Subgroups — #26216822

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245781	C₂	1	2	yes	maximal
3	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243518	C₂	1	2	yes	maximal, complemented
5	#20243414	C₂	1	2	yes	maximal
6	#20243410	C₂	1	2	yes	maximal
7	#107×Dic₄₇	C₂	1	2	yes	maximal, complemented
8	#106×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
19	#15688990	—	2	2	no	—
20	#15688989	C₂²	1	2	yes	—
21	#15688985	—	2	2	no	—
22	#15688980	C₂²	1	4	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
24	#15688970	—	2	2	no	—
25	#15688968	—	2	2	no	—
26	#15688885	C₂²	1	4	yes	—
27	#15688885	C₂²	1	4	yes	—
28	C₂×#107×C₄₇	C₂²	1	2	yes	—
29	C₂×#106×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#382	C₂²	1	2	yes	—
32	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₄₇	—	2	2	no	—
38	C₂×C₈×D₄₇	—	2	2	no	—
39	C₂×C₈×D₄₇	—	2	2	no	—
40	C₂×C₈×D₄₇	—	2	2	no	—
41	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
44	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₄₇	—	2	4	no	—
46	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₄²×D₄₇	C₂×C₄	1	4	yes	—
50	C₄²×D₄₇	Q₈	1	4	yes	—
51	#13093958	—	2	4	no	—
52	#13093956	—	2	4	no	—
53	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
55	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	4	no	—
59	C₄₇⋊#95	—	2	4	no	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#107×C₄₇	—	2	4	no	—
65	#106×C₄₇	—	2	4	no	—
66	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
67	#95×C₄₇	—	2	2	no	—
68	#95×C₄₇	—	2	2	no	—
69	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×C₄×D₄₇	—	2	1	no	—
80	C₂×C₄×D₄₇	—	2	1	no	—
81	C₂×C₄×D₄₇	—	2	1	no	—
82	C₂×C₄×D₄₇	—	2	1	no	—
83	C₂×C₄×D₄₇	—	2	4	no	—
84	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
85	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
86	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₂×C₈×C₄₇	#41	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
98	(C₂×D₄₇)⋊C₄	—	4	8	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	—	4	8	no	—
102	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
103	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
104	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
105	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
106	C₂×C₄₇⋊C₈	—	2	1	no	—
107	C₂×C₄₇⋊C₈	—	2	1	no	—
108	C₂×C₄₇⋊C₈	—	2	1	no	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
112	C₈×D₄₇	—	2	8	no	—
113	C₈×D₄₇	—	2	8	no	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²×D₄₇	—	2	1	no	—
116	C₂²×D₄₇	—	2	1	no	—
117	C₂²×D₄₇	—	2	2	no	—
118	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂²×D₄₇	#135	1	4	yes	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	1	no	—
134	C₂×Dic₄₇	—	2	1	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₄₇	#135	1	4	yes	—
143	C₄×D₄₇	—	2	8	no	—
144	C₄×D₄₇	—	2	2	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	8	no	—
149	C₈×C₄₇	#117	1	4	yes	—
150	C₈×C₄₇	#117	1	4	yes	—
151	C₄₇⋊C₈	—	2	2	no	—
152	C₄₇⋊C₈	—	2	2	no	—
153	#20818	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄₇	#532	1	1	yes	characteristic
156	C₂²×C₄₇	#532	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#135	1	2	yes	—
158	C₂²×C₄₇	C₂×#135	1	2	yes	—
159	C₂×D₄₇	—	2	2	no	—
160	C₂×D₄₇	—	2	4	no	—
161	C₂×D₄₇	—	2	4	no	—
162	C₂×D₄₇	—	2	2	no	—
163	C₂×D₄₇	#427	1	4	yes	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	8	4	no	—
167	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
168	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
169	C₄×C₄₇	#533	1	1	yes	characteristic
170	C₄×C₄₇	#533	1	1	yes	characteristic
171	Dic₄₇	—	2	2	no	—
172	Dic₄₇	—	2	2	no	—
173	Dic₄₇	—	8	4	no	—
174	Dic₄₇	#427	1	4	yes	—
175	C₂²×C₄×C₈	—	47	1	no	—
176	#2283	—	47	2	no	—
177	#1846	—	47	1	no	—
178	#1846	D₄₇	1	1	yes	characteristic, complemented
179	#1835	—	47	2	no	—
180	#1808	—	47	2	no	—
181	#1807	—	47	2	no	—
182	C₄×#107	—	47	2	no	—
183	C₄×#106	—	47	2	no	—
184	C₂×C₄₇	#2285	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
187	C₂×C₄₇	#3168	1	2	yes	—
188	C₂×C₄₇	#3168	1	2	yes	—
189	D₄₇	—	2	4	no	—
190	C₂³×C₈	—	47	1	no	—
191	C₂³×C₈	—	47	1	no	—
192	C₂²×C₄²	—	47	1	no	—
193	C₂×#107	—	47	2	no	—
194	C₂×#107	C₂×D₄₇	1	2	yes	—
195	C₂×#106	—	47	2	no	—
196	C₂×#106	C₂×D₄₇	1	2	yes	—
197	C₂×C₄×C₈	—	47	1	no	—
198	C₂×C₄×C₈	—	47	1	no	—
199	C₂×C₄×C₈	—	47	1	no	—
200	C₂×C₄×C₈	—	47	8	no	—
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#383	—	94	2	no	—
203	#382	—	47	2	no	—
204	#382	C₂×D₄₇	1	2	yes	—
205	#381	—	94	2	no	—
206	#380	—	47	4	no	—
207	C₄×C₄⋊C₄	—	47	4	no	—
208	C₄×C₂²⋊C₄	—	94	2	no	—
209	#376	—	94	2	no	—
210	#340	—	47	4	no	—
211	#340	—	47	4	no	—
212	C₄₇	#20818	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	47	1	no	—
214	C₂³×C₄	—	47	1	no	—
215	C₂³×C₄	—	47	1	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	47	4	no	—
219	C₂²×C₈	—	47	4	no	—
220	C₂²×C₈	—	94	2	no	—
221	C₂²×C₈	—	94	2	no	—
222	C₂²×C₈	—	94	2	no	—
223	C₂²×C₈	—	94	2	no	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	47	4	no	—
227	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
228	C₂×C₂²⋊C₄	—	94	4	no	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	—	47	1	no	—
232	C₂×C₄²	—	47	4	no	—
233	C₂×C₄²	—	47	4	no	—
234	C₂×C₄²	—	94	4	no	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	#107	—	2	4	no	—
237	#107	—	94	4	no	—
238	#106	—	2	4	no	—
239	#106	—	94	4	no	—
240	C₄×C₈	—	47	4	no	—
241	C₄×C₈	—	47	4	no	—
242	C₄×C₈	—	47	4	no	—
243	C₄×C₈	C₄×D₄₇	1	4	yes	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	—	94	4	no	—
247	#95	—	94	4	no	—
248	#95	—	94	2	no	—
249	#95	—	94	2	no	—
250	C₂⁴	—	47	1	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	47	1	no	—
255	C₂²×C₄	—	47	4	no	—
256	C₂²×C₄	—	47	4	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	94	4	no	—
260	C₂²×C₄	—	94	4	no	—
261	C₂²×C₄	—	94	1	no	—
262	C₂²×C₄	—	94	1	no	—
263	C₂²×C₄	—	94	1	no	—
264	C₂²×C₄	—	94	1	no	—
265	C₂²×C₄	—	94	4	no	—
266	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	47	2	no	—
271	C₂×C₈	—	94	1	no	—
272	C₂×C₈	—	94	1	no	—
273	C₂×C₈	—	94	1	no	—
274	C₂×C₈	—	94	1	no	—
275	C₂×C₈	—	94	8	no	—
276	C₂×C₈	—	94	8	no	—
277	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
278	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
279	C₂×C₈	#10625352	1	2	yes	—
280	C₂×C₈	#10625352	1	2	yes	—
281	C₂×C₈	#10625354	1	2	yes	—
282	C₂×C₈	#10625354	1	2	yes	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	94	4	no	—
287	C₂²⋊C₄	—	188	8	no	—
288	C₄²	—	47	4	no	—
289	C₄²	—	47	4	no	—
290	C₄²	—	47	2	no	—
291	C₄²	—	47	2	no	—
292	C₄²	—	188	8	no	—
293	C₄²	C₂×C₄×D₄₇	1	2	yes	—
294	C₄²	Q₈×D₄₇	1	2	yes	—
295	C₂³	—	47	2	no	—
296	C₂³	—	47	4	no	—
297	C₂³	—	94	2	no	—
298	C₂³	—	94	1	no	—
299	C₂³	—	94	1	no	—
300	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	47	2	no	—
307	C₂×C₄	—	47	4	no	—
308	C₂×C₄	—	47	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	94	8	no	—
311	C₂×C₄	—	94	8	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	1	no	—
314	C₂×C₄	—	94	1	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	188	4	no	—
320	C₂×C₄	—	188	4	no	—
321	C₂×C₄	—	188	4	no	—
322	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
323	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
324	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
328	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
329	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
330	C₈	—	94	2	no	—
331	C₈	—	94	2	no	—
332	C₈	#13093982	1	4	yes	—
333	C₈	#13093982	1	4	yes	—
334	C₂²	—	47	4	no	—
335	C₂²	—	94	4	no	—
336	C₂²	—	94	4	no	—
337	C₂²	—	94	2	no	—
338	C₂²	—	94	2	no	—
339	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
340	C₂²	#15689671	1	1	yes	characteristic
341	C₂²	#15689674	1	1	yes	characteristic
342	C₂²	#135×D₄₇	1	2	yes	—
343	C₂²	#135×D₄₇	1	2	yes	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	8	4	no	—
347	C₄	—	47	4	no	—
348	C₄	—	94	2	no	—
349	C₄	—	94	2	no	—
350	C₄	—	376	4	no	—
351	C₄	C₂×#13093982	1	1	yes	characteristic
352	C₄	C₂×#13093982	1	1	yes	characteristic
353	C₄	#15689678	1	1	yes	characteristic
354	C₄	#15689678	1	1	yes	characteristic
355	C₂	—	94	4	no	—
356	C₂	#20245793	1	1	yes	characteristic
357	C₂	#427×D₄₇	1	1	yes	characteristic
358	C₂	#427×D₄₇	1	1	yes	characteristic
359	C₂	#20253955	1	2	yes	—
360	C₂	#20253960	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1846	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#20818
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
3. #20243552	C₂	semidirect
4. #20243518	C₂	semidirect
7. #107×Dic₄₇	C₂	semidirect
8. #106×Dic₄₇	C₂	semidirect
9. C₄₇×#1846	C₂	semidirect
18. C₄×C₈×D₄₇	C₄	semidirect
178. #1846	D₄₇	semidirect
212. C₄₇	#20818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
#20245781	2
#20243552	2
#20243518	2
#20243414	2
#20243410	2
#107×Dic₄₇	2
#106×Dic₄₇	2
C₄₇×#1846	2
#20818	47