Subgroups — C₄₇×#11370 #26051068

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#339	C₂	1	4	yes	maximal
3	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
4	C₄×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
6	C₂×#117×C₄₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#384	C₄	1	1	yes	characteristic, complemented
10	C₄₇×#384	C₄	1	1	yes	characteristic, complemented
11	C₄₇×#382	C₄	1	2	yes	complemented
12	C₄₇×#380	—	2	2	no	—
13	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×#95×C₄₇	—	2	2	no	—
15	C₄₇×#339	—	2	8	no	—
16	C₄₇×#336	—	2	2	no	—
17	C₂³×C₄×C₄₇	—	2	2	no	—
18	C₂³×C₄×C₄₇	D₄	1	2	yes	—
19	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	—	2	2	no	—
21	C₂×M₁₆×C₄₇	—	2	4	no	—
22	C₂×M₁₆×C₄₇	—	2	4	no	—
23	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	#117×C₄₇	—	2	4	no	—
26	#117×C₄₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	—	2	4	no	—
33	C₂×C₄²×C₄₇	—	2	4	no	—
34	C₂×C₄²×C₄₇	—	2	4	no	—
35	C₂×C₄²×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	4	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	#95×C₄₇	—	2	8	no	—
42	#95×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	M₁₆×C₄₇	—	2	8	no	—
64	M₁₆×C₄₇	—	4	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	—	4	4	no	—
71	C₄⋊C₄×C₄₇	—	4	4	no	—
72	C₂²⋊C₄×C₄₇	—	4	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	2	8	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	—	4	2	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂³×C₄₇	#121	1	2	yes	—
84	C₂³×C₄₇	#123	1	2	yes	—
85	C₂³×C₄₇	#124	1	1	yes	characteristic
86	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	—	4	8	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	#104	1	4	yes	—
103	C₂×C₄×C₄₇	#117	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	#123	1	4	yes	—
109	C₂×C₄×C₄₇	#123	1	2	yes	—
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#125	1	4	yes	—
112	C₂×C₄×C₄₇	#126	1	4	yes	—
113	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
114	C₈×C₄₇	—	4	4	no	—
115	#11370	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	#381	1	1	yes	characteristic
122	C₂²×C₄₇	#390	1	1	yes	characteristic
123	C₂²×C₄₇	#391	1	1	yes	characteristic
124	C₂²×C₄₇	#397	1	2	yes	—
125	C₂²×C₄₇	#398	1	2	yes	—
126	C₂²×C₄₇	C₂×#104	1	2	yes	—
127	C₂²×C₄₇	#421	1	2	yes	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	4	8	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	—	8	2	no	—
132	C₄×C₄₇	—	8	2	no	—
133	C₄×C₄₇	#387	1	4	yes	—
134	C₄×C₄₇	#399	1	2	yes	—
135	C₄×C₄₇	#400	1	2	yes	—
136	#2285	C₂×C₄₇	1	1	yes	characteristic
137	C₂×#339	C₂×C₄₇	1	4	yes	—
138	#1728	C₂×C₄₇	1	1	yes	characteristic
139	C₄×#95	C₂×C₄₇	1	1	yes	characteristic
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	#1445	1	1	yes	characteristic
143	C₂×C₄₇	#1826	1	2	yes	—
144	C₂×C₄₇	#1962	1	1	yes	characteristic
145	C₂×C₄₇	#1963	1	1	yes	characteristic
146	C₂×C₄₇	#1986	1	2	yes	—
147	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
148	C₂×#117	C₂²×C₄₇	1	2	yes	—
149	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
150	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
151	#384	C₄×C₄₇	1	1	yes	characteristic, complemented
152	#384	C₄×C₄₇	1	1	yes	characteristic, complemented
153	#382	C₄×C₄₇	1	2	yes	complemented
154	#380	—	2	2	no	—
155	C₄×C₄⋊C₄	—	2	2	no	—
156	C₂×#95	—	2	2	no	—
157	#339	—	2	8	no	—
158	#336	—	2	2	no	—
159	C₄₇	#11370	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₄₇	1	2	yes	—
162	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	Q₈×C₄₇	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	#117	—	2	4	no	—
169	#117	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	2	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₄₇	1	4	yes	—
183	C₂×C₄²	D₄×C₄₇	1	2	yes	—
184	#95	—	2	8	no	—
185	#95	—	2	4	no	—
186	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
203	C₂²×C₄	#41×C₄₇	1	4	yes	—
204	C₂²×C₄	#41×C₄₇	1	2	yes	—
205	C₂²×C₄	#41×C₄₇	1	2	yes	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	4	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	8	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
226	C₂³	#121×C₄₇	1	2	yes	—
227	C₂³	#123×C₄₇	1	2	yes	—
228	C₂³	#124×C₄₇	1	1	yes	characteristic
229	C₂³	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	#104×C₄₇	1	4	yes	—
246	C₂×C₄	#117×C₄₇	1	2	yes	—
247	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
248	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
249	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
250	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
251	C₂×C₄	#123×C₄₇	1	4	yes	—
252	C₂×C₄	#123×C₄₇	1	2	yes	—
253	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
254	C₂×C₄	#125×C₄₇	1	4	yes	—
255	C₂×C₄	#126×C₄₇	1	4	yes	—
256	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
257	C₈	—	4	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₄₇×#381	1	1	yes	characteristic
264	C₂²	C₄₇×#390	1	1	yes	characteristic
265	C₂²	C₄₇×#391	1	1	yes	characteristic
266	C₂²	C₄₇×#397	1	2	yes	—
267	C₂²	C₄₇×#398	1	2	yes	—
268	C₂²	C₂×#104×C₄₇	1	2	yes	—
269	C₂²	C₄₇×#421	1	2	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	2	no	—
274	C₄	—	8	2	no	—
275	C₄	C₄₇×#387	1	4	yes	—
276	C₄	C₄₇×#399	1	2	yes	—
277	C₄	C₄₇×#400	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	C₄₇×#1445	1	1	yes	characteristic
281	C₂	C₄₇×#1826	1	2	yes	—
282	C₂	C₄₇×#1962	1	1	yes	characteristic
283	C₂	C₄₇×#1963	1	1	yes	characteristic
284	C₂	C₄₇×#1986	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄₇×#384	C₄	semidirect
10. C₄₇×#384	C₄	semidirect
11. C₄₇×#382	C₄	semidirect
115. #11370	C₄₇	direct (N × H)
151. #384	C₄×C₄₇	semidirect
152. #384	C₄×C₄₇	semidirect
153. #382	C₄×C₄₇	semidirect
159. C₄₇	#11370	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×#2285	2
C₂×C₄₇×#339	2
C₄₇×#1728	2
C₄×#95×C₄₇	2
#11370	47

Subgroups — C47×#11370 #26051068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11370 #26051068