Subgroups — C₄₇×#33970 #26073668

Order 12032.

528 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3396	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2250	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2211	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2210	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2206	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#559	—	2	2	no	—
11	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
13	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
14	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#495	C₂²	1	2	yes	—
18	C₄₇×#495	C₂²	1	2	yes	—
19	C₄₇×#493	C₂²	1	2	yes	—
20	C₄₇×#463	—	2	2	no	—
21	C₄₇×#463	—	2	2	no	—
22	C₄₇×#460	—	2	2	no	—
23	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₂²	1	2	yes	—
26	C₄×D₈×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#366	C₂²	1	1	yes	characteristic
28	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄₇×#358	C₂²	1	2	yes	—
31	C₄₇×#357	C₂²	1	2	yes	—
32	C₄₇×#357	C₂²	1	2	yes	—
33	C₄₇×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₄₇	—	2	2	no	—
37	C₂×SD₁₆×C₄₇	—	2	2	no	—
38	C₂×D₈×C₄₇	—	2	2	no	—
39	C₂×D₈×C₄₇	C₂³	1	2	yes	—
40	C₂×D₈×C₄₇	C₂³	1	2	yes	—
41	#127×C₄₇	—	2	2	no	—
42	#124×C₄₇	—	2	2	no	—
43	#124×C₄₇	—	2	2	no	—
44	#124×C₄₇	—	4	2	no	—
45	#123×C₄₇	—	4	2	no	—
46	#123×C₄₇	—	4	2	no	—
47	C₄×Q₈×C₄₇	—	2	2	no	—
48	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
49	C₄×D₄×C₄₇	—	2	2	no	—
50	C₄×D₄×C₄₇	—	2	2	no	—
51	C₄×D₄×C₄₇	—	4	2	no	—
52	Q₃₂×C₄₇	—	4	1	no	—
53	SD₃₂×C₄₇	—	2	4	no	—
54	SD₃₂×C₄₇	—	4	2	no	—
55	D₁₆×C₄₇	—	2	2	no	—
56	D₁₆×C₄₇	—	4	2	no	—
57	D₁₆×C₄₇	—	4	1	no	—
58	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
59	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
61	#107×C₄₇	C₂³	1	2	yes	—
62	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
63	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
64	#103×C₄₇	—	2	2	no	—
65	#102×C₄₇	—	2	2	no	—
66	#102×C₄₇	—	2	2	no	—
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
71	C₂×D₄×C₄₇	—	2	2	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	4	2	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	Q₁₆×C₄₇	—	2	2	no	—
78	Q₁₆×C₄₇	—	2	2	no	—
79	SD₁₆×C₄₇	—	8	2	no	—
80	SD₁₆×C₄₇	—	8	2	no	—
81	D₈×C₄₇	—	2	2	no	—
82	D₈×C₄₇	—	2	2	no	—
83	D₈×C₄₇	—	2	2	no	—
84	D₈×C₄₇	—	2	2	no	—
85	D₈×C₄₇	—	8	2	no	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
92	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
93	C₂²⋊C₄×C₄₇	—	4	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	2	no	—
95	C₂²⋊C₄×C₄₇	—	4	2	no	—
96	C₂²⋊C₄×C₄₇	—	4	2	no	—
97	C₂²⋊C₄×C₄₇	—	4	2	no	—
98	C₂²⋊C₄×C₄₇	—	4	2	no	—
99	C₄²×C₄₇	—	4	2	no	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	2	yes	—
102	C₁₆×C₄₇	—	2	2	no	—
103	C₁₆×C₄₇	—	2	1	no	—
104	C₁₆×C₄₇	#41	1	2	yes	—
105	C₂³×C₄₇	—	4	2	no	—
106	C₂³×C₄₇	—	4	2	no	—
107	Q₈×C₄₇	—	4	1	no	—
108	Q₈×C₄₇	—	4	2	no	—
109	Q₈×C₄₇	—	4	2	no	—
110	D₄×C₄₇	—	4	2	no	—
111	D₄×C₄₇	—	4	2	no	—
112	D₄×C₄₇	—	4	2	no	—
113	D₄×C₄₇	—	4	2	no	—
114	D₄×C₄₇	—	8	4	no	—
115	D₄×C₄₇	—	16	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	16	2	no	—
119	C₂×C₄×C₄₇	—	16	2	no	—
120	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	2	1	no	—
127	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
129	#33970	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₇	—	8	2	no	—
131	C₂²×C₄₇	—	8	2	no	—
132	C₂²×C₄₇	—	8	2	no	—
133	C₂²×C₄₇	—	8	2	no	—
134	C₂²×C₄₇	—	8	2	no	—
135	C₂²×C₄₇	—	8	2	no	—
136	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	—	8	2	no	—
143	C₄×C₄₇	#550	1	1	yes	characteristic
144	C₄×C₄₇	#576	1	1	yes	characteristic
145	#3396	C₂×C₄₇	1	2	yes	complemented
146	#2250	C₂×C₄₇	1	2	yes	complemented
147	#2249	C₂×C₄₇	1	1	yes	characteristic, complemented
148	#2211	C₂×C₄₇	1	2	yes	complemented
149	#2210	C₂×C₄₇	1	2	yes	complemented
150	#2206	C₂×C₄₇	1	2	yes	complemented
151	#2183	C₂×C₄₇	1	1	yes	characteristic, complemented
152	#2175	C₂×C₄₇	1	2	yes	complemented
153	C₄×D₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
154	C₂×C₄₇	—	16	2	no	—
155	C₂×C₄₇	—	16	2	no	—
156	C₂×C₄₇	#3391	1	1	yes	characteristic
157	C₂×C₄₇	#3414	1	1	yes	characteristic
158	C₂×C₄₇	#3415	1	1	yes	characteristic
159	#559	—	2	2	no	—
160	C₂×Q₃₂	C₂²×C₄₇	1	1	yes	characteristic
161	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
162	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
163	C₂×D₁₆	C₂²×C₄₇	1	1	yes	characteristic
164	C₂×D₁₆	C₂²×C₄₇	1	1	yes	characteristic
165	C₂×D₁₆	C₂²×C₄₇	1	2	yes	—
166	#495	C₂²×C₄₇	1	2	yes	—
167	#495	C₂²×C₄₇	1	2	yes	—
168	#493	C₂²×C₄₇	1	2	yes	—
169	#463	—	2	2	no	—
170	#463	—	2	2	no	—
171	#460	—	2	2	no	—
172	C₈×Q₈	C₂²×C₄₇	1	1	yes	characteristic
173	C₄×Q₁₆	C₂²×C₄₇	1	2	yes	—
174	C₄×D₈	C₂²×C₄₇	1	2	yes	—
175	C₄×D₈	C₂²×C₄₇	1	2	yes	—
176	#366	C₂²×C₄₇	1	1	yes	characteristic
177	C₄⋊C₁₆	C₂²×C₄₇	1	2	yes	—
178	C₄⋊C₁₆	C₂²×C₄₇	1	1	yes	characteristic
179	#358	C₂²×C₄₇	1	2	yes	—
180	#357	C₂²×C₄₇	1	2	yes	—
181	#357	C₂²×C₄₇	1	2	yes	—
182	#346	C₂²×C₄₇	1	2	yes	—
183	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
184	C₄₇	#33970	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×SD₁₆	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	C₂³×C₄₇	1	2	yes	—
190	C₂×D₈	C₂³×C₄₇	1	2	yes	—
191	#127	—	2	2	no	—
192	#124	—	2	2	no	—
193	#124	—	2	2	no	—
194	#124	—	4	2	no	—
195	#123	—	4	2	no	—
196	#123	—	4	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	4	2	no	—
202	Q₃₂	—	4	1	no	—
203	SD₃₂	—	2	4	no	—
204	SD₃₂	—	4	2	no	—
205	D₁₆	—	2	2	no	—
206	D₁₆	—	4	2	no	—
207	D₁₆	—	4	1	no	—
208	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
209	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
210	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
211	#107	C₂³×C₄₇	1	2	yes	—
212	C₄⋊C₈	D₄×C₄₇	1	2	yes	—
213	C₄⋊C₈	D₄×C₄₇	1	1	yes	characteristic
214	#103	—	2	2	no	—
215	#102	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₄₇	1	2	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	D₈×C₄₇	1	1	yes	characteristic
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	8	2	no	—
230	SD₁₆	—	8	2	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	D₈	—	2	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	8	2	no	—
236	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
238	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	D₈×C₄₇	1	2	yes	—
242	C₄⋊C₄	D₈×C₄₇	1	1	yes	characteristic
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₄₇	1	2	yes	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	2	1	no	—
254	C₁₆	#41×C₄₇	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	16	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	16	2	no	—
269	C₂×C₄	—	16	2	no	—
270	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	1	no	—
277	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
278	C₈	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₄₇×#550	1	1	yes	characteristic
293	C₄	C₄₇×#576	1	1	yes	characteristic
294	C₂	—	16	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₄₇×#3391	1	1	yes	characteristic
297	C₂	C₄₇×#3414	1	1	yes	characteristic
298	C₂	C₄₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄×Q₈×C₄₇ ◁ C₈×Q₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3396	C₂	semidirect
2. C₄₇×#2250	C₂	semidirect
3. C₄₇×#2249	C₂	semidirect
4. C₄₇×#2211	C₂	semidirect
5. C₄₇×#2210	C₂	semidirect
6. C₄₇×#2206	C₂	semidirect
7. C₄₇×#2183	C₂	semidirect
8. C₄₇×#2175	C₂	semidirect
9. C₄×D₁₆×C₄₇	C₂	semidirect
129. #33970	C₄₇	direct (N × H)
145. #3396	C₂×C₄₇	semidirect
146. #2250	C₂×C₄₇	semidirect
147. #2249	C₂×C₄₇	semidirect
148. #2211	C₂×C₄₇	semidirect
149. #2210	C₂×C₄₇	semidirect
150. #2206	C₂×C₄₇	semidirect
151. #2183	C₂×C₄₇	semidirect
152. #2175	C₂×C₄₇	semidirect
153. C₄×D₁₆	C₂×C₄₇	semidirect
184. C₄₇	#33970	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×#3396	2
C₄₇×#2250	2
C₄₇×#2249	2
C₄₇×#2211	2
C₄₇×#2210	2
C₄₇×#2206	2
C₄₇×#2183	2
C₄₇×#2175	2
C₄×D₁₆×C₄₇	2
#33970	47

Subgroups — C47×#33970 #26073668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33970 #26073668