Subgroups — #26207874

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256361	C₂	1	1	yes	maximal, characteristic
2	#20256256	C₂	1	1	yes	maximal, characteristic, complemented
3	#20255729	C₂	1	2	yes	maximal, complemented
4	#20242977	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242920	C₂	1	2	yes	maximal, complemented
6	#20242793	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242516	C₂	1	2	yes	maximal, complemented
8	#20242459	C₂	1	2	yes	maximal, complemented
9	#20242272	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1668	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689833	C₂²	1	2	yes	—
13	#15689793	C₂²	1	2	yes	—
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689525	—	2	1	no	—
16	#15689498	—	2	1	no	—
17	#15689495	C₂²	1	2	yes	—
18	#15689469	C₂²	1	2	yes	—
19	#15689435	—	2	2	no	—
20	#15689424	—	2	2	no	—
21	#15689419	C₂²	1	2	yes	—
22	#15689414	C₂²	1	2	yes	—
23	#15689366	—	2	2	no	—
24	#15689316	—	2	2	no	—
25	#15689315	C₂²	1	2	yes	—
26	#15689282	—	2	2	no	—
27	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
28	C₄×#10625337	C₂²	1	1	yes	characteristic
29	C₄×D₃₇₆	—	2	1	no	—
30	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
31	D₈×Dic₄₇	—	2	1	no	—
32	#15689200	C₂²	1	2	yes	—
33	#15689121	—	2	2	no	—
34	#15688875	C₂²	1	1	yes	characteristic
35	#15688871	C₂²	1	2	yes	—
36	Q₈⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
37	#15688853	C₂²	1	1	yes	characteristic
38	#15688850	C₂²	1	2	yes	—
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₇×#493	C₂²	1	1	yes	characteristic
41	C₄₇×#460	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₄₇×#335	C₂²	1	1	yes	characteristic
44	C₄₇×#331	C₂²	1	2	yes	—
45	C₂×#10625352	—	2	2	no	—
46	C₂×#10625337	D₄	1	1	yes	characteristic
47	C₂×#10625335	—	4	2	no	—
48	C₂×D₃₇₆	D₄	1	1	yes	characteristic
49	C₂×#10625334	—	2	2	no	—
50	C₂×#10625324	—	2	2	no	—
51	#13094024	C₂³	1	2	yes	—
52	#13094020	—	2	2	no	—
53	#13094016	—	2	2	no	—
54	#13094002	—	4	2	no	—
55	#13094000	—	2	2	no	—
56	#13093996	—	4	2	no	—
57	#13093995	—	2	2	no	—
58	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
59	C₄×D₁₈₈	—	2	2	no	—
60	C₄×D₁₈₈	C₂³	1	2	yes	—
61	D₄×Dic₄₇	—	2	2	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	#13093958	—	2	1	no	—
64	#13093958	—	2	1	no	—
65	#13093958	—	2	2	no	—
66	#13093958	D₄	1	1	yes	characteristic
67	#13093957	D₄	1	1	yes	characteristic
68	#13093956	—	2	2	no	—
69	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₄₇⋊C₈	—	2	1	no	—
71	#13093949	—	2	2	no	—
72	#13093948	D₄	1	1	yes	characteristic
73	#13093947	D₄	1	1	yes	characteristic
74	#13093946	—	2	2	no	—
75	#13093946	D₄	1	2	yes	—
76	#13093945	—	2	2	no	—
77	#13093945	—	2	2	no	—
78	#13093945	—	2	2	no	—
79	(C₂×D₄₇)⋊C₈	—	2	2	no	—
80	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₄₇	—	2	1	no	—
83	C₂×SD₁₆×C₄₇	—	2	2	no	—
84	C₂×D₈×C₄₇	—	2	1	no	—
85	C₂×D₈×C₄₇	—	2	1	no	—
86	#127×C₄₇	C₂³	1	2	yes	—
87	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₇	—	2	1	no	—
89	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
90	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
91	#102×C₄₇	—	2	2	no	—
92	C₄×C₈×C₄₇	—	2	1	no	—
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₄₇	—	2	2	no	—
96	C₂×D₄×C₄₇	—	2	2	no	—
97	C₂×D₄×C₄₇	—	2	2	no	—
98	C₂×C₂²⋊D₄₇	—	4	2	no	—
99	C₂×C₂²⋊D₄₇	—	4	2	no	—
100	C₂²×Dic₄₇	—	4	2	no	—
101	#10625352	—	4	4	no	—
102	#10625352	—	8	2	no	—
103	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₄₇	—	2	2	no	—
105	C₂×C₄×D₄₇	—	4	2	no	—
106	C₂×Dic₉₄	—	2	2	no	—
107	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
108	SD₁₆×C₄₇	—	8	2	no	—
109	D₈×C₄₇	—	4	2	no	—
110	D₈×C₄₇	—	8	1	no	—
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₇	—	2	1	no	—
116	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	—	2	1	no	—
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊Dic₄₇	—	4	2	no	—
120	#10625337	—	2	2	no	—
121	#10625335	—	8	4	no	—
122	#10625334	—	8	2	no	—
123	(C₂×D₄₇)⋊C₄	—	2	2	no	—
124	(C₂×D₄₇)⋊C₄	—	4	2	no	—
125	C₄⋊Dic₄₇	—	2	2	no	—
126	C₄⋊Dic₄₇	—	2	2	no	—
127	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
128	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
129	Dic₄₇⋊C₄	—	2	1	no	—
130	Dic₄₇⋊C₄	—	4	2	no	—
131	Dic₄₇⋊C₄	—	4	2	no	—
132	C₄×Dic₄₇	—	2	1	no	—
133	C₂×C₄₇⋊C₈	—	2	1	no	—
134	C₂×C₄₇⋊C₈	—	2	1	no	—
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	D₃₇₆	—	2	2	no	—
138	#10625324	—	8	2	no	—
139	C₂³×C₄₇	—	4	2	no	—
140	C₂²×D₄₇	—	2	2	no	—
141	Q₈×C₄₇	—	2	1	no	—
142	Q₈×C₄₇	—	2	1	no	—
143	D₄×C₄₇	—	4	2	no	—
144	D₄×C₄₇	—	4	2	no	—
145	D₄×C₄₇	—	4	2	no	—
146	D₄×C₄₇	—	4	2	no	—
147	D₄×C₄₇	—	8	2	no	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	2	1	no	—
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²⋊D₄₇	—	8	4	no	—
154	C₂²⋊D₄₇	—	16	2	no	—
155	C₂×Dic₄₇	—	2	1	no	—
156	C₂×Dic₄₇	—	2	1	no	—
157	C₂×Dic₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	C₂×Dic₄₇	—	8	4	no	—
160	D₁₈₈	—	2	2	no	—
161	D₁₈₈	—	2	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	C₄×D₄₇	—	4	2	no	—
164	C₄×D₄₇	—	16	2	no	—
165	Dic₉₄	—	2	1	no	—
166	Dic₉₄	—	8	2	no	—
167	Dic₉₄	#135	1	2	yes	—
168	C₈×C₄₇	—	2	1	no	—
169	C₈×C₄₇	—	4	2	no	—
170	C₈×C₄₇	—	4	1	no	—
171	C₈×C₄₇	#137	1	2	yes	—
172	C₄₇⋊C₈	—	2	2	no	—
173	C₄₇⋊C₈	—	2	1	no	—
174	C₄₇⋊C₈	—	2	1	no	—
175	C₄₇⋊C₈	—	4	1	no	—
176	#19909	—	47	1	no	maximal, Sylow
177	C₂²×C₄₇	—	8	2	no	—
178	C₂²×C₄₇	—	8	2	no	—
179	C₂²×C₄₇	—	8	2	no	—
180	C₂²×C₄₇	D₄²	1	1	yes	characteristic
181	C₂×D₄₇	—	4	2	no	—
182	C₂×D₄₇	—	4	2	no	—
183	C₂×D₄₇	—	4	2	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	—	4	1	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
188	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
189	C₄×C₄₇	#576	1	1	yes	characteristic
190	C₄×C₄₇	#577	1	1	yes	characteristic
191	Dic₄₇	—	2	2	no	—
192	Dic₄₇	—	4	2	no	—
193	Dic₄₇	—	4	1	no	—
194	Dic₄₇	—	8	2	no	—
195	#3391	—	47	1	no	—
196	#3380	—	47	1	no	—
197	#3333	—	47	2	no	—
198	#1706	—	47	1	no	—
199	#1697	—	47	2	no	—
200	#1676	—	47	1	no	—
201	#1668	D₄₇	1	1	yes	characteristic, complemented
202	#1631	—	47	2	no	—
203	#1622	—	47	2	no	—
204	#1589	—	47	1	no	—
205	C₈×Q₁₆	—	47	1	no	—
206	C₂×C₄₇	—	16	2	no	—
207	C₂×C₄₇	#3278	1	1	yes	characteristic
208	C₂×C₄₇	#3280	1	1	yes	characteristic
209	C₂×C₄₇	#3292	1	1	yes	characteristic
210	D₄₇	—	8	2	no	—
211	#554	—	47	2	no	—
212	#550	—	47	2	no	—
213	#500	—	47	1	no	—
214	#499	—	94	1	no	—
215	#493	—	94	1	no	—
216	#493	C₂×D₄₇	1	1	yes	characteristic
217	#492	—	47	2	no	—
218	#486	—	47	2	no	—
219	#480	—	94	2	no	—
220	#478	—	94	2	no	—
221	#477	—	47	2	no	—
222	#476	—	47	2	no	—
223	#468	—	94	2	no	—
224	#460	C₂×D₄₇	1	2	yes	—
225	#459	—	47	2	no	—
226	#459	—	94	2	no	—
227	#452	—	94	2	no	—
228	C₈×Q₈	—	47	1	no	—
229	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
230	C₄×Q₁₆	—	47	1	no	—
231	C₄×D₈	—	47	1	no	—
232	C₄×D₈	—	94	1	no	—
233	C₄×D₈	—	94	1	no	—
234	#436	—	47	2	no	—
235	#417	—	94	2	no	—
236	#335	—	47	1	no	—
237	#335	C₂×D₄₇	1	1	yes	characteristic
238	#333	—	47	2	no	—
239	#331	C₂×D₄₇	1	2	yes	—
240	Q₈⋊C₈	—	47	1	no	—
241	Q₈⋊C₈	—	47	1	no	—
242	D₄⋊C₈	—	47	2	no	—
243	C₈²	—	47	1	no	—
244	C₄₇	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	94	2	no	—
246	C₂×Q₁₆	—	47	1	no	—
247	C₂×SD₁₆	—	2	2	no	—
248	C₂×SD₁₆	—	188	2	no	—
249	C₂×D₈	—	2	1	no	—
250	C₂×D₈	—	2	1	no	—
251	C₂×D₈	—	47	1	no	—
252	C₂×D₈	—	94	2	no	—
253	C₂×M₁₆	—	94	2	no	—
254	C₄⋊Q₈	—	47	2	no	—
255	#127	—	94	2	no	—
256	#127	C₂²×D₄₇	1	2	yes	—
257	#125	—	94	2	no	—
258	C₂²⋊Q₈	—	94	2	no	—
259	C₂²⋊Q₈	—	188	2	no	—
260	#121	—	94	2	no	—
261	#121	—	188	2	no	—
262	C₄×Q₈	—	47	1	no	—
263	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₄×D₄	—	47	2	no	—
265	C₄×D₄	—	94	2	no	—
266	C₄×D₄	—	94	2	no	—
267	C₂×C₄⋊C₄	—	94	2	no	—
268	#107	—	47	1	no	—
269	#107	—	47	1	no	—
270	#107	—	94	2	no	—
271	#107	—	94	1	no	—
272	#107	—	94	1	no	—
273	#106	—	94	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	47	1	no	—
276	C₄⋊C₈	—	94	1	no	—
277	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
278	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
279	#103	—	47	1	no	—
280	#103	—	47	1	no	—
281	#103	—	94	2	no	—
282	#102	—	2	2	no	—
283	#102	—	47	2	no	—
284	#102	—	94	2	no	—
285	#102	—	94	2	no	—
286	#102	—	94	2	no	—
287	#102	—	94	2	no	—
288	C₂²⋊C₈	—	94	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	47	1	no	—
291	C₄×C₈	—	47	1	no	—
292	C₄×C₈	—	94	1	no	—
293	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
294	#41	—	376	2	no	—
295	#41	—	188	4	no	—
296	C₂×Q₈	—	47	1	no	—
297	C₂×Q₈	—	94	2	no	—
298	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	47	2	no	—
303	C₂×D₄	—	188	2	no	—
304	C₂×D₄	—	188	2	no	—
305	C₂²×C₄	—	94	2	no	—
306	C₂²×C₄	—	188	2	no	—
307	C₂²×C₄	—	188	2	no	—
308	Q₁₆	—	94	2	no	—
309	SD₁₆	—	8	2	no	—
310	SD₁₆	—	376	4	no	—
311	D₈	—	4	2	no	—
312	D₈	—	8	1	no	—
313	D₈	—	94	2	no	—
314	D₈	—	376	2	no	—
315	M₁₆	—	376	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	94	1	no	—
321	C₂×C₈	—	94	1	no	—
322	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
323	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	47	2	no	—
326	C₄⋊C₄	—	47	1	no	—
327	C₄⋊C₄	—	94	1	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	—	94	2	no	—
330	C₄⋊C₄	—	188	2	no	—
331	C₄⋊C₄	—	188	2	no	—
332	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
333	C₂²⋊C₄	—	94	2	no	—
334	C₂²⋊C₄	—	188	2	no	—
335	C₂²⋊C₄	—	188	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	94	1	no	—
338	C₄²	C₂³×D₄₇	1	1	yes	characteristic
339	C₂³	—	4	2	no	—
340	C₂³	—	94	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	47	2	no	—
344	Q₈	—	94	1	no	—
345	Q₈	—	376	2	no	—
346	D₄	—	4	2	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	8	2	no	—
351	D₄	—	94	2	no	—
352	D₄	—	94	2	no	—
353	D₄	—	376	4	no	—
354	D₄	—	752	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	94	1	no	—
358	C₂×C₄	—	94	1	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	376	4	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	—	188	2	no	—
363	C₂×C₄	—	188	2	no	—
364	C₂×C₄	—	752	2	no	—
365	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
367	C₂×C₄	#13094070	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	1	no	—
371	C₈	—	94	2	no	—
372	C₈	—	94	1	no	—
373	C₈	—	94	1	no	—
374	C₈	—	188	1	no	—
375	C₈	#13094053	1	2	yes	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	188	2	no	—
380	C₂²	—	188	2	no	—
381	C₂²	—	188	2	no	—
382	C₂²	#15689757	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	94	2	no	—
387	C₄	—	376	2	no	—
388	C₄	—	188	2	no	—
389	C₄	—	188	1	no	—
390	C₄	#135×D₄₇	1	1	yes	characteristic
391	C₄	C₂×#13094053	1	1	yes	characteristic
392	C₄	#15689924	1	1	yes	characteristic
393	C₄	#15689933	1	1	yes	characteristic
394	C₂	—	16	2	no	—
395	C₂	—	376	2	no	—
396	C₂	#20255057	1	1	yes	characteristic
397	C₂	#20255076	1	1	yes	characteristic
398	C₂	#20255186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689757
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1668	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20256361	C₂
hypercenter	#1668	D₄₇
hypocenter	C₄₇	#19909
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1668

Complements

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

N	H ≅ G/N	type
2. #20256256	C₂	semidirect
3. #20255729	C₂	semidirect
4. #20242977	C₂	semidirect
5. #20242920	C₂	semidirect
6. #20242793	C₂	semidirect
7. #20242516	C₂	semidirect
8. #20242459	C₂	semidirect
9. #20242272	C₂	semidirect
10. C₈×#10625337	C₂	semidirect
11. C₄₇×#1668	C₂	semidirect
201. #1668	D₄₇	semidirect
244. C₄₇	#19909	semidirect

Maximal subgroups

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

subgroup	index
#20256361	2
#20256256	2
#20255729	2
#20242977	2
#20242920	2
#20242793	2
#20242516	2
#20242459	2
#20242272	2
C₈×#10625337	2
C₄₇×#1668	2
#19909	47