Subgroups — C₄₇×#14910 #26054608

Order 12032.

1626 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#399	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#389	C₂	1	3	yes	maximal, complemented
3	C₂×C₄₇×#388	C₂	1	3	yes	maximal
4	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1445	C₂	1	3	yes	maximal, complemented
6	C₄₇×#1444	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1439	C₂	1	3	yes	maximal
8	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
13	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
15	C₄₇×#399	—	2	4	no	—
16	C₄₇×#389	—	2	12	no	—
17	C₄₇×#388	—	2	12	no	—
18	C₄₇×#376	—	2	4	no	—
19	C₂×#95×C₄₇	C₂²	1	3	yes	—
20	C₂×#95×C₄₇	C₂²	1	3	yes	—
21	C₂×#95×C₄₇	C₂²	1	3	yes	—
22	C₂×#95×C₄₇	C₂²	1	6	yes	—
23	C₂×#95×C₄₇	C₂²	1	1	yes	characteristic
24	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
31	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
45	C₂×C₄²×C₄₇	—	2	6	no	—
46	C₂×C₄²×C₄₇	—	2	6	no	—
47	C₂×C₄²×C₄₇	D₄	1	12	yes	—
48	#95×C₄₇	—	2	6	no	—
49	#95×C₄₇	—	2	6	no	—
50	#95×C₄₇	—	2	6	no	—
51	#95×C₄₇	—	2	12	no	—
52	#95×C₄₇	D₄	1	4	yes	—
53	C₂⁴×C₄₇	—	2	3	no	—
54	C₂⁴×C₄₇	—	2	1	no	—
55	C₂⁴×C₄₇	—	2	3	no	—
56	C₂⁴×C₄₇	—	2	3	no	—
57	C₂⁴×C₄₇	—	2	1	no	—
58	C₂⁴×C₄₇	—	2	3	no	—
59	C₂⁴×C₄₇	#41	1	2	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	3	no	—
63	C₂²×C₄×C₄₇	—	2	6	no	—
64	C₂²×C₄×C₄₇	—	2	6	no	—
65	C₂²×C₄×C₄₇	—	2	3	no	—
66	C₂²×C₄×C₄₇	—	2	3	no	—
67	C₂²×C₄×C₄₇	—	2	6	no	—
68	C₂²×C₄×C₄₇	—	2	3	no	—
69	C₂²×C₄×C₄₇	—	2	3	no	—
70	C₂²×C₄×C₄₇	—	2	3	no	—
71	C₂²×C₄×C₄₇	—	2	3	no	—
72	C₂²×C₄×C₄₇	—	2	6	no	—
73	C₂²×C₄×C₄₇	—	2	3	no	—
74	C₂²×C₄×C₄₇	—	2	3	no	—
75	C₂²×C₄×C₄₇	—	2	3	no	—
76	C₂²×C₄×C₄₇	—	2	3	no	—
77	C₂²×C₄×C₄₇	—	2	3	no	—
78	C₂²×C₄×C₄₇	—	2	6	no	—
79	C₂²×C₄×C₄₇	—	2	3	no	—
80	C₂²×C₄×C₄₇	—	2	6	no	—
81	C₂²×C₄×C₄₇	—	2	3	no	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₄₇	#41	1	6	yes	—
88	C₂²×C₄×C₄₇	#41	1	6	yes	—
89	C₄⋊C₄×C₄₇	—	2	24	no	—
90	C₄⋊C₄×C₄₇	—	4	12	no	—
91	C₄⋊C₄×C₄₇	—	4	12	no	—
92	C₂²⋊C₄×C₄₇	—	2	24	no	—
93	C₂²⋊C₄×C₄₇	—	4	12	no	—
94	C₂²⋊C₄×C₄₇	—	4	4	no	—
95	C₄²×C₄₇	—	2	24	no	—
96	C₂³×C₄₇	—	2	6	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	6	no	—
99	C₂³×C₄₇	—	4	3	no	—
100	C₂³×C₄₇	—	4	3	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	—	4	3	no	—
103	C₂³×C₄₇	—	4	3	no	—
104	C₂³×C₄₇	—	4	3	no	—
105	C₂³×C₄₇	—	4	3	no	—
106	C₂³×C₄₇	—	4	6	no	—
107	C₂³×C₄₇	—	4	3	no	—
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
110	C₂³×C₄₇	C₂×#41	1	3	yes	—
111	C₂³×C₄₇	C₂×#41	1	3	yes	—
112	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₂³×C₄₇	2₋¹⁺⁴	1	3	yes	—
114	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	6	no	—
116	C₂×C₄×C₄₇	—	2	6	no	—
117	C₂×C₄×C₄₇	—	2	6	no	—
118	C₂×C₄×C₄₇	—	2	6	no	—
119	C₂×C₄×C₄₇	—	2	12	no	—
120	C₂×C₄×C₄₇	—	2	12	no	—
121	C₂×C₄×C₄₇	—	2	6	no	—
122	C₂×C₄×C₄₇	—	2	6	no	—
123	C₂×C₄×C₄₇	—	2	6	no	—
124	C₂×C₄×C₄₇	—	2	12	no	—
125	C₂×C₄×C₄₇	—	2	6	no	—
126	C₂×C₄×C₄₇	—	4	3	no	—
127	C₂×C₄×C₄₇	—	4	6	no	—
128	C₂×C₄×C₄₇	—	4	3	no	—
129	C₂×C₄×C₄₇	—	4	3	no	—
130	C₂×C₄×C₄₇	—	4	3	no	—
131	C₂×C₄×C₄₇	—	4	3	no	—
132	C₂×C₄×C₄₇	—	4	3	no	—
133	C₂×C₄×C₄₇	—	4	3	no	—
134	C₂×C₄×C₄₇	—	4	6	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	3	no	—
137	C₂×C₄×C₄₇	—	4	3	no	—
138	C₂×C₄×C₄₇	—	4	6	no	—
139	C₂×C₄×C₄₇	—	4	3	no	—
140	C₂×C₄×C₄₇	—	4	3	no	—
141	C₂×C₄×C₄₇	#121	1	12	yes	—
142	C₂×C₄×C₄₇	#127	1	4	yes	—
143	#14910	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₇	—	4	6	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	6	no	—
147	C₂²×C₄₇	—	8	1	no	—
148	C₂²×C₄₇	—	8	3	no	—
149	C₂²×C₄₇	—	8	1	no	—
150	C₂²×C₄₇	—	8	3	no	—
151	C₂²×C₄₇	C₂×#121	1	3	yes	—
152	C₂²×C₄₇	#525	1	3	yes	—
153	C₂²×C₄₇	#525	1	3	yes	—
154	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
155	C₂²×C₄₇	#532	1	3	yes	—
156	C₂²×C₄₇	#536	1	3	yes	—
157	C₂²×C₄₇	#536	1	3	yes	—
158	C₂²×C₄₇	#547	1	3	yes	—
159	C₂²×C₄₇	#548	1	1	yes	characteristic
160	C₂²×C₄₇	#552	1	6	yes	—
161	C₂²×C₄₇	#552	1	3	yes	—
162	C₂²×C₄₇	#556	1	3	yes	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	—	2	12	no	—
165	C₄×C₄₇	—	4	6	no	—
166	C₄×C₄₇	—	4	6	no	—
167	C₄×C₄₇	—	8	3	no	—
168	C₄×C₄₇	—	8	1	no	—
169	C₂×#399	C₂×C₄₇	1	1	yes	characteristic
170	C₂×#389	C₂×C₄₇	1	3	yes	complemented
171	C₂×#388	C₂×C₄₇	1	3	yes	—
172	C₂×#376	C₂×C₄₇	1	1	yes	characteristic, complemented
173	#1445	C₂×C₄₇	1	3	yes	complemented
174	#1444	C₂×C₄₇	1	1	yes	characteristic, complemented
175	#1439	C₂×C₄₇	1	3	yes	—
176	C₂×C₄₇	—	8	2	no	—
177	C₂×C₄₇	#2397	1	1	yes	characteristic
178	C₂×C₄₇	#2413	1	3	yes	—
179	C₂×C₄₇	#2415	1	3	yes	—
180	C₂×C₄₇	#2604	1	3	yes	—
181	C₂×C₄₇	#2611	1	3	yes	—
182	C₂×C₄₇	#2658	1	1	yes	characteristic
183	C₂×C₄₇	#2856	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	C₂²×C₄₇	1	3	yes	—
185	C₂²×C₄⋊C₄	C₂²×C₄₇	1	3	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₄₇	1	3	yes	—
187	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	3	yes	—
188	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	3	yes	—
189	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
190	C₂²×C₄²	C₂²×C₄₇	1	3	yes	—
191	#399	—	2	4	no	—
192	#389	—	2	12	no	—
193	#388	—	2	12	no	—
194	#376	—	2	4	no	—
195	C₂×#95	C₂²×C₄₇	1	3	yes	—
196	C₂×#95	C₂²×C₄₇	1	6	yes	—
197	C₂×#95	C₂²×C₄₇	1	3	yes	—
198	C₂×#95	C₂²×C₄₇	1	3	yes	—
199	C₂×#95	C₂²×C₄₇	1	1	yes	characteristic
200	C₄₇	#14910	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
203	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
205	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
206	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
208	C₂×C₄⋊C₄	—	2	6	no	—
209	C₂×C₄⋊C₄	—	2	12	no	—
210	C₂×C₄⋊C₄	—	2	6	no	—
211	C₂×C₄⋊C₄	—	2	6	no	—
212	C₂×C₄⋊C₄	—	2	12	no	—
213	C₂×C₄⋊C₄	—	2	6	no	—
214	C₂×C₄⋊C₄	—	2	6	no	—
215	C₂×C₂²⋊C₄	—	2	6	no	—
216	C₂×C₂²⋊C₄	—	2	6	no	—
217	C₂×C₂²⋊C₄	—	2	6	no	—
218	C₂×C₂²⋊C₄	—	2	6	no	—
219	C₂×C₂²⋊C₄	—	2	6	no	—
220	C₂×C₂²⋊C₄	—	2	6	no	—
221	C₂×C₂²⋊C₄	—	2	6	no	—
222	C₂×C₄²	—	2	6	no	—
223	C₂×C₄²	—	2	6	no	—
224	C₂×C₄²	D₄×C₄₇	1	12	yes	—
225	#95	—	2	6	no	—
226	#95	—	2	6	no	—
227	#95	—	2	6	no	—
228	#95	—	2	12	no	—
229	#95	D₄×C₄₇	1	4	yes	—
230	C₂⁴	—	2	3	no	—
231	C₂⁴	—	2	1	no	—
232	C₂⁴	—	2	3	no	—
233	C₂⁴	—	2	3	no	—
234	C₂⁴	—	2	1	no	—
235	C₂⁴	—	2	3	no	—
236	C₂⁴	#41×C₄₇	1	2	yes	—
237	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	3	no	—
240	C₂²×C₄	—	2	6	no	—
241	C₂²×C₄	—	2	6	no	—
242	C₂²×C₄	—	2	3	no	—
243	C₂²×C₄	—	2	3	no	—
244	C₂²×C₄	—	2	6	no	—
245	C₂²×C₄	—	2	3	no	—
246	C₂²×C₄	—	2	3	no	—
247	C₂²×C₄	—	2	3	no	—
248	C₂²×C₄	—	2	3	no	—
249	C₂²×C₄	—	2	6	no	—
250	C₂²×C₄	—	2	3	no	—
251	C₂²×C₄	—	2	3	no	—
252	C₂²×C₄	—	2	3	no	—
253	C₂²×C₄	—	2	3	no	—
254	C₂²×C₄	—	2	6	no	—
255	C₂²×C₄	—	2	3	no	—
256	C₂²×C₄	—	2	3	no	—
257	C₂²×C₄	—	2	6	no	—
258	C₂²×C₄	—	2	3	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
261	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
262	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
263	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
264	C₂²×C₄	#41×C₄₇	1	6	yes	—
265	C₂²×C₄	#41×C₄₇	1	6	yes	—
266	C₄⋊C₄	—	2	24	no	—
267	C₄⋊C₄	—	4	12	no	—
268	C₄⋊C₄	—	4	12	no	—
269	C₂²⋊C₄	—	2	24	no	—
270	C₂²⋊C₄	—	4	12	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₄²	—	2	24	no	—
273	C₂³	—	2	6	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	6	no	—
276	C₂³	—	4	3	no	—
277	C₂³	—	4	3	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	3	no	—
280	C₂³	—	4	3	no	—
281	C₂³	—	4	3	no	—
282	C₂³	—	4	3	no	—
283	C₂³	—	4	6	no	—
284	C₂³	—	4	3	no	—
285	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
286	C₂³	C₂²×D₄×C₄₇	1	3	yes	—
287	C₂³	C₂×#41×C₄₇	1	3	yes	—
288	C₂³	C₂×#41×C₄₇	1	3	yes	—
289	C₂³	C₂×#41×C₄₇	1	1	yes	characteristic
290	C₂³	2₋¹⁺⁴×C₄₇	1	3	yes	—
291	C₂³	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	2	6	no	—
295	C₂×C₄	—	2	6	no	—
296	C₂×C₄	—	2	12	no	—
297	C₂×C₄	—	2	12	no	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	2	6	no	—
300	C₂×C₄	—	2	6	no	—
301	C₂×C₄	—	2	12	no	—
302	C₂×C₄	—	2	6	no	—
303	C₂×C₄	—	4	3	no	—
304	C₂×C₄	—	4	6	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	3	no	—
307	C₂×C₄	—	4	3	no	—
308	C₂×C₄	—	4	3	no	—
309	C₂×C₄	—	4	3	no	—
310	C₂×C₄	—	4	3	no	—
311	C₂×C₄	—	4	6	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	3	no	—
314	C₂×C₄	—	4	3	no	—
315	C₂×C₄	—	4	6	no	—
316	C₂×C₄	—	4	3	no	—
317	C₂×C₄	—	4	3	no	—
318	C₂×C₄	#121×C₄₇	1	12	yes	—
319	C₂×C₄	#127×C₄₇	1	4	yes	—
320	C₂²	—	4	6	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	6	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	3	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	3	no	—
327	C₂²	C₂×#121×C₄₇	1	3	yes	—
328	C₂²	C₄₇×#525	1	3	yes	—
329	C₂²	C₄₇×#525	1	3	yes	—
330	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
331	C₂²	C₄₇×#532	1	3	yes	—
332	C₂²	C₄₇×#536	1	3	yes	—
333	C₂²	C₄₇×#536	1	3	yes	—
334	C₂²	C₄₇×#547	1	3	yes	—
335	C₂²	C₄₇×#548	1	1	yes	characteristic
336	C₂²	C₄₇×#552	1	6	yes	—
337	C₂²	C₄₇×#552	1	3	yes	—
338	C₂²	C₄₇×#556	1	3	yes	—
339	C₄	—	2	4	no	—
340	C₄	—	2	12	no	—
341	C₄	—	4	6	no	—
342	C₄	—	4	6	no	—
343	C₄	—	8	3	no	—
344	C₄	—	8	1	no	—
345	C₂	—	8	2	no	—
346	C₂	C₄₇×#2397	1	1	yes	characteristic
347	C₂	C₄₇×#2413	1	3	yes	—
348	C₂	C₄₇×#2415	1	3	yes	—
349	C₂	C₄₇×#2604	1	3	yes	—
350	C₂	C₄₇×#2611	1	3	yes	—
351	C₂	C₄₇×#2658	1	1	yes	characteristic
352	C₂	C₄₇×#2856	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#389	C₂	semidirect
4. C₂×C₄₇×#376	C₂	semidirect
5. C₄₇×#1445	C₂	semidirect
6. C₄₇×#1444	C₂	semidirect
143. #14910	C₄₇	direct (N × H)
170. C₂×#389	C₂×C₄₇	semidirect
172. C₂×#376	C₂×C₄₇	semidirect
173. #1445	C₂×C₄₇	semidirect
174. #1444	C₂×C₄₇	semidirect
200. C₄₇	#14910	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×#399	2
C₂×C₄₇×#389	2
C₂×C₄₇×#388	2
C₂×C₄₇×#376	2
C₄₇×#1445	2
C₄₇×#1444	2
C₄₇×#1439	2
#14910	47

Subgroups — C47×#14910 #26054608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#14910 #26054608