Subgroups — C₄₇×#29154 #26068852

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#443	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2031	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1976	C₂	1	2	yes	maximal
5	C₄₇×#1833	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1814	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1812	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₄₇	C₂²	1	2	yes	—
11	C₄₇×#533	C₂²	1	2	yes	—
12	C₄₇×#500	—	2	1	no	—
13	C₄₇×#499	—	2	1	no	—
14	C₄₇×#499	—	2	1	no	—
15	C₄₇×#498	—	2	1	no	—
16	C₄₇×#443	—	2	1	no	—
17	C₄₇×#443	—	2	1	no	—
18	C₄₇×#443	—	2	1	no	—
19	C₄₇×#443	—	2	1	no	—
20	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#427	C₂²	1	1	yes	characteristic
23	C₄₇×#427	C₂²	1	2	yes	—
24	C₄₇×#427	C₂²	1	2	yes	—
25	C₄₇×#421	C₂²	1	4	yes	—
26	C₂×#104×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#410	C₂²	1	4	yes	—
28	C₄₇×#405	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#344	C₂²	1	4	yes	—
31	C₄₇×#339	C₂²	1	2	yes	—
32	C₄₇×H(3,ℤ/4)	C₂²	1	4	yes	—
33	C₂×#41×C₄₇	C₂³	1	2	yes	—
34	#135×C₄₇	—	2	1	no	—
35	#135×C₄₇	—	2	1	no	—
36	#135×C₄₇	—	2	1	no	—
37	#135×C₄₇	—	2	1	no	—
38	C₂×Q₁₆×C₄₇	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₄₇	Q₈	1	2	yes	complemented
40	C₂×D₈×C₄₇	Q₈	1	1	yes	characteristic, complemented
41	#131×C₄₇	—	2	2	no	—
42	#131×C₄₇	—	2	2	no	—
43	#131×C₄₇	—	2	2	no	—
44	#131×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
47	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	2	no	—
52	#125×C₄₇	—	2	2	no	—
53	C₂²⋊Q₈×C₄₇	—	4	2	no	—
54	C₄×Q₈×C₄₇	—	4	2	no	—
55	#117×C₄₇	C₂³	1	4	yes	—
56	#117×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#108×C₄₇	—	2	1	no	—
59	#108×C₄₇	—	2	1	no	—
60	#108×C₄₇	—	2	1	no	—
61	#108×C₄₇	—	2	1	no	—
62	#107×C₄₇	—	2	2	no	—
63	#107×C₄₇	—	2	2	no	—
64	#107×C₄₇	—	2	1	no	—
65	#106×C₄₇	—	2	2	no	—
66	#106×C₄₇	—	2	2	no	—
67	#106×C₄₇	—	2	1	no	—
68	#104×C₄₇	—	2	2	no	—
69	#104×C₄₇	—	2	2	no	—
70	#104×C₄₇	—	4	4	no	—
71	#99×C₄₇	—	4	4	no	—
72	#97×C₄₇	—	2	2	no	—
73	C₄×C₈×C₄₇	—	2	2	no	—
74	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	#41×C₄₇	—	2	2	no	—
79	#41×C₄₇	—	2	2	no	—
80	#41×C₄₇	—	2	2	no	—
81	#41×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	4	2	no	—
83	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₄₇	—	2	1	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₄₇	—	2	1	no	—
90	Q₁₆×C₄₇	—	2	1	no	—
91	SD₁₆×C₄₇	—	2	2	no	—
92	SD₁₆×C₄₇	—	2	2	no	—
93	D₈×C₄₇	—	2	1	no	—
94	D₈×C₄₇	—	2	1	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	2	2	no	—
98	M₁₆×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	4	2	no	—
100	M₁₆×C₄₇	—	4	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	4	2	no	—
104	C₂×C₈×C₄₇	—	4	2	no	—
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
114	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
115	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
116	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
117	C₄⋊C₄×C₄₇	—	2	4	no	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	4	4	no	—
120	C₄⋊C₄×C₄₇	—	4	2	no	—
121	C₂²⋊C₄×C₄₇	—	2	4	no	—
122	C₂²⋊C₄×C₄₇	—	2	2	no	—
123	C₄²×C₄₇	—	2	1	no	—
124	C₄²×C₄₇	—	2	2	no	—
125	C₄²×C₄₇	—	2	4	no	—
126	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₄₇	—	2	2	no	—
129	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₄₇	—	2	2	no	—
131	Q₈×C₄₇	—	2	2	no	—
132	Q₈×C₄₇	—	8	4	no	—
133	D₄×C₄₇	—	2	2	no	—
134	D₄×C₄₇	—	2	2	no	—
135	D₄×C₄₇	—	4	2	no	—
136	D₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	4	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₈×C₄₇	—	4	2	no	—
152	C₈×C₄₇	—	4	2	no	—
153	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
157	C₈×C₄₇	#124	1	1	yes	characteristic
158	C₈×C₄₇	#124	1	1	yes	characteristic
159	C₈×C₄₇	#124	1	1	yes	characteristic
160	C₈×C₄₇	#124	1	1	yes	characteristic
161	#29154	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₄₇	—	4	1	no	—
163	C₂²×C₄₇	—	4	2	no	—
164	C₂²×C₄₇	—	8	2	no	—
165	C₂²×C₄₇	#535	1	1	yes	characteristic
166	C₂²×C₄₇	#554	1	1	yes	characteristic
167	C₂²×C₄₇	#555	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	—	8	4	no	—
173	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
175	C₄×C₄₇	#554	1	1	yes	characteristic
176	C₄×C₄₇	#555	1	1	yes	characteristic
177	#3160	C₂×C₄₇	1	1	yes	characteristic, complemented
178	C₂×#443	C₂×C₄₇	1	1	yes	characteristic
179	#2031	C₂×C₄₇	1	4	yes	complemented
180	#1976	C₂×C₄₇	1	2	yes	—
181	#1833	C₂×C₄₇	1	2	yes	complemented
182	#1814	C₂×C₄₇	1	2	yes	complemented
183	#1812	C₂×C₄₇	1	2	yes	complemented
184	#1772	C₂×C₄₇	1	1	yes	characteristic, complemented
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	8	2	no	—
189	C₂×C₄₇	#2563	1	1	yes	characteristic
190	C₂×#135	C₂²×C₄₇	1	1	yes	characteristic
191	C₂×#131	C₂²×C₄₇	1	2	yes	—
192	#533	C₂²×C₄₇	1	2	yes	—
193	#500	—	2	1	no	—
194	#499	—	2	1	no	—
195	#499	—	2	1	no	—
196	#498	—	2	1	no	—
197	#443	—	2	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	C₂×#108	C₂²×C₄₇	1	1	yes	characteristic
202	C₂×#108	C₂²×C₄₇	1	1	yes	characteristic
203	#427	C₂²×C₄₇	1	1	yes	characteristic
204	#427	C₂²×C₄₇	1	2	yes	—
205	#427	C₂²×C₄₇	1	2	yes	—
206	#421	C₂²×C₄₇	1	4	yes	—
207	C₂×#104	C₂²×C₄₇	1	2	yes	—
208	#410	C₂²×C₄₇	1	4	yes	—
209	#405	C₂²×C₄₇	1	2	yes	—
210	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
211	#344	C₂²×C₄₇	1	4	yes	—
212	#339	C₂²×C₄₇	1	2	yes	—
213	H(3,ℤ/4)	C₂²×C₄₇	1	4	yes	—
214	C₄₇	#29154	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#41	C₂³×C₄₇	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Q₈×C₄₇	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Q₈×C₄₇	1	2	yes	complemented
222	C₂×D₈	Q₈×C₄₇	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
229	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	2	2	no	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	4	2	no	—
236	C₄×Q₈	—	4	2	no	—
237	#117	C₂³×C₄₇	1	4	yes	—
238	#117	C₂³×C₄₇	1	2	yes	—
239	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	2	1	no	—
243	#108	—	2	1	no	—
244	#107	—	2	2	no	—
245	#107	—	2	2	no	—
246	#107	—	2	1	no	—
247	#106	—	2	2	no	—
248	#106	—	2	2	no	—
249	#106	—	2	1	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	4	4	no	—
253	#99	—	4	4	no	—
254	#97	—	2	2	no	—
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
257	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
258	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
259	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	C₂×Q₈×C₄₇	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Q₈×C₄₇	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂×Q₈×C₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×Q₈×C₄₇	1	1	yes	characteristic
293	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
294	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
295	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
296	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
297	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
298	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	4	no	—
308	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
336	C₈	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
337	C₈	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
338	C₈	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
339	C₈	#124×C₄₇	1	1	yes	characteristic
340	C₈	#124×C₄₇	1	1	yes	characteristic
341	C₈	#124×C₄₇	1	1	yes	characteristic
342	C₈	#124×C₄₇	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₄₇×#535	1	1	yes	characteristic
347	C₂²	C₄₇×#554	1	1	yes	characteristic
348	C₂²	C₄₇×#555	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	4	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
355	C₄	C₂×#124×C₄₇	1	1	yes	characteristic
356	C₄	C₄₇×#554	1	1	yes	characteristic
357	C₄	C₄₇×#555	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₄₇×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₇	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
socle	C₂×C₄₇	#2563
prime radical	C₂×#135×C₄₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3160	C₂	semidirect
3. C₄₇×#2031	C₂	semidirect
5. C₄₇×#1833	C₂	semidirect
6. C₄₇×#1814	C₂	semidirect
7. C₄₇×#1812	C₂	semidirect
8. C₄₇×#1772	C₂	semidirect
38. C₂×Q₁₆×C₄₇	Q₈	semidirect
39. C₂×SD₁₆×C₄₇	Q₈	semidirect
40. C₂×D₈×C₄₇	Q₈	semidirect
161. #29154	C₄₇	direct (N × H)
177. #3160	C₂×C₄₇	semidirect
179. #2031	C₂×C₄₇	semidirect
181. #1833	C₂×C₄₇	semidirect
182. #1814	C₂×C₄₇	semidirect
183. #1812	C₂×C₄₇	semidirect
184. #1772	C₂×C₄₇	semidirect
214. C₄₇	#29154	direct (N × H)
220. C₂×Q₁₆	Q₈×C₄₇	semidirect
221. C₂×SD₁₆	Q₈×C₄₇	semidirect
222. C₂×D₈	Q₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3160	2
C₂×C₄₇×#443	2
C₄₇×#2031	2
C₄₇×#1976	2
C₄₇×#1833	2
C₄₇×#1814	2
C₄₇×#1812	2
C₄₇×#1772	2
#29154	47

Subgroups — C47×#29154 #26068852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29154 #26068852