Subgroups — #26316810

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246107	C₂	1	1	yes	maximal, characteristic
2	#20244780	C₂	1	2	yes	maximal
3	#20244766	C₂	1	2	yes	maximal
4	#20244207	C₂	1	1	yes	maximal, characteristic
5	#20244189	C₂	1	1	yes	maximal, characteristic
6	#20243655	C₂	1	2	yes	maximal
7	#20243641	C₂	1	2	yes	maximal
8	#20243550	C₂	1	1	yes	maximal, characteristic
9	#20243523	C₂	1	1	yes	maximal, characteristic
10	#20243245	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2647	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
14	C₂×#13093957	C₂²	1	2	yes	—
15	C₂×#13093955	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#13093947	C₂²	1	2	yes	—
19	C₂×#13093947	C₂²	1	2	yes	—
20	C₂×#13093947	C₂²	1	2	yes	—
21	C₂×#13093947	C₂²	1	2	yes	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	#15689029	C₂²	1	2	yes	—
25	#15689008	—	2	2	no	—
26	#15689008	—	2	2	no	—
27	#15688989	C₂²	1	1	yes	characteristic
28	#15688885	C₂²	1	2	yes	—
29	#15688885	C₂²	1	2	yes	—
30	#15688878	C₂²	1	1	yes	characteristic
31	#15688878	C₂²	1	1	yes	characteristic
32	C₄₇×#400	—	2	2	no	—
33	C₄₇×#398	—	2	2	no	—
34	C₄₇×#389	C₂²	1	2	yes	—
35	C₄₇×#389	C₂²	1	2	yes	—
36	C₄₇×#384	C₂²	1	1	yes	characteristic
37	C₄₇×#382	C₂²	1	1	yes	characteristic
38	C₄₇×#376	C₂²	1	1	yes	characteristic
39	C₂²×Dic₉₄	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	#13094024	—	2	2	no	—
45	#13094024	Q₈	1	4	yes	—
46	C₄×Dic₉₄	—	4	2	no	—
47	C₄×Dic₉₄	—	4	2	no	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×C₄×Dic₄₇	—	2	2	no	—
52	#13093957	—	4	2	no	—
53	#13093955	—	4	2	no	—
54	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
56	#13093947	—	2	4	no	—
57	#13093947	—	2	4	no	—
58	#13093947	—	4	2	no	—
59	#13093947	—	4	2	no	—
60	#13093935	—	2	2	no	—
61	C₄×C₄₇⋊C₈	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₄₇⋊#95	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#95×C₄₇	—	2	2	no	—
74	#95×C₄₇	—	2	2	no	—
75	#95×C₄₇	—	2	2	no	—
76	#95×C₄₇	D₄	1	2	yes	—
77	#95×C₄₇	D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	2	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	C₂×Dic₉₄	—	2	2	no	—
92	C₂×Dic₉₄	C₂×Q₈	1	4	yes	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	8	2	no	—
98	C₄⋊C₄×C₄₇	—	8	2	no	—
99	C₄⋊C₄×C₄₇	—	8	2	no	—
100	C₄²×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	—	2	1	no	—
102	C₄²×C₄₇	—	2	1	no	—
103	C₄²×C₄₇	—	2	1	no	—
104	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
105	C₄²×C₄₇	#41	1	2	yes	—
106	C₄⋊Dic₄₇	—	2	8	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	Dic₄₇⋊C₄	—	4	4	no	—
110	C₄×Dic₄₇	—	4	4	no	—
111	C₂×C₄₇⋊C₈	—	2	1	no	—
112	C₂×C₄₇⋊C₈	—	2	1	no	—
113	C₂×C₄₇⋊C₈	—	2	1	no	—
114	C₂×C₄₇⋊C₈	—	2	1	no	—
115	C₂×C₄₇⋊C₈	—	2	1	no	—
116	C₂×C₄₇⋊C₈	—	2	1	no	—
117	C₂×C₄₇⋊C₈	—	2	1	no	—
118	C₂×C₄₇⋊C₈	—	2	1	no	—
119	C₂×C₄₇⋊C₈	—	2	1	no	—
120	C₂×C₄₇⋊C₈	—	2	1	no	—
121	C₂×C₄₇⋊C₈	#41	1	2	yes	—
122	C₂×C₄₇⋊C₈	#41	1	2	yes	—
123	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
140	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	Dic₉₄	—	2	4	no	—
153	Dic₉₄	—	2	4	no	—
154	Dic₉₄	—	4	2	no	—
155	Dic₉₄	—	4	2	no	—
156	C₄₇⋊C₈	—	2	2	no	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	—	4	1	no	—
159	C₄₇⋊C₈	—	4	1	no	—
160	#29202	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄₇	#532	1	1	yes	characteristic
163	C₂²×C₄₇	#535	1	1	yes	characteristic
164	C₂²×C₄₇	#577	1	2	yes	—
165	C₂²×C₄₇	#578	1	2	yes	—
166	C₄×C₄₇	—	2	4	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	—	4	1	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	#553	1	1	yes	characteristic
173	C₄×C₄₇	#553	1	1	yes	characteristic
174	C₄×C₄₇	#554	1	1	yes	characteristic
175	C₄×C₄₇	#554	1	1	yes	characteristic
176	Dic₄₇	—	4	4	no	—
177	Dic₄₇	—	8	2	no	—
178	#2647	D₄₇	1	1	yes	characteristic
179	#2327	—	47	1	no	—
180	#2068	—	47	2	no	—
181	#2066	—	47	2	no	—
182	#1960	—	47	1	no	—
183	#1956	—	47	1	no	—
184	#1865	—	47	2	no	—
185	#1863	—	47	2	no	—
186	#1846	—	47	1	no	—
187	#1836	—	47	1	no	—
188	#1768	—	47	1	no	—
189	C₂×C₄₇	#2566	1	1	yes	characteristic
190	C₂×C₄₇	#3082	1	2	yes	—
191	C₂×C₄₇	#3174	1	1	yes	characteristic
192	C₂×C₄₇	#3177	1	1	yes	characteristic
193	C₂×C₄₇	#3225	1	2	yes	—
194	C₂×C₄⋊Q₈	—	47	1	no	—
195	C₂×C₄×Q₈	—	47	2	no	—
196	C₂×#107	—	47	2	no	—
197	C₂×#106	—	47	2	no	—
198	C₂×C₄⋊C₈	—	47	1	no	—
199	C₂×C₄⋊C₈	—	47	1	no	—
200	C₂×#103	—	47	2	no	—
201	C₂×#103	—	47	2	no	—
202	C₂×#103	—	47	2	no	—
203	C₂×#103	—	47	2	no	—
204	C₂×#97	—	47	1	no	—
205	C₂×C₄×C₈	—	47	1	no	—
206	#400	—	2	2	no	—
207	#398	—	2	2	no	—
208	#391	—	47	2	no	—
209	#389	C₂×D₄₇	1	2	yes	—
210	#389	C₂×D₄₇	1	2	yes	—
211	#387	—	94	2	no	—
212	#387	—	94	2	no	—
213	#384	C₂×D₄₇	1	1	yes	characteristic
214	#382	—	47	1	no	—
215	#382	C₂×D₄₇	1	1	yes	characteristic
216	#376	C₂×D₄₇	1	1	yes	characteristic
217	#340	—	47	2	no	—
218	#340	—	47	2	no	—
219	#336	—	47	1	no	—
220	#336	—	47	1	no	—
221	C₄₇	#29202	1	1	yes	characteristic, complemented, Sylow
222	C₂²×Q₈	—	47	2	no	—
223	C₂²×C₈	—	47	1	no	—
224	C₂²×C₈	—	47	1	no	—
225	C₂²×C₈	—	47	1	no	—
226	C₂²×C₈	—	47	1	no	—
227	C₄⋊Q₈	—	47	4	no	—
228	C₄⋊Q₈	—	94	2	no	—
229	C₄×Q₈	—	188	2	no	—
230	C₄×Q₈	—	188	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	47	2	no	—
235	C₂×C₄⋊C₄	—	94	2	no	—
236	C₂×C₄⋊C₄	—	94	2	no	—
237	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
238	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
239	C₂×C₄²	—	94	2	no	—
240	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	#107	—	188	2	no	—
244	#106	—	188	2	no	—
245	C₄⋊C₈	—	94	2	no	—
246	C₄⋊C₈	—	94	2	no	—
247	#103	—	94	4	no	—
248	#103	—	94	4	no	—
249	#103	—	188	2	no	—
250	#103	—	188	2	no	—
251	#97	—	94	2	no	—
252	C₄×C₈	—	94	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	94	2	no	—
257	#95	—	94	2	no	—
258	#95	—	94	2	no	—
259	#95	C₂²⋊D₄₇	1	2	yes	—
260	#95	C₂²⋊D₄₇	1	2	yes	—
261	C₂×Q₈	—	47	4	no	—
262	C₂×Q₈	—	94	2	no	—
263	C₂×Q₈	—	94	2	no	—
264	C₂×Q₈	—	94	2	no	—
265	C₂×Q₈	—	94	2	no	—
266	C₂×Q₈	—	94	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	94	2	no	—
271	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
275	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
276	C₂×C₈	—	47	2	no	—
277	C₂×C₈	—	47	2	no	—
278	C₂×C₈	—	94	1	no	—
279	C₂×C₈	—	94	1	no	—
280	C₂×C₈	—	94	1	no	—
281	C₂×C₈	—	94	1	no	—
282	C₂×C₈	—	94	1	no	—
283	C₂×C₈	—	94	1	no	—
284	C₂×C₈	—	94	1	no	—
285	C₂×C₈	—	94	1	no	—
286	C₂×C₈	—	94	1	no	—
287	C₂×C₈	—	94	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	8	2	no	—
293	C₄⋊C₄	—	8	2	no	—
294	C₄⋊C₄	—	8	2	no	—
295	C₄⋊C₄	—	94	8	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	188	4	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	188	4	no	—
304	C₄²	Q₈×D₄₇	1	2	yes	—
305	C₄²	#10625354	1	2	yes	—
306	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
307	Q₈	—	94	4	no	—
308	Q₈	—	94	4	no	—
309	Q₈	—	188	2	no	—
310	Q₈	—	188	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	#13094000	1	4	yes	—
333	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
334	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
335	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
337	C₂×C₄	#13094070	1	1	yes	characteristic
338	C₂×C₄	#13094070	1	1	yes	characteristic
339	C₈	—	94	2	no	—
340	C₈	—	94	2	no	—
341	C₈	—	188	1	no	—
342	C₈	—	188	1	no	—
343	C₂²	C₂×#13094000	1	1	yes	characteristic
344	C₂²	#15689669	1	1	yes	characteristic
345	C₂²	#15689687	1	1	yes	characteristic
346	C₂²	#15689934	1	2	yes	—
347	C₂²	#15689938	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	376	2	no	—
355	C₄	—	188	4	no	—
356	C₄	#15689826	1	1	yes	characteristic
357	C₄	#15689826	1	1	yes	characteristic
358	C₄	#15689829	1	1	yes	characteristic
359	C₄	#15689829	1	1	yes	characteristic
360	C₂	#20248589	1	1	yes	characteristic
361	C₂	#20253224	1	2	yes	—
362	C₂	#20254010	1	1	yes	characteristic
363	C₂	#20254041	1	1	yes	characteristic
364	C₂	#20254491	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2647	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2647	D₄₇
hypocenter	C₄₇	#29202
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2647

Complements

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

N	H ≅ G/N	type
221. C₄₇	#29202	semidirect

Maximal subgroups

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

subgroup	index
#20246107	2
#20244780	2
#20244766	2
#20244207	2
#20244189	2
#20243655	2
#20243641	2
#20243550	2
#20243523	2
#20243245	2
C₄₇×#2647	2
#29202	47