Subgroups — C₄₇×#33776 #26073474

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2211	C₂	1	4	yes	maximal, complemented
8	C₄₇×#2189	C₂	1	4	yes	maximal, complemented
9	C₄₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₄₇	—	2	1	no	—
11	C₂×#135×C₄₇	C₂²	1	2	yes	—
12	C₄₇×#533	—	2	1	no	—
13	C₄₇×#532	—	2	1	no	—
14	C₂×Q₃₂×C₄₇	C₂²	1	4	yes	—
15	C₂×SD₃₂×C₄₇	C₂²	1	4	yes	—
16	C₄₇×#500	C₂²	1	1	yes	characteristic
17	C₄₇×#495	—	2	1	no	—
18	C₄₇×#495	C₂²	1	2	yes	—
19	C₄₇×#494	C₂²	1	1	yes	characteristic
20	C₄₇×#494	C₂²	1	1	yes	characteristic
21	C₄₇×#493	C₂²	1	1	yes	characteristic
22	C₄₇×#492	—	2	1	no	—
23	C₄₇×#487	—	2	1	no	—
24	C₄₇×#467	—	2	1	no	—
25	C₄₇×#467	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₄₇	—	2	1	no	—
27	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄₇×#358	C₂²	1	4	yes	—
31	C₄₇×#357	C₂²	1	4	yes	—
32	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#41×C₄₇	—	2	2	no	—
36	#135×C₄₇	—	4	4	no	—
37	#135×C₄₇	—	4	4	no	—
38	C₂×Q₁₆×C₄₇	—	2	1	no	—
39	C₂×Q₁₆×C₄₇	—	2	1	no	—
40	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
41	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₄₇	—	2	2	no	—
43	C₂×SD₁₆×C₄₇	—	2	2	no	—
44	C₂×D₈×C₄₇	—	2	1	no	—
45	C₂×D₈×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₇	—	2	1	no	—
49	C₄⋊Q₈×C₄₇	—	2	1	no	—
50	C₄⋊Q₈×C₄₇	—	2	1	no	—
51	#127×C₄₇	—	2	1	no	—
52	#125×C₄₇	—	4	1	no	—
53	#124×C₄₇	—	2	2	no	—
54	C₂²⋊Q₈×C₄₇	—	2	2	no	—
55	C₂²⋊Q₈×C₄₇	—	4	1	no	—
56	#121×C₄₇	—	4	1	no	—
57	#121×C₄₇	—	4	1	no	—
58	C₄×Q₈×C₄₇	—	2	2	no	—
59	C₄×Q₈×C₄₇	—	4	1	no	—
60	C₄×D₄×C₄₇	—	4	1	no	—
61	C₄×D₄×C₄₇	—	4	1	no	—
62	#117×C₄₇	—	4	1	no	—
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	Q₃₂×C₄₇	—	4	4	no	—
65	SD₃₂×C₄₇	—	4	4	no	—
66	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
67	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
68	#107×C₄₇	—	2	1	no	—
69	#107×C₄₇	C₂³	1	2	yes	—
70	#103×C₄₇	—	2	2	no	—
71	#103×C₄₇	—	2	2	no	—
72	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	#41×C₄₇	—	4	4	no	—
77	#41×C₄₇	—	4	4	no	—
78	C₂×Q₈×C₄₇	—	2	2	no	—
79	C₂×Q₈×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	4	1	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	—	4	2	no	—
83	C₂×D₄×C₄₇	—	4	1	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂²×C₄×C₄₇	—	4	2	no	—
86	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₄₇	—	2	4	no	—
90	Q₁₆×C₄₇	—	2	4	no	—
91	Q₁₆×C₄₇	—	4	2	no	—
92	Q₁₆×C₄₇	—	4	2	no	—
93	SD₁₆×C₄₇	—	4	4	no	—
94	SD₁₆×C₄₇	—	4	4	no	—
95	D₈×C₄₇	—	2	4	no	—
96	D₈×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	4	1	no	—
105	C₄⋊C₄×C₄₇	—	4	1	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄⋊C₄×C₄₇	—	4	2	no	—
109	C₄⋊C₄×C₄₇	—	4	2	no	—
110	C₄⋊C₄×C₄₇	—	4	1	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₂²⋊C₄×C₄₇	—	4	2	no	—
113	C₂²⋊C₄×C₄₇	—	4	2	no	—
114	C₄²×C₄₇	—	4	2	no	—
115	C₄²×C₄₇	D₈	1	1	yes	characteristic
116	C₄²×C₄₇	D₈	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₄₇	—	2	2	no	—
120	C₁₆×C₄₇	—	2	2	no	—
121	C₂³×C₄₇	—	4	2	no	—
122	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
123	Q₈×C₄₇	—	4	2	no	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	4	2	no	—
126	Q₈×C₄₇	—	4	2	no	—
127	Q₈×C₄₇	—	8	2	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	4	2	no	—
130	D₄×C₄₇	—	8	4	no	—
131	D₄×C₄₇	—	8	2	no	—
132	D₄×C₄₇	—	8	2	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	8	4	no	—
141	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₄₇	—	2	1	no	—
148	C₈×C₄₇	—	2	1	no	—
149	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
150	C₈×C₄₇	H(5,𝔽₂)	1	2	yes	—
151	#33776	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	2	1	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	—	8	2	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#508	1	2	yes	—
166	C₄×C₄₇	#510	1	2	yes	—
167	C₄×C₄₇	#535	1	1	yes	characteristic
168	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	#3170	C₂×C₄₇	1	1	yes	characteristic, complemented
170	#3147	C₂×C₄₇	1	1	yes	characteristic
171	#2248	C₂×C₄₇	1	1	yes	characteristic, complemented
172	#2247	C₂×C₄₇	1	1	yes	characteristic, complemented
173	#2241	C₂×C₄₇	1	1	yes	characteristic, complemented
174	#2240	C₂×C₄₇	1	1	yes	characteristic, complemented
175	#2211	C₂×C₄₇	1	4	yes	complemented
176	#2189	C₂×C₄₇	1	4	yes	complemented
177	#2162	C₂×C₄₇	1	1	yes	characteristic, complemented
178	C₂×C₄₇	—	4	1	no	—
179	C₂×C₄₇	—	16	2	no	—
180	C₂×C₄₇	#3205	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
183	C₂×#135	—	2	1	no	—
184	C₂×#135	C₂²×C₄₇	1	2	yes	—
185	#533	—	2	1	no	—
186	#532	—	2	1	no	—
187	C₂×Q₃₂	C₂²×C₄₇	1	4	yes	—
188	C₂×SD₃₂	C₂²×C₄₇	1	4	yes	—
189	#500	C₂²×C₄₇	1	1	yes	characteristic
190	#495	—	2	1	no	—
191	#495	C₂²×C₄₇	1	2	yes	—
192	#494	C₂²×C₄₇	1	1	yes	characteristic
193	#494	C₂²×C₄₇	1	1	yes	characteristic
194	#493	C₂²×C₄₇	1	1	yes	characteristic
195	#492	—	2	1	no	—
196	#487	—	2	1	no	—
197	#467	—	2	1	no	—
198	#467	C₂²×C₄₇	1	2	yes	—
199	C₄×Q₁₆	—	2	1	no	—
200	C₄×Q₁₆	C₂²×C₄₇	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂²×C₄₇	1	2	yes	—
203	#358	C₂²×C₄₇	1	4	yes	—
204	#357	C₂²×C₄₇	1	4	yes	—
205	C₂²⋊C₁₆	C₂²×C₄₇	1	2	yes	—
206	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
207	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
208	C₄₇	#33776	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×#41	—	2	2	no	—
210	#135	—	4	4	no	—
211	#135	—	4	4	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
215	C₂×Q₁₆	C₂³×C₄₇	1	2	yes	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	C₂³×C₄₇	1	2	yes	—
220	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	2	1	no	—
226	#125	—	4	1	no	—
227	#124	—	2	2	no	—
228	C₂²⋊Q₈	—	2	2	no	—
229	C₂²⋊Q₈	—	4	1	no	—
230	#121	—	4	1	no	—
231	#121	—	4	1	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	4	1	no	—
234	C₄×D₄	—	4	1	no	—
235	C₄×D₄	—	4	1	no	—
236	#117	—	4	1	no	—
237	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
238	Q₃₂	—	4	4	no	—
239	SD₃₂	—	4	4	no	—
240	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
241	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
242	#107	—	2	1	no	—
243	#107	C₂³×C₄₇	1	2	yes	—
244	#103	—	2	2	no	—
245	#103	—	2	2	no	—
246	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
247	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
250	#41	—	4	4	no	—
251	#41	—	4	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
261	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
263	Q₁₆	—	2	4	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	Q₁₆	—	4	2	no	—
267	SD₁₆	—	4	4	no	—
268	SD₁₆	—	4	4	no	—
269	D₈	—	2	4	no	—
270	D₈	—	4	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
276	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	4	2	no	—
289	C₄²	D₈×C₄₇	1	1	yes	characteristic
290	C₄²	D₈×C₄₇	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
293	C₁₆	—	2	2	no	—
294	C₁₆	—	2	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
318	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
324	C₈	H(5,𝔽₂)×C₄₇	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₄₇×#508	1	2	yes	—
339	C₄	C₄₇×#510	1	2	yes	—
340	C₄	C₄₇×#535	1	1	yes	characteristic
341	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₄₇×#3205	1	1	yes	characteristic
345	C₂	C₂×C₄₇×#508	1	1	yes	characteristic
346	C₂	C₂×C₄₇×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₄₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ C₂×C₄×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3170	C₂	semidirect
3. C₄₇×#2248	C₂	semidirect
4. C₄₇×#2247	C₂	semidirect
5. C₄₇×#2241	C₂	semidirect
6. C₄₇×#2240	C₂	semidirect
7. C₄₇×#2211	C₂	semidirect
8. C₄₇×#2189	C₂	semidirect
9. C₄₇×#2162	C₂	semidirect
151. #33776	C₄₇	direct (N × H)
169. #3170	C₂×C₄₇	semidirect
171. #2248	C₂×C₄₇	semidirect
172. #2247	C₂×C₄₇	semidirect
173. #2241	C₂×C₄₇	semidirect
174. #2240	C₂×C₄₇	semidirect
175. #2211	C₂×C₄₇	semidirect
176. #2189	C₂×C₄₇	semidirect
177. #2162	C₂×C₄₇	semidirect
208. C₄₇	#33776	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×#3170	2
C₄₇×#3147	2
C₄₇×#2248	2
C₄₇×#2247	2
C₄₇×#2241	2
C₄₇×#2240	2
C₄₇×#2211	2
C₄₇×#2189	2
C₄₇×#2162	2
#33776	47

Subgroups — C47×#33776 #26073474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33776 #26073474