Subgroups — #26369516

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₄₇	C₂	1	2	yes	maximal, complemented
2	#20252286	C₂	1	1	yes	maximal, characteristic
3	#20245231	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245228	C₂	1	2	yes	maximal, complemented
5	#20245225	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245224	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245222	C₂	1	2	yes	maximal, complemented
8	#20245152	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245014	C₂	1	2	yes	maximal, complemented
10	#20244986	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689920	C₄	1	1	yes	characteristic
13	#131×D₄₇	C₂²	1	2	yes	—
14	#15689541	—	2	2	no	—
15	#15689541	C₂²	1	2	yes	complemented
16	#15689540	C₂²	1	2	yes	complemented
17	#15689539	—	2	2	no	—
18	#15689539	C₂²	1	2	yes	complemented
19	C₂×#13093963	C₂²	1	1	yes	characteristic
20	M₃₂×D₄₇	—	2	2	no	—
21	M₃₂×D₄₇	C₂²	1	2	yes	—
22	C₂×C₁₆×D₄₇	—	2	2	no	—
23	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
24	#15689245	C₄	1	1	yes	characteristic
25	#15689244	C₄	1	1	yes	characteristic
26	#15689242	C₄	1	2	yes	—
27	#15689241	C₂²	1	1	yes	characteristic
28	#15689238	C₂²	1	2	yes	—
29	#15689170	C₄	1	1	yes	characteristic
30	#15689138	C₄	1	2	yes	—
31	#15689077	C₂²	1	1	yes	characteristic
32	#15688944	C₂²	1	1	yes	characteristic, complemented
33	#15688943	C₂²	1	2	yes	complemented
34	#15688908	C₂²	1	2	yes	complemented
35	#15688907	C₂²	1	2	yes	complemented
36	#15688906	C₂²	1	2	yes	complemented
37	#15688898	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#504	C₂²	1	2	yes	complemented
41	C₄₇×#443	C₂²	1	1	yes	characteristic
42	C₄₇×#364	C₂²	1	1	yes	characteristic, complemented
43	C₄₇×#350	C₂²	1	2	yes	complemented
44	C₄₇×#346	C₂²	1	1	yes	characteristic, complemented
45	#41×D₄₇	C₂×C₄	1	2	yes	—
46	#13094056	C₈	1	2	yes	—
47	#13094053	C₈	1	2	yes	—
48	#13094051	C₈	1	2	yes	—
49	#13094048	C₈	1	2	yes	—
50	#13094046	C₂×C₄	1	1	yes	characteristic
51	#13094044	C₂×C₄	1	2	yes	—
52	#13094032	—	2	2	no	—
53	#13094032	C₂×C₄	1	2	yes	—
54	#13094031	C₂×C₄	1	2	yes	—
55	#13094030	—	2	2	no	—
56	#13094030	C₂³	1	2	yes	—
57	C₂×#10625324	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₄₇	—	2	2	no	—
59	M₁₆×D₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₄₇	—	2	2	no	—
61	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
62	#13093981	C₂×C₄	1	1	yes	characteristic
63	#13093964	—	2	2	no	—
64	#13093964	C₂³	1	2	yes	—
65	#13093963	—	2	2	no	—
66	#13093963	—	2	2	no	—
67	#13093963	—	2	2	no	—
68	#13093963	—	2	2	no	—
69	C₂×C₄₇⋊C₁₆	—	2	2	no	—
70	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₁₆×D₄₇	—	2	2	no	—
73	C₁₆×D₄₇	—	2	2	no	—
74	C₁₆×D₄₇	—	2	2	no	—
75	C₁₆×D₄₇	D₄	1	2	yes	—
76	C₁₆×D₄₇	D₄	1	2	yes	—
77	#13093960	C₂×C₄	1	1	yes	characteristic
78	#13093959	C₂×C₄	1	2	yes	—
79	#13093952	C₂×C₄	1	2	yes	—
80	#13093951	C₂×C₄	1	2	yes	—
81	#13093950	C₂×C₄	1	2	yes	—
82	#13093935	C₂×C₄	1	1	yes	characteristic
83	#13093934	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
85	#135×C₄₇	C₂×C₄	1	1	yes	characteristic
86	#131×C₄₇	C₂³	1	2	yes	—
87	M₃₂×C₄₇	—	2	2	no	—
88	M₃₂×C₄₇	C₂³	1	2	yes	—
89	C₂×C₁₆×C₄₇	—	2	2	no	—
90	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
92	#108×C₄₇	C₂×C₄	1	1	yes	characteristic
93	#104×C₄₇	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
95	#41×C₄₇	C₂²×C₄	1	2	yes	—
96	#10625354	C₂×C₈	1	2	yes	—
97	Q₈×D₄₇	C₂×C₈	1	2	yes	—
98	#10625352	—	2	2	no	—
99	#10625352	C₂×C₈	1	2	yes	—
100	D₄×D₄₇	—	2	2	no	—
101	D₄×D₄₇	C₂×C₈	1	2	yes	—
102	#10625350	—	2	2	no	—
103	#10625350	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₄₇	—	2	2	no	—
105	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
106	Q₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
107	SD₁₆×C₄₇	C₂×C₈	1	2	yes	—
108	D₈×C₄₇	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₄₇	—	2	2	no	—
110	M₁₆×C₄₇	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
115	#10625337	C₂×C₈	1	2	yes	—
116	#10625336	C₂×C₈	1	2	yes	—
117	#10625335	C₂×C₈	1	2	yes	—
118	#10625334	C₂×C₈	1	2	yes	—
119	(C₂×D₄₇)⋊C₄	—	2	1	no	—
120	Dic₄₇⋊C₄	—	2	1	no	—
121	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
122	#10625329	—	2	2	no	—
123	#10625329	C₂²×C₄	1	2	yes	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	Dic₁₈₈	C₂×C₈	1	1	yes	characteristic
128	D₃₇₆	C₂×C₈	1	1	yes	characteristic
129	#10625325	C₂×C₈	1	2	yes	—
130	#10625324	—	2	2	no	—
131	#10625324	—	2	2	no	—
132	#10625324	—	2	2	no	—
133	#10625324	C₂×C₈	1	2	yes	—
134	#10625324	C₂×C₈	1	2	yes	—
135	C₈×D₄₇	—	2	2	no	—
136	C₈×D₄₇	—	2	2	no	—
137	C₈×D₄₇	—	2	2	no	—
138	C₈×D₄₇	C₂×D₄	1	2	yes	—
139	C₈×D₄₇	#41	1	2	yes	—
140	C₁₆×C₄₇	—	2	2	no	—
141	C₁₆×C₄₇	—	2	1	no	—
142	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
143	C₄₇⋊C₁₆	—	2	2	no	—
144	C₄₇⋊C₁₆	—	2	1	no	—
145	C₄₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
146	C₄₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
147	C₂²×D₄₇	—	2	2	no	—
148	C₂²×D₄₇	#131	1	1	yes	characteristic
149	Q₈×C₄₇	C₂²×C₈	1	2	yes	—
150	D₄×C₄₇	—	2	2	no	—
151	D₄×C₄₇	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	1	no	—
154	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₄₇	—	2	2	no	—
156	C₂²⋊D₄₇	—	2	2	no	—
157	C₂²⋊D₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	2	1	no	—
160	C₂×Dic₄₇	#131	1	1	yes	characteristic
161	D₁₈₈	—	2	2	no	—
162	D₁₈₈	C₂²×C₈	1	2	yes	—
163	C₄×D₄₇	—	2	2	no	—
164	C₄×D₄₇	—	2	2	no	—
165	C₄×D₄₇	—	2	2	no	—
166	C₄×D₄₇	C₄×D₄	1	2	yes	—
167	C₄×D₄₇	C₂²×C₈	1	2	yes	—
168	Dic₉₄	—	2	2	no	—
169	Dic₉₄	C₂²×C₈	1	2	yes	—
170	C₈×C₄₇	—	2	2	no	—
171	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
175	C₄₇⋊C₈	—	2	2	no	—
176	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
177	C₄₇⋊C₈	C₂²×C₈	1	2	yes	—
178	#33441	—	47	1	no	maximal, Sylow
179	C₂²×C₄₇	—	2	2	no	—
180	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
181	C₂×D₄₇	—	2	2	no	—
182	C₂×D₄₇	—	4	1	no	—
183	C₂×D₄₇	—	4	4	no	—
184	C₂×D₄₇	C₈×D₄	1	2	yes	—
185	C₄×C₄₇	—	2	2	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
189	Dic₄₇	—	2	2	no	—
190	Dic₄₇	—	4	1	no	—
191	Dic₄₇	C₈×D₄	1	2	yes	—
192	C₂×#504	—	47	2	no	—
193	#2955	—	47	1	no	—
194	#2171	—	47	2	no	—
195	#2171	—	47	1	no	—
196	#2171	D₄₇	1	1	yes	characteristic, complemented
197	#2170	—	47	1	no	—
198	#2170	—	47	1	no	—
199	#2170	—	47	2	no	—
200	#2153	—	47	1	no	—
201	#2117	—	47	2	no	—
202	#2108	—	47	1	no	—
203	C₂×C₄₇	—	2	1	no	—
204	C₂×C₄₇	—	4	2	no	—
205	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
206	D₄₇	—	2	2	no	—
207	D₄₇	—	4	2	no	—
208	#575	—	47	1	no	—
209	C₂×#131	—	47	2	no	—
210	#504	—	47	2	no	—
211	#504	—	47	2	no	—
212	#504	—	47	2	no	—
213	#504	—	94	2	no	—
214	#504	—	94	2	no	—
215	#504	C₂×D₄₇	1	2	yes	complemented
216	C₂×M₃₂	—	47	1	no	—
217	C₂×M₃₂	—	47	2	no	—
218	C₂×M₃₂	—	94	2	no	—
219	C₂²×C₁₆	—	47	1	no	—
220	C₂²×C₁₆	—	94	2	no	—
221	#444	—	47	1	no	—
222	#444	—	47	1	no	—
223	#444	—	47	2	no	—
224	#443	—	47	2	no	—
225	#443	—	47	1	no	—
226	#443	C₂×D₄₇	1	1	yes	characteristic
227	#430	—	47	1	no	—
228	#421	—	47	2	no	—
229	#405	—	47	1	no	—
230	#364	—	47	2	no	—
231	#364	—	47	1	no	—
232	#364	C₂×D₄₇	1	1	yes	characteristic, complemented
233	#350	—	47	2	no	—
234	#350	—	47	2	no	—
235	#350	—	47	2	no	—
236	#350	C₂×D₄₇	1	2	yes	complemented
237	#346	—	47	1	no	—
238	#346	C₂×D₄₇	1	1	yes	characteristic, complemented
239	C₄×C₁₆	—	47	1	no	—
240	C₄×C₁₆	—	47	1	no	—
241	C₄₇	#33441	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	47	2	no	—
243	#137	—	47	2	no	—
244	#137	—	47	2	no	—
245	AGL(1,ℤ/8)	—	47	2	no	—
246	AGL(1,ℤ/8)	—	47	2	no	—
247	#135	—	47	2	no	—
248	#135	—	47	1	no	—
249	#135	C₄×D₄₇	1	1	yes	characteristic
250	#131	—	47	2	no	—
251	#131	—	47	2	no	—
252	#131	—	47	2	no	—
253	#131	—	94	2	no	—
254	#131	—	94	2	no	—
255	#131	C₂²×D₄₇	1	2	yes	—
256	C₂×M₁₆	—	47	1	no	—
257	C₂×M₁₆	—	47	2	no	—
258	C₂×M₁₆	—	94	2	no	—
259	C₂²×C₈	—	47	1	no	—
260	C₂²×C₈	—	94	2	no	—
261	#117	—	47	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	47	2	no	—
264	M₃₂	—	94	2	no	—
265	M₃₂	—	94	2	no	—
266	M₃₂	—	94	2	no	—
267	M₃₂	—	94	2	no	—
268	M₃₂	—	94	2	no	—
269	M₃₂	C₂²×D₄₇	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	47	1	no	—
272	C₂×C₁₆	—	47	2	no	—
273	C₂×C₁₆	—	47	2	no	—
274	C₂×C₁₆	—	47	1	no	—
275	C₂×C₁₆	—	94	2	no	—
276	C₂×C₁₆	—	94	2	no	—
277	C₂×C₁₆	—	94	2	no	—
278	C₂×C₁₆	—	94	2	no	—
279	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
281	#108	—	47	2	no	—
282	#108	—	47	1	no	—
283	#108	C₄×D₄₇	1	1	yes	characteristic
284	#104	—	47	2	no	—
285	#104	—	47	2	no	—
286	#104	—	47	2	no	—
287	#104	C₄×D₄₇	1	2	yes	—
288	#97	—	47	1	no	—
289	#97	—	47	1	no	—
290	C₄×C₈	—	47	1	no	—
291	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
292	#41	—	47	2	no	—
293	#41	—	47	2	no	—
294	#41	—	47	2	no	—
295	#41	—	94	2	no	—
296	#41	—	94	2	no	—
297	#41	C₂×C₄×D₄₇	1	2	yes	—
298	C₂×Q₈	—	47	2	no	—
299	C₂×D₄	—	47	2	no	—
300	C₂×D₄	—	94	2	no	—
301	C₂²×C₄	—	47	1	no	—
302	C₂²×C₄	—	94	2	no	—
303	Q₁₆	—	47	2	no	—
304	Q₁₆	—	47	1	no	—
305	Q₁₆	C₈×D₄₇	1	1	yes	characteristic
306	SD₁₆	—	47	2	no	—
307	SD₁₆	—	47	2	no	—
308	SD₁₆	—	47	2	no	—
309	SD₁₆	C₈×D₄₇	1	2	yes	—
310	D₈	—	47	2	no	—
311	D₈	—	47	1	no	—
312	D₈	C₈×D₄₇	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	47	2	no	—
315	M₁₆	—	47	2	no	—
316	M₁₆	—	47	2	no	—
317	M₁₆	—	94	2	no	—
318	M₁₆	—	94	2	no	—
319	M₁₆	—	94	2	no	—
320	M₁₆	—	94	2	no	—
321	M₁₆	C₂×C₄×D₄₇	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	47	1	no	—
324	C₂×C₈	—	47	1	no	—
325	C₂×C₈	—	47	2	no	—
326	C₂×C₈	—	47	2	no	—
327	C₂×C₈	—	94	2	no	—
328	C₂×C₈	—	94	2	no	—
329	C₂×C₈	—	94	2	no	—
330	C₂×C₈	—	94	2	no	—
331	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
333	C₄⋊C₄	—	94	1	no	—
334	C₂²⋊C₄	—	94	1	no	—
335	C₄²	—	47	1	no	—
336	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	47	1	no	—
340	C₁₆	—	47	1	no	—
341	C₁₆	—	94	2	no	—
342	C₁₆	—	94	1	no	—
343	C₁₆	D₄×D₄₇	1	2	yes	—
344	C₂³	—	47	1	no	—
345	C₂³	—	94	2	no	—
346	Q₈	—	47	2	no	—
347	Q₈	—	94	2	no	—
348	Q₈	C₂×C₈×D₄₇	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	47	2	no	—
351	D₄	—	94	2	no	—
352	D₄	—	94	2	no	—
353	D₄	—	94	2	no	—
354	D₄	—	94	2	no	—
355	D₄	C₂×C₈×D₄₇	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	47	1	no	—
359	C₂×C₄	—	47	2	no	—
360	C₂×C₄	—	47	2	no	—
361	C₂×C₄	—	94	2	no	—
362	C₂×C₄	—	94	2	no	—
363	C₂×C₄	—	94	2	no	—
364	C₂×C₄	—	94	2	no	—
365	C₂×C₄	—	94	1	no	—
366	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	47	2	no	—
369	C₈	—	47	2	no	—
370	C₈	—	94	2	no	—
371	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
372	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
373	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
374	C₈	#41×D₄₇	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	47	2	no	—
377	C₂²	—	94	2	no	—
378	C₂²	—	188	1	no	—
379	C₂²	—	188	4	no	—
380	C₂²	#131×D₄₇	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	47	2	no	—
384	C₄	—	94	2	no	—
385	C₄	—	188	1	no	—
386	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	94	2	no	—
391	C₂	—	188	2	no	—
392	C₂	C₈×D₄×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2171	C₂
socle	C₂×C₄₇	C₂×C₈×D₄
prime radical	#15689920	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2171	D₄₇
hypocenter	C₄₇	#33441
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #504×D₄₇	C₂	semidirect
3. #20245231	C₂	semidirect
4. #20245228	C₂	semidirect
5. #20245225	C₂	semidirect
6. #20245224	C₂	semidirect
7. #20245222	C₂	semidirect
8. #20245152	C₂	semidirect
9. #20245014	C₂	semidirect
10. #20244986	C₂	semidirect
11. C₄₇×#2171	C₂	semidirect
15. #15689541	C₂²	semidirect
16. #15689540	C₂²	semidirect
18. #15689539	C₂²	semidirect
32. #15688944	C₂²	semidirect
33. #15688943	C₂²	semidirect
34. #15688908	C₂²	semidirect
35. #15688907	C₂²	semidirect
36. #15688906	C₂²	semidirect
37. #15688898	C₂²	semidirect
38. C₄×C₄₇⋊C₁₆	C₂²	semidirect
39. C₁₆×Dic₄₇	C₂²	semidirect
40. C₄₇×#504	C₂²	semidirect
42. C₄₇×#364	C₂²	semidirect
43. C₄₇×#350	C₂²	semidirect
44. C₄₇×#346	C₂²	semidirect
196. #2171	D₄₇	semidirect
215. #504	C₂×D₄₇	semidirect
232. #364	C₂×D₄₇	semidirect
236. #350	C₂×D₄₇	semidirect
238. #346	C₂×D₄₇	semidirect
241. C₄₇	#33441	semidirect

Maximal subgroups

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

subgroup	index
#504×D₄₇	2
#20252286	2
#20245231	2
#20245228	2
#20245225	2
#20245224	2
#20245222	2
#20245152	2
#20245014	2
#20244986	2
C₄₇×#2171	2
#33441	47