Subgroups — #26119071

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689496	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
4	#20244349	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244194	C₂	1	1	yes	maximal, characteristic
6	#20244190	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625325	—	2	2	no	—
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	#15689528	—	2	2	no	—
13	#15689528	—	2	2	no	—
14	#15689496	—	2	2	no	—
15	#15689496	—	2	2	no	—
16	#15689470	—	2	2	no	—
17	#15689470	—	2	2	no	—
18	C₂×#13093958	—	2	2	no	—
19	C₂×#13093949	—	2	2	no	—
20	C₂×#13093946	—	2	2	no	—
21	C₂×#13093946	—	2	2	no	—
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688886	—	2	2	no	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×Dic₉₄	—	2	1	no	—
28	C₂²×Dic₉₄	—	2	1	no	—
29	C₂²×D₁₈₈	—	2	1	no	—
30	C₂²×D₁₈₈	—	2	1	no	—
31	C₂²×D₁₈₈	—	2	1	no	—
32	C₂×#10625325	—	2	8	no	—
33	C₂×#10625325	—	4	4	no	—
34	C₂×#10625325	—	4	4	no	—
35	#13094024	—	2	1	no	—
36	#13094024	—	2	1	no	—
37	#13094024	—	2	1	no	—
38	#13094024	—	2	1	no	—
39	#13094024	—	2	1	no	—
40	#13094024	—	2	1	no	—
41	#13094024	D₄	1	2	yes	—
42	#13094024	D₄	1	2	yes	—
43	#13094021	—	2	1	no	—
44	#13094021	—	2	1	no	—
45	#13094021	—	2	1	no	—
46	#13094021	D₄	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	—	2	1	no	—
48	C₂×C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	C₂×C₄⋊Dic₄₇	—	2	1	no	—
54	#13093958	—	2	8	no	—
55	#13093949	—	4	4	no	—
56	#13093946	—	2	8	no	—
57	#13093946	—	4	4	no	—
58	C₂²×C₈×C₄₇	D₄	1	2	yes	—
59	C₂²×C₈×C₄₇	D₄	1	2	yes	—
60	C₂²×C₈×C₄₇	D₄	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	—	2	1	no	—
64	C₄×C₈×C₄₇	—	2	1	no	—
65	C₄×C₈×C₄₇	—	2	1	no	—
66	C₄×C₈×C₄₇	—	2	1	no	—
67	C₄×C₈×C₄₇	—	2	1	no	—
68	C₂³×D₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	4	1	no	—
73	C₂²×Dic₄₇	—	4	1	no	—
74	C₂²×Dic₄₇	—	4	1	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	—	2	2	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×D₁₈₈	—	4	2	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×D₁₈₈	—	4	2	no	—
84	C₂×D₁₈₈	—	4	1	no	—
85	C₂×D₁₈₈	—	4	1	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	C₂×Dic₉₄	—	4	2	no	—
90	C₂×Dic₉₄	—	4	2	no	—
91	C₂×Dic₉₄	—	4	2	no	—
92	C₂×Dic₉₄	—	4	2	no	—
93	C₂×Dic₉₄	—	4	1	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	D₈	1	4	yes	—
103	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	#41	1	1	yes	characteristic
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	4	1	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₄⋊Dic₄₇	—	4	1	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	C₄⋊Dic₄₇	—	4	2	no	—
117	C₄⋊Dic₄₇	—	4	2	no	—
118	C₄⋊Dic₄₇	—	4	2	no	—
119	#10625325	—	4	16	no	—
120	C₂³×C₄₇	#127	1	1	yes	characteristic
121	C₂²×D₄₇	—	4	2	no	—
122	C₂²×D₄₇	—	8	1	no	—
123	C₂²×D₄₇	—	8	1	no	—
124	C₂²×D₄₇	—	8	1	no	—
125	C₂²×D₄₇	—	8	1	no	—
126	C₂²×D₄₇	—	8	1	no	—
127	C₂²×D₄₇	—	8	1	no	—
128	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₄₇	#135	1	2	yes	—
137	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₄₇	#137	1	2	yes	—
139	C₂×C₄×C₄₇	#137	1	2	yes	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	1	no	—
143	C₂×Dic₄₇	—	8	1	no	—
144	C₂×Dic₄₇	—	8	1	no	—
145	C₂×Dic₄₇	—	8	1	no	—
146	C₂×Dic₄₇	—	8	1	no	—
147	C₂×Dic₄₇	—	8	1	no	—
148	C₂×Dic₄₇	—	8	1	no	—
149	D₁₈₈	—	4	4	no	—
150	D₁₈₈	—	4	4	no	—
151	D₁₈₈	—	4	4	no	—
152	D₁₈₈	—	8	2	no	—
153	D₁₈₈	—	8	2	no	—
154	D₁₈₈	—	8	2	no	—
155	Dic₉₄	—	4	4	no	—
156	Dic₉₄	—	4	4	no	—
157	Dic₉₄	—	8	2	no	—
158	Dic₉₄	—	8	2	no	—
159	C₈×C₄₇	—	2	4	no	—
160	C₈×C₄₇	—	2	4	no	—
161	C₈×C₄₇	—	4	2	no	—
162	#10308	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	#392	1	1	yes	characteristic
164	C₂²×C₄₇	#492	1	1	yes	characteristic
165	C₂²×C₄₇	#493	1	1	yes	characteristic
166	C₂²×C₄₇	#495	1	1	yes	characteristic
167	C₂²×C₄₇	#496	1	1	yes	characteristic
168	C₂²×C₄₇	#497	1	1	yes	characteristic
169	C₂²×C₄₇	#497	1	1	yes	characteristic
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	8	2	no	—
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	16	1	no	—
174	C₂×D₄₇	—	16	1	no	—
175	C₂×D₄₇	—	16	1	no	—
176	C₂×D₄₇	—	16	1	no	—
177	C₄×C₄₇	#447	1	2	yes	—
178	C₄×C₄₇	#448	1	2	yes	—
179	C₄×C₄₇	#452	1	2	yes	—
180	C₄×C₄₇	#459	1	2	yes	—
181	C₄×C₄₇	#461	1	2	yes	—
182	C₄×C₄₇	#464	1	2	yes	—
183	Dic₄₇	—	8	2	no	—
184	Dic₄₇	—	8	2	no	—
185	Dic₄₇	—	16	1	no	—
186	C₂×#500	—	47	1	no	—
187	C₂×#492	—	47	1	no	—
188	C₂×#486	—	47	1	no	—
189	#1990	—	47	1	no	—
190	#1957	—	47	1	no	—
191	#1956	—	47	1	no	—
192	#1273	D₄₇	1	1	yes	characteristic, complemented
193	C₂×C₄₇	#1706	1	1	yes	characteristic
194	C₂×C₄₇	#1714	1	1	yes	characteristic
195	C₂×C₄₇	#1719	1	1	yes	characteristic
196	C₂×C₄₇	#1722	1	1	yes	characteristic
197	C₂×C₄₇	#2011	1	1	yes	characteristic
198	C₂×C₄₇	#2013	1	1	yes	characteristic
199	C₂×C₄₇	#2015	1	1	yes	characteristic
200	D₄₇	—	16	2	no	—
201	C₂²×SD₁₆	—	94	2	no	—
202	C₂×C₄⋊Q₈	—	47	1	no	—
203	C₂×C₄⋊Q₈	—	47	1	no	—
204	C₂×#127	—	47	1	no	—
205	#500	—	94	2	no	—
206	#500	—	94	2	no	—
207	#492	—	94	2	no	—
208	#492	—	94	2	no	—
209	#486	—	94	2	no	—
210	#486	—	94	2	no	—
211	C₂×#107	—	94	2	no	—
212	C₂×#103	—	94	2	no	—
213	C₂×#102	—	94	2	no	—
214	C₂×#102	—	94	2	no	—
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
218	#384	—	47	1	no	—
219	#340	—	94	2	no	—
220	C₄₇	#10308	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	—	94	1	no	—
222	C₂²×Q₈	—	94	1	no	—
223	C₂²×D₄	—	94	1	no	—
224	C₂²×D₄	—	94	1	no	—
225	C₂²×D₄	—	94	1	no	—
226	C₂×SD₁₆	—	94	8	no	—
227	C₂×SD₁₆	—	188	4	no	—
228	C₂×SD₁₆	—	188	4	no	—
229	C₂²×C₈	D₁₈₈	1	2	yes	—
230	C₂²×C₈	D₁₈₈	1	2	yes	—
231	C₂²×C₈	D₁₈₈	1	2	yes	—
232	C₄⋊Q₈	—	47	2	no	—
233	C₄⋊Q₈	—	47	2	no	—
234	C₄⋊Q₈	—	94	1	no	—
235	C₄⋊Q₈	—	94	1	no	—
236	C₄⋊Q₈	—	94	1	no	—
237	C₄⋊Q₈	—	94	1	no	—
238	C₄⋊Q₈	—	94	1	no	—
239	C₄⋊Q₈	—	94	1	no	—
240	#127	—	47	2	no	—
241	#127	—	94	1	no	—
242	#127	—	94	1	no	—
243	#127	—	94	1	no	—
244	C₂×C₄⋊C₄	—	94	1	no	—
245	C₂×C₄⋊C₄	—	94	1	no	—
246	C₂×C₄⋊C₄	—	94	1	no	—
247	C₂×C₄⋊C₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	—	94	1	no	—
250	C₂×C₄⋊C₄	—	94	1	no	—
251	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
252	#107	—	94	8	no	—
253	#103	—	188	4	no	—
254	#102	—	94	8	no	—
255	#102	—	188	4	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₂⁴	—	188	1	no	—
263	C₂×Q₈	—	94	2	no	—
264	C₂×Q₈	—	94	2	no	—
265	C₂×Q₈	—	188	2	no	—
266	C₂×Q₈	—	188	2	no	—
267	C₂×Q₈	—	188	2	no	—
268	C₂×Q₈	—	188	2	no	—
269	C₂×Q₈	—	188	1	no	—
270	C₂×Q₈	—	188	1	no	—
271	C₂×D₄	—	94	2	no	—
272	C₂×D₄	—	94	2	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	—	188	1	no	—
275	C₂×D₄	—	188	1	no	—
276	C₂×D₄	—	188	1	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	—	188	2	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂×D₄	—	188	2	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂²×C₄	—	188	1	no	—
284	C₂²×C₄	—	188	1	no	—
285	C₂²×C₄	—	188	1	no	—
286	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
289	SD₁₆	—	188	16	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	#10625325	1	4	yes	complemented
298	C₂×C₈	D₃₇₆	1	4	yes	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	188	2	no	—
304	C₄⋊C₄	—	188	2	no	—
305	C₄⋊C₄	—	188	2	no	—
306	C₄⋊C₄	—	188	1	no	—
307	C₄⋊C₄	—	188	1	no	—
308	C₄⋊C₄	—	188	1	no	—
309	C₄⋊C₄	—	188	1	no	—
310	C₄²	D₄×D₄₇	1	1	yes	characteristic
311	C₄²	D₄×D₄₇	1	1	yes	characteristic
312	C₄²	D₄×D₄₇	1	1	yes	characteristic
313	C₄²	#10625354	1	1	yes	characteristic
314	C₂³	—	376	1	no	—
315	C₂³	—	376	1	no	—
316	C₂³	—	376	1	no	—
317	C₂³	—	376	1	no	—
318	C₂³	—	376	1	no	—
319	C₂³	—	376	1	no	—
320	C₂³	—	188	2	no	—
321	C₂³	#13094021	1	1	yes	characteristic
322	Q₈	—	376	2	no	—
323	Q₈	—	376	2	no	—
324	Q₈	—	188	4	no	—
325	Q₈	—	188	4	no	—
326	D₄	—	376	2	no	—
327	D₄	—	376	2	no	—
328	D₄	—	376	2	no	—
329	D₄	—	188	4	no	—
330	D₄	—	188	4	no	—
331	D₄	—	188	4	no	—
332	C₂×C₄	—	376	1	no	—
333	C₂×C₄	—	376	1	no	—
334	C₂×C₄	—	376	1	no	—
335	C₂×C₄	—	376	1	no	—
336	C₂×C₄	—	376	1	no	—
337	C₂×C₄	—	376	1	no	—
338	C₂×C₄	—	376	1	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	#13093994	1	1	yes	characteristic
342	C₂×C₄	#13093994	1	1	yes	characteristic
343	C₂×C₄	#13093994	1	1	yes	characteristic
344	C₂×C₄	#13093999	1	1	yes	characteristic
345	C₂×C₄	#13093999	1	1	yes	characteristic
346	C₂×C₄	#13093999	1	1	yes	characteristic
347	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
348	C₂×C₄	C₂×#10625325	1	2	yes	—
349	C₂×C₄	#13094046	1	2	yes	—
350	C₂×C₄	#13094052	1	2	yes	—
351	C₂×C₄	#13094057	1	2	yes	—
352	C₂×C₄	#13094057	1	2	yes	—
353	C₈	—	2	4	no	—
354	C₈	—	2	4	no	—
355	C₈	—	4	2	no	—
356	C₂²	—	376	2	no	—
357	C₂²	—	376	2	no	—
358	C₂²	—	376	2	no	—
359	C₂²	—	752	1	no	—
360	C₂²	—	752	1	no	—
361	C₂²	—	752	1	no	—
362	C₂²	—	752	1	no	—
363	C₂²	#15689033	1	1	yes	characteristic
364	C₂²	#15689496	1	1	yes	characteristic
365	C₂²	#15689500	1	1	yes	characteristic
366	C₂²	#15689507	1	1	yes	characteristic
367	C₂²	#15689512	1	1	yes	characteristic
368	C₂²	#15689517	1	1	yes	characteristic
369	C₂²	#15689517	1	1	yes	characteristic
370	C₄	—	376	2	no	—
371	C₄	—	376	2	no	—
372	C₄	—	752	1	no	—
373	C₄	#15689259	1	2	yes	—
374	C₄	#15689264	1	2	yes	—
375	C₄	#15689287	1	2	yes	—
376	C₄	#15689319	1	2	yes	—
377	C₄	#15689329	1	2	yes	—
378	C₄	#15689346	1	2	yes	—
379	C₂	—	752	2	no	—
380	C₂	#20242980	1	1	yes	characteristic
381	C₂	#20243010	1	1	yes	characteristic
382	C₂	#20243041	1	1	yes	characteristic
383	C₂	#20243059	1	1	yes	characteristic
384	C₂	#20244461	1	1	yes	characteristic
385	C₂	#20244473	1	1	yes	characteristic
386	C₂	#20244485	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10308
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
4. #20244349	C₂	semidirect
6. #20244190	C₂	semidirect
7. C₄₇×#1273	C₂	semidirect
103. C₂×C₈×C₄₇	SD₁₆	semidirect
192. #1273	D₄₇	semidirect
220. C₄₇	#10308	semidirect
297. C₂×C₈	#10625325	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
C₂×#15689496	2
C₂×#15689470	2
#20244349	2
#20244194	2
#20244190	2
C₄₇×#1273	2
#10308	47