Subgroups — #26199446

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254518	C₂	1	1	yes	maximal, characteristic
2	#20254511	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
4	#20252101	C₂	1	1	yes	maximal, characteristic
5	#20242964	C₂	1	4	yes	maximal, complemented
6	#20242840	C₂	1	4	yes	maximal, complemented
7	#20242068	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	—	2	1	no	—
11	#15689673	C₂²	1	1	yes	characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	C₂×#13094024	C₂²	1	1	yes	characteristic
14	#15689528	—	2	1	no	—
15	#15689521	—	2	1	no	—
16	#15689474	—	2	1	no	—
17	#15689474	—	2	1	no	—
18	#15689474	C₂²	1	2	yes	—
19	#15689474	C₂²	1	2	yes	—
20	#15689470	—	2	1	no	—
21	#15689470	—	2	1	no	—
22	#15689442	—	2	2	no	—
23	#15689437	—	2	2	no	—
24	#15689414	C₂²	1	4	yes	—
25	#15689404	C₂²	1	4	yes	—
26	#15689202	—	2	1	no	—
27	#15689202	C₂²	1	2	yes	—
28	C₈×D₁₈₈	—	2	1	no	—
29	C₈×D₁₈₈	C₂²	1	2	yes	—
30	#15689147	C₂²	1	1	yes	characteristic
31	C₂×#13093949	—	2	1	no	—
32	C₂×#13093949	—	2	1	no	—
33	#15689089	—	2	1	no	—
34	#15688871	C₂²	1	4	yes	—
35	#15688868	C₂²	1	4	yes	—
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₄₇×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#335	C₂²	1	2	yes	—
40	C₄₇×#334	C₂²	1	2	yes	—
41	C₂×#10625350	—	2	1	no	—
42	C₂²×Dic₉₄	—	2	1	no	—
43	C₂²×Dic₉₄	—	2	1	no	—
44	#13094030	—	4	4	no	—
45	C₂×#10625329	—	2	1	no	—
46	C₂×#10625324	—	2	2	no	—
47	C₂×C₈×D₄₇	—	2	2	no	—
48	#13094024	—	2	1	no	—
49	#13094024	—	2	1	no	—
50	#13094024	—	2	1	no	—
51	#13094024	—	2	1	no	—
52	#13094024	—	2	1	no	—
53	#13094024	—	2	1	no	—
54	#13094024	D₄	1	1	yes	characteristic
55	#13094024	C₂³	1	2	yes	—
56	#13094024	C₂³	1	2	yes	—
57	#13094021	D₄	1	1	yes	characteristic
58	#13094012	—	2	1	no	—
59	#13093998	—	2	1	no	—
60	#13093998	D₄	1	2	yes	—
61	C₄×D₁₈₈	—	2	1	no	—
62	C₄×D₁₈₈	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₄₇	—	2	1	no	—
64	C₂×C₄⋊Dic₄₇	—	2	1	no	—
65	C₂×C₄⋊Dic₄₇	—	2	1	no	—
66	C₂×C₄⋊Dic₄₇	—	2	1	no	—
67	#13093958	—	2	4	no	—
68	#13093956	—	2	4	no	—
69	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
71	#13093949	—	2	2	no	—
72	#13093949	—	2	2	no	—
73	#13093949	—	2	2	no	—
74	#13093949	—	2	2	no	—
75	#13093946	—	2	4	no	—
76	#13093946	—	2	4	no	—
77	(C₂×D₄₇)⋊C₈	—	2	2	no	—
78	(C₂×D₄₇)⋊C₈	—	2	2	no	—
79	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₄₇	—	2	1	no	—
85	C₂²⋊C₈×C₄₇	—	2	1	no	—
86	C₄×C₈×C₄₇	—	2	1	no	—
87	C₄×C₈×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₇	—	2	2	no	—
96	C₂²×Dic₄₇	—	4	1	no	—
97	C₂²×Dic₄₇	—	4	1	no	—
98	#10625350	—	4	2	no	—
99	#10625350	—	4	2	no	—
100	C₂×D₁₈₈	—	2	1	no	—
101	C₂×D₁₈₈	—	2	1	no	—
102	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₄₇	—	2	2	no	—
104	C₂×Dic₉₄	—	2	1	no	—
105	C₂×Dic₉₄	—	2	2	no	—
106	C₂×Dic₉₄	—	2	2	no	—
107	C₂×Dic₉₄	—	4	2	no	—
108	C₂×Dic₉₄	—	4	2	no	—
109	C₂×Dic₉₄	—	4	1	no	—
110	C₂×Dic₉₄	—	4	1	no	—
111	C₂×Dic₉₄	—	8	2	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₂×C₈×C₄₇	—	2	2	no	—
115	C₂×C₈×C₄₇	—	2	2	no	—
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₄₇	#41	1	1	yes	characteristic
123	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
124	(C₂×D₄₇)⋊C₄	—	2	2	no	—
125	(C₂×D₄₇)⋊C₄	—	2	2	no	—
126	C₄⋊Dic₄₇	—	2	1	no	—
127	C₄⋊Dic₄₇	—	2	2	no	—
128	C₄⋊Dic₄₇	—	2	2	no	—
129	C₄⋊Dic₄₇	—	2	2	no	—
130	C₄⋊Dic₄₇	—	2	2	no	—
131	C₄⋊Dic₄₇	—	4	1	no	—
132	C₄⋊Dic₄₇	—	4	1	no	—
133	C₄⋊Dic₄₇	—	4	1	no	—
134	C₄⋊Dic₄₇	—	4	1	no	—
135	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
136	#10625329	—	4	2	no	—
137	C₂×C₄₇⋊C₈	—	2	2	no	—
138	#10625324	—	4	4	no	—
139	C₈×D₄₇	—	4	4	no	—
140	C₂³×C₄₇	#121	1	1	yes	characteristic
141	C₂²×D₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#135	1	2	yes	—
148	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₄₇	—	4	4	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	8	2	no	—
157	C₂×Dic₄₇	—	8	2	no	—
158	D₁₈₈	—	2	4	no	—
159	D₁₈₈	—	4	2	no	—
160	D₁₈₈	—	4	2	no	—
161	C₄×D₄₇	—	4	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	Dic₉₄	—	4	2	no	—
164	Dic₉₄	—	4	2	no	—
165	Dic₉₄	—	4	2	no	—
166	Dic₉₄	—	8	2	no	—
167	Dic₉₄	—	8	2	no	—
168	Dic₉₄	—	8	2	no	—
169	C₈×C₄₇	—	2	1	no	—
170	C₈×C₄₇	—	2	1	no	—
171	C₈×C₄₇	—	2	1	no	—
172	C₈×C₄₇	—	2	1	no	—
173	C₈×C₄₇	—	4	2	no	—
174	C₈×C₄₇	—	4	2	no	—
175	C₄₇⋊C₈	—	4	2	no	—
176	#19199	—	47	1	no	maximal, Sylow
177	C₂²×C₄₇	—	2	1	no	—
178	C₂²×C₄₇	—	2	1	no	—
179	C₂²×C₄₇	#463	1	2	yes	—
180	C₂²×C₄₇	C₂×#121	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	—	2	1	no	—
186	C₄×C₄₇	#463	1	2	yes	—
187	C₄×C₄₇	#471	1	2	yes	—
188	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
189	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₄₇	#578	1	2	yes	—
191	Dic₄₇	—	4	2	no	—
192	Dic₄₇	—	8	2	no	—
193	Dic₄₇	—	8	2	no	—
194	#3227	—	47	1	no	—
195	#3226	—	47	1	no	—
196	C₂×#487	—	47	1	no	—
197	#2931	—	47	1	no	—
198	#1703	—	47	4	no	—
199	#1683	—	47	4	no	—
200	#1564	D₄₇	1	1	yes	characteristic, complemented
201	#1548	—	47	1	no	—
202	#1546	—	47	1	no	—
203	C₂×C₄₇	—	2	2	no	—
204	C₂×C₄₇	C₂×#463	1	1	yes	characteristic
205	C₂×C₄₇	#3059	1	1	yes	characteristic
206	C₂×C₄₇	C₂×#471	1	1	yes	characteristic
207	D₄₇	—	8	2	no	—
208	C₂×#131	—	94	1	no	—
209	#532	—	47	1	no	—
210	C₂×C₄⋊Q₈	—	47	1	no	—
211	C₂×C₄⋊Q₈	—	47	1	no	—
212	#500	—	94	1	no	—
213	#498	—	94	1	no	—
214	#487	—	47	2	no	—
215	#487	—	47	2	no	—
216	#487	—	94	1	no	—
217	#487	—	94	1	no	—
218	#486	—	94	1	no	—
219	#486	—	94	1	no	—
220	#481	—	94	2	no	—
221	#480	—	94	2	no	—
222	#476	—	47	4	no	—
223	#474	—	47	4	no	—
224	#436	—	47	2	no	—
225	#436	—	94	1	no	—
226	C₈×D₄	—	47	2	no	—
227	C₈×D₄	—	94	1	no	—
228	#431	C₂×D₄₇	1	1	yes	characteristic
229	#431	C₂×D₄₇	1	1	yes	characteristic
230	#423	—	47	1	no	—
231	C₂×#103	—	94	1	no	—
232	C₂×#103	—	94	1	no	—
233	#408	—	94	1	no	—
234	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
235	#335	C₂×D₄₇	1	2	yes	—
236	#334	C₂×D₄₇	1	2	yes	—
237	#333	—	47	4	no	—
238	#332	—	47	4	no	—
239	C₄₇	#19199	1	1	yes	characteristic, complemented, Sylow
240	C₂×#41	—	94	1	no	—
241	C₂²×Q₈	—	94	1	no	—
242	C₂²×Q₈	—	94	1	no	—
243	#131	—	188	4	no	—
244	C₂×M₁₆	—	94	1	no	—
245	C₂×M₁₆	—	94	2	no	—
246	C₂²×C₈	—	94	2	no	—
247	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
248	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
249	C₄⋊Q₈	—	47	1	no	—
250	C₄⋊Q₈	—	47	2	no	—
251	C₄⋊Q₈	—	47	2	no	—
252	C₄⋊Q₈	—	94	1	no	—
253	C₄⋊Q₈	—	94	1	no	—
254	C₄⋊Q₈	—	94	1	no	—
255	C₄⋊Q₈	—	94	1	no	—
256	C₄⋊Q₈	—	94	1	no	—
257	C₄⋊Q₈	—	94	1	no	—
258	#127	—	47	1	no	—
259	#124	—	94	1	no	—
260	#121	—	47	2	no	—
261	#121	—	94	1	no	—
262	C₄×D₄	—	47	2	no	—
263	C₄×D₄	—	94	1	no	—
264	C₂×C₄⋊C₄	—	94	1	no	—
265	C₂×C₄⋊C₄	—	94	1	no	—
266	C₂×C₄⋊C₄	—	94	1	no	—
267	C₂×C₄⋊C₄	—	94	1	no	—
268	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
269	#107	—	94	4	no	—
270	#106	—	94	4	no	—
271	C₄⋊C₈	—	47	2	no	—
272	C₄⋊C₈	—	94	1	no	—
273	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
274	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
275	#103	—	94	2	no	—
276	#103	—	94	2	no	—
277	#103	—	94	2	no	—
278	#103	—	94	2	no	—
279	#102	—	94	4	no	—
280	#102	—	94	4	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	94	2	no	—
284	C₂²⋊C₈	—	94	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	D₁₈₈	1	1	yes	characteristic
288	C₄×C₈	D₁₈₈	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
291	#41	—	188	2	no	—
292	#41	—	188	2	no	—
293	C₂×Q₈	—	94	1	no	—
294	C₂×Q₈	—	94	2	no	—
295	C₂×Q₈	—	94	2	no	—
296	C₂×Q₈	—	376	2	no	—
297	C₂×Q₈	—	188	2	no	—
298	C₂×Q₈	—	188	2	no	—
299	C₂×Q₈	—	188	1	no	—
300	C₂×Q₈	—	188	1	no	—
301	C₂×D₄	—	47	2	no	—
302	C₂×D₄	—	94	2	no	—
303	C₂×D₄	—	94	1	no	—
304	C₂×D₄	—	94	1	no	—
305	C₂²×C₄	—	94	2	no	—
306	C₂²×C₄	—	188	1	no	—
307	C₂²×C₄	—	188	1	no	—
308	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
309	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
310	C₂²×C₄	#10625354	1	1	yes	characteristic
311	M₁₆	—	188	4	no	—
312	M₁₆	—	188	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	94	2	no	—
318	C₂×C₈	—	188	4	no	—
319	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
320	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
321	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
322	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
323	C₄⋊C₄	—	47	2	no	—
324	C₄⋊C₄	—	94	1	no	—
325	C₄⋊C₄	—	94	2	no	—
326	C₄⋊C₄	—	94	2	no	—
327	C₄⋊C₄	—	94	2	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	—	188	1	no	—
330	C₄⋊C₄	—	188	1	no	—
331	C₄⋊C₄	—	188	1	no	—
332	C₄⋊C₄	—	188	1	no	—
333	C₂²⋊C₄	—	94	2	no	—
334	C₂²⋊C₄	—	94	2	no	—
335	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
336	C₄²	D₄×D₄₇	1	1	yes	characteristic
337	C₄²	#10625354	1	1	yes	characteristic
338	C₄²	C₂³×D₄₇	1	1	yes	characteristic
339	C₂³	—	94	2	no	—
340	C₂³	#13093999	1	1	yes	characteristic
341	Q₈	—	376	2	no	—
342	Q₈	—	376	2	no	—
343	Q₈	—	376	2	no	—
344	Q₈	—	188	2	no	—
345	Q₈	—	188	2	no	—
346	Q₈	—	188	2	no	—
347	D₄	—	94	4	no	—
348	D₄	—	188	4	no	—
349	D₄	—	188	2	no	—
350	D₄	—	188	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	94	2	no	—
354	C₂×C₄	—	376	2	no	—
355	C₂×C₄	—	376	2	no	—
356	C₂×C₄	—	188	2	no	—
357	C₂×C₄	—	188	2	no	—
358	C₂×C₄	—	188	2	no	—
359	C₂×C₄	—	188	2	no	—
360	C₂×C₄	#13093999	1	1	yes	characteristic
361	C₂×C₄	#13093999	1	1	yes	characteristic
362	C₂×C₄	#13093999	1	1	yes	characteristic
363	C₂×C₄	#13094046	1	2	yes	—
364	C₂×C₄	#13094052	1	2	yes	—
365	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
367	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₈	—	188	2	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	188	2	no	—
378	C₂²	—	188	2	no	—
379	C₂²	—	188	2	no	—
380	C₂²	#15689339	1	2	yes	—
381	C₂²	C₂×#13093999	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	376	2	no	—
385	C₄	—	376	2	no	—
386	C₄	—	188	2	no	—
387	C₄	#15689339	1	2	yes	—
388	C₄	#15689387	1	2	yes	—
389	C₄	C₂×#13094046	1	1	yes	characteristic
390	C₄	C₂×#13094052	1	1	yes	characteristic
391	C₄	#15689937	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	376	2	no	—
394	C₂	C₂×#15689339	1	1	yes	characteristic
395	C₂	#20253044	1	1	yes	characteristic
396	C₂	C₂×#15689387	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	#15689673	C₂²
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#19199
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#15689474	C₂	semidirect
5. #20242964	C₂	semidirect
6. #20242840	C₂	semidirect
7. #20242068	C₂	semidirect
8. #20242059	C₂	semidirect
9. C₄₇×#1564	C₂	semidirect
200. #1564	D₄₇	semidirect
239. C₄₇	#19199	semidirect

Maximal subgroups

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

subgroup	index
#20254518	2
#20254511	2
C₂×#15689474	2
#20252101	2
#20242964	2
#20242840	2
#20242068	2
#20242059	2
C₄₇×#1564	2
#19199	47