Subgroups — #26186957

Order 12032.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253971	C₂	1	1	yes	maximal, characteristic
2	#20252484	C₂	1	2	yes	maximal
3	#20242295	C₂	1	2	yes	maximal, complemented
4	#20242286	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242204	C₂	1	2	yes	maximal, complemented
6	Q₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242194	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242150	C₂	1	1	yes	maximal, characteristic
9	#20241859	C₂	1	2	yes	maximal
10	#20241670	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689677	C₂²	1	2	yes	—
13	#15689527	C₄	1	1	yes	characteristic
14	#15689501	C₄	1	1	yes	characteristic
15	#15689472	C₄	1	2	yes	—
16	#15689365	C₄	1	1	yes	characteristic
17	#15689363	C₄	1	2	yes	—
18	#15689361	C₄	1	1	yes	characteristic
19	#15689253	C₂²	1	2	yes	—
20	#15689251	—	2	2	no	—
21	#15689250	—	2	2	no	—
22	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
23	#15689247	C₂²	1	2	yes	—
24	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
25	C₄×#10625337	C₂²	1	2	yes	—
26	#15689185	—	2	2	no	—
27	#15689183	—	2	2	no	—
28	#15689179	C₂²	1	2	yes	—
29	#15689174	C₂²	1	1	yes	characteristic
30	#15689174	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688876	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	Q₈⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
36	#15688853	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688845	C₂²	1	1	yes	characteristic
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₇×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₄₇×#335	C₂²	1	1	yes	characteristic
43	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625337	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
49	#13094023	C₂×C₄	1	2	yes	—
50	#13094022	C₂×C₄	1	2	yes	—
51	#13094014	—	2	2	no	—
52	#13094004	C₂×C₄	1	2	yes	—
53	#13094002	—	2	2	no	—
54	#13094000	C₂×C₄	1	2	yes	—
55	C₄×Dic₉₄	—	2	2	no	—
56	C₄×Dic₉₄	C₂³	1	2	yes	—
57	#13093981	—	2	2	no	—
58	C₄²×D₄₇	C₂³	1	1	yes	characteristic
59	#13093958	C₂×C₄	1	1	yes	characteristic
60	#13093957	C₂×C₄	1	2	yes	—
61	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₇⋊C₈	—	2	2	no	—
64	Dic₄₇⋊C₈	—	2	1	no	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	D₄	1	1	yes	characteristic
67	Dic₄₇⋊C₈	D₄	1	1	yes	characteristic
68	#13093949	C₂×C₄	1	2	yes	—
69	#13093948	C₂×C₄	1	2	yes	—
70	#13093947	C₂×C₄	1	2	yes	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	1	no	—
73	(C₂×D₄₇)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₄₇)⋊C₈	D₄	1	1	yes	characteristic
75	#13093934	—	2	2	no	—
76	#13093934	—	2	2	no	—
77	C₄×C₄₇⋊C₈	—	2	2	no	—
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₄₇	—	2	2	no	—
82	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
86	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	—	2	2	no	—
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	#103×C₄₇	C₂×C₄	1	2	yes	—
90	#97×C₄₇	—	2	2	no	—
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
98	C₂×Dic₉₄	—	2	2	no	—
99	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
111	#10625337	—	2	4	no	—
112	(C₂×D₄₇)⋊C₄	—	2	2	no	—
113	(C₂×D₄₇)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
116	Dic₄₇⋊C₄	—	2	2	no	—
117	Dic₄₇⋊C₄	—	2	2	no	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	Dic₄₇⋊C₄	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₇	#41	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Dic₁₈₈	—	2	2	no	—
132	C₈×D₄₇	—	2	2	no	—
133	C₈×D₄₇	—	2	2	no	—
134	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₄₇	—	2	2	no	—
136	Q₈×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
147	C₄×D₄₇	—	2	1	no	—
148	C₄×D₄₇	—	2	1	no	—
149	C₄×D₄₇	—	2	1	no	—
150	C₄×D₄₇	—	2	1	no	—
151	C₄×D₄₇	—	4	1	no	—
152	Dic₉₄	—	2	2	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	—	8	2	no	—
155	C₈×C₄₇	—	2	1	no	—
156	C₈×C₄₇	—	2	1	no	—
157	C₈×C₄₇	—	2	1	no	—
158	C₈×C₄₇	—	4	1	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	#131	1	2	yes	—
161	C₄₇⋊C₈	—	2	1	no	—
162	C₄₇⋊C₈	—	2	1	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	C₄₇⋊C₈	—	4	1	no	—
165	C₄₇⋊C₈	—	4	2	no	—
166	C₄₇⋊C₈	#131	1	2	yes	—
167	#17972	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₄₇	—	2	1	no	—
170	C₂×D₄₇	—	2	1	no	—
171	C₂×D₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	4	2	no	—
174	C₄×C₄₇	—	4	2	no	—
175	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
177	C₄×C₄₇	#578	1	1	yes	characteristic
178	C₄×C₄₇	#578	1	1	yes	characteristic
179	Dic₄₇	—	2	1	no	—
180	Dic₄₇	—	2	1	no	—
181	Dic₄₇	—	2	1	no	—
182	Dic₄₇	—	4	2	no	—
183	Dic₄₇	—	4	2	no	—
184	#3170	—	47	1	no	—
185	#2985	—	47	2	no	—
186	#1593	—	47	2	no	—
187	#1591	—	47	1	no	—
188	#1591	D₄₇	1	1	yes	characteristic, complemented
189	#1579	—	47	2	no	—
190	C₈×Q₁₆	—	47	1	no	—
191	C₈×Q₁₆	—	47	1	no	—
192	#1568	—	47	1	no	—
193	#1497	—	47	2	no	—
194	#1450	—	47	1	no	—
195	C₂×C₄₇	#2931	1	1	yes	characteristic
196	C₂×C₄₇	#2950	1	1	yes	characteristic
197	C₂×C₄₇	#2957	1	1	yes	characteristic
198	D₄₇	—	4	1	no	—
199	#533	—	47	2	no	—
200	#500	—	47	1	no	—
201	#494	—	47	1	no	—
202	#487	—	47	2	no	—
203	#467	—	47	1	no	—
204	#467	—	47	1	no	—
205	#467	—	47	2	no	—
206	#446	—	47	2	no	—
207	#446	—	94	2	no	—
208	#446	—	94	2	no	—
209	#446	C₂×D₄₇	1	2	yes	—
210	C₈×Q₈	—	47	2	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₄×Q₁₆	—	47	2	no	—
213	C₄×Q₁₆	—	47	1	no	—
214	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
215	#433	—	94	2	no	—
216	#433	—	94	2	no	—
217	#432	—	47	2	no	—
218	#431	—	47	1	no	—
219	#431	—	47	1	no	—
220	C₂×C₄×C₈	—	47	1	no	—
221	#335	—	47	1	no	—
222	#335	C₂×D₄₇	1	1	yes	characteristic
223	#334	—	47	2	no	—
224	Q₈⋊C₈	—	47	2	no	—
225	Q₈⋊C₈	—	47	2	no	—
226	Q₈⋊C₈	—	47	2	no	—
227	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
228	#322	—	47	1	no	—
229	#322	—	47	1	no	—
230	C₈²	—	47	1	no	—
231	C₈²	C₂×D₄₇	1	1	yes	characteristic
232	C₄₇	#17972	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	—	47	2	no	—
234	C₂×Q₁₆	—	47	1	no	—
235	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₈	—	47	1	no	—
237	C₂²×C₈	—	47	1	no	—
238	C₄⋊Q₈	—	47	2	no	—
239	C₄⋊Q₈	—	47	2	no	—
240	#125	—	94	2	no	—
241	C₂²⋊Q₈	—	47	2	no	—
242	C₂²⋊Q₈	—	47	2	no	—
243	C₂²⋊Q₈	—	94	2	no	—
244	C₄×Q₈	—	47	2	no	—
245	C₄×Q₈	—	94	2	no	—
246	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
247	#117	—	94	2	no	—
248	C₂×C₄²	—	47	1	no	—
249	#107	—	47	1	no	—
250	#107	—	47	2	no	—
251	#107	C₄×D₄₇	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	47	1	no	—
254	C₄⋊C₈	—	47	1	no	—
255	C₄⋊C₈	—	47	2	no	—
256	C₄⋊C₈	—	94	2	no	—
257	C₄⋊C₈	—	94	1	no	—
258	C₄⋊C₈	—	94	2	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
261	#103	—	47	2	no	—
262	#103	—	47	2	no	—
263	#103	—	47	2	no	—
264	#103	C₄×D₄₇	1	2	yes	—
265	C₂²⋊C₈	—	47	1	no	—
266	C₂²⋊C₈	—	47	1	no	—
267	C₂²⋊C₈	—	94	1	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	#97	—	2	2	no	—
270	#97	—	94	2	no	—
271	#97	—	94	2	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	—	47	1	no	—
276	C₄×C₈	—	47	1	no	—
277	C₄×C₈	—	94	2	no	—
278	C₄×C₈	—	94	2	no	—
279	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
282	C₂×Q₈	—	47	2	no	—
283	C₂×Q₈	—	94	2	no	—
284	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
285	C₂²×C₄	—	47	1	no	—
286	C₂²×C₄	—	47	1	no	—
287	C₂²×C₄	—	47	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	94	4	no	—
290	Q₁₆	—	94	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	47	1	no	—
294	C₂×C₈	—	47	1	no	—
295	C₂×C₈	—	47	1	no	—
296	C₂×C₈	—	47	1	no	—
297	C₂×C₈	—	94	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	94	2	no	—
300	C₂×C₈	—	94	1	no	—
301	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
304	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	47	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	94	2	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
314	C₂²⋊C₄	—	94	2	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	47	1	no	—
318	C₄²	—	47	1	no	—
319	C₄²	—	47	1	no	—
320	C₄²	—	94	2	no	—
321	C₄²	C₂³×D₄₇	1	1	yes	characteristic
322	C₂³	—	47	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	94	2	no	—
326	Q₈	—	94	2	no	—
327	Q₈	—	376	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	47	1	no	—
331	C₂×C₄	—	47	1	no	—
332	C₂×C₄	—	47	1	no	—
333	C₂×C₄	—	94	1	no	—
334	C₂×C₄	—	94	1	no	—
335	C₂×C₄	—	94	1	no	—
336	C₂×C₄	—	94	1	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	188	1	no	—
340	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	47	2	no	—
349	C₈	—	94	1	no	—
350	C₈	—	94	1	no	—
351	C₈	—	94	1	no	—
352	C₈	—	188	1	no	—
353	C₈	—	188	2	no	—
354	C₈	#13094032	1	2	yes	—
355	C₂²	—	94	1	no	—
356	C₂²	—	94	1	no	—
357	C₂²	—	94	1	no	—
358	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	94	1	no	—
363	C₄	—	94	1	no	—
364	C₄	—	94	1	no	—
365	C₄	—	188	2	no	—
366	C₄	—	188	2	no	—
367	C₄	#131×D₄₇	1	1	yes	characteristic
368	C₄	C₂×#13094032	1	1	yes	characteristic
369	C₄	#15689936	1	1	yes	characteristic
370	C₄	#15689937	1	1	yes	characteristic
371	C₂	—	188	1	no	—
372	C₂	#20252097	1	1	yes	characteristic
373	C₂	#20252251	1	1	yes	characteristic
374	C₂	#20252300	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1591	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1591	D₄₇
hypocenter	C₄₇	#17972
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #20242295	C₂	semidirect
4. #20242286	C₂	semidirect
5. #20242204	C₂	semidirect
6. Q₁₆×C₄₇⋊C₈	C₂	semidirect
7. #20242194	C₂	semidirect
11. C₄₇×#1591	C₂	semidirect
188. #1591	D₄₇	semidirect
232. C₄₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20253971	2
#20252484	2
#20242295	2
#20242286	2
#20242204	2
Q₁₆×C₄₇⋊C₈	2
#20242194	2
#20242150	2
#20241859	2
#20241670	2
C₄₇×#1591	2
#17972	47