Subgroups — #26199430

Order 12032.

662 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	#20254511	C₂	1	1	yes	maximal, characteristic
2	#20254313	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689475	C₂	1	1	yes	maximal, characteristic, complemented
4	#20252099	C₂	1	1	yes	maximal, characteristic
5	#20242938	C₂	1	4	yes	maximal, complemented
6	#20242814	C₂	1	4	yes	maximal, complemented
7	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242061	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094032	—	2	1	no	—
11	#15689673	C₂²	1	1	yes	characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	#15689528	—	2	1	no	—
14	#15689497	—	2	1	no	—
15	#15689475	—	2	1	no	—
16	#15689475	—	2	1	no	—
17	#15689475	C₂²	1	2	yes	—
18	#15689475	C₂²	1	2	yes	—
19	#15689473	—	2	1	no	—
20	#15689470	—	2	1	no	—
21	#15689437	—	2	2	no	—
22	#15689404	C₂²	1	4	yes	—
23	#15689325	C₂²	1	4	yes	—
24	#15689262	—	2	2	no	—
25	#15689200	—	2	1	no	—
26	#15689200	C₂²	1	2	yes	—
27	#15689188	—	2	1	no	—
28	#15689188	C₂²	1	2	yes	—
29	#15689176	C₂²	1	1	yes	characteristic
30	C₂×#13093947	—	2	1	no	—
31	C₂×#13093947	—	2	1	no	—
32	#15689091	—	2	1	no	—
33	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	#15688875	C₂²	1	2	yes	—
35	#15688873	C₂²	1	2	yes	—
36	#15688870	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#431	C₂²	1	1	yes	characteristic
39	C₄₇×#423	C₂²	1	1	yes	characteristic
40	C₄₇×#332	C₂²	1	4	yes	—
41	C₂×#10625350	—	2	1	no	—
42	C₂²×Dic₉₄	—	2	1	no	—
43	C₂×#10625336	—	2	4	no	—
44	#13094032	—	4	4	no	—
45	C₂×#10625324	—	2	2	no	—
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₈×D₄₇	—	2	2	no	—
49	#13094024	—	2	1	no	—
50	#13094024	—	2	1	no	—
51	#13094024	—	2	1	no	—
52	#13094024	D₄	1	1	yes	characteristic
53	#13094024	C₂³	1	2	yes	—
54	#13094021	D₄	1	1	yes	characteristic
55	#13094012	—	2	1	no	—
56	#13093998	—	2	1	no	—
57	#13093998	D₄	1	2	yes	—
58	C₄×D₁₈₈	—	2	1	no	—
59	C₄×D₁₈₈	C₂³	1	2	yes	—
60	C₂×C₄⋊Dic₄₇	—	2	1	no	—
61	C₂×C₄⋊Dic₄₇	—	2	1	no	—
62	#13093958	—	2	4	no	—
63	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
64	#13093948	—	2	4	no	—
65	#13093947	—	2	2	no	—
66	#13093947	—	2	2	no	—
67	#13093947	—	2	2	no	—
68	#13093947	—	2	2	no	—
69	#13093946	—	2	4	no	—
70	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	C₄×C₄₇⋊C₈	—	2	1	no	—
74	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₄₇	—	2	1	no	—
79	C₂×M₁₆×C₄₇	—	2	1	no	—
80	C₄⋊Q₈×C₄₇	—	2	1	no	—
81	C₄⋊Q₈×C₄₇	—	2	1	no	—
82	C₄⋊Q₈×C₄₇	—	2	1	no	—
83	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
86	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	—	2	1	no	—
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
90	C₂²⋊C₈×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	—	2	1	no	—
92	C₂×Q₈×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	4	1	no	—
94	C₂×Q₈×C₄₇	—	8	2	no	—
95	C₂²×C₄×C₄₇	—	4	1	no	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂×C₂²⋊D₄₇	—	2	2	no	—
100	C₂²×Dic₄₇	—	4	1	no	—
101	#10625350	—	4	2	no	—
102	#10625350	—	4	2	no	—
103	C₂×D₁₈₈	—	2	1	no	—
104	C₂×D₁₈₈	—	2	1	no	—
105	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
106	C₂×C₄×D₄₇	—	2	2	no	—
107	C₂×Dic₉₄	—	2	1	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	4	2	no	—
110	C₂×Dic₉₄	—	4	2	no	—
111	C₂×Dic₉₄	—	4	1	no	—
112	M₁₆×C₄₇	—	4	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₂×C₈×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	4	1	no	—
118	C₄⋊C₄×C₄₇	—	4	1	no	—
119	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₇	#41	1	1	yes	characteristic
122	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
123	#10625336	—	8	4	no	—
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	—	4	1	no	—
130	C₄⋊Dic₄₇	—	4	1	no	—
131	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	C₂×C₄₇⋊C₈	—	2	2	no	—
134	C₂×C₄₇⋊C₈	—	2	2	no	—
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
139	#10625324	—	4	4	no	—
140	C₈×D₄₇	—	4	4	no	—
141	C₂³×C₄₇	#121	1	1	yes	characteristic
142	C₂²×D₄₇	—	2	2	no	—
143	Q₈×C₄₇	—	4	4	no	—
144	Q₈×C₄₇	—	8	2	no	—
145	C₂×C₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	4	2	no	—
148	C₂×C₄×C₄₇	—	8	2	no	—
149	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#135	1	2	yes	—
153	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₄₇	—	4	4	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	4	2	no	—
160	C₂×Dic₄₇	—	8	2	no	—
161	D₁₈₈	—	2	4	no	—
162	D₁₈₈	—	4	2	no	—
163	D₁₈₈	—	4	2	no	—
164	C₄×D₄₇	—	4	2	no	—
165	C₄×D₄₇	—	4	2	no	—
166	Dic₉₄	—	4	2	no	—
167	Dic₉₄	—	4	2	no	—
168	Dic₉₄	—	4	2	no	—
169	Dic₉₄	—	8	2	no	—
170	C₈×C₄₇	—	4	2	no	—
171	C₈×C₄₇	—	4	2	no	—
172	C₄₇⋊C₈	—	2	1	no	—
173	C₄₇⋊C₈	—	2	1	no	—
174	C₄₇⋊C₈	—	2	1	no	—
175	C₄₇⋊C₈	—	2	1	no	—
176	C₄₇⋊C₈	—	4	2	no	—
177	#19197	—	47	1	no	maximal, Sylow
178	C₂²×C₄₇	—	2	1	no	—
179	C₂²×C₄₇	—	2	1	no	—
180	C₂²×C₄₇	#463	1	2	yes	—
181	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
182	C₂×D₄₇	—	4	2	no	—
183	C₂×D₄₇	—	4	2	no	—
184	C₂×D₄₇	—	4	2	no	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	2	1	no	—
187	C₄×C₄₇	—	8	2	no	—
188	C₄×C₄₇	#463	1	2	yes	—
189	C₄×C₄₇	#473	1	2	yes	—
190	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
191	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₄₇	#577	1	2	yes	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	2	no	—
195	#3226	—	47	1	no	—
196	#3206	—	47	1	no	—
197	C₂×#487	—	47	1	no	—
198	#2931	—	47	1	no	—
199	#1699	—	47	4	no	—
200	#1679	—	47	4	no	—
201	#1564	—	47	1	no	—
202	#1549	D₄₇	1	1	yes	characteristic, complemented
203	#1546	—	47	1	no	—
204	C₂×C₄₇	—	2	2	no	—
205	C₂×C₄₇	C₂×#463	1	1	yes	characteristic
206	C₂×C₄₇	#3057	1	1	yes	characteristic
207	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
208	D₄₇	—	8	2	no	—
209	C₂×#131	—	94	1	no	—
210	#532	—	47	1	no	—
211	C₂×C₄⋊Q₈	—	47	1	no	—
212	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
213	#500	—	94	1	no	—
214	#492	—	94	1	no	—
215	#487	—	47	2	no	—
216	#487	—	47	2	no	—
217	#487	—	94	1	no	—
218	#487	—	94	1	no	—
219	#487	—	94	1	no	—
220	#486	—	94	1	no	—
221	#480	—	94	2	no	—
222	#474	—	47	4	no	—
223	#461	—	47	4	no	—
224	#448	—	94	2	no	—
225	#436	—	47	2	no	—
226	#436	—	94	1	no	—
227	C₈×D₄	—	47	2	no	—
228	C₈×D₄	—	94	1	no	—
229	#431	—	47	1	no	—
230	#431	C₂×D₄₇	1	1	yes	characteristic
231	#423	C₂×D₄₇	1	1	yes	characteristic
232	C₂×#103	—	94	1	no	—
233	C₂×#103	—	94	1	no	—
234	#408	—	94	1	no	—
235	C₂×C₄×C₈	—	47	1	no	—
236	#335	—	47	2	no	—
237	#334	—	47	2	no	—
238	#332	—	47	4	no	—
239	#332	C₂×D₄₇	1	4	yes	—
240	C₄₇	#19197	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	94	1	no	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×Q₈	—	94	1	no	—
244	C₂×SD₁₆	—	94	4	no	—
245	#131	—	188	4	no	—
246	C₂×M₁₆	—	2	1	no	—
247	C₂×M₁₆	—	94	2	no	—
248	C₂²×C₈	—	47	1	no	—
249	C₂²×C₈	—	47	1	no	—
250	C₂²×C₈	—	94	2	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	47	1	no	—
255	C₄⋊Q₈	—	47	2	no	—
256	C₄⋊Q₈	—	94	1	no	—
257	C₄⋊Q₈	—	94	1	no	—
258	C₄⋊Q₈	—	94	1	no	—
259	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
260	#127	—	47	1	no	—
261	#124	—	94	1	no	—
262	#121	—	47	2	no	—
263	#121	—	94	1	no	—
264	C₄×D₄	—	47	2	no	—
265	C₄×D₄	—	94	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	94	1	no	—
269	C₂×C₄⋊C₄	—	94	1	no	—
270	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
271	#107	—	94	4	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	47	2	no	—
274	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
275	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
276	#103	—	94	2	no	—
277	#103	—	94	2	no	—
278	#103	—	94	2	no	—
279	#103	—	94	2	no	—
280	#103	—	94	4	no	—
281	#102	—	94	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	94	2	no	—
284	C₂²⋊C₈	—	94	2	no	—
285	C₂²⋊C₈	—	94	1	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	—	47	1	no	—
288	C₄×C₈	—	47	1	no	—
289	C₄×C₈	—	47	1	no	—
290	C₄×C₈	—	47	1	no	—
291	C₄×C₈	—	94	1	no	—
292	#41	—	188	2	no	—
293	#41	—	188	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	8	2	no	—
297	C₂×Q₈	—	94	1	no	—
298	C₂×Q₈	—	94	2	no	—
299	C₂×Q₈	—	188	2	no	—
300	C₂×Q₈	—	188	2	no	—
301	C₂×Q₈	—	188	1	no	—
302	C₂×D₄	—	47	2	no	—
303	C₂×D₄	—	94	2	no	—
304	C₂×D₄	—	94	1	no	—
305	C₂×D₄	—	94	1	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	94	2	no	—
308	C₂²×C₄	—	188	1	no	—
309	C₂²×C₄	#10625350	1	1	yes	characteristic
310	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
311	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
312	SD₁₆	—	376	4	no	—
313	M₁₆	—	4	2	no	—
314	M₁₆	—	188	4	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	47	1	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	47	1	no	—
321	C₂×C₈	—	94	2	no	—
322	C₂×C₈	—	94	2	no	—
323	C₂×C₈	—	94	2	no	—
324	C₂×C₈	—	188	4	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	47	2	no	—
330	C₄⋊C₄	—	94	1	no	—
331	C₄⋊C₄	—	94	2	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₄⋊C₄	—	188	1	no	—
334	C₄⋊C₄	—	188	1	no	—
335	C₂²⋊C₄	—	94	2	no	—
336	C₂²⋊C₄	—	94	2	no	—
337	C₄²	#10625350	1	1	yes	characteristic
338	C₄²	D₄×D₄₇	1	1	yes	characteristic
339	C₄²	D₄×D₄₇	1	1	yes	characteristic
340	C₄²	C₂³×D₄₇	1	1	yes	characteristic
341	C₂³	—	94	2	no	—
342	C₂³	#13093997	1	1	yes	characteristic
343	Q₈	—	4	4	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	376	2	no	—
346	Q₈	—	188	2	no	—
347	Q₈	—	188	2	no	—
348	Q₈	—	188	2	no	—
349	D₄	—	94	4	no	—
350	D₄	—	188	4	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	8	2	no	—
357	C₂×C₄	—	94	2	no	—
358	C₂×C₄	—	376	2	no	—
359	C₂×C₄	—	188	2	no	—
360	C₂×C₄	—	188	2	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	#13093997	1	1	yes	characteristic
363	C₂×C₄	#13093997	1	1	yes	characteristic
364	C₂×C₄	#13093997	1	1	yes	characteristic
365	C₂×C₄	#13094047	1	2	yes	—
366	C₂×C₄	#13094051	1	2	yes	—
367	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
368	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
369	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
370	C₈	—	4	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	94	1	no	—
373	C₈	—	94	1	no	—
374	C₈	—	94	1	no	—
375	C₈	—	94	1	no	—
376	C₈	—	188	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	188	2	no	—
380	C₂²	—	188	2	no	—
381	C₂²	—	188	2	no	—
382	C₂²	#15689341	1	2	yes	—
383	C₂²	C₂×#13093997	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	376	2	no	—
388	C₄	—	188	2	no	—
389	C₄	#15689341	1	2	yes	—
390	C₄	#15689397	1	2	yes	—
391	C₄	C₂×#13094047	1	1	yes	characteristic
392	C₄	C₂×#13094051	1	1	yes	characteristic
393	C₄	#15689933	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	376	2	no	—
396	C₂	C₂×#15689341	1	1	yes	characteristic
397	C₂	#20253026	1	1	yes	characteristic
398	C₂	C₂×#15689397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1549	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	#15689673	C₂²
hypercenter	#1549	D₄₇
hypocenter	C₄₇	#19197
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

Complements

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

N	H ≅ G/N	type
3. C₂×#15689475	C₂	semidirect
5. #20242938	C₂	semidirect
6. #20242814	C₂	semidirect
7. #20242134	C₂	semidirect
8. #20242061	C₂	semidirect
9. C₄₇×#1549	C₂	semidirect
202. #1549	D₄₇	semidirect
240. C₄₇	#19197	semidirect

Maximal subgroups

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

subgroup	index
#20254511	2
#20254313	2
C₂×#15689475	2
#20252099	2
#20242938	2
#20242814	2
#20242134	2
#20242061	2
C₄₇×#1549	2
#19197	47