Subgroups — C₄₇×#33969 #26073667

Order 12032.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3393	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2251	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2249	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2211	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2210	C₂	1	2	yes	maximal
6	C₄₇×#2209	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×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#556	—	2	2	no	—
11	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
14	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
16	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#495	C₂²	1	2	yes	—
18	C₄₇×#495	C₂²	1	2	yes	—
19	C₄₇×#494	C₂²	1	2	yes	—
20	C₄₇×#464	—	2	2	no	—
21	C₄₇×#464	—	2	2	no	—
22	C₄₇×#461	—	2	2	no	—
23	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×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×#358	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	—	2	2	no	—
36	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₄₇	—	2	2	no	—
39	C₂×SD₁₆×C₄₇	—	2	2	no	—
40	C₂×D₈×C₄₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₄₇	—	2	2	no	—
42	#126×C₄₇	—	4	2	no	—
43	#126×C₄₇	—	4	2	no	—
44	#124×C₄₇	—	2	2	no	—
45	#124×C₄₇	—	2	2	no	—
46	C₂²⋊Q₈×C₄₇	—	4	2	no	—
47	C₄×Q₈×C₄₇	—	2	2	no	—
48	C₄×Q₈×C₄₇	—	2	2	no	—
49	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×D₄×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	4	2	no	—
52	Q₃₂×C₄₇	—	2	2	no	—
53	Q₃₂×C₄₇	—	4	1	no	—
54	Q₃₂×C₄₇	—	4	2	no	—
55	SD₃₂×C₄₇	—	2	4	no	—
56	SD₃₂×C₄₇	—	4	2	no	—
57	D₁₆×C₄₇	—	4	1	no	—
58	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
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	#103×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	—	2	2	no	—
71	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	Q₁₆×C₄₇	—	2	2	no	—
75	Q₁₆×C₄₇	—	2	2	no	—
76	Q₁₆×C₄₇	—	2	2	no	—
77	Q₁₆×C₄₇	—	2	2	no	—
78	Q₁₆×C₄₇	—	8	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	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	4	2	no	—
88	C₄⋊C₄×C₄₇	—	4	2	no	—
89	C₄⋊C₄×C₄₇	—	4	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	C₄²×C₄₇	—	4	2	no	—
99	C₄²×C₄₇	—	4	2	no	—
100	C₄²×C₄₇	C₂×D₄	1	2	yes	—
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₄₇	—	2	1	no	—
103	C₁₆×C₄₇	—	2	2	no	—
104	C₁₆×C₄₇	#41	1	2	yes	—
105	C₂³×C₄₇	—	4	2	no	—
106	Q₈×C₄₇	—	4	1	no	—
107	Q₈×C₄₇	—	4	2	no	—
108	Q₈×C₄₇	—	4	2	no	—
109	Q₈×C₄₇	—	4	2	no	—
110	Q₈×C₄₇	—	4	2	no	—
111	D₄×C₄₇	—	4	2	no	—
112	D₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	8	4	no	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₄₇	—	2	1	no	—
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
126	#33969	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₇	—	8	2	no	—
128	C₂²×C₄₇	—	8	2	no	—
129	C₂²×C₄₇	—	8	2	no	—
130	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	#550	1	1	yes	characteristic
139	C₄×C₄₇	#576	1	1	yes	characteristic
140	#3393	C₂×C₄₇	1	2	yes	complemented
141	#2251	C₂×C₄₇	1	2	yes	complemented
142	#2249	C₂×C₄₇	1	1	yes	characteristic
143	#2211	C₂×C₄₇	1	2	yes	complemented
144	#2210	C₂×C₄₇	1	2	yes	—
145	#2209	C₂×C₄₇	1	2	yes	complemented
146	#2183	C₂×C₄₇	1	1	yes	characteristic, complemented
147	#2175	C₂×C₄₇	1	2	yes	complemented
148	C₄×Q₃₂	C₂×C₄₇	1	1	yes	characteristic, complemented
149	C₂×C₄₇	—	16	2	no	—
150	C₂×C₄₇	#3391	1	1	yes	characteristic
151	C₂×C₄₇	#3414	1	1	yes	characteristic
152	C₂×C₄₇	#3417	1	1	yes	characteristic
153	#556	—	2	2	no	—
154	C₂×Q₃₂	C₂²×C₄₇	1	1	yes	characteristic
155	C₂×Q₃₂	C₂²×C₄₇	1	1	yes	characteristic
156	C₂×Q₃₂	C₂²×C₄₇	1	2	yes	—
157	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
158	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
159	C₂×D₁₆	C₂²×C₄₇	1	1	yes	characteristic
160	#495	C₂²×C₄₇	1	2	yes	—
161	#495	C₂²×C₄₇	1	2	yes	—
162	#494	C₂²×C₄₇	1	2	yes	—
163	#464	—	2	2	no	—
164	#464	—	2	2	no	—
165	#461	—	2	2	no	—
166	C₈×Q₈	C₂²×C₄₇	1	1	yes	characteristic
167	C₄×Q₁₆	C₂²×C₄₇	1	2	yes	—
168	C₄×Q₁₆	C₂²×C₄₇	1	2	yes	—
169	C₄×D₈	C₂²×C₄₇	1	2	yes	—
170	#366	C₂²×C₄₇	1	1	yes	characteristic
171	C₄⋊C₁₆	C₂²×C₄₇	1	2	yes	—
172	C₄⋊C₁₆	C₂²×C₄₇	1	1	yes	characteristic
173	#358	C₂²×C₄₇	1	2	yes	—
174	#358	C₂²×C₄₇	1	2	yes	—
175	#357	C₂²×C₄₇	1	2	yes	—
176	#346	C₂²×C₄₇	1	2	yes	—
177	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
178	C₄₇	#33969	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
181	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂³×C₄₇	1	2	yes	—
185	C₄⋊Q₈	—	2	2	no	—
186	#126	—	4	2	no	—
187	#126	—	4	2	no	—
188	#124	—	2	2	no	—
189	#124	—	2	2	no	—
190	C₂²⋊Q₈	—	4	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
194	C₄×D₄	—	2	2	no	—
195	#117	—	4	2	no	—
196	Q₃₂	—	2	2	no	—
197	Q₃₂	—	4	1	no	—
198	Q₃₂	—	4	2	no	—
199	SD₃₂	—	2	4	no	—
200	SD₃₂	—	4	2	no	—
201	D₁₆	—	4	1	no	—
202	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
204	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
205	#107	C₂³×C₄₇	1	2	yes	—
206	C₄⋊C₈	D₄×C₄₇	1	2	yes	—
207	C₄⋊C₈	D₄×C₄₇	1	1	yes	characteristic
208	#103	—	2	2	no	—
209	#103	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₄₇	1	2	yes	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	D₈×C₄₇	1	1	yes	characteristic
216	C₂×D₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	Q₁₆	—	8	2	no	—
223	SD₁₆	—	8	2	no	—
224	SD₁₆	—	8	2	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	D₈×C₄₇	1	2	yes	—
236	C₄⋊C₄	D₈×C₄₇	1	1	yes	characteristic
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	C₂×D₄×C₄₇	1	2	yes	—
245	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	#41×C₄₇	1	2	yes	—
249	C₂³	—	4	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
269	C₈	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₄₇×#550	1	1	yes	characteristic
282	C₄	C₄₇×#576	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₄₇×#3391	1	1	yes	characteristic
285	C₂	C₄₇×#3414	1	1	yes	characteristic
286	C₂	C₄₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₂×D₁₆×C₄₇	C₂²
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×#3393	C₂	semidirect
2. C₄₇×#2251	C₂	semidirect
4. C₄₇×#2211	C₂	semidirect
6. C₄₇×#2209	C₂	semidirect
7. C₄₇×#2183	C₂	semidirect
8. C₄₇×#2175	C₂	semidirect
9. C₄×Q₃₂×C₄₇	C₂	semidirect
126. #33969	C₄₇	direct (N × H)
140. #3393	C₂×C₄₇	semidirect
141. #2251	C₂×C₄₇	semidirect
143. #2211	C₂×C₄₇	semidirect
145. #2209	C₂×C₄₇	semidirect
146. #2183	C₂×C₄₇	semidirect
147. #2175	C₂×C₄₇	semidirect
148. C₄×Q₃₂	C₂×C₄₇	semidirect
178. C₄₇	#33969	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×#3393	2
C₄₇×#2251	2
C₄₇×#2249	2
C₄₇×#2211	2
C₄₇×#2210	2
C₄₇×#2209	2
C₄₇×#2183	2
C₄₇×#2175	2
C₄×Q₃₂×C₄₇	2
#33969	47

Subgroups — C47×#33969 #26073667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33969 #26073667