Subgroups — #26199355

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 414 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	#20254509	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689502	C₂	1	1	yes	maximal, characteristic, complemented
4	#20252099	C₂	1	1	yes	maximal, characteristic
5	#20242876	C₂	1	4	yes	maximal, complemented
6	#20242752	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×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625337	—	2	1	no	—
11	C₂²×#10625337	—	2	1	no	—
12	C₂×#13094032	—	2	1	no	—
13	#15689673	C₂²	1	1	yes	characteristic
14	C₂×#13094024	C₂²	1	1	yes	characteristic
15	#15689528	—	2	1	no	—
16	#15689526	—	2	1	no	—
17	#15689502	—	2	1	no	—
18	#15689502	—	2	1	no	—
19	#15689502	C₂²	1	2	yes	—
20	#15689502	C₂²	1	2	yes	—
21	#15689471	—	2	1	no	—
22	#15689470	—	2	1	no	—
23	#15689437	—	2	2	no	—
24	#15689436	—	2	2	no	—
25	#15689404	C₂²	1	4	yes	—
26	#15689400	C₂²	1	4	yes	—
27	#15689200	—	2	1	no	—
28	#15689200	C₂²	1	2	yes	—
29	#15689188	—	2	1	no	—
30	#15689188	C₂²	1	2	yes	—
31	#15689176	C₂²	1	1	yes	characteristic
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×#10625337	—	2	2	no	—
44	C₂×#10625337	—	2	2	no	—
45	C₂×#10625337	—	2	2	no	—
46	C₂×#10625337	—	2	2	no	—
47	C₂×#10625337	—	4	4	no	—
48	C₂×#10625337	—	4	4	no	—
49	#13094032	—	4	4	no	—
50	C₂×#10625324	—	2	2	no	—
51	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
52	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
53	C₂×C₈×D₄₇	—	2	2	no	—
54	#13094024	—	2	1	no	—
55	#13094024	—	2	1	no	—
56	#13094024	—	2	1	no	—
57	#13094024	D₄	1	1	yes	characteristic
58	#13094024	C₂³	1	2	yes	—
59	#13094021	D₄	1	1	yes	characteristic
60	#13094012	—	2	1	no	—
61	#13093998	—	2	1	no	—
62	#13093998	D₄	1	2	yes	—
63	C₄×D₁₈₈	—	2	1	no	—
64	C₄×D₁₈₈	C₂³	1	2	yes	—
65	C₂×C₄⋊Dic₄₇	—	2	1	no	—
66	C₂×C₄⋊Dic₄₇	—	2	1	no	—
67	#13093958	—	2	4	no	—
68	#13093957	—	2	4	no	—
69	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
70	#13093946	—	2	4	no	—
71	#13093944	—	2	4	no	—
72	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
73	(C₂×D₄₇)⋊C₈	—	2	2	no	—
74	(C₂×D₄₇)⋊C₈	—	2	2	no	—
75	C₄×C₄₇⋊C₈	—	2	1	no	—
76	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₂²×Q₈×C₄₇	—	2	1	no	—
81	C₂×M₁₆×C₄₇	—	2	1	no	—
82	C₄⋊Q₈×C₄₇	—	2	1	no	—
83	C₄⋊Q₈×C₄₇	—	2	1	no	—
84	C₄⋊Q₈×C₄₇	—	2	1	no	—
85	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
88	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₄₇	—	2	1	no	—
90	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
91	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
92	C₂²⋊C₈×C₄₇	—	2	1	no	—
93	C₄×C₈×C₄₇	—	2	1	no	—
94	C₂×Q₈×C₄₇	—	2	2	no	—
95	C₂×Q₈×C₄₇	—	4	1	no	—
96	C₂×Q₈×C₄₇	—	4	2	no	—
97	C₂×Q₈×C₄₇	—	4	2	no	—
98	C₂²×C₄×C₄₇	—	4	1	no	—
99	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₄₇	—	2	2	no	—
103	C₂²×Dic₄₇	—	4	1	no	—
104	#10625350	—	4	2	no	—
105	#10625350	—	4	2	no	—
106	C₂×D₁₈₈	—	2	1	no	—
107	C₂×D₁₈₈	—	2	1	no	—
108	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
109	C₂×C₄×D₄₇	—	2	2	no	—
110	C₂×Dic₉₄	—	2	1	no	—
111	C₂×Dic₉₄	—	2	2	no	—
112	C₂×Dic₉₄	—	4	2	no	—
113	C₂×Dic₉₄	—	4	2	no	—
114	C₂×Dic₉₄	—	4	1	no	—
115	M₁₆×C₄₇	—	4	2	no	—
116	C₂×C₈×C₄₇	—	2	2	no	—
117	C₂×C₈×C₄₇	—	2	2	no	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	2	2	no	—
120	C₄⋊C₄×C₄₇	—	4	1	no	—
121	C₄⋊C₄×C₄₇	—	4	1	no	—
122	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₄₇	#41	1	1	yes	characteristic
125	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
126	#10625337	—	4	4	no	—
127	#10625337	—	4	4	no	—
128	#10625337	—	4	4	no	—
129	#10625337	—	4	4	no	—
130	(C₂×D₄₇)⋊C₄	—	2	2	no	—
131	(C₂×D₄₇)⋊C₄	—	2	2	no	—
132	C₄⋊Dic₄₇	—	2	1	no	—
133	C₄⋊Dic₄₇	—	2	2	no	—
134	C₄⋊Dic₄₇	—	2	2	no	—
135	C₄⋊Dic₄₇	—	4	1	no	—
136	C₄⋊Dic₄₇	—	4	1	no	—
137	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
138	C₂×C₄₇⋊C₈	—	2	2	no	—
139	C₂×C₄₇⋊C₈	—	2	2	no	—
140	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
141	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
145	#10625324	—	4	4	no	—
146	C₈×D₄₇	—	4	4	no	—
147	C₂³×C₄₇	#121	1	1	yes	characteristic
148	C₂²×D₄₇	—	2	2	no	—
149	Q₈×C₄₇	—	4	2	no	—
150	Q₈×C₄₇	—	4	2	no	—
151	Q₈×C₄₇	—	8	2	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	2	no	—
154	C₂×C₄×C₄₇	—	4	2	no	—
155	C₂×C₄×C₄₇	—	8	2	no	—
156	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
164	C₂²⋊D₄₇	—	4	4	no	—
165	C₂×Dic₄₇	—	2	2	no	—
166	C₂×Dic₄₇	—	4	2	no	—
167	C₂×Dic₄₇	—	8	2	no	—
168	D₁₈₈	—	2	4	no	—
169	D₁₈₈	—	4	2	no	—
170	D₁₈₈	—	4	2	no	—
171	C₄×D₄₇	—	4	2	no	—
172	C₄×D₄₇	—	4	2	no	—
173	Dic₉₄	—	4	2	no	—
174	Dic₉₄	—	4	2	no	—
175	Dic₉₄	—	4	2	no	—
176	Dic₉₄	—	8	2	no	—
177	C₈×C₄₇	—	4	2	no	—
178	C₈×C₄₇	—	4	2	no	—
179	C₄₇⋊C₈	—	2	1	no	—
180	C₄₇⋊C₈	—	2	1	no	—
181	C₄₇⋊C₈	—	4	2	no	—
182	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
183	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
184	#19188	—	47	1	no	maximal, Sylow
185	C₂²×C₄₇	—	2	1	no	—
186	C₂²×C₄₇	—	2	1	no	—
187	C₂²×C₄₇	#459	1	2	yes	—
188	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
189	C₂×D₄₇	—	4	2	no	—
190	C₂×D₄₇	—	4	2	no	—
191	C₂×D₄₇	—	4	2	no	—
192	C₄×C₄₇	—	2	1	no	—
193	C₄×C₄₇	—	2	1	no	—
194	C₄×C₄₇	—	8	2	no	—
195	C₄×C₄₇	#459	1	2	yes	—
196	C₄×C₄₇	#473	1	2	yes	—
197	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	C₄×C₄₇	#578	1	2	yes	—
200	Dic₄₇	—	4	2	no	—
201	Dic₄₇	—	8	2	no	—
202	#3226	—	47	1	no	—
203	#3226	—	47	1	no	—
204	C₂×#494	—	47	1	no	—
205	#2931	—	47	1	no	—
206	#1689	—	47	4	no	—
207	#1669	—	47	4	no	—
208	#1564	—	47	1	no	—
209	#1546	—	47	1	no	—
210	#1546	D₄₇	1	1	yes	characteristic, complemented
211	C₂×C₄₇	—	2	2	no	—
212	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
213	C₂×C₄₇	#3059	1	1	yes	characteristic
214	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
215	D₄₇	—	8	2	no	—
216	C₂²×Q₁₆	—	94	1	no	—
217	C₂²×Q₁₆	—	94	1	no	—
218	C₂×#131	—	94	1	no	—
219	#532	—	47	1	no	—
220	C₂×C₄⋊Q₈	—	47	1	no	—
221	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
222	#500	—	94	1	no	—
223	#500	—	94	1	no	—
224	#494	—	47	2	no	—
225	#494	—	47	2	no	—
226	#494	—	94	1	no	—
227	#494	—	94	1	no	—
228	#486	—	94	1	no	—
229	#486	—	94	1	no	—
230	#480	—	94	2	no	—
231	#480	—	94	2	no	—
232	#474	—	47	4	no	—
233	#474	—	47	4	no	—
234	#436	—	47	2	no	—
235	#436	—	94	1	no	—
236	C₈×D₄	—	47	2	no	—
237	C₈×D₄	—	94	1	no	—
238	#431	—	47	1	no	—
239	#431	C₂×D₄₇	1	1	yes	characteristic
240	#423	C₂×D₄₇	1	1	yes	characteristic
241	#408	—	94	1	no	—
242	C₂×C₄×C₈	—	47	1	no	—
243	#335	—	47	2	no	—
244	#334	—	47	2	no	—
245	#332	—	47	4	no	—
246	#332	C₂×D₄₇	1	4	yes	—
247	C₄₇	#19188	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	94	1	no	—
249	C₂²×Q₈	—	2	1	no	—
250	C₂²×Q₈	—	94	1	no	—
251	C₂×Q₁₆	—	94	2	no	—
252	C₂×Q₁₆	—	94	2	no	—
253	C₂×Q₁₆	—	94	2	no	—
254	C₂×Q₁₆	—	94	2	no	—
255	C₂×Q₁₆	—	188	4	no	—
256	C₂×Q₁₆	—	188	4	no	—
257	#131	—	188	4	no	—
258	C₂×M₁₆	—	2	1	no	—
259	C₂×M₁₆	—	94	2	no	—
260	C₂²×C₈	—	47	1	no	—
261	C₂²×C₈	—	47	1	no	—
262	C₂²×C₈	—	94	2	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	47	1	no	—
267	C₄⋊Q₈	—	47	2	no	—
268	C₄⋊Q₈	—	94	1	no	—
269	C₄⋊Q₈	—	94	1	no	—
270	C₄⋊Q₈	—	94	1	no	—
271	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
272	#127	—	47	1	no	—
273	#124	—	94	1	no	—
274	#121	—	47	2	no	—
275	#121	—	94	1	no	—
276	C₄×D₄	—	47	2	no	—
277	C₄×D₄	—	94	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	94	1	no	—
281	C₂×C₄⋊C₄	—	94	1	no	—
282	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
283	#107	—	94	4	no	—
284	#107	—	94	4	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	47	2	no	—
287	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
288	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
289	#102	—	94	4	no	—
290	#102	—	94	4	no	—
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	94	2	no	—
293	C₂²⋊C₈	—	94	2	no	—
294	C₂²⋊C₈	—	94	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	47	1	no	—
297	C₄×C₈	—	47	1	no	—
298	C₄×C₈	—	47	1	no	—
299	C₄×C₈	—	47	1	no	—
300	C₄×C₈	—	94	1	no	—
301	#41	—	188	2	no	—
302	#41	—	188	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	4	1	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	—	94	1	no	—
308	C₂×Q₈	—	94	2	no	—
309	C₂×Q₈	—	188	2	no	—
310	C₂×Q₈	—	188	2	no	—
311	C₂×Q₈	—	188	1	no	—
312	C₂×D₄	—	47	2	no	—
313	C₂×D₄	—	94	2	no	—
314	C₂×D₄	—	94	1	no	—
315	C₂×D₄	—	94	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	94	2	no	—
318	C₂²×C₄	—	188	1	no	—
319	C₂²×C₄	#10625350	1	1	yes	characteristic
320	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
321	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
322	Q₁₆	—	188	4	no	—
323	Q₁₆	—	188	4	no	—
324	Q₁₆	—	188	4	no	—
325	Q₁₆	—	188	4	no	—
326	M₁₆	—	4	2	no	—
327	M₁₆	—	188	4	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	47	4	no	—
331	C₂×C₈	—	47	1	no	—
332	C₂×C₈	—	47	1	no	—
333	C₂×C₈	—	47	1	no	—
334	C₂×C₈	—	47	1	no	—
335	C₂×C₈	—	94	2	no	—
336	C₂×C₈	—	94	2	no	—
337	C₂×C₈	—	188	4	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	47	2	no	—
343	C₄⋊C₄	—	94	1	no	—
344	C₄⋊C₄	—	94	2	no	—
345	C₄⋊C₄	—	94	2	no	—
346	C₄⋊C₄	—	188	1	no	—
347	C₄⋊C₄	—	188	1	no	—
348	C₂²⋊C₄	—	94	2	no	—
349	C₂²⋊C₄	—	94	2	no	—
350	C₄²	#10625350	1	1	yes	characteristic
351	C₄²	D₄×D₄₇	1	1	yes	characteristic
352	C₄²	D₄×D₄₇	1	1	yes	characteristic
353	C₄²	C₂³×D₄₇	1	1	yes	characteristic
354	C₂³	—	94	2	no	—
355	C₂³	#13093997	1	1	yes	characteristic
356	Q₈	—	4	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	8	2	no	—
359	Q₈	—	376	2	no	—
360	Q₈	—	188	2	no	—
361	Q₈	—	188	2	no	—
362	Q₈	—	188	2	no	—
363	D₄	—	94	4	no	—
364	D₄	—	188	4	no	—
365	D₄	—	188	2	no	—
366	D₄	—	188	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	2	no	—
371	C₂×C₄	—	94	2	no	—
372	C₂×C₄	—	376	2	no	—
373	C₂×C₄	—	188	2	no	—
374	C₂×C₄	—	188	2	no	—
375	C₂×C₄	—	188	2	no	—
376	C₂×C₄	#13093997	1	1	yes	characteristic
377	C₂×C₄	#13093997	1	1	yes	characteristic
378	C₂×C₄	#13093997	1	1	yes	characteristic
379	C₂×C₄	D₈×D₄₇	1	2	yes	—
380	C₂×C₄	#13094051	1	2	yes	—
381	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
382	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	47	2	no	—
387	C₈	—	47	2	no	—
388	C₈	—	94	1	no	—
389	C₈	—	94	1	no	—
390	C₈	—	188	2	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	188	2	no	—
394	C₂²	—	188	2	no	—
395	C₂²	—	188	2	no	—
396	C₂²	#15689317	1	2	yes	—
397	C₂²	C₂×#13093997	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	376	2	no	—
402	C₄	—	188	2	no	—
403	C₄	#15689317	1	2	yes	—
404	C₄	#15689397	1	2	yes	—
405	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
406	C₄	C₂×#13094051	1	1	yes	characteristic
407	C₄	#15689939	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	376	2	no	—
410	C₂	C₂×#15689317	1	1	yes	characteristic
411	C₂	#20253042	1	1	yes	characteristic
412	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×#1546	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	#15689673	C₂²
hypercenter	#1546	D₄₇
hypocenter	C₄₇	#19188
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1546

Complements

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

N	H ≅ G/N	type
3. C₂×#15689502	C₂	semidirect
5. #20242876	C₂	semidirect
6. #20242752	C₂	semidirect
7. #20242134	C₂	semidirect
8. #20242061	C₂	semidirect
9. C₄₇×#1546	C₂	semidirect
210. #1546	D₄₇	semidirect
247. C₄₇	#19188	semidirect

Maximal subgroups

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

subgroup	index
#20254511	2
#20254509	2
C₂×#15689502	2
#20252099	2
#20242876	2
#20242752	2
#20242134	2
#20242061	2
C₄₇×#1546	2
#19188	47