Subgroups — #26111770

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252411	C₂	1	1	yes	maximal, characteristic
2	#20246095	C₂	1	1	yes	maximal, characteristic
3	#20241783	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241708	C₂	1	1	yes	maximal, characteristic
5	#20241216	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094013	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₈×D₁₈₈	—	2	2	no	—
11	#15689188	—	2	2	no	—
12	#15689176	C₂²	1	1	yes	characteristic
13	#15689176	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×D₄₇	—	2	2	no	—
15	#15689030	C₄	1	1	yes	characteristic
16	#15689024	C₄	1	1	yes	characteristic
17	#15689014	—	2	2	no	—
18	#15688994	—	2	2	no	—
19	#15688992	C₄	1	1	yes	characteristic
20	#15688969	C₂²	1	1	yes	characteristic
21	Q₈⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#15688846	—	2	2	no	—
23	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₄₇×#431	C₂²	1	1	yes	characteristic
26	C₄₇×Q₈⋊C₈	—	2	2	no	—
27	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	D₄	1	2	yes	—
29	C₂×C₈×D₄₇	—	2	4	no	—
30	#13094013	—	2	2	no	—
31	#13094013	C₈	1	4	yes	complemented
32	C₄×D₁₈₈	—	2	1	no	—
33	C₄×D₁₈₈	—	2	1	no	—
34	C₄×D₁₈₈	—	2	1	no	—
35	C₄×D₁₈₈	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₄₇⋊C₄	—	2	2	no	—
38	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
40	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
47	(C₂×D₄₇)⋊C₈	—	2	4	no	—
48	C₄×C₄₇⋊C₈	—	2	1	no	—
49	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
50	C₄₇⋊#95	—	2	1	no	—
51	C₄₇⋊#95	—	2	1	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₄₇	—	2	1	no	—
55	C₄×Q₈×C₄₇	—	2	1	no	—
56	C₄×Q₈×C₄₇	—	2	1	no	—
57	C₄×Q₈×C₄₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	—	2	1	no	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄⋊C₈×C₄₇	—	2	1	no	—
64	C₂²⋊C₈×C₄₇	—	2	1	no	—
65	C₂²⋊C₈×C₄₇	—	2	1	no	—
66	C₄×C₈×C₄₇	—	2	1	no	—
67	C₄×C₈×C₄₇	D₄	1	2	yes	—
68	C₂³×D₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×Q₈×C₄₇	—	2	1	no	—
70	C₂×Q₈×C₄₇	—	2	1	no	—
71	C₂×Q₈×C₄₇	—	2	1	no	—
72	C₂×Q₈×C₄₇	—	2	1	no	—
73	C₂×Q₈×C₄₇	—	2	1	no	—
74	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	complemented
75	C₂²×C₄×C₄₇	—	2	1	no	—
76	C₂²×C₄×C₄₇	—	2	1	no	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	1	no	—
81	C₂²×Dic₄₇	—	2	1	no	—
82	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×D₁₈₈	—	2	1	no	—
84	C₂×D₁₈₈	—	2	1	no	—
85	C₂×D₁₈₈	—	2	1	no	—
86	C₂×D₁₈₈	—	2	1	no	—
87	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
88	C₂×D₁₈₈	M₁₆	1	2	yes	—
89	C₂×C₄×D₄₇	—	2	2	no	—
90	C₂×C₄×D₄₇	—	2	2	no	—
91	C₂×C₄×D₄₇	—	8	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	4	1	no	—
94	C₄⋊C₄×C₄₇	—	2	1	no	—
95	C₄⋊C₄×C₄₇	—	2	1	no	—
96	C₄⋊C₄×C₄₇	—	4	1	no	—
97	C₄⋊C₄×C₄₇	—	4	1	no	—
98	C₄²×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₇	#41	1	1	yes	characteristic
103	C₄²×C₄₇	#41	1	1	yes	characteristic
104	(C₂×D₄₇)⋊C₄	—	2	8	no	—
105	(C₂×D₄₇)⋊C₄	—	2	2	no	—
106	(C₂×D₄₇)⋊C₄	—	2	2	no	—
107	(C₂×D₄₇)⋊C₄	—	8	2	no	—
108	C₄⋊Dic₄₇	—	2	1	no	—
109	C₄⋊Dic₄₇	—	2	1	no	—
110	Dic₄₇⋊C₄	—	8	2	no	—
111	C₄×Dic₄₇	—	2	1	no	—
112	C₄×Dic₄₇	—	2	1	no	—
113	C₄×Dic₄₇	—	2	1	no	—
114	C₄×Dic₄₇	C₂×C₈	1	2	yes	complemented
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	4	1	no	—
117	C₈×D₄₇	—	8	4	no	—
118	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×D₄₇	—	2	2	no	—
120	C₂²×D₄₇	—	2	2	no	—
121	C₂²×D₄₇	—	4	2	no	—
122	C₂²×D₄₇	—	4	2	no	—
123	C₂²×D₄₇	C₂²⋊C₈	1	4	yes	—
124	Q₈×C₄₇	—	2	2	no	—
125	Q₈×C₄₇	—	2	2	no	—
126	Q₈×C₄₇	—	4	1	no	—
127	Q₈×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	C₂×Dic₄₇	—	4	1	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	4	1	no	—
152	C₂×Dic₄₇	—	4	1	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	2	2	no	—
155	D₁₈₈	—	2	2	no	—
156	D₁₈₈	—	2	2	no	—
157	C₄×D₄₇	—	4	4	no	—
158	C₄×D₄₇	—	8	2	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	—	8	1	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	C₄₇⋊C₈	—	8	1	no	—
163	#9099	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	#386	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₄₇	#408	1	1	yes	characteristic
169	C₂×D₄₇	—	2	4	no	—
170	C₂×D₄₇	—	4	2	no	—
171	C₂×D₄₇	—	4	2	no	—
172	C₂×D₄₇	—	4	4	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	4	2	no	—
179	C₄×C₄₇	—	8	1	no	—
180	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
181	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
182	C₄×C₄₇	#435	1	1	yes	characteristic
183	C₄×C₄₇	#435	1	1	yes	characteristic
184	Dic₄₇	—	4	2	no	—
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	8	1	no	—
187	#2972	—	47	1	no	—
188	#2326	—	47	1	no	—
189	#1479	—	47	1	no	—
190	#1479	D₄₇	1	1	yes	characteristic, complemented
191	#1460	—	47	1	no	—
192	#1274	—	47	1	no	—
193	#1274	—	47	1	no	—
194	C₂×C₄₇	—	2	1	no	—
195	C₂×C₄₇	—	2	1	no	—
196	C₂×C₄₇	#1869	1	1	yes	characteristic
197	C₂×C₄₇	#1909	1	1	yes	characteristic
198	C₂×C₄₇	#1911	1	1	yes	characteristic
199	D₄₇	—	4	4	no	—
200	C₂×#124	—	47	1	no	—
201	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
202	C₂×C₄×D₄	—	47	1	no	—
203	C₈×D₄	—	94	2	no	—
204	C₈×D₄	—	94	2	no	—
205	#431	—	47	1	no	—
206	#431	—	47	1	no	—
207	#431	C₂×D₄₇	1	1	yes	characteristic
208	#431	C₂×D₄₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₈	—	94	2	no	—
210	#391	—	47	1	no	—
211	#390	—	47	1	no	—
212	#388	—	94	2	no	—
213	#385	—	94	2	no	—
214	#384	—	47	1	no	—
215	#376	—	47	1	no	—
216	Q₈⋊C₈	—	2	2	no	—
217	Q₈⋊C₈	—	94	2	no	—
218	#323	—	94	2	no	—
219	C₄₇	#9099	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	C₄×D₄₇	1	1	yes	characteristic
221	C₂²×D₄	—	47	1	no	—
222	C₂³×C₄	—	47	2	no	—
223	C₂²×C₈	—	94	4	no	—
224	#124	—	47	4	no	—
225	#124	—	94	2	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×D₄	—	94	1	no	—
231	C₄×D₄	—	94	1	no	—
232	C₄×D₄	—	94	1	no	—
233	C₄×D₄	—	94	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	47	1	no	—
236	C₂×C₄⋊C₄	—	94	2	no	—
237	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	—	47	2	no	—
239	C₂×C₂²⋊C₄	—	94	2	no	—
240	C₂×C₂²⋊C₄	—	94	2	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	94	1	no	—
248	C₄⋊C₈	—	94	1	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	94	4	no	—
252	C₂²⋊C₈	—	94	1	no	—
253	C₂²⋊C₈	—	94	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	47	2	no	—
256	C₄×C₈	—	94	1	no	—
257	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
258	#95	—	94	2	no	—
259	#95	—	94	1	no	—
260	#95	—	94	1	no	—
261	C₂⁴	—	47	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	C₈×D₄₇	1	2	yes	complemented
268	C₂×D₄	—	47	2	no	—
269	C₂×D₄	—	47	2	no	—
270	C₂×D₄	—	94	1	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂×D₄	—	94	1	no	—
273	C₂×D₄	—	94	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	94	1	no	—
278	C₂²×C₄	—	94	2	no	—
279	C₂²×C₄	—	94	2	no	—
280	C₂²×C₄	—	94	1	no	—
281	C₂²×C₄	—	376	2	no	—
282	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
283	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	376	4	no	—
289	C₂×C₈	—	188	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	94	1	no	—
295	C₄⋊C₄	—	94	1	no	—
296	C₄⋊C₄	—	376	2	no	—
297	C₂²⋊C₄	—	94	8	no	—
298	C₂²⋊C₄	—	94	2	no	—
299	C₂²⋊C₄	—	94	2	no	—
300	C₂²⋊C₄	—	376	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	47	2	no	—
304	C₄²	—	94	1	no	—
305	C₄²	—	94	1	no	—
306	C₄²	—	94	1	no	—
307	C₄²	#10625350	1	1	yes	characteristic
308	C₄²	D₄×D₄₇	1	1	yes	characteristic
309	C₄²	#10625352	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₂³	—	47	4	no	—
312	C₂³	—	94	2	no	—
313	C₂³	—	94	2	no	—
314	C₂³	—	188	2	no	—
315	C₂³	—	188	2	no	—
316	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	4	1	no	—
320	Q₈	—	4	1	no	—
321	D₄	—	94	2	no	—
322	D₄	—	94	2	no	—
323	D₄	—	94	2	no	—
324	D₄	—	94	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	376	2	no	—
335	C₂×C₄	—	188	4	no	—
336	C₂×C₄	—	188	1	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	188	1	no	—
339	C₂×C₄	—	188	1	no	—
340	C₂×C₄	—	188	1	no	—
341	C₂×C₄	—	188	2	no	—
342	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
344	C₂×C₄	#13093992	1	1	yes	characteristic
345	C₂×C₄	#13093995	1	1	yes	characteristic
346	C₂×C₄	#13093997	1	1	yes	characteristic
347	C₂×C₄	#13094006	1	1	yes	characteristic
348	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
349	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
350	C₂×C₄	#13094030	1	1	yes	characteristic
351	C₂×C₄	#13094032	1	1	yes	characteristic
352	C₈	—	4	2	no	—
353	C₈	—	8	1	no	—
354	C₈	—	376	1	no	—
355	C₈	—	188	2	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	94	4	no	—
359	C₂²	—	376	2	no	—
360	C₂²	—	188	4	no	—
361	C₂²	—	188	2	no	—
362	C₂²	—	188	2	no	—
363	C₂²	#15688999	1	1	yes	characteristic
364	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
365	C₂²	#15689088	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	8	1	no	—
372	C₄	—	376	1	no	—
373	C₄	—	188	2	no	—
374	C₄	—	188	2	no	—
375	C₄	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
376	C₄	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
377	C₄	#15689192	1	1	yes	characteristic
378	C₄	#15689197	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	188	4	no	—
382	C₂	#20243677	1	1	yes	characteristic
383	C₂	#20243907	1	1	yes	characteristic
384	C₂	#20243917	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688999
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1479	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1479	D₄₇
hypocenter	C₄₇	#9099
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
3. #20241783	C₂	semidirect
5. #20241216	C₂	semidirect
6. #20241216	C₂	semidirect
7. C₄₇×#1479	C₂	semidirect
31. #13094013	C₈	semidirect
74. C₂×Q₈×C₄₇	C₂×C₈	semidirect
114. C₄×Dic₄₇	C₂×C₈	semidirect
190. #1479	D₄₇	semidirect
219. C₄₇	#9099	semidirect
267. C₂×Q₈	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252411	2
#20246095	2
#20241783	2
#20241708	2
#20241216	2
C₄₇×#1479	2
#9099	47