Subgroups — C₄₇×#19054 #26058752

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3235	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3150	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1660	C₂	1	4	yes	maximal, complemented
5	C₄₇×#1657	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1531	C₂	1	2	yes	maximal, complemented
8	C₂×#135×C₄₇	—	2	2	no	—
9	C₄₇×#563	C₂²	1	4	yes	—
10	C₄₇×#559	—	2	1	no	—
11	Q₈²×C₄₇	—	2	1	no	—
12	C₄₇×#556	—	2	2	no	—
13	D₄×Q₈×C₄₇	—	2	2	no	—
14	C₄₇×#542	—	2	4	no	—
15	C₄₇×#536	—	2	2	no	—
16	C₄₇×#533	C₂²	1	2	yes	—
17	C₄₇×#532	C₂²	1	1	yes	characteristic
18	C₄₇×#532	C₂²	1	1	yes	characteristic
19	C₄₇×#501	C₂²	1	4	yes	—
20	C₄₇×#497	—	2	1	no	—
21	C₄₇×#497	C₂²	1	2	yes	—
22	C₄₇×#496	—	2	1	no	—
23	C₄₇×#496	C₂²	1	2	yes	—
24	C₄₇×#490	C₂²	1	4	yes	—
25	C₄₇×#482	—	2	4	no	—
26	C₄₇×#476	—	2	2	no	—
27	C₄₇×#474	—	2	2	no	—
28	C₄₇×#432	C₂²	1	2	yes	—
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#330	C₂²	1	8	yes	—
31	C₄₇×#329	C₂²	1	4	yes	—
32	C₂×#41×C₄₇	—	2	1	no	—
33	C₂×#41×C₄₇	D₄	1	2	yes	—
34	C₂²×Q₈×C₄₇	—	2	2	no	—
35	#135×C₄₇	—	4	8	no	—
36	C₂×Q₁₆×C₄₇	—	2	2	no	—
37	C₂×SD₁₆×C₄₇	—	2	2	no	—
38	C₂×SD₁₆×C₄₇	—	2	2	no	—
39	C₂×D₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	—	2	1	no	—
41	C₄⋊Q₈×C₄₇	—	2	4	no	—
42	C₄⋊Q₈×C₄₇	—	2	4	no	—
43	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
46	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
47	#127×C₄₇	D₄	1	1	yes	characteristic
48	#127×C₄₇	C₂³	1	1	yes	characteristic
49	#126×C₄₇	—	2	8	no	—
50	#125×C₄₇	C₂³	1	4	yes	—
51	#124×C₄₇	—	2	4	no	—
52	#124×C₄₇	—	2	4	no	—
53	#124×C₄₇	—	2	1	no	—
54	#124×C₄₇	C₂³	1	2	yes	—
55	#123×C₄₇	—	2	4	no	—
56	#123×C₄₇	—	2	4	no	—
57	C₂²⋊Q₈×C₄₇	—	2	4	no	—
58	C₂²⋊Q₈×C₄₇	—	2	2	no	—
59	C₂²⋊Q₈×C₄₇	—	2	2	no	—
60	C₂²⋊Q₈×C₄₇	—	2	8	no	—
61	#121×C₄₇	—	2	2	no	—
62	#121×C₄₇	—	2	2	no	—
63	C₄×Q₈×C₄₇	—	2	4	no	—
64	C₄×Q₈×C₄₇	—	2	2	no	—
65	C₄×Q₈×C₄₇	—	2	2	no	—
66	C₄×D₄×C₄₇	—	2	4	no	—
67	C₄×D₄×C₄₇	—	2	2	no	—
68	C₄×D₄×C₄₇	—	2	2	no	—
69	#117×C₄₇	—	2	2	no	—
70	#117×C₄₇	D₄	1	4	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	—	2	4	no	—
73	#106×C₄₇	—	2	4	no	—
74	C₄⋊C₈×C₄₇	—	2	2	no	—
75	#102×C₄₇	—	2	4	no	—
76	#102×C₄₇	—	2	4	no	—
77	C₂²⋊C₈×C₄₇	—	2	2	no	—
78	#97×C₄₇	—	2	1	no	—
79	#97×C₄₇	C₂³	1	4	yes	—
80	#97×C₄₇	C₂³	1	2	yes	—
81	#41×C₄₇	—	4	4	no	—
82	#41×C₄₇	—	4	4	no	—
83	C₂×Q₈×C₄₇	—	2	4	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×Q₈×C₄₇	—	2	2	no	—
86	C₂×Q₈×C₄₇	—	2	1	no	—
87	C₂×Q₈×C₄₇	—	8	4	no	—
88	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×D₄×C₄₇	—	2	1	no	—
93	C₂×D₄×C₄₇	—	4	1	no	—
94	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₇	—	2	4	no	—
96	C₂²×C₄×C₄₇	—	2	2	no	—
97	C₂²×C₄×C₄₇	—	4	1	no	—
98	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
100	Q₁₆×C₄₇	—	4	4	no	—
101	SD₁₆×C₄₇	—	4	4	no	—
102	SD₁₆×C₄₇	—	4	4	no	—
103	D₈×C₄₇	—	4	4	no	—
104	C₂×C₈×C₄₇	—	2	4	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	—	2	4	no	—
107	C₄⋊C₄×C₄₇	—	2	8	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	4	no	—
112	C₄⋊C₄×C₄₇	—	2	4	no	—
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
116	C₂²⋊C₄×C₄₇	—	2	4	no	—
117	C₂²⋊C₄×C₄₇	—	2	8	no	—
118	C₂²⋊C₄×C₄₇	—	2	4	no	—
119	C₂²⋊C₄×C₄₇	—	2	4	no	—
120	C₂²⋊C₄×C₄₇	—	4	2	no	—
121	C₄²×C₄₇	—	2	4	no	—
122	C₄²×C₄₇	—	2	4	no	—
123	C₄²×C₄₇	C₂×D₄	1	2	yes	—
124	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
126	C₂³×C₄₇	—	2	2	no	—
127	C₂³×C₄₇	—	4	1	no	—
128	C₂³×C₄₇	#120	1	1	yes	characteristic
129	Q₈×C₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	4	4	no	—
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	4	4	no	—
133	Q₈×C₄₇	—	8	4	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	4	4	no	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	4	no	—
138	D₄×C₄₇	—	8	2	no	—
139	D₄×C₄₇	—	8	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	4	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	4	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	4	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	8	4	no	—
148	C₂×C₄×C₄₇	—	8	2	no	—
149	C₂×C₄×C₄₇	#120	1	2	yes	—
150	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₄₇	—	2	4	no	—
154	C₈×C₄₇	—	4	4	no	—
155	#19054	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₄₇	—	2	1	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	4	2	no	—
159	C₂²×C₄₇	—	4	4	no	—
160	C₂²×C₄₇	—	8	1	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	4	2	no	—
167	C₄×C₄₇	—	4	4	no	—
168	C₄×C₄₇	—	4	4	no	—
169	C₄×C₄₇	—	8	1	no	—
170	C₄×C₄₇	#453	1	2	yes	—
171	#3550	C₂×C₄₇	1	1	yes	characteristic, complemented
172	#3235	C₂×C₄₇	1	2	yes	complemented
173	#3150	C₂×C₄₇	1	1	yes	characteristic
174	#1660	C₂×C₄₇	1	4	yes	complemented
175	#1657	C₂×C₄₇	1	4	yes	complemented
176	#1536	C₂×C₄₇	1	1	yes	characteristic, complemented
177	#1531	C₂×C₄₇	1	2	yes	complemented
178	C₂×C₄₇	—	4	1	no	—
179	C₂×C₄₇	—	8	2	no	—
180	C₂×C₄₇	—	16	1	no	—
181	C₂×C₄₇	C₂×#453	1	1	yes	characteristic
182	C₂×C₄₇	#3020	1	2	yes	—
183	C₂×#135	—	2	2	no	—
184	#563	C₂²×C₄₇	1	4	yes	—
185	#559	—	2	1	no	—
186	Q₈²	—	2	1	no	—
187	#556	—	2	2	no	—
188	D₄×Q₈	—	2	2	no	—
189	#542	—	2	4	no	—
190	#536	—	2	2	no	—
191	#533	C₂²×C₄₇	1	2	yes	—
192	#532	C₂²×C₄₇	1	1	yes	characteristic
193	#532	C₂²×C₄₇	1	1	yes	characteristic
194	#501	C₂²×C₄₇	1	4	yes	—
195	#497	—	2	1	no	—
196	#497	C₂²×C₄₇	1	2	yes	—
197	#496	—	2	1	no	—
198	#496	C₂²×C₄₇	1	2	yes	—
199	#490	C₂²×C₄₇	1	4	yes	—
200	#482	—	2	4	no	—
201	#476	—	2	2	no	—
202	#474	—	2	2	no	—
203	#432	C₂²×C₄₇	1	2	yes	—
204	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
205	#330	C₂²×C₄₇	1	8	yes	—
206	#329	C₂²×C₄₇	1	4	yes	—
207	C₄₇	#19054	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×#41	—	2	1	no	—
209	C₂×#41	D₄×C₄₇	1	2	yes	—
210	C₂²×Q₈	—	2	2	no	—
211	#135	—	4	8	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	D₄×C₄₇	1	1	yes	characteristic
220	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
221	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
222	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
223	#127	D₄×C₄₇	1	1	yes	characteristic
224	#127	C₂³×C₄₇	1	1	yes	characteristic
225	#126	—	2	8	no	—
226	#125	C₂³×C₄₇	1	4	yes	—
227	#124	—	2	4	no	—
228	#124	—	2	4	no	—
229	#124	—	2	1	no	—
230	#124	C₂³×C₄₇	1	2	yes	—
231	#123	—	2	4	no	—
232	#123	—	2	4	no	—
233	C₂²⋊Q₈	—	2	4	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	8	no	—
237	#121	—	2	2	no	—
238	#121	—	2	2	no	—
239	C₄×Q₈	—	2	4	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×D₄	—	2	4	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	#117	—	2	2	no	—
246	#117	D₄×C₄₇	1	4	yes	—
247	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
248	#107	—	2	4	no	—
249	#106	—	2	4	no	—
250	C₄⋊C₈	—	2	2	no	—
251	#102	—	2	4	no	—
252	#102	—	2	4	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	#97	—	2	1	no	—
255	#97	C₂³×C₄₇	1	4	yes	—
256	#97	C₂³×C₄₇	1	2	yes	—
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	8	4	no	—
264	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	4	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	4	no	—
283	C₄⋊C₄	—	2	8	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×D₄×C₄₇	1	4	yes	—
291	C₄⋊C₄	C₂×D₄×C₄₇	1	4	yes	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	8	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	4	no	—
299	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₄₇	1	2	yes	—
301	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	#120×C₄₇	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	8	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	#120×C₄₇	1	2	yes	—
326	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	1	no	—
345	C₄	C₄₇×#453	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	8	2	no	—
348	C₂	—	16	1	no	—
349	C₂	C₂×C₄₇×#453	1	1	yes	characteristic
350	C₂	C₄₇×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#3150	C₂
LCS	G ▷ C₄² ▷ 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×#3550	C₂	semidirect
2. C₄₇×#3235	C₂	semidirect
4. C₄₇×#1660	C₂	semidirect
5. C₄₇×#1657	C₂	semidirect
6. C₄₇×#1536	C₂	semidirect
7. C₄₇×#1531	C₂	semidirect
155. #19054	C₄₇	direct (N × H)
171. #3550	C₂×C₄₇	semidirect
172. #3235	C₂×C₄₇	semidirect
174. #1660	C₂×C₄₇	semidirect
175. #1657	C₂×C₄₇	semidirect
176. #1536	C₂×C₄₇	semidirect
177. #1531	C₂×C₄₇	semidirect
207. C₄₇	#19054	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×#3550	2
C₄₇×#3235	2
C₄₇×#3150	2
C₄₇×#1660	2
C₄₇×#1657	2
C₄₇×#1536	2
C₄₇×#1531	2
#19054	47

Subgroups — C47×#19054 #26058752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19054 #26058752