Subgroups — C₄₇×#15176 #26054874

Order 12032.

1842 conjugacy classes of subgroups.

The subgroup lattice has 362 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₇×#393	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#388	C₂	1	3	yes	maximal
5	C₄₇×#1441	C₂	1	3	yes	maximal, complemented
6	C₄₇×#1439	C₂	1	3	yes	maximal
7	C₄₇×#1434	C₂	1	3	yes	maximal, complemented
8	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₄₇	—	2	12	no	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
14	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
16	C₄₇×#395	—	2	4	no	—
17	C₄₇×#393	—	2	4	no	—
18	C₄₇×#388	—	2	12	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	3	yes	—
23	C₂×#95×C₄₇	C₂²	1	6	yes	—
24	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Q₈×C₄₇	—	2	6	no	—
26	C₂²×Q₈×C₄₇	—	2	6	no	—
27	C₂²×Q₈×C₄₇	D₄	1	4	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
34	C₄×Q₈×C₄₇	—	2	32	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×C₄⋊C₄×C₄₇	—	2	6	no	—
39	C₂×C₄⋊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×C₂²⋊C₄×C₄₇	—	2	6	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
47	C₂×C₄²×C₄₇	—	2	12	no	—
48	C₂×C₄²×C₄₇	D₄	1	12	yes	—
49	#95×C₄₇	—	2	6	no	—
50	#95×C₄₇	—	2	6	no	—
51	#95×C₄₇	—	2	6	no	—
52	#95×C₄₇	—	2	6	no	—
53	#95×C₄₇	—	2	12	no	—
54	C₂⁴×C₄₇	—	2	3	no	—
55	C₂⁴×C₄₇	—	2	1	no	—
56	C₂⁴×C₄₇	—	2	3	no	—
57	C₂⁴×C₄₇	—	2	3	no	—
58	C₂⁴×C₄₇	—	2	1	no	—
59	C₂⁴×C₄₇	—	2	3	no	—
60	C₂⁴×C₄₇	#41	1	2	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×Q₈×C₄₇	—	2	24	no	—
64	C₂×Q₈×C₄₇	—	2	24	no	—
65	C₂²×C₄×C₄₇	—	2	6	no	—
66	C₂²×C₄×C₄₇	—	2	6	no	—
67	C₂²×C₄×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	—	2	3	no	—
69	C₂²×C₄×C₄₇	—	2	6	no	—
70	C₂²×C₄×C₄₇	—	2	3	no	—
71	C₂²×C₄×C₄₇	—	2	3	no	—
72	C₂²×C₄×C₄₇	—	2	3	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	6	no	—
77	C₂²×C₄×C₄₇	—	2	3	no	—
78	C₂²×C₄×C₄₇	—	2	3	no	—
79	C₂²×C₄×C₄₇	—	2	3	no	—
80	C₂²×C₄×C₄₇	—	2	3	no	—
81	C₂²×C₄×C₄₇	—	2	3	no	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	—	2	3	no	—
84	C₂²×C₄×C₄₇	—	2	3	no	—
85	C₂²×C₄×C₄₇	—	2	6	no	—
86	C₂²×C₄×C₄₇	—	2	6	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
91	C₂²×C₄×C₄₇	#41	1	6	yes	—
92	C₂²×C₄×C₄₇	#41	1	6	yes	—
93	C₄⋊C₄×C₄₇	—	2	24	no	—
94	C₄⋊C₄×C₄₇	—	4	12	no	—
95	C₂²⋊C₄×C₄₇	—	2	24	no	—
96	C₂²⋊C₄×C₄₇	—	4	4	no	—
97	C₂²⋊C₄×C₄₇	—	4	12	no	—
98	C₄²×C₄₇	—	2	24	no	—
99	C₂³×C₄₇	—	2	6	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	6	no	—
102	C₂³×C₄₇	—	4	1	no	—
103	C₂³×C₄₇	—	4	3	no	—
104	C₂³×C₄₇	—	4	3	no	—
105	C₂³×C₄₇	—	4	3	no	—
106	C₂³×C₄₇	—	4	3	no	—
107	C₂³×C₄₇	—	4	3	no	—
108	C₂³×C₄₇	—	4	3	no	—
109	C₂³×C₄₇	—	4	3	no	—
110	C₂³×C₄₇	—	4	6	no	—
111	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂³×C₄₇	C₂×#41	1	3	yes	—
114	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂³×C₄₇	C₂×#41	1	3	yes	—
116	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂³×C₄₇	2₋¹⁺⁴	1	3	yes	—
118	Q₈×C₄₇	—	2	32	no	—
119	C₂×C₄×C₄₇	—	2	6	no	—
120	C₂×C₄×C₄₇	—	2	12	no	—
121	C₂×C₄×C₄₇	—	2	6	no	—
122	C₂×C₄×C₄₇	—	2	12	no	—
123	C₂×C₄×C₄₇	—	2	6	no	—
124	C₂×C₄×C₄₇	—	2	6	no	—
125	C₂×C₄×C₄₇	—	2	6	no	—
126	C₂×C₄×C₄₇	—	2	6	no	—
127	C₂×C₄×C₄₇	—	2	6	no	—
128	C₂×C₄×C₄₇	—	2	6	no	—
129	C₂×C₄×C₄₇	—	2	6	no	—
130	C₂×C₄×C₄₇	—	2	6	no	—
131	C₂×C₄×C₄₇	—	4	3	no	—
132	C₂×C₄×C₄₇	—	4	6	no	—
133	C₂×C₄×C₄₇	—	4	6	no	—
134	C₂×C₄×C₄₇	—	4	3	no	—
135	C₂×C₄×C₄₇	—	4	3	no	—
136	C₂×C₄×C₄₇	—	4	3	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	3	no	—
139	C₂×C₄×C₄₇	—	4	3	no	—
140	C₂×C₄×C₄₇	—	4	3	no	—
141	C₂×C₄×C₄₇	—	4	3	no	—
142	C₂×C₄×C₄₇	—	4	3	no	—
143	C₂×C₄×C₄₇	—	4	6	no	—
144	C₂×C₄×C₄₇	—	4	6	no	—
145	C₂×C₄×C₄₇	#120	1	4	yes	—
146	C₂×C₄×C₄₇	#121	1	12	yes	—
147	#15176	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₇	—	4	6	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	6	no	—
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	—	8	3	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	—	8	3	no	—
155	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#121	1	3	yes	—
157	C₂²×C₄₇	#525	1	3	yes	—
158	C₂²×C₄₇	#525	1	3	yes	—
159	C₂²×C₄₇	#532	1	3	yes	—
160	C₂²×C₄₇	#536	1	3	yes	—
161	C₂²×C₄₇	#536	1	3	yes	—
162	C₂²×C₄₇	#548	1	1	yes	characteristic
163	C₂²×C₄₇	#548	1	3	yes	—
164	C₂²×C₄₇	#552	1	3	yes	—
165	C₂²×C₄₇	#552	1	3	yes	—
166	C₂²×C₄₇	#556	1	6	yes	—
167	C₄×C₄₇	—	2	12	no	—
168	C₄×C₄₇	—	2	4	no	—
169	C₄×C₄₇	—	4	6	no	—
170	C₄×C₄₇	—	4	6	no	—
171	C₄×C₄₇	—	8	1	no	—
172	C₄×C₄₇	—	8	3	no	—
173	C₂²×C₄×Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
174	C₂×#395	C₂×C₄₇	1	1	yes	characteristic, complemented
175	C₂×#393	C₂×C₄₇	1	1	yes	characteristic
176	C₂×#388	C₂×C₄₇	1	3	yes	—
177	#1441	C₂×C₄₇	1	3	yes	complemented
178	#1439	C₂×C₄₇	1	3	yes	—
179	#1434	C₂×C₄₇	1	3	yes	complemented
180	C₂×C₄₇	—	8	2	no	—
181	C₂×C₄₇	#2388	1	1	yes	characteristic
182	C₂×C₄₇	#2536	1	3	yes	—
183	C₂×C₄₇	#2545	1	3	yes	—
184	C₂×C₄₇	#2599	1	3	yes	—
185	C₂×C₄₇	#2604	1	3	yes	—
186	C₂×C₄₇	#2614	1	1	yes	characteristic
187	C₂×C₄₇	#2847	1	1	yes	characteristic
188	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
189	C₂×C₄×Q₈	—	2	12	no	—
190	C₂²×C₄⋊C₄	C₂²×C₄₇	1	3	yes	—
191	C₂²×C₄⋊C₄	C₂²×C₄₇	1	3	yes	—
192	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	3	yes	—
193	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	3	yes	—
194	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
195	C₂²×C₄²	C₂²×C₄₇	1	3	yes	—
196	#395	—	2	4	no	—
197	#393	—	2	4	no	—
198	#388	—	2	12	no	—
199	C₂×#95	C₂²×C₄₇	1	3	yes	—
200	C₂×#95	C₂²×C₄₇	1	3	yes	—
201	C₂×#95	C₂²×C₄₇	1	3	yes	—
202	C₂×#95	C₂²×C₄₇	1	6	yes	—
203	C₂×#95	C₂²×C₄₇	1	3	yes	—
204	C₄₇	#15176	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
206	C₂²×Q₈	—	2	6	no	—
207	C₂²×Q₈	—	2	6	no	—
208	C₂²×Q₈	D₄×C₄₇	1	4	yes	—
209	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
210	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
211	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
212	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
213	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
214	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
215	C₄×Q₈	—	2	32	no	—
216	C₂×C₄⋊C₄	—	2	12	no	—
217	C₂×C₄⋊C₄	—	2	6	no	—
218	C₂×C₄⋊C₄	—	2	6	no	—
219	C₂×C₄⋊C₄	—	2	6	no	—
220	C₂×C₄⋊C₄	—	2	6	no	—
221	C₂×C₂²⋊C₄	—	2	6	no	—
222	C₂×C₂²⋊C₄	—	2	6	no	—
223	C₂×C₂²⋊C₄	—	2	6	no	—
224	C₂×C₂²⋊C₄	—	2	6	no	—
225	C₂×C₂²⋊C₄	—	2	6	no	—
226	C₂×C₂²⋊C₄	—	2	6	no	—
227	C₂×C₂²⋊C₄	—	2	6	no	—
228	C₂×C₄²	—	2	12	no	—
229	C₂×C₄²	D₄×C₄₇	1	12	yes	—
230	#95	—	2	6	no	—
231	#95	—	2	6	no	—
232	#95	—	2	6	no	—
233	#95	—	2	6	no	—
234	#95	—	2	12	no	—
235	C₂⁴	—	2	3	no	—
236	C₂⁴	—	2	1	no	—
237	C₂⁴	—	2	3	no	—
238	C₂⁴	—	2	3	no	—
239	C₂⁴	—	2	1	no	—
240	C₂⁴	—	2	3	no	—
241	C₂⁴	#41×C₄₇	1	2	yes	—
242	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	2	24	no	—
245	C₂×Q₈	—	2	24	no	—
246	C₂²×C₄	—	2	6	no	—
247	C₂²×C₄	—	2	6	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	3	no	—
250	C₂²×C₄	—	2	3	no	—
251	C₂²×C₄	—	2	3	no	—
252	C₂²×C₄	—	2	6	no	—
253	C₂²×C₄	—	2	6	no	—
254	C₂²×C₄	—	2	3	no	—
255	C₂²×C₄	—	2	3	no	—
256	C₂²×C₄	—	2	3	no	—
257	C₂²×C₄	—	2	3	no	—
258	C₂²×C₄	—	2	3	no	—
259	C₂²×C₄	—	2	3	no	—
260	C₂²×C₄	—	2	3	no	—
261	C₂²×C₄	—	2	3	no	—
262	C₂²×C₄	—	2	3	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	6	no	—
265	C₂²×C₄	—	2	3	no	—
266	C₂²×C₄	—	2	3	no	—
267	C₂²×C₄	—	2	6	no	—
268	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
269	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
270	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
271	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
272	C₂²×C₄	#41×C₄₇	1	6	yes	—
273	C₂²×C₄	#41×C₄₇	1	6	yes	—
274	C₄⋊C₄	—	2	24	no	—
275	C₄⋊C₄	—	4	12	no	—
276	C₂²⋊C₄	—	2	24	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	12	no	—
279	C₄²	—	2	24	no	—
280	C₂³	—	2	6	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	6	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	3	no	—
285	C₂³	—	4	3	no	—
286	C₂³	—	4	3	no	—
287	C₂³	—	4	3	no	—
288	C₂³	—	4	3	no	—
289	C₂³	—	4	3	no	—
290	C₂³	—	4	3	no	—
291	C₂³	—	4	6	no	—
292	C₂³	C₂²×D₄×C₄₇	1	3	yes	—
293	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
294	C₂³	C₂×#41×C₄₇	1	3	yes	—
295	C₂³	C₂×#41×C₄₇	1	1	yes	characteristic
296	C₂³	C₂×#41×C₄₇	1	3	yes	—
297	C₂³	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
298	C₂³	2₋¹⁺⁴×C₄₇	1	3	yes	—
299	Q₈	—	2	32	no	—
300	C₂×C₄	—	2	6	no	—
301	C₂×C₄	—	2	12	no	—
302	C₂×C₄	—	2	6	no	—
303	C₂×C₄	—	2	12	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	2	6	no	—
306	C₂×C₄	—	2	6	no	—
307	C₂×C₄	—	2	6	no	—
308	C₂×C₄	—	2	6	no	—
309	C₂×C₄	—	2	6	no	—
310	C₂×C₄	—	2	6	no	—
311	C₂×C₄	—	2	6	no	—
312	C₂×C₄	—	4	3	no	—
313	C₂×C₄	—	4	6	no	—
314	C₂×C₄	—	4	6	no	—
315	C₂×C₄	—	4	3	no	—
316	C₂×C₄	—	4	3	no	—
317	C₂×C₄	—	4	3	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	3	no	—
320	C₂×C₄	—	4	3	no	—
321	C₂×C₄	—	4	3	no	—
322	C₂×C₄	—	4	3	no	—
323	C₂×C₄	—	4	3	no	—
324	C₂×C₄	—	4	6	no	—
325	C₂×C₄	—	4	6	no	—
326	C₂×C₄	#120×C₄₇	1	4	yes	—
327	C₂×C₄	#121×C₄₇	1	12	yes	—
328	C₂²	—	4	6	no	—
329	C₂²	—	4	6	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	8	3	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	3	no	—
334	C₂²	—	8	1	no	—
335	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
336	C₂²	C₂×#121×C₄₇	1	3	yes	—
337	C₂²	C₄₇×#525	1	3	yes	—
338	C₂²	C₄₇×#525	1	3	yes	—
339	C₂²	C₄₇×#532	1	3	yes	—
340	C₂²	C₄₇×#536	1	3	yes	—
341	C₂²	C₄₇×#536	1	3	yes	—
342	C₂²	C₄₇×#548	1	1	yes	characteristic
343	C₂²	C₄₇×#548	1	3	yes	—
344	C₂²	C₄₇×#552	1	3	yes	—
345	C₂²	C₄₇×#552	1	3	yes	—
346	C₂²	C₄₇×#556	1	6	yes	—
347	C₄	—	2	12	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	6	no	—
350	C₄	—	4	6	no	—
351	C₄	—	8	3	no	—
352	C₄	—	8	1	no	—
353	C₂	—	8	2	no	—
354	C₂	C₄₇×#2388	1	1	yes	characteristic
355	C₂	C₄₇×#2536	1	3	yes	—
356	C₂	C₄₇×#2545	1	3	yes	—
357	C₂	C₄₇×#2599	1	3	yes	—
358	C₂	C₄₇×#2604	1	3	yes	—
359	C₂	C₄₇×#2614	1	1	yes	characteristic
360	C₂	C₄₇×#2847	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
1. C₂²×C₄×Q₈×C₄₇	C₂	semidirect
2. C₂×C₄₇×#395	C₂	semidirect
5. C₄₇×#1441	C₂	semidirect
7. C₄₇×#1434	C₂	semidirect
147. #15176	C₄₇	direct (N × H)
173. C₂²×C₄×Q₈	C₂×C₄₇	semidirect
174. C₂×#395	C₂×C₄₇	semidirect
177. #1441	C₂×C₄₇	semidirect
179. #1434	C₂×C₄₇	semidirect
204. C₄₇	#15176	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×#395	2
C₂×C₄₇×#393	2
C₂×C₄₇×#388	2
C₄₇×#1441	2
C₄₇×#1439	2
C₄₇×#1434	2
#15176	47

Subgroups — C47×#15176 #26054874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#15176 #26054874