Subgroups — #26187007

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253958	C₂	1	1	yes	maximal, characteristic
2	#20252357	C₂	1	2	yes	maximal, complemented
3	#20242237	C₂	1	2	yes	maximal, complemented
4	#20242230	C₂	1	2	yes	maximal, complemented
5	#20242223	C₂	1	2	yes	maximal, complemented
6	#20242216	C₂	1	2	yes	maximal, complemented
7	#20241846	C₂	1	2	yes	maximal, complemented
8	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689657	C₂²	1	2	yes	—
11	#15689525	C₄	1	1	yes	characteristic
12	#15689507	C₄	1	1	yes	characteristic
13	#15689478	C₄	1	2	yes	—
14	#15689364	C₄	1	1	yes	characteristic
15	#15689359	C₄	1	1	yes	characteristic
16	#15689352	C₄	1	2	yes	—
17	#15689250	C₂²	1	2	yes	—
18	#15689247	C₂²	1	2	yes	—
19	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
20	C₄×#10625325	C₂²	1	2	yes	—
21	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
22	#15689200	C₂²	1	2	yes	—
23	#15689191	—	2	2	no	—
24	#15689188	C₂²	1	2	yes	—
25	(C₂²⋊D₄₇)⋊C₈	—	2	2	no	—
26	#15689185	—	2	2	no	—
27	#15689080	—	2	2	no	—
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	#15688853	C₂²	1	2	yes	—
31	#15688853	C₂²	1	2	yes	—
32	#15688850	C₂²	1	2	yes	—
33	#15688850	C₂²	1	2	yes	—
34	#15688845	C₂²	1	2	yes	—
35	#15688845	C₂²	1	2	yes	—
36	C₄₇×#432	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#335	C₂²	1	2	yes	—
39	C₄₇×#334	C₂²	1	1	yes	characteristic
40	C₄₇×#334	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
42	C₂×#10625325	C₂×C₄	1	2	yes	—
43	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
44	C₂×#10625324	—	2	2	no	—
45	C₂²×C₄₇⋊C₈	—	2	1	no	—
46	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
47	C₂×C₈×D₄₇	—	2	2	no	—
48	#13094018	—	2	2	no	—
49	#13094016	C₂×C₄	1	2	yes	—
50	#13094012	C₂×C₄	1	2	yes	—
51	#13094007	—	2	2	no	—
52	#13094003	C₂×C₄	1	2	yes	—
53	#13093998	C₂×C₄	1	2	yes	—
54	C₄×Dic₉₄	C₂³	1	2	yes	—
55	C₄×D₁₈₈	C₂³	1	2	yes	—
56	C₄×C₂²⋊D₄₇	—	2	2	no	—
57	#13093981	—	2	2	no	—
58	#13093958	C₂×C₄	1	1	yes	characteristic
59	#13093958	C₂×C₄	1	1	yes	characteristic
60	#13093956	C₂×C₄	1	2	yes	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	#13093949	C₂×C₄	1	2	yes	—
64	#13093949	C₂×C₄	1	2	yes	—
65	#13093946	C₂×C₄	1	2	yes	—
66	#13093946	C₂×C₄	1	2	yes	—
67	(C₂×D₄₇)⋊C₈	—	2	2	no	—
68	(C₂×D₄₇)⋊C₈	—	2	2	no	—
69	#13093934	—	2	2	no	—
70	C₄×C₄₇⋊C₈	—	2	1	no	—
71	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
74	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
75	C₈×Dic₄₇	—	2	2	no	—
76	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₄₇	—	2	2	no	—
82	#97×C₄₇	—	2	2	no	—
83	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	—	2	2	no	—
91	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₄₇	—	2	2	no	—
93	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	—	4	1	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₇	#41	1	1	yes	characteristic
105	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊Dic₄₇	—	2	2	no	—
107	(C₂×D₄₇)⋊C₄	—	2	2	no	—
108	(C₂×D₄₇)⋊C₄	—	2	2	no	—
109	(C₂×D₄₇)⋊C₄	—	2	2	no	—
110	(C₂×D₄₇)⋊C₄	—	2	2	no	—
111	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
113	Dic₄₇⋊C₄	—	2	2	no	—
114	Dic₄₇⋊C₄	—	2	2	no	—
115	Dic₄₇⋊C₄	—	2	2	no	—
116	Dic₄₇⋊C₄	—	2	2	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	4	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	4	2	no	—
121	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	Dic₁₈₈	—	4	1	no	—
124	D₃₇₆	—	4	1	no	—
125	#10625325	—	4	2	no	—
126	#10625324	—	4	4	no	—
127	C₈×D₄₇	—	4	4	no	—
128	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂²⋊D₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	D₁₈₈	—	2	2	no	—
145	D₁₈₈	—	2	2	no	—
146	C₄×D₄₇	—	4	2	no	—
147	C₄×D₄₇	—	4	2	no	—
148	Dic₉₄	—	2	2	no	—
149	Dic₉₄	—	2	2	no	—
150	C₈×C₄₇	—	2	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	C₄₇⋊C₈	#135	1	4	yes	—
159	#17978	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	#444	1	2	yes	—
163	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₄₇	—	4	2	no	—
165	C₂×D₄₇	—	4	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	#568	1	1	yes	characteristic
172	C₄×C₄₇	#568	1	1	yes	characteristic
173	C₄×C₄₇	C₂×#135	1	2	yes	—
174	Dic₄₇	—	4	2	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	#3168	—	47	1	no	—
178	#2966	—	47	2	no	—
179	#1584	—	47	2	no	—
180	#1583	—	47	2	no	—
181	#1582	—	47	2	no	—
182	#1581	—	47	2	no	—
183	#1568	D₄₇	1	1	yes	characteristic, complemented
184	#1494	—	47	2	no	—
185	C₂×#322	—	47	1	no	—
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	#2932	1	1	yes	characteristic
188	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
189	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
190	D₄₇	—	8	2	no	—
191	#529	—	47	2	no	—
192	#499	—	47	1	no	—
193	#495	—	47	1	no	—
194	#488	—	47	2	no	—
195	#467	—	47	1	no	—
196	#466	—	47	1	no	—
197	#465	—	47	2	no	—
198	#446	—	47	2	no	—
199	C₈×Q₈	—	47	2	no	—
200	C₄×Q₁₆	—	47	1	no	—
201	C₄×SD₁₆	—	47	2	no	—
202	C₄×D₈	—	47	1	no	—
203	#436	—	47	2	no	—
204	#435	—	94	2	no	—
205	C₈×D₄	—	47	2	no	—
206	C₈×D₄	—	94	2	no	—
207	#433	—	94	2	no	—
208	#432	C₂×D₄₇	1	2	yes	—
209	#405	—	94	2	no	—
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
213	#335	C₂×D₄₇	1	2	yes	—
214	#334	C₂×D₄₇	1	1	yes	characteristic
215	#334	C₂×D₄₇	1	1	yes	characteristic
216	Q₈⋊C₈	—	47	2	no	—
217	Q₈⋊C₈	—	47	2	no	—
218	D₄⋊C₈	—	47	2	no	—
219	D₄⋊C₈	—	47	2	no	—
220	#322	—	47	2	no	—
221	#322	—	47	2	no	—
222	C₄₇	#17978	1	1	yes	characteristic, complemented, Sylow
223	C₂×Q₁₆	—	47	1	no	—
224	C₂×SD₁₆	—	47	2	no	—
225	C₂×D₈	—	47	1	no	—
226	C₂×M₁₆	—	94	2	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	—	94	2	no	—
229	C₂²×C₈	—	94	1	no	—
230	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
232	#126	—	94	2	no	—
233	#125	—	47	2	no	—
234	#124	—	47	2	no	—
235	#123	—	94	2	no	—
236	C₂²⋊Q₈	—	47	2	no	—
237	#121	—	47	2	no	—
238	C₄×Q₈	—	47	2	no	—
239	C₄×D₄	—	47	2	no	—
240	C₄×D₄	—	94	2	no	—
241	#117	—	94	2	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	#107	—	47	1	no	—
244	#107	—	47	1	no	—
245	#106	—	47	2	no	—
246	C₄⋊C₈	—	94	2	no	—
247	C₄⋊C₈	—	94	2	no	—
248	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
249	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
250	#103	—	47	2	no	—
251	#103	—	47	2	no	—
252	#102	—	47	2	no	—
253	#102	—	47	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	94	2	no	—
256	C₂²⋊C₈	—	94	2	no	—
257	#97	—	2	2	no	—
258	#97	—	94	2	no	—
259	C₄×C₈	—	47	1	no	—
260	C₄×C₈	—	47	1	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	47	2	no	—
263	C₄×C₈	—	94	2	no	—
264	C₄×C₈	—	94	1	no	—
265	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
266	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
269	C₂×Q₈	—	47	2	no	—
270	C₂×D₄	—	47	2	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂²×C₄	—	94	2	no	—
273	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	#10625354	1	1	yes	characteristic
276	Q₁₆	—	188	1	no	—
277	SD₁₆	—	188	2	no	—
278	D₈	—	188	1	no	—
279	M₁₆	—	188	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	94	4	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	188	4	no	—
289	C₂×C₈	—	188	2	no	—
290	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
294	C₄⋊C₄	—	47	2	no	—
295	C₄⋊C₄	—	47	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₂²⋊C₄	—	94	2	no	—
301	C₂²⋊C₄	—	94	2	no	—
302	C₂²⋊C₄	—	94	2	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₄²	—	94	2	no	—
306	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
307	C₄²	D₄×D₄₇	1	1	yes	characteristic
308	C₄²	#10625354	1	1	yes	characteristic
309	C₄²	C₂³×D₄₇	1	1	yes	characteristic
310	C₂³	—	94	2	no	—
311	C₂³	#13093986	1	1	yes	characteristic
312	Q₈	—	94	2	no	—
313	Q₈	—	94	2	no	—
314	D₄	—	94	2	no	—
315	D₄	—	94	2	no	—
316	D₄	—	188	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	#13093986	1	1	yes	characteristic
327	C₂×C₄	#13093986	1	1	yes	characteristic
328	C₂×C₄	#13093986	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	47	4	no	—
339	C₈	—	94	2	no	—
340	C₈	—	188	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	188	2	no	—
344	C₂²	—	188	2	no	—
345	C₂²	—	188	2	no	—
346	C₂²	#15689244	1	2	yes	—
347	C₂²	C₂×#13093986	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	188	2	no	—
353	C₄	—	188	2	no	—
354	C₄	—	188	2	no	—
355	C₄	#15689889	1	1	yes	characteristic
356	C₄	#15689889	1	1	yes	characteristic
357	C₄	#135×D₄₇	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	376	2	no	—
360	C₂	#20252105	1	1	yes	characteristic
361	C₂	#20252159	1	1	yes	characteristic
362	C₂	C₂×#15689244	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689359	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#17978
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. #20252357	C₂	semidirect
3. #20242237	C₂	semidirect
4. #20242230	C₂	semidirect
5. #20242223	C₂	semidirect
6. #20242216	C₂	semidirect
7. #20241846	C₂	semidirect
8. C₂×#15688845	C₂	semidirect
9. C₄₇×#1568	C₂	semidirect
183. #1568	D₄₇	semidirect
222. C₄₇	#17978	semidirect

Maximal subgroups

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

subgroup	index
#20253958	2
#20252357	2
#20242237	2
#20242230	2
#20242223	2
#20242216	2
#20241846	2
C₂×#15688845	2
C₄₇×#1568	2
#17978	47