Subgroups — C₄₇×#29546 #26069244

Order 12032.

746 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2319	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2087	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2067	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2054	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1925	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1921	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1879	C₂	1	2	yes	maximal
8	C₄₇×#1862	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
12	C₂×#107×C₄₇	C₂²	1	2	yes	—
13	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#102×C₄₇	C₂²	1	2	yes	—
17	C₂×#102×C₄₇	C₂²	1	2	yes	—
18	C₂×#102×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#390	C₂²	1	2	yes	—
20	C₄₇×#390	C₂²	1	2	yes	—
21	C₄₇×#389	C₂²	1	2	yes	—
22	C₄₇×#389	C₂²	1	2	yes	—
23	C₄₇×#388	—	2	2	no	—
24	C₄₇×#388	—	2	2	no	—
25	C₄₇×#381	—	2	2	no	—
26	C₄₇×#380	—	2	2	no	—
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₄₇×#340	C₂²	1	1	yes	characteristic
30	C₄₇×#340	C₂²	1	2	yes	—
31	C₄₇×#340	C₂²	1	1	yes	characteristic
32	C₄₇×#340	C₂²	1	2	yes	—
33	C₄₇×#336	C₂²	1	2	yes	—
34	C₄₇×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
36	C₂³×C₄×C₄₇	—	2	2	no	—
37	C₂×D₈×C₄₇	—	2	4	no	—
38	C₂×D₈×C₄₇	—	4	1	no	—
39	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₄₇	—	2	4	no	—
43	C₄×D₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#107×C₄₇	—	2	4	no	—
58	#107×C₄₇	—	4	1	no	—
59	C₄⋊C₈×C₄₇	—	4	1	no	—
60	C₄⋊C₈×C₄₇	—	4	1	no	—
61	#102×C₄₇	—	2	4	no	—
62	#102×C₄₇	—	2	4	no	—
63	#102×C₄₇	—	4	2	no	—
64	#95×C₄₇	—	2	2	no	—
65	#95×C₄₇	—	2	2	no	—
66	#95×C₄₇	—	2	2	no	—
67	#95×C₄₇	D₄	1	1	yes	characteristic
68	#95×C₄₇	D₄	1	2	yes	—
69	#95×C₄₇	D₄	1	1	yes	characteristic
70	C₂⁴×C₄₇	—	2	2	no	—
71	C₂×D₄×C₄₇	—	2	2	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	2	2	no	—
75	C₂×D₄×C₄₇	—	2	2	no	—
76	C₂×D₄×C₄₇	#41	1	4	yes	—
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	8	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	8	2	no	—
82	C₂²×C₄×C₄₇	—	8	2	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	D₈×C₄₇	—	4	4	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	8	2	no	—
102	C₄⋊C₄×C₄₇	—	8	2	no	—
103	C₄⋊C₄×C₄₇	#41	1	4	yes	—
104	C₂²⋊C₄×C₄₇	—	2	8	no	—
105	C₂²⋊C₄×C₄₇	—	8	2	no	—
106	C₂²⋊C₄×C₄₇	—	8	2	no	—
107	C₂²⋊C₄×C₄₇	—	8	2	no	—
108	C₂²⋊C₄×C₄₇	—	8	2	no	—
109	C₄²×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	—	2	2	no	—
111	C₄²×C₄₇	—	2	1	no	—
112	C₄²×C₄₇	—	2	1	no	—
113	C₄²×C₄₇	—	8	2	no	—
114	C₂³×C₄₇	—	2	4	no	—
115	C₂³×C₄₇	—	4	2	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	—	4	2	no	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂³×C₄₇	—	4	2	no	—
121	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	D₄×C₄₇	—	2	4	no	—
123	D₄×C₄₇	—	2	4	no	—
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	8	8	no	—
147	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
148	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
149	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₈×C₄₇	—	4	1	no	—
152	C₈×C₄₇	—	4	1	no	—
153	C₈×C₄₇	—	4	2	no	—
154	#29546	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₇	—	4	4	no	—
156	C₂²×C₄₇	—	4	4	no	—
157	C₂²×C₄₇	—	4	4	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	#525	1	2	yes	—
163	C₂²×C₄₇	#537	1	1	yes	characteristic
164	C₂²×C₄₇	#576	1	1	yes	characteristic
165	C₂²×C₄₇	#576	1	2	yes	—
166	C₂²×C₄₇	#578	1	1	yes	characteristic
167	C₄×C₄₇	—	4	2	no	—
168	C₄×C₄₇	—	4	4	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	—	4	1	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#551	1	1	yes	characteristic
175	C₄×C₄₇	#552	1	2	yes	—
176	C₄×C₄₇	#555	1	1	yes	characteristic
177	#2319	C₂×C₄₇	1	2	yes	complemented
178	#2087	C₂×C₄₇	1	1	yes	characteristic, complemented
179	#2067	C₂×C₄₇	1	2	yes	complemented
180	#2054	C₂×C₄₇	1	1	yes	characteristic
181	#1925	C₂×C₄₇	1	2	yes	complemented
182	#1921	C₂×C₄₇	1	2	yes	complemented
183	#1879	C₂×C₄₇	1	2	yes	—
184	#1862	C₂×C₄₇	1	2	yes	complemented
185	#1832	C₂×C₄₇	1	1	yes	characteristic, complemented
186	C₂×C₄₇	—	8	4	no	—
187	C₂×C₄₇	#2609	1	1	yes	characteristic
188	C₂×C₄₇	#3122	1	2	yes	—
189	C₂×C₄₇	#3124	1	2	yes	—
190	C₂×C₄₇	#3259	1	1	yes	characteristic
191	C₂×C₄₇	#3262	1	1	yes	characteristic
192	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
193	C₂×C₄×D₄	C₂²×C₄₇	1	2	yes	—
194	C₂×#107	C₂²×C₄₇	1	2	yes	—
195	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
198	C₂×#102	C₂²×C₄₇	1	2	yes	—
199	C₂×#102	C₂²×C₄₇	1	2	yes	—
200	C₂×#102	C₂²×C₄₇	1	2	yes	—
201	#390	C₂²×C₄₇	1	2	yes	—
202	#390	C₂²×C₄₇	1	2	yes	—
203	#389	C₂²×C₄₇	1	2	yes	—
204	#389	C₂²×C₄₇	1	2	yes	—
205	#388	—	2	2	no	—
206	#388	—	2	2	no	—
207	#381	—	2	2	no	—
208	#380	—	2	2	no	—
209	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
210	#376	C₂²×C₄₇	1	1	yes	characteristic
211	#340	C₂²×C₄₇	1	1	yes	characteristic
212	#340	C₂²×C₄₇	1	2	yes	—
213	#340	C₂²×C₄₇	1	1	yes	characteristic
214	#340	C₂²×C₄₇	1	2	yes	—
215	#336	C₂²×C₄₇	1	2	yes	—
216	#336	C₂²×C₄₇	1	2	yes	—
217	C₄₇	#29546	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
219	C₂³×C₄	—	2	2	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×D₈	—	4	1	no	—
222	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
225	C₄×D₄	—	2	4	no	—
226	C₄×D₄	—	2	4	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
239	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
240	#107	—	2	4	no	—
241	#107	—	4	1	no	—
242	C₄⋊C₈	—	4	1	no	—
243	C₄⋊C₈	—	4	1	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	4	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	D₄×C₄₇	1	1	yes	characteristic
251	#95	D₄×C₄₇	1	2	yes	—
252	#95	D₄×C₄₇	1	1	yes	characteristic
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	#41×C₄₇	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	8	2	no	—
265	C₂²×C₄	—	8	2	no	—
266	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
271	D₈	—	4	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	8	2	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	#41×C₄₇	1	4	yes	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	8	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
305	D₄	—	2	4	no	—
306	D₄	—	2	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	8	no	—
330	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
331	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
332	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
333	C₂×C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₄₇×#525	1	2	yes	—
345	C₂²	C₄₇×#537	1	1	yes	characteristic
346	C₂²	C₄₇×#576	1	1	yes	characteristic
347	C₂²	C₄₇×#576	1	2	yes	—
348	C₂²	C₄₇×#578	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₄₇×#551	1	1	yes	characteristic
357	C₄	C₄₇×#552	1	2	yes	—
358	C₄	C₄₇×#555	1	1	yes	characteristic
359	C₂	—	8	4	no	—
360	C₂	C₄₇×#2609	1	1	yes	characteristic
361	C₂	C₄₇×#3122	1	2	yes	—
362	C₂	C₄₇×#3124	1	2	yes	—
363	C₂	C₄₇×#3259	1	1	yes	characteristic
364	C₂	C₄₇×#3262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂²×D₈×C₄₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₄₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#2319	C₂	semidirect
2. C₄₇×#2087	C₂	semidirect
3. C₄₇×#2067	C₂	semidirect
5. C₄₇×#1925	C₂	semidirect
6. C₄₇×#1921	C₂	semidirect
8. C₄₇×#1862	C₂	semidirect
9. C₄₇×#1832	C₂	semidirect
154. #29546	C₄₇	direct (N × H)
177. #2319	C₂×C₄₇	semidirect
178. #2087	C₂×C₄₇	semidirect
179. #2067	C₂×C₄₇	semidirect
181. #1925	C₂×C₄₇	semidirect
182. #1921	C₂×C₄₇	semidirect
184. #1862	C₂×C₄₇	semidirect
185. #1832	C₂×C₄₇	semidirect
217. C₄₇	#29546	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2319	2
C₄₇×#2087	2
C₄₇×#2067	2
C₄₇×#2054	2
C₄₇×#1925	2
C₄₇×#1921	2
C₄₇×#1879	2
C₄₇×#1862	2
C₄₇×#1832	2
#29546	47

Subgroups — C47×#29546 #26069244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29546 #26069244