Subgroups — #26131203

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689160	C₂	1	1	yes	maximal, characteristic
4	#20241498	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241497	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241496	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₇	—	2	1	no	—
9	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093982	—	2	1	no	—
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂×#13093963	—	2	1	no	—
13	M₃₂×D₄₇	—	2	2	no	—
14	M₃₂×D₄₇	C₄	1	4	yes	complemented
15	C₂×C₁₆×D₄₇	—	2	1	no	—
16	C₂×#13093960	C₂²	1	1	yes	characteristic
17	C₂×#13093959	—	2	1	no	—
18	#108×D₄₇	—	2	2	no	—
19	#108×D₄₇	C₄	1	4	yes	complemented
20	#15689160	—	2	1	no	—
21	#15689160	—	2	1	no	—
22	#15689160	C₄	1	2	yes	—
23	#15689160	C₄	1	2	yes	—
24	C₂×#13093958	C₂²	1	1	yes	characteristic
25	C₂×#13093955	—	2	1	no	—
26	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
30	C₂×#10625329	—	2	1	no	—
31	C₂×#10625324	—	4	1	no	—
32	M₁₆×D₄₇	—	2	4	no	—
33	M₁₆×D₄₇	—	4	2	no	—
34	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₄₇	—	4	1	no	—
36	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
41	C₂×C₈×D₄₇	D₄	1	2	yes	—
42	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₇	Q₈	1	2	yes	—
44	#13093982	—	2	2	no	—
45	#13093982	—	2	2	no	—
46	#13093982	—	4	1	no	—
47	#13093982	—	4	1	no	—
48	C₂×C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
50	C₂×C₄×Dic₄₇	—	4	1	no	—
51	#13093964	—	2	1	no	—
52	#13093964	C₂×C₄	1	2	yes	—
53	#13093963	—	2	4	no	—
54	C₂×C₄₇⋊C₁₆	—	2	1	no	—
55	C₁₆×D₄₇	—	2	4	no	—
56	#13093960	—	2	1	no	—
57	#13093960	C₂×C₄	1	2	yes	—
58	#13093959	—	2	4	no	—
59	#13093958	—	2	1	no	—
60	#13093958	C₂×C₄	1	2	yes	—
61	#13093955	—	2	2	no	—
62	#13093955	—	2	2	no	—
63	C₂×M₁₆×C₄₇	—	2	1	no	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
66	M₃₂×C₄₇	—	2	1	no	—
67	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₁₆×C₄₇	—	2	1	no	—
69	#108×C₄₇	—	2	1	no	—
70	#108×C₄₇	C₂×C₄	1	2	yes	—
71	#107×C₄₇	—	2	1	no	—
72	#107×C₄₇	C₂×C₄	1	2	yes	—
73	C₂³×D₄₇	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	4	1	no	—
77	C₂²×Dic₄₇	SD₁₆	1	1	yes	characteristic
78	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
83	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
84	C₂×C₄×D₄₇	SD₁₆	1	2	yes	—
85	C₂×C₄×D₄₇	Q₁₆	1	2	yes	—
86	M₁₆×C₄₇	—	2	2	no	—
87	M₁₆×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	—	4	1	no	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	4	1	no	—
97	(C₂×D₄₇)⋊C₄	—	4	2	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	4	1	no	—
101	C₄×Dic₄₇	—	4	2	no	—
102	C₄×Dic₄₇	—	4	2	no	—
103	#10625329	—	2	2	no	—
104	#10625329	—	4	1	no	—
105	C₂×C₄₇⋊C₈	—	4	1	no	—
106	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
109	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
110	#10625324	—	4	4	no	—
111	C₈×D₄₇	—	4	4	no	—
112	C₈×D₄₇	C₄²	1	2	yes	—
113	C₈×D₄₇	C₄²	1	2	yes	—
114	C₈×D₄₇	C₂²⋊C₄	1	2	yes	—
115	C₈×D₄₇	C₂²⋊C₄	1	2	yes	—
116	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
117	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
118	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
119	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
120	C₁₆×C₄₇	—	2	2	no	—
121	C₄₇⋊C₁₆	—	2	2	no	—
122	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×D₄₇	—	4	1	no	—
124	C₂²×D₄₇	—	4	1	no	—
125	C₂²×D₄₇	#102	1	1	yes	characteristic
126	C₂²×D₄₇	#102	1	1	yes	characteristic
127	C₂²×D₄₇	#103	1	2	yes	—
128	C₂²×D₄₇	#106	1	2	yes	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	8	2	no	—
139	C₂×Dic₄₇	#102	1	1	yes	characteristic
140	C₂×Dic₄₇	#102	1	1	yes	characteristic
141	C₂×Dic₄₇	#103	1	2	yes	—
142	C₂×Dic₄₇	#106	1	2	yes	—
143	C₄×D₄₇	#95	1	2	yes	—
144	C₄×D₄₇	#95	1	2	yes	—
145	C₄×D₄₇	#102	1	2	yes	—
146	C₄×D₄₇	#102	1	2	yes	—
147	C₄×D₄₇	#103	1	2	yes	—
148	C₄×D₄₇	#103	1	2	yes	—
149	C₄×D₄₇	#107	1	2	yes	—
150	C₄×D₄₇	#107	1	2	yes	—
151	C₈×C₄₇	—	4	2	no	—
152	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
153	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₄₇⋊C₈	—	4	2	no	—
157	C₄₇⋊C₈	#95	1	2	yes	—
158	C₄₇⋊C₈	#95	1	2	yes	—
159	#12569	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
165	C₂×D₄₇	—	2	1	no	—
166	C₂×D₄₇	—	2	1	no	—
167	C₂×D₄₇	—	2	1	no	—
168	C₂×D₄₇	—	2	1	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₂×D₄₇	—	4	2	no	—
171	C₂×D₄₇	#340	1	2	yes	—
172	C₂×D₄₇	#340	1	2	yes	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
175	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
177	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
178	Dic₄₇	—	8	2	no	—
179	Dic₄₇	#340	1	2	yes	—
180	Dic₄₇	#340	1	2	yes	—
181	C₂²×M₃₂	—	47	1	no	—
182	C₂²×#108	—	47	1	no	—
183	C₂×#427	—	47	1	no	—
184	#1387	—	47	1	no	—
185	#1387	—	47	1	no	—
186	#1387	—	47	1	no	—
187	#1387	D₄₇	1	1	yes	characteristic, complemented
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	—	2	1	no	—
190	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
191	C₂×C₄₇	#2148	1	1	yes	characteristic
192	C₂×C₄₇	#2148	1	1	yes	characteristic
193	D₄₇	—	2	2	no	—
194	D₄₇	—	2	2	no	—
195	C₂²×M₁₆	—	94	1	no	—
196	C₂³×C₈	—	47	1	no	—
197	C₂×#117	—	94	1	no	—
198	C₂×M₃₂	—	47	1	no	—
199	C₂×M₃₂	—	47	4	no	—
200	C₂×M₃₂	—	94	1	no	—
201	C₂×M₃₂	—	94	2	no	—
202	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
203	C₂²×C₁₆	—	94	1	no	—
204	C₂×#108	—	47	1	no	—
205	C₂×#108	—	47	4	no	—
206	C₂×#108	—	94	2	no	—
207	C₂×#108	—	94	1	no	—
208	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
209	#427	—	47	2	no	—
210	#427	—	47	2	no	—
211	#427	—	94	1	no	—
212	#427	—	94	1	no	—
213	C₂×#107	—	47	1	no	—
214	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
215	C₂×#106	—	94	1	no	—
216	C₄₇	#12569	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	—	47	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	94	4	no	—
220	C₂×M₁₆	—	94	1	no	—
221	C₂×M₁₆	—	188	1	no	—
222	C₂×M₁₆	—	188	2	no	—
223	C₂²×C₈	—	47	1	no	—
224	C₂²×C₈	—	47	1	no	—
225	C₂²×C₈	—	47	1	no	—
226	C₂²×C₈	—	47	2	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	—	47	1	no	—
229	C₂²×C₈	—	47	1	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	—	188	1	no	—
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	#117	—	94	2	no	—
235	#117	—	94	2	no	—
236	#117	—	188	1	no	—
237	#117	—	188	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	94	1	no	—
240	C₂×C₂²⋊C₄	—	188	1	no	—
241	C₂×C₄²	—	188	1	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	47	2	no	—
244	M₃₂	—	94	4	no	—
245	M₃₂	—	94	1	no	—
246	M₃₂	C₄×D₄₇	1	2	yes	—
247	C₂×C₁₆	—	2	1	no	—
248	C₂×C₁₆	—	94	4	no	—
249	C₂×C₁₆	—	94	1	no	—
250	#108	—	2	1	no	—
251	#108	—	47	2	no	—
252	#108	—	94	4	no	—
253	#108	—	94	1	no	—
254	#108	C₄×D₄₇	1	2	yes	—
255	#107	—	2	1	no	—
256	#107	—	47	2	no	—
257	#107	—	94	1	no	—
258	#107	C₄×D₄₇	1	2	yes	—
259	#106	—	94	2	no	—
260	#106	—	94	2	no	—
261	C₂⁴	—	47	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	47	1	no	—
264	C₂²×C₄	—	47	1	no	—
265	C₂²×C₄	—	47	1	no	—
266	C₂²×C₄	—	47	2	no	—
267	C₂²×C₄	—	47	2	no	—
268	C₂²×C₄	—	47	1	no	—
269	C₂²×C₄	—	47	1	no	—
270	C₂²×C₄	—	47	2	no	—
271	C₂²×C₄	—	47	2	no	—
272	C₂²×C₄	—	188	1	no	—
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	M₁₆	—	2	2	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	94	2	no	—
277	M₁₆	—	188	4	no	—
278	M₁₆	—	188	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	47	1	no	—
281	C₂×C₈	—	47	1	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	47	2	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	47	2	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	188	4	no	—
293	C₂×C₈	—	188	1	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	188	1	no	—
304	C₂²⋊C₄	—	188	2	no	—
305	C₂²⋊C₄	—	188	2	no	—
306	C₄²	—	188	2	no	—
307	C₄²	—	188	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	94	2	no	—
310	C₂³	—	47	1	no	—
311	C₂³	—	47	1	no	—
312	C₂³	—	47	2	no	—
313	C₂³	—	47	2	no	—
314	C₂³	—	188	1	no	—
315	C₂³	—	188	1	no	—
316	C₂³	SD₁₆×D₄₇	1	1	yes	characteristic
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	47	1	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	47	2	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	47	2	no	—
326	C₂×C₄	—	47	2	no	—
327	C₂×C₄	—	47	2	no	—
328	C₂×C₄	—	47	2	no	—
329	C₂×C₄	—	47	2	no	—
330	C₂×C₄	—	47	2	no	—
331	C₂×C₄	—	376	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
336	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
338	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	47	2	no	—
341	C₈	—	47	2	no	—
342	C₈	—	188	2	no	—
343	C₈	C₄²×D₄₇	1	1	yes	characteristic
344	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
345	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
346	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	47	2	no	—
350	C₂²	—	47	2	no	—
351	C₂²	—	94	1	no	—
352	C₂²	—	94	1	no	—
353	C₂²	—	94	1	no	—
354	C₂²	—	94	1	no	—
355	C₂²	—	188	2	no	—
356	C₂²	—	188	2	no	—
357	C₂²	#102×D₄₇	1	1	yes	characteristic
358	C₂²	#103×D₄₇	1	1	yes	characteristic
359	C₂²	#106×D₄₇	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	47	2	no	—
362	C₄	—	47	2	no	—
363	C₄	—	376	2	no	—
364	C₄	#95×D₄₇	1	1	yes	characteristic
365	C₄	#102×D₄₇	1	1	yes	characteristic
366	C₄	#103×D₄₇	1	1	yes	characteristic
367	C₄	#107×D₄₇	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	94	2	no	—
371	C₂	—	94	2	no	—
372	C₂	#340×D₄₇	1	1	yes	characteristic
373	C₂	#20245134	1	1	yes	characteristic
374	C₂	#20245136	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1387	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×D₄₇	SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1387	D₄₇
hypocenter	C₄₇	#12569
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. #20241498	C₂	semidirect
5. #20241497	C₂	semidirect
6. #20241496	C₂	semidirect
7. C₄₇×#1387	C₂	semidirect
14. M₃₂×D₄₇	C₄	semidirect
19. #108×D₄₇	C₄	semidirect
187. #1387	D₄₇	semidirect
216. C₄₇	#12569	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₄₇	2
C₂×#108×D₄₇	2
C₂×#15689160	2
#20241498	2
#20241497	2
#20241496	2
C₄₇×#1387	2
#12569	47