Subgroups — #26131181

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#15689158	C₂	1	1	yes	maximal, characteristic
4	#20241488	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241487	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1383	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	C₄	1	2	yes	complemented
13	M₃₂×D₄₇	—	2	2	no	—
14	M₃₂×D₄₇	—	2	2	no	—
15	C₂×C₁₆×D₄₇	C₄	1	2	yes	complemented
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	#15689158	—	2	2	no	—
21	#15689158	C₄	1	4	yes	—
22	C₂×#13093957	—	2	1	no	—
23	C₂×#13093956	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
28	C₂×#10625329	—	2	1	no	—
29	C₂×#10625324	—	4	1	no	—
30	M₁₆×D₄₇	—	2	4	no	—
31	M₁₆×D₄₇	—	4	2	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₄₇	—	4	1	no	—
34	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₄₇	D₄	1	2	yes	—
39	C₂×C₈×D₄₇	D₄	1	2	yes	—
40	C₂×C₈×D₄₇	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₄₇	Q₈	1	1	yes	characteristic
42	#13093982	—	2	4	no	—
43	#13093982	—	4	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
46	C₂×C₄×Dic₄₇	—	4	1	no	—
47	#13093964	—	2	1	no	—
48	#13093964	—	2	1	no	—
49	#13093963	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₁₆×D₄₇	—	2	4	no	—
52	#13093960	—	2	1	no	—
53	#13093960	C₂×C₄	1	2	yes	—
54	#13093959	—	2	4	no	—
55	#13093957	—	2	4	no	—
56	#13093956	—	2	1	no	—
57	#13093956	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
61	M₃₂×C₄₇	—	2	1	no	—
62	M₃₂×C₄₇	—	2	1	no	—
63	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
64	#108×C₄₇	—	2	1	no	—
65	#108×C₄₇	C₂×C₄	1	2	yes	—
66	#106×C₄₇	—	2	1	no	—
67	#106×C₄₇	C₂×C₄	1	2	yes	—
68	C₂³×D₄₇	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	4	1	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	4	1	no	—
72	C₂²×Dic₄₇	SD₁₆	1	1	yes	characteristic
73	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₄₇	D₈	1	2	yes	—
78	C₂×C₄×D₄₇	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₄₇	SD₁₆	1	1	yes	characteristic
80	C₂×C₄×D₄₇	Q₁₆	1	2	yes	—
81	M₁₆×C₄₇	—	2	2	no	—
82	M₁₆×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	(C₂×D₄₇)⋊C₄	—	4	4	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	4	1	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	#10625329	—	2	2	no	—
97	#10625329	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	4	1	no	—
99	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
103	#10625324	—	4	4	no	—
104	C₈×D₄₇	—	4	4	no	—
105	C₈×D₄₇	C₄²	1	2	yes	—
106	C₈×D₄₇	C₄²	1	2	yes	—
107	C₈×D₄₇	C₂²⋊C₄	1	2	yes	—
108	C₈×D₄₇	C₂²⋊C₄	1	2	yes	—
109	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
110	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
111	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
112	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
113	C₁₆×C₄₇	—	2	2	no	—
114	C₄₇⋊C₁₆	—	2	2	no	—
115	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×D₄₇	—	4	2	no	—
117	C₂²×D₄₇	#102	1	2	yes	—
118	C₂²×D₄₇	#103	1	2	yes	—
119	C₂²×D₄₇	#106	1	1	yes	characteristic
120	C₂²×D₄₇	#106	1	1	yes	characteristic
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	8	2	no	—
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	8	2	no	—
131	C₂×Dic₄₇	#102	1	2	yes	—
132	C₂×Dic₄₇	#103	1	2	yes	—
133	C₂×Dic₄₇	#106	1	1	yes	characteristic
134	C₂×Dic₄₇	#106	1	1	yes	characteristic
135	C₄×D₄₇	#95	1	2	yes	—
136	C₄×D₄₇	#95	1	2	yes	—
137	C₄×D₄₇	#102	1	2	yes	—
138	C₄×D₄₇	#102	1	2	yes	—
139	C₄×D₄₇	#103	1	2	yes	—
140	C₄×D₄₇	#103	1	2	yes	—
141	C₄×D₄₇	#107	1	2	yes	—
142	C₄×D₄₇	#107	1	2	yes	—
143	C₈×C₄₇	—	4	2	no	—
144	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
145	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₄₇⋊C₈	—	4	2	no	—
149	C₄₇⋊C₈	#95	1	2	yes	—
150	C₄₇⋊C₈	#95	1	2	yes	—
151	#12563	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
156	C₂×D₄₇	—	2	2	no	—
157	C₂×D₄₇	—	2	2	no	—
158	C₂×D₄₇	—	4	2	no	—
159	C₂×D₄₇	—	4	2	no	—
160	C₂×D₄₇	#340	1	2	yes	—
161	C₂×D₄₇	#340	1	2	yes	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
166	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
167	Dic₄₇	—	8	2	no	—
168	Dic₄₇	#340	1	2	yes	—
169	Dic₄₇	#340	1	2	yes	—
170	C₂²×M₃₂	—	47	1	no	—
171	C₂²×#108	—	47	1	no	—
172	C₂×#427	—	47	1	no	—
173	#1383	—	47	1	no	—
174	#1383	—	47	1	no	—
175	#1383	—	47	1	no	—
176	#1383	D₄₇	1	1	yes	characteristic, complemented
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
180	C₂×C₄₇	#2155	1	2	yes	—
181	D₄₇	—	2	2	no	—
182	D₄₇	—	2	2	no	—
183	C₂²×M₁₆	—	94	1	no	—
184	C₂³×C₈	—	47	1	no	—
185	C₂×#117	—	94	1	no	—
186	C₂×M₃₂	—	47	1	no	—
187	C₂×M₃₂	—	47	2	no	—
188	C₂×M₃₂	—	94	2	no	—
189	C₂×M₃₂	—	94	2	no	—
190	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
191	C₂²×C₁₆	—	47	2	no	—
192	C₂×#108	—	47	1	no	—
193	C₂×#108	—	47	4	no	—
194	C₂×#108	—	94	2	no	—
195	C₂×#108	—	94	1	no	—
196	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
197	#427	—	47	4	no	—
198	#427	—	94	2	no	—
199	C₂×#107	—	94	1	no	—
200	C₂×#106	—	47	1	no	—
201	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
202	C₄₇	#12563	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	47	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	94	4	no	—
206	C₂×M₁₆	—	94	1	no	—
207	C₂×M₁₆	—	188	1	no	—
208	C₂×M₁₆	—	188	2	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	2	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	1	no	—
213	C₂²×C₈	—	47	1	no	—
214	C₂²×C₈	—	47	2	no	—
215	C₂²×C₈	—	47	2	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	188	1	no	—
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#117	—	94	4	no	—
221	#117	—	188	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	C₂×C₂²⋊C₄	—	188	1	no	—
225	C₂×C₄²	—	188	1	no	—
226	M₃₂	—	2	1	no	—
227	M₃₂	—	2	1	no	—
228	M₃₂	—	94	4	no	—
229	M₃₂	—	94	1	no	—
230	M₃₂	—	94	1	no	—
231	C₂×C₁₆	—	47	2	no	—
232	C₂×C₁₆	—	94	4	no	—
233	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
234	#108	—	2	1	no	—
235	#108	—	47	2	no	—
236	#108	—	94	4	no	—
237	#108	—	94	1	no	—
238	#108	C₄×D₄₇	1	2	yes	—
239	#107	—	94	4	no	—
240	#106	—	2	1	no	—
241	#106	—	47	2	no	—
242	#106	—	94	1	no	—
243	#106	C₄×D₄₇	1	2	yes	—
244	C₂⁴	—	47	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	47	2	no	—
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	—	47	1	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	188	1	no	—
256	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	94	2	no	—
260	M₁₆	—	188	4	no	—
261	M₁₆	—	188	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	47	1	no	—
266	C₂×C₈	—	47	1	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	47	2	no	—
271	C₂×C₈	—	47	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	—	47	2	no	—
274	C₂×C₈	—	47	2	no	—
275	C₂×C₈	—	188	4	no	—
276	C₂×C₈	—	188	1	no	—
277	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
282	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	94	2	no	—
286	C₄⋊C₄	—	188	1	no	—
287	C₂²⋊C₄	—	188	4	no	—
288	C₄²	—	188	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	94	2	no	—
291	C₂³	—	47	1	no	—
292	C₂³	—	47	1	no	—
293	C₂³	—	47	2	no	—
294	C₂³	—	47	2	no	—
295	C₂³	—	188	2	no	—
296	C₂³	SD₁₆×D₄₇	1	1	yes	characteristic
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	1	no	—
302	C₂×C₄	—	47	1	no	—
303	C₂×C₄	—	47	2	no	—
304	C₂×C₄	—	47	2	no	—
305	C₂×C₄	—	47	2	no	—
306	C₂×C₄	—	47	2	no	—
307	C₂×C₄	—	47	2	no	—
308	C₂×C₄	—	47	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	376	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
317	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
318	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	47	2	no	—
321	C₈	—	47	2	no	—
322	C₈	—	188	2	no	—
323	C₈	C₄²×D₄₇	1	1	yes	characteristic
324	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
325	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
326	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
327	C₂²	—	2	2	no	—
328	C₂²	—	47	2	no	—
329	C₂²	—	47	2	no	—
330	C₂²	—	94	2	no	—
331	C₂²	—	94	2	no	—
332	C₂²	—	188	2	no	—
333	C₂²	—	188	2	no	—
334	C₂²	#102×D₄₇	1	1	yes	characteristic
335	C₂²	#103×D₄₇	1	1	yes	characteristic
336	C₂²	#106×D₄₇	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	47	2	no	—
339	C₄	—	47	2	no	—
340	C₄	—	376	2	no	—
341	C₄	#95×D₄₇	1	1	yes	characteristic
342	C₄	#102×D₄₇	1	1	yes	characteristic
343	C₄	#103×D₄₇	1	1	yes	characteristic
344	C₄	#107×D₄₇	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	94	2	no	—
348	C₂	—	94	2	no	—
349	C₂	#340×D₄₇	1	1	yes	characteristic
350	C₂	#20245159	1	2	yes	—

	subgroup	G/H
center	C₂²	#106×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1383	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂³×D₄₇	SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1383	D₄₇
hypocenter	C₄₇	#12563
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
4. #20241488	C₂	semidirect
5. #20241487	C₂	semidirect
6. #20241486	C₂	semidirect
7. C₄₇×#1383	C₂	semidirect
12. C₂×#13093963	C₄	semidirect
15. C₂×C₁₆×D₄₇	C₄	semidirect
19. #108×D₄₇	C₄	semidirect
176. #1383	D₄₇	semidirect
202. C₄₇	#12563	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×#15689158	2
#20241488	2
#20241487	2
#20241486	2
C₄₇×#1383	2
#12563	47