Subgroups — C₄₇×#8394 #26048092

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#556	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#1548	C₂	1	2	yes	maximal, complemented
4	C₄₇×#1486	C₂	1	2	yes	maximal
5	C₄₇×#1461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#556	—	2	4	no	—
7	C₄₇×#556	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#126×C₄₇	—	2	2	no	—
10	C₂×#124×C₄₇	C₄	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
12	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₄₇×#431	C₂²	1	2	yes	—
17	C₄₇×#423	C₂²	1	2	yes	—
18	C₂×#101×C₄₇	—	2	1	no	—
19	C₄₇×#343	—	2	1	no	—
20	C₄₇×#333	—	2	4	no	—
21	C₄₇×D₄⋊C₈	—	2	4	no	—
22	C₄₇×#324	—	2	2	no	—
23	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	—	2	1	no	—
27	C₂×M₁₆×C₄₇	—	2	2	no	—
28	C₄⋊Q₈×C₄₇	—	2	2	no	—
29	C₄⋊Q₈×C₄₇	D₄	1	4	yes	—
30	#126×C₄₇	—	2	16	no	—
31	#124×C₄₇	—	2	4	no	—
32	#124×C₄₇	—	2	4	no	—
33	C₂²⋊Q₈×C₄₇	—	2	8	no	—
34	C₄×Q₈×C₄₇	—	2	4	no	—
35	C₄×Q₈×C₄₇	—	2	4	no	—
36	C₄×Q₈×C₄₇	—	2	2	no	—
37	C₄×Q₈×C₄₇	—	2	2	no	—
38	C₄×D₄×C₄₇	—	2	2	no	—
39	C₄×D₄×C₄₇	D₄	1	4	yes	—
40	#117×C₄₇	—	2	8	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	2	no	—
54	C₄⋊C₈×C₄₇	—	2	2	no	—
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	#101×C₄₇	—	4	2	no	—
57	C₂²⋊C₈×C₄₇	—	2	2	no	—
58	C₄×C₈×C₄₇	—	2	2	no	—
59	C₂⁴×C₄₇	—	2	1	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	1	no	—
64	C₂×Q₈×C₄₇	—	2	1	no	—
65	C₂×Q₈×C₄₇	—	4	4	no	—
66	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₂×D₄×C₄₇	—	2	1	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	—	2	1	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	M₁₆×C₄₇	—	4	2	no	—
82	M₁₆×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	1	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
92	C₂²⋊C₄×C₄₇	—	2	8	no	—
93	C₂²⋊C₄×C₄₇	—	2	8	no	—
94	C₂²⋊C₄×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	8	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	C₂×D₄	1	2	yes	—
99	C₄²×C₄₇	C₂×D₄	1	2	yes	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	1	no	—
104	C₂³×C₄₇	—	4	1	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈×C₄₇	—	4	4	no	—
107	Q₈×C₄₇	—	4	4	no	—
108	Q₈×C₄₇	—	4	2	no	—
109	Q₈×C₄₇	—	4	2	no	—
110	D₄×C₄₇	—	2	4	no	—
111	D₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	1	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#120	1	2	yes	—
129	C₂×C₄×C₄₇	#120	1	2	yes	—
130	C₂×C₄×C₄₇	#135	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
133	C₈×C₄₇	—	8	2	no	—
134	C₈×C₄₇	—	8	2	no	—
135	#8394	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	4	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	—	8	2	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	#379	1	1	yes	characteristic
144	C₂²×C₄₇	#416	1	1	yes	characteristic
145	C₂²×C₄₇	#417	1	1	yes	characteristic
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	—	4	2	no	—
152	C₄×C₄₇	#413	1	2	yes	—
153	C₄×C₄₇	#449	1	2	yes	—
154	C₄×C₄₇	#452	1	2	yes	—
155	C₂×#556	C₂×C₄₇	1	1	yes	characteristic
156	#1801	C₂×C₄₇	1	1	yes	characteristic, complemented
157	#1548	C₂×C₄₇	1	2	yes	complemented
158	#1486	C₂×C₄₇	1	2	yes	—
159	#1461	C₂×C₄₇	1	1	yes	characteristic, complemented
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	—	8	2	no	—
163	C₂×C₄₇	#1785	1	1	yes	characteristic
164	C₂×C₄₇	#1788	1	1	yes	characteristic
165	C₂×C₄₇	#1796	1	1	yes	characteristic
166	#556	—	2	4	no	—
167	#556	—	2	4	no	—
168	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
169	C₂×#126	—	2	2	no	—
170	C₂×#124	C₄×C₄₇	1	2	yes	—
171	C₂×C₂²⋊Q₈	—	2	1	no	—
172	C₂×C₄×Q₈	C₄×C₄₇	1	2	yes	—
173	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
175	C₂×#117	—	2	1	no	—
176	#431	C₂²×C₄₇	1	2	yes	—
177	#423	C₂²×C₄₇	1	2	yes	—
178	C₂×#101	—	2	1	no	—
179	#343	—	2	1	no	—
180	#333	—	2	4	no	—
181	D₄⋊C₈	—	2	4	no	—
182	#324	—	2	2	no	—
183	C₄₇	#8394	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×Q₈	C₂×C₄×C₄₇	1	2	yes	—
185	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
186	C₂²×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	D₄×C₄₇	1	4	yes	—
191	#126	—	2	16	no	—
192	#124	—	2	4	no	—
193	#124	—	2	4	no	—
194	C₂²⋊Q₈	—	2	8	no	—
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	D₄×C₄₇	1	4	yes	—
201	#117	—	2	8	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₄₇	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	#101	—	4	2	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	C₂²⋊C₄×C₄₇	1	4	yes	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	C₂²⋊C₄×C₄₇	1	2	yes	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
238	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	C₂×D₄×C₄₇	1	2	yes	—
260	C₄²	C₂×D₄×C₄₇	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
289	C₂×C₄	#120×C₄₇	1	2	yes	—
290	C₂×C₄	#120×C₄₇	1	2	yes	—
291	C₂×C₄	#135×C₄₇	1	2	yes	—
292	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
293	C₂×C₄	#137×C₄₇	1	1	yes	characteristic
294	C₈	—	8	2	no	—
295	C₈	—	8	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₄₇×#379	1	1	yes	characteristic
304	C₂²	C₄₇×#416	1	1	yes	characteristic
305	C₂²	C₄₇×#417	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	C₄₇×#413	1	2	yes	—
313	C₄	C₄₇×#449	1	2	yes	—
314	C₄	C₄₇×#452	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	2	no	—
318	C₂	C₄₇×#1785	1	1	yes	characteristic
319	C₂	C₄₇×#1788	1	1	yes	characteristic
320	C₂	C₄₇×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₂²×D₄×C₄₇	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#1801	C₂	semidirect
3. C₄₇×#1548	C₂	semidirect
5. C₄₇×#1461	C₂	semidirect
135. #8394	C₄₇	direct (N × H)
156. #1801	C₂×C₄₇	semidirect
157. #1548	C₂×C₄₇	semidirect
159. #1461	C₂×C₄₇	semidirect
183. C₄₇	#8394	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×#556	2
C₄₇×#1801	2
C₄₇×#1548	2
C₄₇×#1486	2
C₄₇×#1461	2
#8394	47

Subgroups — C47×#8394 #26048092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#8394 #26048092