Subgroups — C₄₇×#9197 #26048895

Order 12032.

624 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#2619	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1545	C₂	1	2	yes	maximal, complemented
4	C₄₇×#1458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1286	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₄₇	—	2	1	no	—
7	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×#121×C₄₇	—	2	1	no	—
9	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#431	C₂²	1	2	yes	—
11	C₄₇×#424	—	2	4	no	—
12	C₄₇×#423	—	2	1	no	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₄₇×#394	—	2	1	no	—
15	C₄₇×#394	C₄	1	2	yes	—
16	C₄₇×#392	—	2	1	no	—
17	C₄₇×#392	C₄	1	2	yes	—
18	C₄₇×#390	—	2	1	no	—
19	C₄₇×#382	C₂²	1	1	yes	characteristic
20	C₄₇×#331	—	2	4	no	—
21	C₄₇×#324	—	2	1	no	—
22	C₄₇×#323	—	2	1	no	—
23	C₂²×D₄×C₄₇	—	2	2	no	—
24	C₂²×D₄×C₄₇	—	2	1	no	—
25	C₂²×D₄×C₄₇	—	2	1	no	—
26	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂×M₁₆×C₄₇	—	4	2	no	—
31	C₂×M₁₆×C₄₇	—	4	2	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	#127×C₄₇	—	2	2	no	—
34	#127×C₄₇	—	2	2	no	—
35	#121×C₄₇	—	4	2	no	—
36	#121×C₄₇	—	4	2	no	—
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₄⋊C₈×C₄₇	—	2	2	no	—
51	C₄⋊C₈×C₄₇	—	2	2	no	—
52	C₄⋊C₈×C₄₇	D₄	1	4	yes	complemented
53	C₂²⋊C₈×C₄₇	—	2	2	no	—
54	C₄×C₈×C₄₇	—	2	2	no	—
55	#95×C₄₇	—	2	1	no	—
56	#95×C₄₇	C₂×C₄	1	2	yes	—
57	C₂⁴×C₄₇	—	2	2	no	—
58	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₄×C₄₇	—	2	4	no	—
60	C₂×D₄×C₄₇	—	4	4	no	—
61	C₂×D₄×C₄₇	—	4	1	no	—
62	C₂×D₄×C₄₇	—	4	2	no	—
63	C₂×D₄×C₄₇	—	4	1	no	—
64	C₂×D₄×C₄₇	—	4	1	no	—
65	C₂×D₄×C₄₇	—	4	1	no	—
66	C₂×D₄×C₄₇	—	4	1	no	—
67	C₂×D₄×C₄₇	—	4	1	no	—
68	C₂×D₄×C₄₇	—	8	2	no	—
69	C₂×D₄×C₄₇	—	8	2	no	—
70	C₂×D₄×C₄₇	—	8	2	no	—
71	C₂×D₄×C₄₇	—	8	2	no	—
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	—	2	1	no	—
75	C₂²×C₄×C₄₇	—	2	1	no	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	D₈	1	2	yes	—
81	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₄₇	—	4	4	no	—
86	M₁₆×C₄₇	—	4	2	no	—
87	M₁₆×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	—	2	4	no	—
89	C₂×C₈×C₄₇	—	4	4	no	—
90	C₂×C₈×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	4	1	no	—
94	C₄⋊C₄×C₄₇	—	4	1	no	—
95	C₂²⋊C₄×C₄₇	—	2	2	no	—
96	C₂²⋊C₄×C₄₇	—	2	2	no	—
97	C₂²⋊C₄×C₄₇	—	4	4	no	—
98	C₂²⋊C₄×C₄₇	—	8	2	no	—
99	C₂²⋊C₄×C₄₇	—	8	2	no	—
100	C₄²×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	—	2	1	no	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	#41	1	2	yes	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	4	1	no	—
107	C₂³×C₄₇	—	4	1	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	—	4	2	no	—
110	C₂³×C₄₇	—	4	2	no	—
111	C₂³×C₄₇	—	4	2	no	—
112	C₂³×C₄₇	—	4	2	no	—
113	C₂³×C₄₇	—	4	2	no	—
114	C₂³×C₄₇	—	4	2	no	—
115	C₂³×C₄₇	#102	1	2	yes	—
116	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	D₄×C₄₇	—	8	4	no	—
118	D₄×C₄₇	—	8	4	no	—
119	D₄×C₄₇	—	8	8	no	—
120	D₄×C₄₇	—	8	1	no	—
121	D₄×C₄₇	—	8	1	no	—
122	D₄×C₄₇	—	8	1	no	—
123	D₄×C₄₇	—	8	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	#102	1	2	yes	—
140	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#121	1	2	yes	—
143	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
148	C₈×C₄₇	—	4	4	no	—
149	C₈×C₄₇	—	8	2	no	—
150	#9197	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	—	4	1	no	—
156	C₂²×C₄₇	—	4	1	no	—
157	C₂²×C₄₇	—	8	1	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	—	8	2	no	—
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	—	8	2	no	—
165	C₂²×C₄₇	#386	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
167	C₂²×C₄₇	#417	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	—	4	1	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#447	1	1	yes	characteristic
175	C₄×C₄₇	#448	1	1	yes	characteristic
176	C₄×C₄₇	#480	1	1	yes	characteristic
177	C₄×C₄₇	#481	1	1	yes	characteristic
178	#2905	C₂×C₄₇	1	1	yes	characteristic, complemented
179	#2619	C₂×C₄₇	1	1	yes	characteristic
180	#1545	C₂×C₄₇	1	2	yes	complemented
181	#1458	C₂×C₄₇	1	1	yes	characteristic, complemented
182	#1286	C₂×C₄₇	1	2	yes	complemented
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	—	4	2	no	—
186	C₂×C₄₇	—	16	2	no	—
187	C₂×C₄₇	#1874	1	1	yes	characteristic
188	C₂×C₄₇	#1910	1	1	yes	characteristic
189	C₂×C₄₇	#1914	1	1	yes	characteristic
190	C₂²×M₁₆	—	2	1	no	—
191	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
192	C₂×#121	—	2	1	no	—
193	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
194	#431	C₂²×C₄₇	1	2	yes	—
195	#424	—	2	4	no	—
196	#423	—	2	1	no	—
197	#423	C₂²×C₄₇	1	2	yes	—
198	#394	—	2	1	no	—
199	#394	C₄×C₄₇	1	2	yes	—
200	#392	—	2	1	no	—
201	#392	C₄×C₄₇	1	2	yes	—
202	#390	—	2	1	no	—
203	#382	C₂²×C₄₇	1	1	yes	characteristic
204	#331	—	2	4	no	—
205	#324	—	2	1	no	—
206	#323	—	2	1	no	—
207	C₄₇	#9197	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	C₂×C₄×C₄₇	1	2	yes	—
212	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂×M₁₆	—	4	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	#127	—	2	2	no	—
219	#127	—	2	2	no	—
220	#121	—	4	2	no	—
221	#121	—	4	2	no	—
222	#117	—	2	2	no	—
223	#117	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
234	C₄⋊C₈	—	2	4	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	D₄×C₄₇	1	4	yes	complemented
238	C₂²⋊C₈	—	2	2	no	—
239	C₄×C₈	—	2	2	no	—
240	#95	—	2	1	no	—
241	#95	C₂×C₄×C₄₇	1	2	yes	—
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂×D₄	—	8	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
265	C₂²×C₄	D₈×C₄₇	1	2	yes	—
266	C₂²×C₄	SD₁₆×C₄₇	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
270	M₁₆	—	4	4	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×D₄×C₄₇	1	2	yes	—
288	C₄²	#41×C₄₇	1	2	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	#102×C₄₇	1	2	yes	—
301	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
302	D₄	—	8	4	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#102×C₄₇	1	2	yes	—
325	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
326	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
327	C₂×C₄	#121×C₄₇	1	2	yes	—
328	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
331	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
332	C₂×C₄	#137×C₄₇	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	8	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₄₇×#386	1	1	yes	characteristic
350	C₂²	C₂×#102×C₄₇	1	1	yes	characteristic
351	C₂²	C₄₇×#417	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	8	2	no	—
358	C₄	C₄₇×#447	1	1	yes	characteristic
359	C₄	C₄₇×#448	1	1	yes	characteristic
360	C₄	C₄₇×#480	1	1	yes	characteristic
361	C₄	C₄₇×#481	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₄₇×#1874	1	1	yes	characteristic
367	C₂	C₄₇×#1910	1	1	yes	characteristic
368	C₂	C₄₇×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₄₇×#2619	C₂
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
1. C₄₇×#2905	C₂	semidirect
3. C₄₇×#1545	C₂	semidirect
4. C₄₇×#1458	C₂	semidirect
5. C₄₇×#1286	C₂	semidirect
52. C₄⋊C₈×C₄₇	D₄	semidirect
150. #9197	C₄₇	direct (N × H)
178. #2905	C₂×C₄₇	semidirect
180. #1545	C₂×C₄₇	semidirect
181. #1458	C₂×C₄₇	semidirect
182. #1286	C₂×C₄₇	semidirect
207. C₄₇	#9197	direct (N × H)
237. C₄⋊C₈	D₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#2905	2
C₄₇×#2619	2
C₄₇×#1545	2
C₄₇×#1458	2
C₄₇×#1286	2
#9197	47

Subgroups — C47×#9197 #26048895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#9197 #26048895