Subgroups — #26133159

Order 12032.

488 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256490	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252082	C₂	1	1	yes	maximal, characteristic
3	#20245008	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244999	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241475	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241473	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2112	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	#15689634	C₄	1	1	yes	characteristic
10	#15689541	—	2	2	no	—
11	#15689539	—	2	2	no	—
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	C₂×#13093964	C₂²	1	1	yes	characteristic
14	M₃₂×D₄₇	—	2	2	no	—
15	#15689194	—	2	2	no	—
16	C₈×C₂²⋊D₄₇	—	2	2	no	—
17	#15689149	C₄	1	1	yes	characteristic
18	#15689089	C₄	1	1	yes	characteristic
19	#15689087	C₄	1	1	yes	characteristic
20	#15689078	C₂²	1	1	yes	characteristic
21	#15688904	—	2	2	no	—
22	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	#13094030	—	2	1	no	—
29	#13094030	—	2	1	no	—
30	#13094030	—	2	1	no	—
31	#13094030	—	2	1	no	—
32	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
33	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
34	C₂×#10625324	D₄	1	2	yes	—
35	C₂×C₈×D₄₇	D₄	1	2	yes	—
36	#13094007	C₈	1	2	yes	—
37	#13094003	—	2	1	no	—
38	#13093998	—	2	1	no	—
39	#13093993	C₈	1	2	yes	—
40	C₄×C₂²⋊D₄₇	—	2	2	no	—
41	#13093983	C₂×C₄	1	1	yes	characteristic
42	#13093964	—	2	1	no	—
43	#13093964	—	2	1	no	—
44	#13093964	—	2	1	no	—
45	#13093964	—	2	1	no	—
46	#13093963	—	2	4	no	—
47	C₂×C₄₇⋊C₁₆	—	2	1	no	—
48	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	complemented
49	C₁₆×D₄₇	—	2	4	no	—
50	Dic₄₇⋊C₈	—	2	2	no	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
52	(C₂×D₄₇)⋊C₈	—	2	2	no	—
53	#13093934	—	2	1	no	—
54	C₈×Dic₄₇	—	2	1	no	—
55	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	—	2	1	no	—
60	C₂²×C₈×C₄₇	—	2	1	no	—
61	C₂²×C₈×C₄₇	—	2	1	no	—
62	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	M₃₂×C₄₇	—	2	1	no	—
65	M₃₂×C₄₇	—	2	1	no	—
66	M₃₂×C₄₇	—	2	1	no	—
67	M₃₂×C₄₇	—	2	1	no	—
68	C₂×C₁₆×C₄₇	—	2	1	no	—
69	C₂×C₁₆×C₄₇	D₄	1	2	yes	complemented
70	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	—	2	1	no	—
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	—	2	1	no	—
75	C₂²×C₄×C₄₇	—	2	1	no	—
76	C₂²×C₄×C₄₇	—	2	1	no	—
77	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	—	2	2	no	—
80	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
81	#10625350	—	2	1	no	—
82	#10625350	—	2	1	no	—
83	#10625350	—	2	1	no	—
84	#10625350	—	2	1	no	—
85	C₂×D₁₈₈	M₁₆	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
87	C₂×Dic₉₄	M₁₆	1	1	yes	characteristic
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	4	1	no	—
92	C₂×C₈×C₄₇	—	4	1	no	—
93	C₂×C₈×C₄₇	—	4	1	no	—
94	C₂×C₈×C₄₇	—	4	1	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
100	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
101	C₂²⋊Dic₄₇	—	2	2	no	—
102	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
103	(C₂×D₄₇)⋊C₄	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	1	no	—
105	Dic₄₇⋊C₄	—	2	2	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	#10625329	—	2	1	no	—
108	#10625329	—	2	1	no	—
109	#10625329	—	2	1	no	—
110	#10625329	—	2	1	no	—
111	C₂×C₄₇⋊C₈	—	2	1	no	—
112	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
113	#10625324	—	2	2	no	—
114	#10625324	—	2	2	no	—
115	C₈×D₄₇	—	2	2	no	—
116	C₈×D₄₇	—	2	2	no	—
117	C₁₆×C₄₇	—	2	2	no	—
118	C₁₆×C₄₇	—	4	1	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₄₇⋊C₁₆	—	4	1	no	—
121	C₂³×C₄₇	—	2	1	no	—
122	C₂³×C₄₇	—	2	1	no	—
123	C₂³×C₄₇	—	2	1	no	—
124	C₂³×C₄₇	—	4	2	no	—
125	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂²×D₄₇	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	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	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
140	C₂²⋊D₄₇	—	2	2	no	—
141	C₂²⋊D₄₇	—	2	2	no	—
142	C₂²⋊D₄₇	—	4	4	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
145	D₁₈₈	—	2	1	no	—
146	D₁₈₈	—	2	1	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	Dic₉₄	—	2	1	no	—
150	Dic₉₄	—	2	1	no	—
151	C₈×C₄₇	—	4	1	no	—
152	C₈×C₄₇	#120	1	1	yes	characteristic
153	C₈×C₄₇	#121	1	1	yes	characteristic
154	C₈×C₄₇	#121	1	1	yes	characteristic
155	C₈×C₄₇	#123	1	1	yes	characteristic
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	4	1	no	—
158	#12981	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	2	2	no	—
162	C₂²×C₄₇	—	4	1	no	—
163	C₂²×C₄₇	—	4	1	no	—
164	C₂²×C₄₇	—	4	1	no	—
165	C₂²×C₄₇	—	4	1	no	—
166	C₂²×C₄₇	—	4	1	no	—
167	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
169	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
170	C₂×D₄₇	—	2	2	no	—
171	C₂×D₄₇	—	4	2	no	—
172	C₄×C₄₇	—	4	1	no	—
173	C₄×C₄₇	#386	1	1	yes	characteristic
174	C₄×C₄₇	#408	1	1	yes	characteristic
175	C₄×C₄₇	#435	1	1	yes	characteristic
176	C₄×C₄₇	#435	1	1	yes	characteristic
177	Dic₄₇	—	2	2	no	—
178	Dic₄₇	—	4	2	no	—
179	#3407	—	47	1	no	—
180	#2929	—	47	1	no	—
181	#2115	—	47	1	no	—
182	#2112	—	47	1	no	—
183	#2112	D₄₇	1	1	yes	characteristic, complemented
184	#1378	—	47	1	no	—
185	#1378	—	47	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	—	4	2	no	—
190	C₂×C₄₇	#1869	1	1	yes	characteristic
191	D₄₇	—	4	2	no	—
192	C₂×#131	—	47	1	no	—
193	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
194	#525	—	47	1	no	—
195	#504	—	94	2	no	—
196	#504	—	94	2	no	—
197	C₂×M₃₂	—	47	1	no	—
198	C₂×M₃₂	—	47	1	no	—
199	C₂×M₃₂	—	94	2	no	—
200	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
201	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
202	#435	—	94	2	no	—
203	C₈×D₄	—	94	2	no	—
204	#424	—	47	1	no	—
205	#408	—	47	1	no	—
206	#407	—	47	1	no	—
207	#405	—	47	1	no	—
208	#349	—	94	2	no	—
209	C₂²⋊C₁₆	—	2	2	no	—
210	C₂²⋊C₁₆	—	94	2	no	—
211	C₄₇	#12981	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	47	1	no	—
213	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
214	#131	—	94	1	no	—
215	#131	—	94	1	no	—
216	#131	—	94	1	no	—
217	#131	—	94	1	no	—
218	C₂×M₁₆	—	47	1	no	—
219	C₂×M₁₆	—	47	2	no	—
220	C₂×M₁₆	—	47	1	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	#123	—	47	2	no	—
231	C₂²⋊Q₈	—	94	1	no	—
232	#121	—	94	1	no	—
233	#120	—	47	2	no	—
234	C₄×D₄	—	94	2	no	—
235	#117	—	47	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	94	4	no	—
241	M₃₂	—	94	1	no	—
242	M₃₂	—	94	1	no	—
243	M₃₂	—	94	1	no	—
244	M₃₂	—	94	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	47	2	no	—
247	C₂×C₁₆	—	94	4	no	—
248	C₂×C₁₆	—	94	1	no	—
249	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	complemented
250	C₄⋊C₈	—	94	2	no	—
251	C₂²⋊C₈	—	94	2	no	—
252	C₂²⋊C₈	—	94	2	no	—
253	#97	—	94	1	no	—
254	C₄×C₈	—	94	1	no	—
255	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
256	#41	—	94	1	no	—
257	#41	—	94	1	no	—
258	#41	—	94	1	no	—
259	#41	—	94	1	no	—
260	C₂×Q₈	—	47	1	no	—
261	C₂×D₄	—	47	2	no	—
262	C₂×D₄	—	47	1	no	—
263	C₂×D₄	—	94	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	47	2	no	—
271	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	M₁₆	—	94	1	no	—
274	M₁₆	—	94	1	no	—
275	M₁₆	—	94	2	no	—
276	M₁₆	—	94	2	no	—
277	M₁₆	—	94	1	no	—
278	M₁₆	—	94	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	47	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	—	94	1	no	—
290	C₂×C₈	#10625350	1	1	yes	characteristic
291	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
292	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
293	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
294	C₂×C₈	#10625352	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	1	no	—
298	C₂²⋊C₄	—	47	2	no	—
299	C₂²⋊C₄	—	94	2	no	—
300	C₂²⋊C₄	—	94	2	no	—
301	C₄²	—	94	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	4	1	no	—
304	C₁₆	—	94	2	no	—
305	C₁₆	—	188	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	47	2	no	—
311	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
312	Q₈	—	94	1	no	—
313	Q₈	—	94	1	no	—
314	D₄	—	94	2	no	—
315	D₄	—	94	2	no	—
316	D₄	—	94	1	no	—
317	D₄	—	94	1	no	—
318	D₄	—	188	4	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	47	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
334	C₂×C₄	#13094030	1	1	yes	characteristic
335	C₂×C₄	#13094032	1	1	yes	characteristic
336	C₈	—	4	1	no	—
337	C₈	—	94	2	no	—
338	C₈	—	188	1	no	—
339	C₈	#13093992	1	1	yes	characteristic
340	C₈	#13093995	1	1	yes	characteristic
341	C₈	#13093997	1	1	yes	characteristic
342	C₈	#13094006	1	1	yes	characteristic
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	94	2	no	—
352	C₂²	—	188	2	no	—
353	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
354	C₂²	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
355	C₂²	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
356	C₄	—	4	1	no	—
357	C₄	—	94	2	no	—
358	C₄	—	188	2	no	—
359	C₄	#15688999	1	1	yes	characteristic
360	C₄	#15689088	1	1	yes	characteristic
361	C₄	#15689192	1	1	yes	characteristic
362	C₄	#15689197	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	4	2	no	—
367	C₂	—	188	2	no	—
368	C₂	#20243677	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13093992
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2112	C₂
socle	C₂×C₄₇	#1869
prime radical	#15689634	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2112	D₄₇
hypocenter	C₄₇	#12981
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
1. #20256490	C₂	semidirect
3. #20245008	C₂	semidirect
4. #20244999	C₂	semidirect
5. #20241475	C₂	semidirect
6. #20241473	C₂	semidirect
7. C₄₇×#2112	C₂	semidirect
48. C₂×C₄₇⋊C₁₆	D₄	semidirect
69. C₂×C₁₆×C₄₇	D₄	semidirect
183. #2112	D₄₇	semidirect
211. C₄₇	#12981	semidirect
249. C₂×C₁₆	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20256490	2
#20252082	2
#20245008	2
#20244999	2
#20241475	2
#20241473	2
C₄₇×#2112	2
#12981	47