Subgroups — C₄₇×#29551 #26069249

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2320	C₂	1	2	yes	maximal
2	C₄₇×#2086	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2068	C₂	1	2	yes	maximal
4	C₄₇×#2056	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1928	C₂	1	2	yes	maximal
6	C₄₇×#1918	C₂	1	2	yes	maximal
7	C₄₇×#1878	C₂	1	2	yes	maximal
8	C₄₇×#1863	C₂	1	2	yes	maximal
9	C₄₇×#1832	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×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×#103×C₄₇	C₂²	1	2	yes	—
17	C₂×#103×C₄₇	C₂²	1	2	yes	—
18	C₂×#103×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#391	C₂²	1	2	yes	—
20	C₄₇×#391	C₂²	1	2	yes	—
21	C₄₇×#389	C₂²	1	2	yes	—
22	C₄₇×#389	C₂²	1	2	yes	—
23	C₄₇×#387	—	2	2	no	—
24	C₄₇×#387	—	2	2	no	—
25	C₄₇×#383	—	2	2	no	—
26	C₄₇×#382	—	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×Q₈×C₄₇	C₂³	1	2	yes	—
36	C₂×Q₁₆×C₄₇	—	2	4	no	—
37	C₂×Q₁₆×C₄₇	—	4	1	no	—
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₄×Q₈×C₄₇	—	2	4	no	—
42	C₄×Q₈×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	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×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#107×C₄₇	—	2	4	no	—
54	#107×C₄₇	—	4	1	no	—
55	C₄⋊C₈×C₄₇	—	4	1	no	—
56	C₄⋊C₈×C₄₇	—	4	1	no	—
57	#103×C₄₇	—	2	4	no	—
58	#103×C₄₇	—	2	4	no	—
59	#103×C₄₇	—	4	2	no	—
60	#95×C₄₇	—	2	2	no	—
61	#95×C₄₇	—	2	2	no	—
62	#95×C₄₇	—	2	2	no	—
63	#95×C₄₇	—	2	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	2	yes	—
68	#95×C₄₇	D₄	1	1	yes	characteristic
69	#95×C₄₇	D₄	1	1	yes	characteristic
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	2	2	no	—
74	C₂×Q₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	#41	1	4	yes	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₄₇	—	4	4	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₄⋊C₄×C₄₇	—	2	8	no	—
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×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	—	8	2	no	—
100	C₄⋊C₄×C₄₇	—	8	2	no	—
101	C₄⋊C₄×C₄₇	#41	1	4	yes	—
102	C₄²×C₄₇	—	2	8	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	2	1	no	—
106	C₄²×C₄₇	—	2	1	no	—
107	C₄²×C₄₇	—	8	2	no	—
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₄₇	—	2	4	no	—
110	Q₈×C₄₇	—	2	4	no	—
111	Q₈×C₄₇	—	4	2	no	—
112	Q₈×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	2	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	1	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₄₇	—	4	1	no	—
141	C₈×C₄₇	—	4	1	no	—
142	C₈×C₄₇	—	4	2	no	—
143	#29551	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₇	#525	1	2	yes	—
145	C₂²×C₄₇	#537	1	1	yes	characteristic
146	C₂²×C₄₇	#576	1	1	yes	characteristic
147	C₂²×C₄₇	#578	1	2	yes	—
148	C₂²×C₄₇	#578	1	1	yes	characteristic
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	1	no	—
154	C₄×C₄₇	—	4	1	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	#551	1	1	yes	characteristic
158	C₄×C₄₇	#552	1	2	yes	—
159	C₄×C₄₇	#555	1	1	yes	characteristic
160	#2320	C₂×C₄₇	1	2	yes	—
161	#2086	C₂×C₄₇	1	1	yes	characteristic
162	#2068	C₂×C₄₇	1	2	yes	—
163	#2056	C₂×C₄₇	1	1	yes	characteristic
164	#1928	C₂×C₄₇	1	2	yes	—
165	#1918	C₂×C₄₇	1	2	yes	—
166	#1878	C₂×C₄₇	1	2	yes	—
167	#1863	C₂×C₄₇	1	2	yes	—
168	#1832	C₂×C₄₇	1	1	yes	characteristic
169	C₂×C₄₇	#2609	1	1	yes	characteristic
170	C₂×C₄₇	#3124	1	2	yes	—
171	C₂×C₄₇	#3127	1	2	yes	—
172	C₂×C₄₇	#3261	1	1	yes	characteristic
173	C₂×C₄₇	#3262	1	1	yes	characteristic
174	C₂²×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
175	C₂×C₄×Q₈	C₂²×C₄₇	1	2	yes	—
176	C₂×#107	C₂²×C₄₇	1	2	yes	—
177	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
180	C₂×#103	C₂²×C₄₇	1	2	yes	—
181	C₂×#103	C₂²×C₄₇	1	2	yes	—
182	C₂×#103	C₂²×C₄₇	1	2	yes	—
183	#391	C₂²×C₄₇	1	2	yes	—
184	#391	C₂²×C₄₇	1	2	yes	—
185	#389	C₂²×C₄₇	1	2	yes	—
186	#389	C₂²×C₄₇	1	2	yes	—
187	#387	—	2	2	no	—
188	#387	—	2	2	no	—
189	#383	—	2	2	no	—
190	#382	—	2	2	no	—
191	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
192	#376	C₂²×C₄₇	1	1	yes	characteristic
193	#340	C₂²×C₄₇	1	1	yes	characteristic
194	#340	C₂²×C₄₇	1	2	yes	—
195	#340	C₂²×C₄₇	1	1	yes	characteristic
196	#340	C₂²×C₄₇	1	2	yes	—
197	#336	C₂²×C₄₇	1	2	yes	—
198	#336	C₂²×C₄₇	1	2	yes	—
199	C₄₇	#29551	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×Q₁₆	—	4	1	no	—
203	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×Q₈	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
217	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
218	#107	—	2	4	no	—
219	#107	—	4	1	no	—
220	C₄⋊C₈	—	4	1	no	—
221	C₄⋊C₈	—	4	1	no	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	#103	—	4	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	D₄×C₄₇	1	2	yes	—
233	#95	D₄×C₄₇	1	1	yes	characteristic
234	#95	D₄×C₄₇	1	1	yes	characteristic
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	#41×C₄₇	1	4	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
250	Q₁₆	—	4	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	8	2	no	—
265	C₄⋊C₄	—	8	2	no	—
266	C₄⋊C₄	#41×C₄₇	1	4	yes	—
267	C₄²	—	2	8	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	8	2	no	—
273	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
274	Q₈	—	2	4	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
302	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
303	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
304	C₂×C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	2	no	—
308	C₂²	C₄₇×#525	1	2	yes	—
309	C₂²	C₄₇×#537	1	1	yes	characteristic
310	C₂²	C₄₇×#576	1	1	yes	characteristic
311	C₂²	C₄₇×#578	1	2	yes	—
312	C₂²	C₄₇×#578	1	1	yes	characteristic
313	C₄	—	4	4	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₄₇×#551	1	1	yes	characteristic
322	C₄	C₄₇×#552	1	2	yes	—
323	C₄	C₄₇×#555	1	1	yes	characteristic
324	C₂	C₄₇×#2609	1	1	yes	characteristic
325	C₂	C₄₇×#3124	1	2	yes	—
326	C₂	C₄₇×#3127	1	2	yes	—
327	C₂	C₄₇×#3261	1	1	yes	characteristic
328	C₂	C₄₇×#3262	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
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
143. #29551	C₄₇	direct (N × H)
199. C₄₇	#29551	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×#2320	2
C₄₇×#2086	2
C₄₇×#2068	2
C₄₇×#2056	2
C₄₇×#1928	2
C₄₇×#1918	2
C₄₇×#1878	2
C₄₇×#1863	2
C₄₇×#1832	2
#29551	47

Subgroups — C47×#29551 #26069249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29551 #26069249