Subgroups — C₄₇×#14705 #26054403

Order 12032.

2430 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#385	C₂	1	2	yes	maximal
3	C₂×C₄₇×#381	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#381	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1436	C₂	1	8	yes	maximal, complemented
6	C₄₇×C₂²⋊#95	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	—	2	8	no	—
14	C₂²×C₄²×C₄₇	—	2	4	no	—
15	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
16	C₄₇×#385	—	2	8	no	—
17	C₄₇×#381	—	2	4	no	—
18	C₄₇×#381	C₄	1	8	yes	complemented
19	C₂×#95×C₄₇	C₂²	1	4	yes	—
20	C₂×#95×C₄₇	C₂²	1	4	yes	—
21	C₂×#95×C₄₇	C₂²	1	8	yes	—
22	C₂×#95×C₄₇	C₂²	1	4	yes	—
23	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	4	no	—
26	C₂³×C₄×C₄₇	—	2	4	no	—
27	C₂³×C₄×C₄₇	—	2	4	no	—
28	C₂³×C₄×C₄₇	—	2	16	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₄₇	D₄	1	8	yes	—
31	C₂³×C₄×C₄₇	D₄	1	4	yes	—
32	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₇	—	2	32	no	—
48	C₂×C₄²×C₄₇	—	2	8	no	—
49	C₂×C₄²×C₄₇	—	2	16	no	—
50	#95×C₄₇	—	2	8	no	—
51	#95×C₄₇	—	2	8	no	—
52	#95×C₄₇	—	2	16	no	—
53	#95×C₄₇	C₂×C₄	1	16	yes	complemented
54	C₂⁴×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
58	C₂⁴×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂⁴×C₄₇	#41	1	4	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	32	no	—
62	C₂²×C₄×C₄₇	—	2	32	no	—
63	C₂²×C₄×C₄₇	—	2	16	no	—
64	C₂²×C₄×C₄₇	—	2	16	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	16	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	8	no	—
73	C₂²×C₄×C₄₇	—	2	32	no	—
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	—	2	8	no	—
77	C₂²×C₄×C₄₇	—	2	8	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	4	no	—
83	C₂²×C₄×C₄₇	—	2	8	no	—
84	C₂²×C₄×C₄₇	—	2	8	no	—
85	C₂²×C₄×C₄₇	C₄⋊C₄	1	16	yes	complemented
86	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
87	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₄₇	#41	1	16	yes	—
92	C₄⋊C₄×C₄₇	—	4	16	no	—
93	C₂²⋊C₄×C₄₇	—	2	16	no	—
94	C₂²⋊C₄×C₄₇	—	4	4	no	—
95	C₂²⋊C₄×C₄₇	—	4	4	no	—
96	C₄²×C₄₇	—	2	32	no	—
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	—	2	8	no	—
99	C₂³×C₄₇	—	2	8	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	2	16	no	—
102	C₂³×C₄₇	—	2	8	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂³×C₄₇	C₄×D₄	1	8	yes	—
105	C₂³×C₄₇	#120	1	4	yes	—
106	C₂³×C₄₇	C₂²⋊Q₈	1	8	yes	—
107	C₂³×C₄₇	#123	1	4	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
111	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂³×C₄₇	C₂×#41	1	2	yes	—
113	C₂³×C₄₇	C₂×#41	1	8	yes	—
114	C₂×C₄×C₄₇	—	2	64	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	16	no	—
122	C₂×C₄×C₄₇	—	2	16	no	—
123	C₂×C₄×C₄₇	—	2	32	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	16	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	8	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	4	8	no	—
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
137	C₂×C₄×C₄₇	#123	1	16	yes	—
138	#14705	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₇	—	2	8	no	—
140	C₂²×C₄₇	—	2	8	no	—
141	C₂²×C₄₇	—	2	16	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	16	no	—
144	C₂²×C₄₇	#385	1	8	yes	—
145	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
147	C₂²×C₄₇	C₄×#41	1	4	yes	—
148	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
150	C₂²×C₄₇	C₂×#123	1	4	yes	—
151	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
152	C₂²×C₄₇	#525	1	4	yes	—
153	C₂²×C₄₇	#525	1	4	yes	—
154	C₂²×C₄₇	#529	1	8	yes	—
155	C₂²×C₄₇	#533	1	4	yes	—
156	C₄×C₄₇	—	2	16	no	—
157	C₄×C₄₇	—	2	8	no	—
158	C₄×C₄₇	—	4	8	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₂³×C₄²	C₂×C₄₇	1	1	yes	characteristic
162	C₂×#385	C₂×C₄₇	1	2	yes	—
163	C₂×#381	C₂×C₄₇	1	1	yes	characteristic
164	C₂×#381	C₂×C₄₇	1	1	yes	characteristic
165	#1436	C₂×C₄₇	1	8	yes	complemented
166	C₂²⋊#95	C₂×C₄₇	1	2	yes	—
167	C₂×C₄₇	—	2	8	no	—
168	C₂×C₄₇	C₂×#385	1	1	yes	characteristic
169	C₂×C₄₇	#2296	1	2	yes	—
170	C₂×C₄₇	C₄×#123	1	4	yes	—
171	C₂×C₄₇	#2849	1	2	yes	—
172	C₂×C₄₇	#2853	1	2	yes	—
173	C₂×C₄₇	#2855	1	4	yes	—
174	C₂⁴×C₄	C₂²×C₄₇	1	2	yes	—
175	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₂²×C₄₇	1	4	yes	—
177	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
180	C₂²×C₄²	—	2	8	no	—
181	C₂²×C₄²	—	2	4	no	—
182	C₂²×C₄²	C₂²×C₄₇	1	4	yes	—
183	#385	—	2	8	no	—
184	#381	—	2	4	no	—
185	#381	C₄×C₄₇	1	8	yes	complemented
186	C₂×#95	C₂²×C₄₇	1	8	yes	—
187	C₂×#95	C₂²×C₄₇	1	4	yes	—
188	C₂×#95	C₂²×C₄₇	1	4	yes	—
189	C₂×#95	C₂²×C₄₇	1	4	yes	—
190	C₄₇	#14705	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
192	C₂³×C₄	—	2	4	no	—
193	C₂³×C₄	—	2	4	no	—
194	C₂³×C₄	—	2	4	no	—
195	C₂³×C₄	—	2	4	no	—
196	C₂³×C₄	—	2	16	no	—
197	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
198	C₂³×C₄	D₄×C₄₇	1	8	yes	—
199	C₂³×C₄	D₄×C₄₇	1	4	yes	—
200	C₂³×C₄	Q₈×C₄₇	1	4	yes	—
201	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
202	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
203	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
204	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
205	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	16	no	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	4	no	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
215	C₂×C₄²	—	2	32	no	—
216	C₂×C₄²	—	2	8	no	—
217	C₂×C₄²	—	2	16	no	—
218	#95	—	2	8	no	—
219	#95	—	2	8	no	—
220	#95	—	2	16	no	—
221	#95	C₂×C₄×C₄₇	1	16	yes	complemented
222	C₂⁴	—	2	8	no	—
223	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
224	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
225	C₂⁴	C₂×D₄×C₄₇	1	4	yes	—
226	C₂⁴	C₂×Q₈×C₄₇	1	2	yes	—
227	C₂⁴	#41×C₄₇	1	4	yes	—
228	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	32	no	—
230	C₂²×C₄	—	2	32	no	—
231	C₂²×C₄	—	2	16	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	16	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	32	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	C₄⋊C₄×C₄₇	1	16	yes	complemented
254	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
255	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
256	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
257	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
258	C₂²×C₄	C₂×Q₈×C₄₇	1	4	yes	—
259	C₂²×C₄	#41×C₄₇	1	16	yes	—
260	C₄⋊C₄	—	4	16	no	—
261	C₂²⋊C₄	—	2	16	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	2	32	no	—
265	C₂³	—	2	8	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	2	16	no	—
270	C₂³	—	2	8	no	—
271	C₂³	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
272	C₂³	C₄×D₄×C₄₇	1	8	yes	—
273	C₂³	#120×C₄₇	1	4	yes	—
274	C₂³	C₂²⋊Q₈×C₄₇	1	8	yes	—
275	C₂³	#123×C₄₇	1	4	yes	—
276	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
277	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
278	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
279	C₂³	C₂²×Q₈×C₄₇	1	1	yes	characteristic
280	C₂³	C₂×#41×C₄₇	1	2	yes	—
281	C₂³	C₂×#41×C₄₇	1	8	yes	—
282	C₂×C₄	—	2	64	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	16	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	—	2	32	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	16	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	8	yes	—
305	C₂×C₄	#123×C₄₇	1	16	yes	—
306	C₂²	—	2	8	no	—
307	C₂²	—	2	16	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	2	8	no	—
310	C₂²	—	2	16	no	—
311	C₂²	C₄₇×#385	1	8	yes	—
312	C₂²	C₂²×C₄⋊C₄×C₄₇	1	1	yes	characteristic
313	C₂²	C₂×C₄×D₄×C₄₇	1	2	yes	—
314	C₂²	C₄×#41×C₄₇	1	4	yes	—
315	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
316	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	2	yes	—
317	C₂²	C₂×#123×C₄₇	1	1	yes	characteristic
318	C₂²	C₂×#123×C₄₇	1	4	yes	—
319	C₂²	C₄₇×#525	1	4	yes	—
320	C₂²	C₄₇×#525	1	4	yes	—
321	C₂²	C₄₇×#529	1	8	yes	—
322	C₂²	C₄₇×#533	1	4	yes	—
323	C₄	—	2	8	no	—
324	C₄	—	2	16	no	—
325	C₄	—	4	8	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₂	—	2	8	no	—
329	C₂	C₂×C₄₇×#385	1	1	yes	characteristic
330	C₂	C₄₇×#2296	1	2	yes	—
331	C₂	C₄×#123×C₄₇	1	4	yes	—
332	C₂	C₄₇×#2849	1	2	yes	—
333	C₂	C₄₇×#2853	1	2	yes	—
334	C₂	C₄₇×#2855	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₇
Frattini, cosocle	C₂⁴	C₂⁴×C₄₇
prime radical	C₂⁵×C₄₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₇×#1436	C₂	semidirect
18. C₄₇×#381	C₄	semidirect
53. #95×C₄₇	C₂×C₄	semidirect
85. C₂²×C₄×C₄₇	C₄⋊C₄	semidirect
138. #14705	C₄₇	direct (N × H)
165. #1436	C₂×C₄₇	semidirect
185. #381	C₄×C₄₇	semidirect
190. C₄₇	#14705	direct (N × H)
221. #95	C₂×C₄×C₄₇	semidirect
253. C₂²×C₄	C₄⋊C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₄₇	2
C₂×C₄₇×#385	2
C₂×C₄₇×#381	2
C₄₇×#1436	2
C₄₇×C₂²⋊#95	2
#14705	47

Subgroups — C47×#14705 #26054403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#14705 #26054403