Subgroups — C₄₇×#2933 #20240543

Order 6016.

502 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	C₂×#131×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#525	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#436	C₂	1	2	yes	maximal, complemented
5	C₄₇×#435	C₂	1	2	yes	maximal, complemented
6	C₄₇×#435	C₂	1	2	yes	maximal, complemented
7	C₈×D₄×C₄₇	C₂	1	2	yes	maximal, complemented
8	C₄₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	#131×C₄₇	—	2	4	no	—
15	C₂×M₁₆×C₄₇	—	2	4	no	—
16	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₄₇	C₂²	1	2	yes	complemented
20	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂²	1	2	yes	complemented
24	#123×C₄₇	C₄	1	2	yes	—
25	C₂²⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
26	C₂²⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
27	#121×C₄₇	C₄	1	1	yes	characteristic
28	#121×C₄₇	C₄	1	1	yes	characteristic
29	#120×C₄₇	C₄	1	2	yes	—
30	C₄×D₄×C₄₇	C₂²	1	2	yes	—
31	C₄×D₄×C₄₇	C₂²	1	2	yes	—
32	#117×C₄₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	C₂²	1	2	yes	complemented
34	C₄⋊C₈×C₄₇	C₂²	1	2	yes	complemented
35	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	complemented
41	#97×C₄₇	C₂²	1	2	yes	complemented
42	C₄×C₈×C₄₇	C₂²	1	2	yes	complemented
43	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
44	#41×C₄₇	—	2	2	no	—
45	#41×C₄₇	—	2	2	no	—
46	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
60	M₁₆×C₄₇	—	2	2	no	—
61	M₁₆×C₄₇	—	2	4	no	—
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	D₄	1	4	yes	complemented
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
77	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
78	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
80	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
81	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂³	1	2	yes	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₄₇	—	2	2	no	—
92	D₄×C₄₇	—	2	4	no	—
93	D₄×C₄₇	—	2	4	no	—
94	D₄×C₄₇	—	2	2	no	—
95	D₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	#41	1	2	yes	—
113	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₈×C₄₇	—	2	1	no	—
115	C₈×C₄₇	—	2	1	no	—
116	C₈×C₄₇	—	2	2	no	—
117	C₈×C₄₇	—	2	1	no	—
118	C₈×C₄₇	—	2	1	no	—
119	C₈×C₄₇	C₂×D₄	1	4	yes	—
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	C₄×D₄	1	2	yes	—
132	C₂²×C₄₇	#131	1	2	yes	complemented
133	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	C₄×D₄	1	2	yes	—
140	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	#2933	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	—	4	2	no	—
146	C₂×C₄₇	—	4	1	no	—
147	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
149	C₂×C₄₇	#568	1	1	yes	characteristic
150	C₂×#131	C₂×C₄₇	1	1	yes	characteristic, complemented
151	C₂²×M₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
152	#525	C₂×C₄₇	1	1	yes	characteristic
153	#436	C₂×C₄₇	1	2	yes	complemented
154	#435	C₂×C₄₇	1	2	yes	complemented
155	#435	C₂×C₄₇	1	2	yes	complemented
156	C₈×D₄	C₂×C₄₇	1	2	yes	complemented
157	#424	C₂×C₄₇	1	1	yes	characteristic, complemented
158	#408	C₂×C₄₇	1	1	yes	characteristic, complemented
159	C₂×C₂²⋊C₈	C₂×C₄₇	1	1	yes	characteristic, complemented
160	#405	C₂×C₄₇	1	1	yes	characteristic, complemented
161	C₄₇	#2933	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	C₂²×C₄₇	1	1	yes	characteristic
163	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
164	#131	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂²×C₄₇	1	2	yes	—
168	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂²×C₄₇	1	2	yes	complemented
170	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
171	C₂²×C₈	C₂²×C₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×C₄₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×C₄₇	1	2	yes	complemented
174	#123	C₄×C₄₇	1	2	yes	—
175	C₂²⋊Q₈	C₄×C₄₇	1	1	yes	characteristic
176	C₂²⋊Q₈	C₄×C₄₇	1	1	yes	characteristic
177	#121	C₄×C₄₇	1	1	yes	characteristic
178	#121	C₄×C₄₇	1	1	yes	characteristic
179	#120	C₄×C₄₇	1	2	yes	—
180	C₄×D₄	C₂²×C₄₇	1	2	yes	—
181	C₄×D₄	C₂²×C₄₇	1	2	yes	—
182	#117	C₂²×C₄₇	1	1	yes	characteristic
183	C₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
184	C₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
185	C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
186	C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
187	C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
188	C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
189	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	complemented
190	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	complemented
191	#97	C₂²×C₄₇	1	2	yes	complemented
192	C₄×C₈	C₂²×C₄₇	1	2	yes	complemented
193	C₂⁴	C₂×C₄×C₄₇	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
197	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
198	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
199	C₂×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
200	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
209	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	D₄×C₄₇	1	4	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
219	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₄₇	1	2	yes	—
223	C₂×C₈	C₂³×C₄₇	1	2	yes	—
224	C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
225	C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
227	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
228	C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
229	C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
230	C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
231	C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
232	C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
233	C₄²	C₂³×C₄₇	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×C₄×C₄₇	1	1	yes	characteristic
238	C₂³	C₂²×C₄×C₄₇	1	2	yes	—
239	C₂³	C₂²×C₄×C₄₇	1	1	yes	characteristic
240	C₂³	C₂²×C₄×C₄₇	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	C₂²×C₄×C₄₇	1	2	yes	—
256	C₂×C₄	C₂²×C₄×C₄₇	1	2	yes	—
257	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×C₄₇	1	2	yes	—
260	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
262	C₂×C₄	#41×C₄₇	1	2	yes	—
263	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	C₂×D₄×C₄₇	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₄×D₄×C₄₇	1	2	yes	—
282	C₂²	#131×C₄₇	1	2	yes	complemented
283	C₂²	C₂³×C₄×C₄₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	C₄×D₄×C₄₇	1	2	yes	—
290	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
291	C₄	C₂×#41×C₄₇	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
297	C₂	C₂×#131×C₄₇	1	1	yes	characteristic
298	C₂	C₄₇×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₄₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
socle	C₂²×C₄₇	C₂³×C₄
prime radical	C₄₇×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#131×C₄₇	C₂	semidirect
2. C₂²×M₁₆×C₄₇	C₂	semidirect
4. C₄₇×#436	C₂	semidirect
5. C₄₇×#435	C₂	semidirect
6. C₄₇×#435	C₂	semidirect
7. C₈×D₄×C₄₇	C₂	semidirect
8. C₄₇×#424	C₂	semidirect
9. C₄₇×#408	C₂	semidirect
10. C₂×C₂²⋊C₈×C₄₇	C₂	semidirect
11. C₄₇×#405	C₂	semidirect
19. C₂×M₁₆×C₄₇	C₂²	semidirect
23. C₂²×C₈×C₄₇	C₂²	semidirect
33. C₄⋊C₈×C₄₇	C₂²	semidirect
34. C₄⋊C₈×C₄₇	C₂²	semidirect
39. C₂²⋊C₈×C₄₇	C₂²	semidirect
40. C₂²⋊C₈×C₄₇	C₂²	semidirect
41. #97×C₄₇	C₂²	semidirect
42. C₄×C₈×C₄₇	C₂²	semidirect
63. M₁₆×C₄₇	D₄	semidirect
132. C₂²×C₄₇	#131	semidirect
142. #2933	C₄₇	direct (N × H)
150. C₂×#131	C₂×C₄₇	semidirect
151. C₂²×M₁₆	C₂×C₄₇	semidirect
153. #436	C₂×C₄₇	semidirect
154. #435	C₂×C₄₇	semidirect
155. #435	C₂×C₄₇	semidirect
156. C₈×D₄	C₂×C₄₇	semidirect
157. #424	C₂×C₄₇	semidirect
158. #408	C₂×C₄₇	semidirect
159. C₂×C₂²⋊C₈	C₂×C₄₇	semidirect
160. #405	C₂×C₄₇	semidirect
161. C₄₇	#2933	direct (N × H)
169. C₂×M₁₆	C₂²×C₄₇	semidirect
173. C₂²×C₈	C₂²×C₄₇	semidirect
183. C₄⋊C₈	C₂²×C₄₇	semidirect
184. C₄⋊C₈	C₂²×C₄₇	semidirect
189. C₂²⋊C₈	C₂²×C₄₇	semidirect
190. C₂²⋊C₈	C₂²×C₄₇	semidirect
191. #97	C₂²×C₄₇	semidirect
192. C₄×C₈	C₂²×C₄₇	semidirect
213. M₁₆	D₄×C₄₇	semidirect
282. C₂²	#131×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₄₇	2
C₂²×M₁₆×C₄₇	2
C₄₇×#525	2
C₄₇×#436	2
C₄₇×#435	2
C₈×D₄×C₄₇	2
C₄₇×#424	2
C₄₇×#408	2
C₂×C₂²⋊C₈×C₄₇	2
C₄₇×#405	2
#2933	47

Subgroups — C47×#2933 #20240543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#2933 #20240543