Subgroups — #26126187

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245186	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245185	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241344	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241342	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	—	2	1	no	—
9	C₄×D₄×D₄₇	C₄	1	1	yes	characteristic
10	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
11	D₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	#15689188	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₄₇	—	2	1	no	—
14	(C₂²⋊D₄₇)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×D₄₇	C₄	1	1	yes	characteristic
16	C₂²⋊C₈×D₄₇	—	2	1	no	—
17	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
19	Dic₄₇⋊C₁₆	—	2	1	no	—
20	(C₂×D₄₇)⋊C₁₆	—	2	1	no	—
21	(C₂×D₄₇)⋊C₁₆	—	2	1	no	—
22	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₁₆×Dic₄₇	—	2	1	no	—
24	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₄₇	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂×C₈×D₄₇	—	2	1	no	—
31	C₂×C₈×D₄₇	—	2	1	no	—
32	C₂×C₈×D₄₇	—	2	1	no	—
33	C₂×C₈×D₄₇	—	4	2	no	—
34	C₂×C₈×D₄₇	—	4	2	no	—
35	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
36	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
37	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₄₇	C₈	1	1	yes	characteristic
39	#13093986	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₄₇	—	2	1	no	—
41	(C₂²⋊D₄₇)⋊C₄	—	2	1	no	—
42	C₄⋊C₄×D₄₇	C₈	1	1	yes	characteristic
43	C₂²⋊C₄×D₄₇	—	2	1	no	—
44	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄₇⋊C₁₆	—	2	1	no	—
46	C₂×C₄₇⋊C₁₆	—	2	1	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	Dic₄₇⋊C₈	—	2	1	no	—
49	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
50	(C₂×D₄₇)⋊C₈	—	2	1	no	—
51	(C₂×D₄₇)⋊C₈	—	2	1	no	—
52	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₄₇	D₄	1	1	yes	characteristic
54	C₈×Dic₄₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	—	2	1	no	—
56	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₄₇	—	2	1	no	—
58	C₂×C₁₆×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₈×C₄₇	—	2	1	no	—
61	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂³×D₄₇	—	2	1	no	—
63	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂×C₂²⋊D₄₇	—	2	1	no	—
66	C₂²×Dic₄₇	—	2	1	no	—
67	D₄×D₄₇	—	2	2	no	—
68	D₄×D₄₇	C₁₆	1	4	yes	complemented
69	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×C₄×D₄₇	—	2	1	no	—
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×C₄×D₄₇	—	4	2	no	—
74	C₂×C₄×D₄₇	—	4	2	no	—
75	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	4	1	no	—
80	C₂×C₈×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊C₄×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₄₇	—	2	1	no	—
87	(C₂×D₄₇)⋊C₄	—	2	1	no	—
88	(C₂×D₄₇)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
90	Dic₄₇⋊C₄	—	2	1	no	—
91	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₄₇	M₁₆	1	1	yes	characteristic
94	C₂×C₄₇⋊C₈	—	2	1	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₈×D₄₇	—	2	1	no	—
99	C₈×D₄₇	—	2	1	no	—
100	C₈×D₄₇	—	4	1	no	—
101	C₈×D₄₇	—	4	2	no	—
102	C₈×D₄₇	—	4	2	no	—
103	C₈×D₄₇	—	4	1	no	—
104	C₈×D₄₇	—	4	1	no	—
105	C₁₆×C₄₇	—	4	1	no	—
106	C₁₆×C₄₇	—	4	1	no	—
107	C₄₇⋊C₁₆	—	4	1	no	—
108	C₄₇⋊C₁₆	—	4	1	no	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂²×D₄₇	—	2	1	no	—
111	C₂²×D₄₇	—	2	4	no	—
112	C₂²×D₄₇	—	4	2	no	—
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
115	D₄×C₄₇	—	2	1	no	—
116	D₄×C₄₇	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	—	4	1	no	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
123	C₂²⋊D₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	1	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
129	D₁₈₈	—	2	1	no	—
130	D₁₈₈	C₂×C₁₆	1	2	yes	—
131	C₄×D₄₇	—	2	1	no	—
132	C₄×D₄₇	—	2	1	no	—
133	C₄×D₄₇	—	4	1	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	C₄×D₄₇	—	4	1	no	—
137	C₄×D₄₇	—	4	1	no	—
138	C₄×D₄₇	C₂×C₁₆	1	2	yes	—
139	C₄×D₄₇	M₃₂	1	2	yes	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	—	4	1	no	—
142	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₄₇⋊C₈	—	2	1	no	—
145	C₄₇⋊C₈	—	2	1	no	—
146	C₄₇⋊C₈	—	4	1	no	—
147	#11629	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	4	1	no	—
150	C₂²×C₄₇	—	4	1	no	—
151	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×D₄₇	—	2	1	no	—
153	C₂×D₄₇	—	2	1	no	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	4	8	no	—
156	C₂×D₄₇	—	4	1	no	—
157	C₂×D₄₇	—	4	1	no	—
158	C₂×D₄₇	C₂²⋊C₁₆	1	2	yes	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	#418	1	1	yes	characteristic
162	C₄×C₄₇	#421	1	1	yes	characteristic
163	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
164	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
165	Dic₄₇	—	2	1	no	—
166	Dic₄₇	—	2	1	no	—
167	Dic₄₇	—	4	1	no	—
168	Dic₄₇	C₂²⋊C₁₆	1	2	yes	—
169	C₂×C₈×D₄	—	47	1	no	—
170	#2162	—	47	1	no	—
171	#2162	—	47	1	no	—
172	D₄⋊C₁₆	—	47	1	no	—
173	D₄⋊C₁₆	—	47	1	no	—
174	D₄⋊C₁₆	—	47	1	no	—
175	D₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
176	C₂×C₄₇	—	4	2	no	—
177	C₂×C₄₇	#1478	1	1	yes	characteristic
178	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
179	C₂×C₄₇	#2117	1	1	yes	characteristic
180	D₄₇	—	2	2	no	—
181	D₄₇	—	4	2	no	—
182	C₂³×C₈	—	94	1	no	—
183	C₂×C₄×D₄	—	47	1	no	—
184	C₈×D₄	—	47	1	no	—
185	C₈×D₄	—	47	1	no	—
186	C₈×D₄	—	47	1	no	—
187	C₈×D₄	—	94	1	no	—
188	C₈×D₄	—	94	1	no	—
189	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	47	1	no	—
191	C₂×C₂²⋊C₈	—	94	1	no	—
192	C₂×C₄×C₈	—	47	1	no	—
193	C₄⋊C₁₆	—	47	1	no	—
194	C₄⋊C₁₆	—	94	1	no	—
195	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
196	C₂²⋊C₁₆	—	94	1	no	—
197	C₂²⋊C₁₆	—	94	1	no	—
198	C₄×C₁₆	—	47	1	no	—
199	C₄×C₁₆	—	94	1	no	—
200	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
201	C₄₇	#11629	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	47	1	no	—
203	C₂³×C₄	—	94	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	47	1	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	94	1	no	—
208	C₂²×C₈	—	94	1	no	—
209	C₂²×C₈	—	94	1	no	—
210	C₂²×C₈	—	94	1	no	—
211	C₂²×C₈	—	188	2	no	—
212	C₂²×C₈	—	188	2	no	—
213	C₄×D₄	—	47	1	no	—
214	C₄×D₄	—	47	1	no	—
215	C₄×D₄	—	47	1	no	—
216	C₄×D₄	—	94	1	no	—
217	C₄×D₄	—	94	1	no	—
218	C₄×D₄	C₄×D₄₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	47	1	no	—
220	C₂×C₂²⋊C₄	—	94	1	no	—
221	C₂×C₄²	—	47	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	94	1	no	—
225	C₂×C₁₆	—	94	1	no	—
226	C₄⋊C₈	—	47	1	no	—
227	C₄⋊C₈	—	94	1	no	—
228	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	94	1	no	—
231	C₂²⋊C₈	—	94	1	no	—
232	C₂²⋊C₈	—	94	1	no	—
233	C₄×C₈	—	47	1	no	—
234	C₄×C₈	—	47	1	no	—
235	C₄×C₈	—	47	1	no	—
236	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₂⁴	—	94	1	no	—
238	C₂×D₄	—	47	4	no	—
239	C₂×D₄	—	47	1	no	—
240	C₂×D₄	—	94	2	no	—
241	C₂×D₄	—	94	1	no	—
242	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	47	1	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	94	1	no	—
248	C₂²×C₄	—	94	1	no	—
249	C₂²×C₄	—	94	1	no	—
250	C₂²×C₄	—	94	1	no	—
251	C₂²×C₄	—	188	2	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	47	1	no	—
257	C₂×C₈	—	47	1	no	—
258	C₂×C₈	—	94	1	no	—
259	C₂×C₈	—	94	1	no	—
260	C₂×C₈	—	94	1	no	—
261	C₂×C₈	—	188	1	no	—
262	C₂×C₈	—	188	2	no	—
263	C₂×C₈	—	188	2	no	—
264	C₂×C₈	—	188	2	no	—
265	C₂×C₈	—	188	1	no	—
266	C₂×C₈	—	188	1	no	—
267	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
269	C₄⋊C₄	—	47	1	no	—
270	C₄⋊C₄	—	94	1	no	—
271	C₄⋊C₄	C₈×D₄₇	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	94	1	no	—
274	C₂²⋊C₄	—	94	1	no	—
275	C₂²⋊C₄	—	94	1	no	—
276	C₄²	—	47	1	no	—
277	C₄²	—	47	1	no	—
278	C₄²	—	47	1	no	—
279	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₁₆	—	4	1	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	188	1	no	—
283	C₁₆	—	188	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	47	1	no	—
286	C₂³	—	94	4	no	—
287	C₂³	—	94	1	no	—
288	C₂³	—	188	2	no	—
289	C₂³	—	188	2	no	—
290	D₄	—	2	1	no	—
291	D₄	—	47	2	no	—
292	D₄	—	94	4	no	—
293	D₄	—	94	1	no	—
294	D₄	C₁₆×D₄₇	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	47	1	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	1	no	—
302	C₂×C₄	—	47	1	no	—
303	C₂×C₄	—	94	1	no	—
304	C₂×C₄	—	94	1	no	—
305	C₂×C₄	—	94	1	no	—
306	C₂×C₄	—	188	1	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	1	no	—
311	C₂×C₄	—	188	1	no	—
312	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
314	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	94	1	no	—
318	C₈	—	94	1	no	—
319	C₈	—	188	1	no	—
320	C₈	#13094048	1	1	yes	characteristic
321	C₈	#13094051	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	47	2	no	—
326	C₂²	—	94	1	no	—
327	C₂²	—	94	1	no	—
328	C₂²	—	94	2	no	—
329	C₂²	—	188	8	no	—
330	C₂²	—	188	1	no	—
331	C₂²	—	188	1	no	—
332	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	47	2	no	—
336	C₄	—	94	1	no	—
337	C₄	—	94	1	no	—
338	C₄	—	188	1	no	—
339	C₄	#15689126	1	1	yes	characteristic
340	C₄	#15689138	1	1	yes	characteristic
341	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
342	C₄	M₃₂×D₄₇	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	94	2	no	—
345	C₂	—	188	2	no	—
346	C₂	#20241777	1	1	yes	characteristic
347	C₂	C₂²⋊C₁₆×D₄₇	1	1	yes	characteristic
348	C₂	#20245014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×D₄⋊C₁₆	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂×D₄×D₄₇	C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	D₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#11629
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20245186	C₂	semidirect
3. #20245185	C₂	semidirect
4. #20241344	C₂	semidirect
5. D₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. #20241342	C₂	semidirect
7. C₄₇×D₄⋊C₁₆	C₂	semidirect
18. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
22. C₄×C₄₇⋊C₁₆	C₂²	semidirect
25. C₄₇×C₄⋊C₁₆	C₂²	semidirect
26. C₄×C₁₆×C₄₇	C₂²	semidirect
68. D₄×D₄₇	C₁₆	semidirect
175. D₄⋊C₁₆	D₄₇	semidirect
195. C₄⋊C₁₆	C₂×D₄₇	semidirect
200. C₄×C₁₆	C₂×D₄₇	semidirect
201. C₄₇	#11629	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₄₇	2
#20245186	2
#20245185	2
#20241344	2
D₄⋊(C₄₇⋊C₁₆)	2
#20241342	2
C₄₇×D₄⋊C₁₆	2
#11629	47