×#1817	C₂	semidirect
145. #29638	C₄₇	direct (N × H)
168. C₂²×C₄⋊Q₈	C₂×C₄₇	semidirect
171. #2057	C₂×C₄₇	semidirect
172. #2023	C₂×C₄₇	semidirect
173. #1823	C₂×C₄₇	semidirect
174. #1817	C₂×C₄₇	semidirect
202. C₄₇	#29638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₄₇	2
C₂×C₄₇×#480	2
C₄₇×#2854	2
C₄₇×#2057	2
C₄₇×#2023	2
C₄₇×#1823	2
C₄₇×#1817	2
#29638	47

Subgroups — C47×#29638 #26069336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29638 #26069336