Subgroups — C₄₇×#12981 #26052679

Order 12032.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#2929	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2112	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#525	C₄	1	1	yes	characteristic
9	C₄₇×#504	—	2	4	no	—
10	C₂×M₃₂×C₄₇	—	2	2	no	—
11	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
12	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#435	—	2	2	no	—
14	C₈×D₄×C₄₇	—	2	2	no	—
15	C₄₇×#424	C₄	1	1	yes	characteristic
16	C₄₇×#408	C₄	1	1	yes	characteristic
17	C₄₇×#407	C₄	1	1	yes	characteristic
18	C₄₇×#405	C₂²	1	1	yes	characteristic
19	C₄₇×#349	—	2	2	no	—
20	C₄₇×C₂²⋊C₁₆	—	2	4	no	—
21	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	#131×C₄₇	—	2	2	no	—
24	#131×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	—	2	1	no	—
30	C₂²×C₈×C₄₇	—	2	1	no	—
31	C₂²×C₈×C₄₇	—	2	2	no	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	#123×C₄₇	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₄₇	—	2	1	no	—
37	#121×C₄₇	—	2	1	no	—
38	#120×C₄₇	C₈	1	2	yes	—
39	C₄×D₄×C₄₇	—	2	2	no	—
40	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₄₇	—	2	4	no	—
42	M₃₂×C₄₇	—	2	2	no	—
43	M₃₂×C₄₇	—	2	2	no	—
44	M₃₂×C₄₇	—	2	2	no	—
45	M₃₂×C₄₇	—	2	2	no	—
46	C₂×C₁₆×C₄₇	—	2	4	no	—
47	C₂×C₁₆×C₄₇	—	2	2	no	—
48	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
49	C₄⋊C₈×C₄₇	—	2	2	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	—	2	2	no	—
52	#97×C₄₇	—	2	1	no	—
53	C₄×C₈×C₄₇	—	2	1	no	—
54	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
55	#41×C₄₇	—	2	2	no	—
56	#41×C₄₇	—	2	2	no	—
57	C₂×Q₈×C₄₇	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₄₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₄₇	—	2	2	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	4	1	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	#41	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	1	no	—
86	C₂²⋊C₄×C₄₇	—	2	2	no	—
87	C₂²⋊C₄×C₄₇	—	2	2	no	—
88	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
89	C₄²×C₄₇	—	2	1	no	—
90	C₁₆×C₄₇	—	2	4	no	—
91	C₁₆×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
97	Q₈×C₄₇	—	2	2	no	—
98	D₄×C₄₇	—	2	4	no	—
99	D₄×C₄₇	—	2	2	no	—
100	D₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#131	1	2	yes	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	—	4	1	no	—
116	C₈×C₄₇	—	4	1	no	—
117	C₈×C₄₇	#120	1	1	yes	characteristic
118	C₈×C₄₇	#121	1	2	yes	—
119	C₈×C₄₇	#123	1	1	yes	characteristic
120	#12981	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	4	1	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₄₇	C₈×D₄	1	2	yes	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	4	2	no	—
133	C₄×C₄₇	—	4	1	no	—
134	C₄×C₄₇	#386	1	1	yes	characteristic
135	C₄×C₄₇	#408	1	1	yes	characteristic
136	C₄×C₄₇	#435	1	2	yes	—
137	#3407	C₂×C₄₇	1	1	yes	characteristic, complemented
138	#2929	C₂×C₄₇	1	1	yes	characteristic
139	#2115	C₂×C₄₇	1	1	yes	characteristic, complemented
140	#2112	C₂×C₄₇	1	2	yes	complemented
141	#1378	C₂×C₄₇	1	2	yes	complemented
142	C₂×C₄₇	—	2	1	no	—
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	4	2	no	—
145	C₂×C₄₇	—	4	2	no	—
146	C₂×C₄₇	#1869	1	1	yes	characteristic
147	C₂×#131	C₂²×C₄₇	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
149	#525	C₄×C₄₇	1	1	yes	characteristic
150	#504	—	2	4	no	—
151	C₂×M₃₂	—	2	2	no	—
152	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
153	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
154	#435	—	2	2	no	—
155	C₈×D₄	—	2	2	no	—
156	#424	C₄×C₄₇	1	1	yes	characteristic
157	#408	C₄×C₄₇	1	1	yes	characteristic
158	#407	C₄×C₄₇	1	1	yes	characteristic
159	#405	C₂²×C₄₇	1	1	yes	characteristic
160	#349	—	2	2	no	—
161	C₂²⋊C₁₆	—	2	4	no	—
162	C₄₇	#12981	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#41	C₂×C₄×C₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	C₂×M₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
169	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₄₇	1	2	yes	—
176	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
177	#123	C₈×C₄₇	1	2	yes	—
178	C₂²⋊Q₈	—	2	1	no	—
179	#121	—	2	1	no	—
180	#120	C₈×C₄₇	1	2	yes	—
181	C₄×D₄	—	2	2	no	—
182	#117	C₂×C₄×C₄₇	1	1	yes	characteristic
183	M₃₂	—	2	4	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	M₃₂	—	2	2	no	—
188	C₂×C₁₆	—	2	4	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	D₄×C₄₇	1	4	yes	complemented
191	C₄⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₂⁴	C₂×C₈×C₄₇	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	M₁₆×C₄₇	1	1	yes	characteristic
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂×C₈×C₄₇	1	2	yes	—
202	C₂×D₄	M₁₆×C₄₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
208	C₂²×C₄	C₂×C₈×C₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₄₇	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	C₂×C₈×C₄₇	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂²⋊C₈×C₄₇	1	2	yes	—
238	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂²⋊C₈×C₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
254	C₂×C₄	C₂×M₁₆×C₄₇	1	1	yes	characteristic
255	C₂×C₄	#131×C₄₇	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	#120×C₄₇	1	1	yes	characteristic
260	C₈	#121×C₄₇	1	2	yes	—
261	C₈	#123×C₄₇	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₂²⋊C₈×C₄₇	1	1	yes	characteristic
271	C₂²	C₈×D₄×C₄₇	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	C₄₇×#386	1	1	yes	characteristic
276	C₄	C₄₇×#408	1	1	yes	characteristic
277	C₄	C₄₇×#435	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	C₄₇×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₄₇	#120
commutator	C₂³	C₂²×C₈×C₄₇
Frattini, cosocle	C₂²×C₈	C₂³×C₄₇
socle	C₂×C₄₇	#1869
prime radical	C₄₇×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₄₇ ◁ C₂²×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3407	C₂	semidirect
3. C₄₇×#2115	C₂	semidirect
4. C₄₇×#2112	C₂	semidirect
5. C₄₇×#1378	C₂	semidirect
48. C₂×C₁₆×C₄₇	D₄	semidirect
120. #12981	C₄₇	direct (N × H)
137. #3407	C₂×C₄₇	semidirect
139. #2115	C₂×C₄₇	semidirect
140. #2112	C₂×C₄₇	semidirect
141. #1378	C₂×C₄₇	semidirect
162. C₄₇	#12981	direct (N × H)
190. C₂×C₁₆	D₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3407	2
C₄₇×#2929	2
C₄₇×#2115	2
C₄₇×#2112	2
C₄₇×#1378	2
#12981	47

Subgroups — C47×#12981 #26052679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#12981 #26052679