Subgroups — C₄₇×#33609 #26073307

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3058	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2220	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2217	C₂	1	4	yes	maximal, complemented
7	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#536	—	2	1	no	—
14	C₄₇×#535	—	2	1	no	—
15	C₂×SD₃₂×C₄₇	—	2	4	no	—
16	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
17	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#504	—	2	4	no	—
19	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₄₇	C₂²	1	2	yes	complemented
21	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	complemented
23	C₄₇×#470	—	2	2	no	—
24	C₄₇×#469	—	2	2	no	—
25	C₄₇×#467	C₂²	1	4	yes	complemented
26	C₄₇×#466	C₂²	1	4	yes	—
27	C₄₇×#427	C₂²	1	1	yes	characteristic
28	C₄₇×#427	C₂²	1	1	yes	characteristic
29	C₄₇×#419	—	2	1	no	—
30	C₄₇×#418	—	2	1	no	—
31	C₄₇×#367	C₂²	1	2	yes	complemented
32	C₄₇×#366	C₂²	1	2	yes	complemented
33	C₄₇×#358	C₂²	1	4	yes	complemented
34	C₄₇×#357	C₂²	1	4	yes	—
35	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₄₇	—	2	1	no	—
37	C₂²×D₄×C₄₇	—	2	1	no	—
38	C₂×Q₁₆×C₄₇	—	2	2	no	—
39	C₂×Q₁₆×C₄₇	—	2	1	no	—
40	C₂×Q₁₆×C₄₇	—	2	2	no	—
41	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
42	C₂×D₈×C₄₇	—	2	2	no	—
43	C₂×D₈×C₄₇	—	2	1	no	—
44	C₂×D₈×C₄₇	—	2	2	no	—
45	C₂×D₈×C₄₇	C₂³	1	2	yes	—
46	#131×C₄₇	—	2	2	no	—
47	#131×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
49	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₇	—	4	1	no	—
53	#127×C₄₇	—	4	1	no	—
54	#124×C₄₇	—	4	1	no	—
55	#124×C₄₇	—	4	1	no	—
56	#123×C₄₇	—	4	2	no	—
57	C₂²⋊Q₈×C₄₇	—	2	4	no	—
58	#121×C₄₇	—	2	4	no	—
59	#120×C₄₇	—	4	2	no	—
60	#117×C₄₇	—	2	1	no	—
61	#117×C₄₇	—	2	1	no	—
62	SD₃₂×C₄₇	—	2	4	no	—
63	SD₃₂×C₄₇	—	2	4	no	—
64	M₃₂×C₄₇	—	2	2	no	—
65	M₃₂×C₄₇	—	2	4	no	—
66	C₂×C₁₆×C₄₇	—	2	4	no	—
67	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
68	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
71	#107×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	C₂³	1	2	yes	—
73	#106×C₄₇	—	2	1	no	—
74	#106×C₄₇	—	2	1	no	—
75	#103×C₄₇	—	2	4	no	—
76	#102×C₄₇	—	2	4	no	—
77	C₂⁴×C₄₇	—	4	1	no	—
78	#41×C₄₇	—	2	2	no	—
79	#41×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	4	1	no	—
82	C₂×Q₈×C₄₇	—	4	2	no	—
83	C₂×Q₈×C₄₇	—	4	2	no	—
84	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	4	4	no	—
87	C₂×D₄×C₄₇	—	4	2	no	—
88	C₂×D₄×C₄₇	—	4	1	no	—
89	C₂×D₄×C₄₇	—	4	2	no	—
90	C₂×D₄×C₄₇	D₈	1	2	yes	—
91	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	—	4	1	no	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₄₇	—	2	2	no	—
96	Q₁₆×C₄₇	—	2	2	no	—
97	Q₁₆×C₄₇	—	4	2	no	—
98	D₈×C₄₇	—	2	2	no	—
99	D₈×C₄₇	—	2	2	no	—
100	D₈×C₄₇	—	4	2	no	—
101	M₁₆×C₄₇	—	2	2	no	—
102	M₁₆×C₄₇	—	2	4	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄⋊C₄×C₄₇	—	4	4	no	—
113	C₂²⋊C₄×C₄₇	—	4	1	no	—
114	C₂²⋊C₄×C₄₇	—	4	1	no	—
115	C₂²⋊C₄×C₄₇	—	4	4	no	—
116	C₂²⋊C₄×C₄₇	—	4	4	no	—
117	C₄²×C₄₇	—	4	1	no	—
118	C₄²×C₄₇	—	4	1	no	—
119	C₁₆×C₄₇	—	2	2	no	—
120	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
121	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	8	2	no	—
124	C₂³×C₄₇	—	16	2	no	—
125	C₂³×C₄₇	C₂×D₈	1	2	yes	—
126	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₄₇	—	2	2	no	—
128	Q₈×C₄₇	—	4	2	no	—
129	Q₈×C₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	8	2	no	—
131	D₄×C₄₇	—	2	4	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	4	2	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	8	2	no	—
136	D₄×C₄₇	—	16	4	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	#121	1	2	yes	—
144	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#135	1	2	yes	—
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	#121	1	2	yes	—
150	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
152	#33609	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	4	no	—
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	8	4	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	16	4	no	—
159	C₂²×C₄₇	#466	1	2	yes	—
160	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	—	8	2	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#466	1	2	yes	—
166	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
167	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
168	C₂²×SD₃₂	C₂×C₄₇	1	1	yes	characteristic, complemented
169	C₂×#504	C₂×C₄₇	1	1	yes	characteristic, complemented
170	#3059	C₂×C₄₇	1	1	yes	characteristic, complemented
171	#3058	C₂×C₄₇	1	1	yes	characteristic
172	#2220	C₂×C₄₇	1	4	yes	complemented
173	#2217	C₂×C₄₇	1	4	yes	complemented
174	#2158	C₂×C₄₇	1	1	yes	characteristic, complemented
175	#2141	C₂×C₄₇	1	1	yes	characteristic, complemented
176	#2140	C₂×C₄₇	1	1	yes	characteristic, complemented
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	4	2	no	—
179	C₂×C₄₇	—	16	2	no	—
180	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
181	C₂×C₄₇	#3416	1	2	yes	—
182	C₂²×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
183	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
184	C₂×#131	C₂²×C₄₇	1	1	yes	characteristic
185	#536	—	2	1	no	—
186	#535	—	2	1	no	—
187	C₂×SD₃₂	—	2	4	no	—
188	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
189	C₂×SD₃₂	C₂²×C₄₇	1	2	yes	—
190	#504	—	2	4	no	—
191	C₂×M₃₂	C₂²×C₄₇	1	1	yes	characteristic
192	C₂×M₃₂	C₂²×C₄₇	1	2	yes	complemented
193	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
194	C₂²×C₁₆	C₂²×C₄₇	1	2	yes	complemented
195	#470	—	2	2	no	—
196	#469	—	2	2	no	—
197	#467	C₂²×C₄₇	1	4	yes	complemented
198	#466	C₂²×C₄₇	1	4	yes	—
199	#427	C₂²×C₄₇	1	1	yes	characteristic
200	#427	C₂²×C₄₇	1	1	yes	characteristic
201	#419	—	2	1	no	—
202	#418	—	2	1	no	—
203	#367	C₂²×C₄₇	1	2	yes	complemented
204	#366	C₂²×C₄₇	1	2	yes	complemented
205	#358	C₂²×C₄₇	1	4	yes	complemented
206	#357	C₂²×C₄₇	1	4	yes	—
207	C₄₇	#33609	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	C₂³×C₄₇	1	2	yes	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₄×C₄₇	1	1	yes	characteristic
222	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
224	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	4	1	no	—
226	#127	—	4	1	no	—
227	#124	—	4	1	no	—
228	#124	—	4	1	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	2	4	no	—
231	#121	—	2	4	no	—
232	#120	—	4	2	no	—
233	#117	—	2	1	no	—
234	#117	—	2	1	no	—
235	SD₃₂	—	2	4	no	—
236	SD₃₂	—	2	4	no	—
237	M₃₂	—	2	2	no	—
238	M₃₂	—	2	4	no	—
239	C₂×C₁₆	—	2	4	no	—
240	C₂×C₁₆	D₄×C₄₇	1	4	yes	complemented
241	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
242	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
244	#107	C₂³×C₄₇	1	2	yes	—
245	#107	C₂³×C₄₇	1	2	yes	—
246	#106	—	2	1	no	—
247	#106	—	2	1	no	—
248	#103	—	2	4	no	—
249	#102	—	2	4	no	—
250	C₂⁴	—	4	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	D₈×C₄₇	1	1	yes	characteristic
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	D₈×C₄₇	1	2	yes	—
264	C₂×D₄	D₈×C₄₇	1	1	yes	characteristic
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	4	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	4	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
281	C₂×C₈	#41×C₄₇	1	2	yes	—
282	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×D₄×C₄₇	1	2	yes	—
294	C₁₆	C₂×D₄×C₄₇	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	16	2	no	—
298	C₂³	C₂×D₈×C₄₇	1	2	yes	—
299	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	#121×C₄₇	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
319	C₂×C₄	#135×C₄₇	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#121×C₄₇	1	2	yes	—
323	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
324	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	16	4	no	—
331	C₂²	C₄₇×#466	1	2	yes	—
332	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₄₇×#466	1	2	yes	—
338	C₄	C₂×#121×C₄₇	1	1	yes	characteristic
339	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×C₄₇×#466	1	1	yes	characteristic
344	C₂	C₄₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₄₇×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×#41×C₄₇ ◁ C₂×#131×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₃₂×C₄₇	C₂	semidirect
2. C₂×C₄₇×#504	C₂	semidirect
3. C₄₇×#3059	C₂	semidirect
5. C₄₇×#2220	C₂	semidirect
6. C₄₇×#2217	C₂	semidirect
7. C₄₇×#2158	C₂	semidirect
8. C₄₇×#2141	C₂	semidirect
9. C₄₇×#2140	C₂	semidirect
20. C₂×M₃₂×C₄₇	C₂²	semidirect
22. C₂²×C₁₆×C₄₇	C₂²	semidirect
25. C₄₇×#467	C₂²	semidirect
31. C₄₇×#367	C₂²	semidirect
32. C₄₇×#366	C₂²	semidirect
33. C₄₇×#358	C₂²	semidirect
67. C₂×C₁₆×C₄₇	D₄	semidirect
152. #33609	C₄₇	direct (N × H)
168. C₂²×SD₃₂	C₂×C₄₇	semidirect
169. C₂×#504	C₂×C₄₇	semidirect
170. #3059	C₂×C₄₇	semidirect
172. #2220	C₂×C₄₇	semidirect
173. #2217	C₂×C₄₇	semidirect
174. #2158	C₂×C₄₇	semidirect
175. #2141	C₂×C₄₇	semidirect
176. #2140	C₂×C₄₇	semidirect
192. C₂×M₃₂	C₂²×C₄₇	semidirect
194. C₂²×C₁₆	C₂²×C₄₇	semidirect
197. #467	C₂²×C₄₇	semidirect
203. #367	C₂²×C₄₇	semidirect
204. #366	C₂²×C₄₇	semidirect
205. #358	C₂²×C₄₇	semidirect
207. C₄₇	#33609	direct (N × H)
240. 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
C₂²×SD₃₂×C₄₇	2
C₂×C₄₇×#504	2
C₄₇×#3059	2
C₄₇×#3058	2
C₄₇×#2220	2
C₄₇×#2217	2
C₄₇×#2158	2
C₄₇×#2141	2
C₄₇×#2140	2
#33609	47

Subgroups — C47×#33609 #26073307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33609 #26073307