Subgroups — C₄₇×#12741 #26052439

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#576	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2145	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2140	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1812	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×H(5,𝔽₂)×C₄₇	—	2	1	no	—
7	C₄₇×#576	—	2	4	no	—
8	C₄₇×#576	—	2	4	no	—
9	C₂×AGL(1,ℤ/8)×C₄₇	—	2	1	no	—
10	C₂×AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
11	C₂×#135×C₄₇	C₄	1	2	yes	complemented
12	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₄₇	C₄	1	2	yes	complemented
14	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₄₇	C₂²	1	2	yes	complemented
17	C₄₇×#427	C₂²	1	2	yes	complemented
18	C₄₇×#410	—	2	1	no	—
19	C₄₇×#360	—	2	4	no	—
20	C₄₇×#357	—	2	4	no	—
21	C₄₇×#349	—	2	2	no	—
22	C₄₇×H(3,ℤ/4)	—	2	1	no	—
23	H(5,𝔽₂)×C₄₇	—	2	8	no	—
24	H(5,𝔽₂)×C₄₇	—	4	4	no	—
25	C₂×#41×C₄₇	—	2	2	no	—
26	C₂×#41×C₄₇	—	2	1	no	—
27	C₂×#41×C₄₇	—	4	1	no	—
28	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₄₇	—	2	2	no	—
30	C₂²×D₄×C₄₇	—	2	1	no	—
31	C₂²×D₄×C₄₇	—	4	2	no	—
32	C₂²×D₄×C₄₇	—	4	1	no	—
33	AGL(1,ℤ/8)×C₄₇	—	2	8	no	—
34	AGL(1,ℤ/8)×C₄₇	—	2	16	no	—
35	#135×C₄₇	—	2	4	no	—
36	#135×C₄₇	—	2	4	no	—
37	#135×C₄₇	—	2	2	no	—
38	#135×C₄₇	D₄	1	4	yes	—
39	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×SD₁₆×C₄₇	—	2	2	no	—
41	C₂×SD₁₆×C₄₇	—	2	1	no	—
42	C₂×D₈×C₄₇	—	2	4	no	—
43	C₂×D₈×C₄₇	—	2	4	no	—
44	C₂×D₈×C₄₇	—	2	2	no	—
45	C₂×D₈×C₄₇	—	2	1	no	—
46	C₂×D₈×C₄₇	—	2	2	no	—
47	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×D₈×C₄₇	D₄	1	4	yes	—
50	#131×C₄₇	—	2	2	no	—
51	#131×C₄₇	—	2	2	no	—
52	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	#117×C₄₇	—	2	2	no	—
57	M₃₂×C₄₇	—	2	2	no	—
58	M₃₂×C₄₇	—	2	2	no	—
59	C₂×C₁₆×C₄₇	—	2	2	no	—
60	#107×C₄₇	—	2	2	no	—
61	#106×C₄₇	—	2	2	no	—
62	#99×C₄₇	—	4	2	no	—
63	C₂⁴×C₄₇	—	4	2	no	—
64	C₂⁴×C₄₇	—	4	1	no	—
65	#41×C₄₇	—	2	2	no	—
66	#41×C₄₇	—	2	2	no	—
67	#41×C₄₇	—	2	8	no	—
68	#41×C₄₇	—	2	4	no	—
69	#41×C₄₇	—	4	4	no	—
70	#41×C₄₇	—	4	8	no	—
71	#41×C₄₇	—	4	2	no	—
72	C₂×Q₈×C₄₇	—	2	1	no	—
73	C₂×Q₈×C₄₇	SD₁₆	1	1	yes	characteristic
74	C₂×D₄×C₄₇	—	2	8	no	—
75	C₂×D₄×C₄₇	—	2	1	no	—
76	C₂×D₄×C₄₇	—	2	2	no	—
77	C₂×D₄×C₄₇	—	2	4	no	—
78	C₂×D₄×C₄₇	—	4	4	no	—
79	C₂×D₄×C₄₇	—	4	16	no	—
80	C₂×D₄×C₄₇	—	4	8	no	—
81	C₂×D₄×C₄₇	—	4	2	no	—
82	C₂×D₄×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	4	2	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂×D₄×C₄₇	—	4	2	no	—
86	C₂×D₄×C₄₇	D₈	1	2	yes	—
87	C₂×D₄×C₄₇	SD₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	—	4	2	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₄₇	—	2	1	no	—
93	Q₁₆×C₄₇	C₂²⋊C₄	1	2	yes	—
94	SD₁₆×C₄₇	—	2	8	no	—
95	SD₁₆×C₄₇	—	2	4	no	—
96	D₈×C₄₇	—	2	8	no	—
97	D₈×C₄₇	—	2	2	no	—
98	D₈×C₄₇	—	2	1	no	—
99	D₈×C₄₇	—	2	4	no	—
100	D₈×C₄₇	C₂²⋊C₄	1	2	yes	—
101	D₈×C₄₇	C₂²⋊C₄	1	4	yes	—
102	M₁₆×C₄₇	—	2	2	no	—
103	M₁₆×C₄₇	—	2	4	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₄₇	—	4	2	no	—
112	C₂²⋊C₄×C₄₇	—	4	2	no	—
113	C₄²×C₄₇	—	4	2	no	—
114	C₁₆×C₄₇	—	4	2	no	—
115	C₂³×C₄₇	—	4	8	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	4	1	no	—
118	C₂³×C₄₇	—	4	4	no	—
119	C₂³×C₄₇	—	16	2	no	—
120	C₂³×C₄₇	—	16	4	no	—
121	C₂³×C₄₇	#102	1	2	yes	—
122	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
123	Q₈×C₄₇	—	2	2	no	—
124	Q₈×C₄₇	—	2	2	no	—
125	Q₈×C₄₇	—	4	1	no	—
126	D₄×C₄₇	—	2	4	no	—
127	D₄×C₄₇	—	2	2	no	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	2	2	no	—
130	D₄×C₄₇	—	4	8	no	—
131	D₄×C₄₇	—	4	8	no	—
132	D₄×C₄₇	—	4	4	no	—
133	D₄×C₄₇	—	4	1	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	4	4	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	4	8	no	—
139	C₂×C₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	4	4	no	—
141	C₂×C₄×C₄₇	—	8	2	no	—
142	C₂×C₄×C₄₇	#102	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#120	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₈×C₄₇	—	2	2	no	—
148	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₈×C₄₇	#120	1	2	yes	—
151	#12741	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	4	1	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	—	4	4	no	—
156	C₂²×C₄₇	—	4	2	no	—
157	C₂²×C₄₇	—	8	4	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	16	4	no	—
160	C₂²×C₄₇	—	16	8	no	—
161	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
162	C₂²×C₄₇	#447	1	2	yes	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	—	16	2	no	—
166	C₄×C₄₇	#379	1	1	yes	characteristic
167	C₄×C₄₇	#418	1	1	yes	characteristic
168	C₄×C₄₇	#450	1	2	yes	—
169	C₂×#576	C₂×C₄₇	1	1	yes	characteristic
170	#2145	C₂×C₄₇	1	2	yes	complemented
171	#2140	C₂×C₄₇	1	2	yes	complemented
172	#2115	C₂×C₄₇	1	1	yes	characteristic, complemented
173	#1812	C₂×C₄₇	1	1	yes	characteristic, complemented
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	2	1	no	—
176	C₂×C₄₇	—	4	2	no	—
177	C₂×C₄₇	—	8	4	no	—
178	C₂×C₄₇	—	8	2	no	—
179	C₂×C₄₇	#1786	1	1	yes	characteristic
180	C₂×H(5,𝔽₂)	—	2	1	no	—
181	#576	—	2	4	no	—
182	#576	—	2	4	no	—
183	C₂×AGL(1,ℤ/8)	—	2	1	no	—
184	C₂×AGL(1,ℤ/8)	—	2	2	no	—
185	C₂×#135	C₄×C₄₇	1	2	yes	complemented
186	C₂×#135	C₂²×C₄₇	1	1	yes	characteristic
187	C₂²×D₈	C₄×C₄₇	1	2	yes	complemented
188	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
189	C₂×#131	C₂²×C₄₇	1	1	yes	characteristic
190	C₂×M₃₂	C₂²×C₄₇	1	2	yes	complemented
191	#427	C₂²×C₄₇	1	2	yes	complemented
192	#410	—	2	1	no	—
193	#360	—	2	4	no	—
194	#357	—	2	4	no	—
195	#349	—	2	2	no	—
196	H(3,ℤ/4)	—	2	1	no	—
197	C₄₇	#12741	1	1	yes	characteristic, direct product factor, Sylow
198	H(5,𝔽₂)	—	2	8	no	—
199	H(5,𝔽₂)	—	4	4	no	—
200	C₂×#41	—	2	2	no	—
201	C₂×#41	—	2	1	no	—
202	C₂×#41	—	4	1	no	—
203	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	4	2	no	—
207	C₂²×D₄	—	4	1	no	—
208	AGL(1,ℤ/8)	—	2	8	no	—
209	AGL(1,ℤ/8)	—	2	16	no	—
210	#135	—	2	4	no	—
211	#135	—	2	4	no	—
212	#135	—	2	2	no	—
213	#135	D₄×C₄₇	1	4	yes	—
214	C₂×Q₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	—	2	4	no	—
218	C₂×D₈	—	2	4	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	C₂×C₄×C₄₇	1	1	yes	characteristic
223	C₂×D₈	C₂×C₄×C₄₇	1	2	yes	—
224	C₂×D₈	D₄×C₄₇	1	4	yes	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	D₄×C₄₇	1	1	yes	characteristic
228	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
229	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
230	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
231	#117	—	2	2	no	—
232	M₃₂	—	2	2	no	—
233	M₃₂	—	2	2	no	—
234	C₂×C₁₆	—	2	2	no	—
235	#107	—	2	2	no	—
236	#106	—	2	2	no	—
237	#99	—	4	2	no	—
238	C₂⁴	—	4	1	no	—
239	C₂⁴	—	4	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	2	8	no	—
243	#41	—	2	4	no	—
244	#41	—	4	4	no	—
245	#41	—	4	8	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	SD₁₆×C₄₇	1	1	yes	characteristic
249	C₂×D₄	—	2	8	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	16	no	—
255	C₂×D₄	—	4	8	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	D₈×C₄₇	1	2	yes	—
262	C₂×D₄	SD₁₆×C₄₇	1	1	yes	characteristic
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
267	Q₁₆	—	2	1	no	—
268	Q₁₆	C₂²⋊C₄×C₄₇	1	2	yes	—
269	SD₁₆	—	2	8	no	—
270	SD₁₆	—	2	4	no	—
271	D₈	—	2	8	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	4	no	—
275	D₈	C₂²⋊C₄×C₄₇	1	2	yes	—
276	D₈	C₂²⋊C₄×C₄₇	1	4	yes	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
282	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
284	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
286	C₄⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	4	2	no	—
289	C₁₆	—	4	2	no	—
290	C₂³	—	4	8	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	16	4	no	—
296	C₂³	#102×C₄₇	1	2	yes	—
297	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
298	Q₈	—	2	2	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	4	1	no	—
301	D₄	—	2	4	no	—
302	D₄	—	2	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	4	8	no	—
306	D₄	—	4	8	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	1	no	—
310	D₄	—	4	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	8	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#102×C₄₇	1	2	yes	—
318	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
319	C₂×C₄	#120×C₄₇	1	2	yes	—
320	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
321	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
324	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
325	C₈	#120×C₄₇	1	2	yes	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	8	4	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	—	16	8	no	—
335	C₂²	C₂×#102×C₄₇	1	1	yes	characteristic
336	C₂²	C₄₇×#447	1	2	yes	—
337	C₄	—	2	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	16	2	no	—
340	C₄	C₄₇×#379	1	1	yes	characteristic
341	C₄	C₄₇×#418	1	1	yes	characteristic
342	C₄	C₄₇×#450	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	8	2	no	—
348	C₂	C₄₇×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#1786
commutator	C₂×C₈	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₈	C₂³×C₄₇
prime radical	C₂×C₄₇×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂×#131×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2145	C₂	semidirect
3. C₄₇×#2140	C₂	semidirect
4. C₄₇×#2115	C₂	semidirect
5. C₄₇×#1812	C₂	semidirect
11. C₂×#135×C₄₇	C₄	semidirect
13. C₂²×D₈×C₄₇	C₄	semidirect
16. C₂×M₃₂×C₄₇	C₂²	semidirect
17. C₄₇×#427	C₂²	semidirect
151. #12741	C₄₇	direct (N × H)
170. #2145	C₂×C₄₇	semidirect
171. #2140	C₂×C₄₇	semidirect
172. #2115	C₂×C₄₇	semidirect
173. #1812	C₂×C₄₇	semidirect
185. C₂×#135	C₄×C₄₇	semidirect
187. C₂²×D₈	C₄×C₄₇	semidirect
190. C₂×M₃₂	C₂²×C₄₇	semidirect
191. #427	C₂²×C₄₇	semidirect
197. C₄₇	#12741	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#576	2
C₄₇×#2145	2
C₄₇×#2140	2
C₄₇×#2115	2
C₄₇×#1812	2
#12741	47

Subgroups — C47×#12741 #26052439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#12741 #26052439