Subgroups — #26186832

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 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	#20252317	C₂	1	2	yes	maximal
3	#20242297	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242296	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242294	C₂	1	2	yes	maximal, complemented
6	#20242208	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242205	C₂	1	2	yes	maximal, complemented
8	#20242142	C₂	1	1	yes	maximal, characteristic
9	#20241841	C₂	1	2	yes	maximal
10	#322×D₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1579	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	#15689254	C₂²	1	2	yes	—
20	#15689252	—	2	2	no	—
21	#15689250	C₂²	1	2	yes	—
22	C₈×Dic₉₄	—	2	2	no	—
23	C₈×Dic₉₄	C₂²	1	2	yes	—
24	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
25	C₄×#10625337	C₂²	1	2	yes	—
26	#15689148	—	2	2	no	—
27	#15689145	C₂²	1	2	yes	—
28	#15689077	—	2	2	no	—
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688874	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	#15688844	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625337	C₂×C₄	1	2	yes	—
47	C₂×#10625324	—	2	2	no	—
48	C₂×C₈×D₄₇	—	2	1	no	—
49	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₇	D₄	1	2	yes	—
52	#13094023	C₂×C₄	1	2	yes	—
53	#13094022	C₂×C₄	1	2	yes	—
54	#13094014	—	2	2	no	—
55	#13094004	C₂×C₄	1	2	yes	—
56	#13094002	—	2	2	no	—
57	#13094000	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₄	—	2	2	no	—
59	C₄×Dic₉₄	C₂³	1	2	yes	—
60	#13093981	—	2	2	no	—
61	C₄²×D₄₇	C₂³	1	1	yes	characteristic
62	#13093958	C₂×C₄	1	1	yes	characteristic
63	#13093957	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093948	C₂×C₄	1	2	yes	—
71	#13093947	C₂×C₄	1	2	yes	—
72	#13093935	—	2	2	no	—
73	#13093934	—	2	2	no	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	—	2	2	no	—
78	C₈×Dic₄₇	—	2	1	no	—
79	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₇	D₄	1	2	yes	—
82	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
84	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₄₇	—	2	2	no	—
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	#103×C₄₇	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₄₇	—	2	2	no	—
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
96	C₂×Dic₉₄	—	2	2	no	—
97	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	—	2	2	no	—
107	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
108	#10625337	—	2	4	no	—
109	(C₂×D₄₇)⋊C₄	—	2	2	no	—
110	(C₂×D₄₇)⋊C₄	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
113	Dic₄₇⋊C₄	—	2	2	no	—
114	Dic₄₇⋊C₄	—	2	2	no	—
115	Dic₄₇⋊C₄	—	2	2	no	—
116	Dic₄₇⋊C₄	—	2	2	no	—
117	Dic₄₇⋊C₄	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₇	#41	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₈₈	—	2	2	no	—
128	#10625324	—	4	4	no	—
129	C₈×D₄₇	—	2	4	no	—
130	C₈×D₄₇	—	2	2	no	—
131	C₈×D₄₇	—	2	2	no	—
132	C₈×D₄₇	—	4	2	no	—
133	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₄₇	—	2	2	no	—
135	Q₈×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
146	C₄×D₄₇	—	2	1	no	—
147	C₄×D₄₇	—	2	1	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	#137	1	4	yes	—
150	Dic₉₄	—	2	2	no	—
151	Dic₉₄	—	2	2	no	—
152	Dic₉₄	—	4	4	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	—	4	1	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₈×C₄₇	#131	1	2	yes	—
158	C₄₇⋊C₈	—	2	1	no	—
159	C₄₇⋊C₈	—	2	2	no	—
160	C₄₇⋊C₈	—	4	1	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	C₄₇⋊C₈	#131	1	2	yes	—
163	#17961	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₄₇	—	2	1	no	—
166	C₂×D₄₇	—	2	1	no	—
167	C₂×D₄₇	#441	1	2	yes	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
172	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
173	C₄×C₄₇	C₂×#137	1	2	yes	—
174	Dic₄₇	—	2	1	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	#441	1	2	yes	—
178	Dic₄₇	#444	1	2	yes	—
179	#3170	—	47	1	no	—
180	#2960	—	47	2	no	—
181	#1593	—	47	1	no	—
182	#1593	—	47	1	no	—
183	#1593	—	47	2	no	—
184	#1579	—	47	2	no	—
185	#1579	—	47	1	no	—
186	#1579	D₄₇	1	1	yes	characteristic, complemented
187	#1566	—	47	1	no	—
188	#1493	—	47	2	no	—
189	C₂×#322	—	47	1	no	—
190	C₂×C₄₇	#2931	1	1	yes	characteristic
191	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
192	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
193	D₄₇	—	2	2	no	—
194	#533	—	47	2	no	—
195	#500	—	47	1	no	—
196	#494	—	47	1	no	—
197	#487	—	47	2	no	—
198	#467	—	47	1	no	—
199	#467	—	47	1	no	—
200	#467	—	47	2	no	—
201	#446	—	47	2	no	—
202	#446	—	47	2	no	—
203	#446	—	94	2	no	—
204	C₈×Q₈	—	47	2	no	—
205	C₈×Q₈	—	94	2	no	—
206	C₈×Q₈	C₂×D₄₇	1	2	yes	—
207	C₄×Q₁₆	—	47	2	no	—
208	C₄×Q₁₆	—	47	1	no	—
209	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
210	#424	—	94	2	no	—
211	#423	—	47	2	no	—
212	#405	—	94	2	no	—
213	C₂×C₄×C₈	—	47	1	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	C₂×C₄×C₈	—	47	1	no	—
216	#334	—	47	2	no	—
217	#334	—	47	1	no	—
218	#334	C₂×D₄₇	1	1	yes	characteristic
219	Q₈⋊C₈	—	47	2	no	—
220	Q₈⋊C₈	—	47	2	no	—
221	Q₈⋊C₈	—	47	2	no	—
222	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
223	#322	—	47	1	no	—
224	#322	—	47	1	no	—
225	#322	—	47	1	no	—
226	#322	C₂×D₄₇	1	1	yes	characteristic
227	C₄₇	#17961	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₁₆	—	47	2	no	—
229	C₂×Q₁₆	—	47	1	no	—
230	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
231	C₂×M₁₆	—	94	2	no	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	—	47	2	no	—
235	C₂²×C₈	—	94	1	no	—
236	C₄⋊Q₈	—	47	2	no	—
237	C₄⋊Q₈	—	47	2	no	—
238	#125	—	94	2	no	—
239	C₂²⋊Q₈	—	47	2	no	—
240	C₂²⋊Q₈	—	47	2	no	—
241	C₂²⋊Q₈	—	94	2	no	—
242	C₄×Q₈	—	47	2	no	—
243	C₄×Q₈	—	94	2	no	—
244	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
245	#117	—	94	2	no	—
246	C₂×C₄²	—	47	1	no	—
247	#107	—	47	1	no	—
248	#107	—	47	2	no	—
249	#107	C₄×D₄₇	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	47	2	no	—
252	C₄⋊C₈	—	94	2	no	—
253	C₄⋊C₈	—	94	2	no	—
254	C₄⋊C₈	—	94	2	no	—
255	C₄⋊C₈	—	94	2	no	—
256	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
257	#103	—	47	2	no	—
258	#103	—	47	2	no	—
259	#103	—	47	2	no	—
260	#103	C₄×D₄₇	1	2	yes	—
261	#97	—	94	2	no	—
262	#97	—	94	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	47	1	no	—
265	C₄×C₈	—	47	1	no	—
266	C₄×C₈	—	47	2	no	—
267	C₄×C₈	—	47	1	no	—
268	C₄×C₈	—	47	1	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	94	1	no	—
271	C₄×C₈	—	94	2	no	—
272	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
275	C₂×Q₈	—	47	2	no	—
276	C₂×Q₈	—	94	2	no	—
277	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	47	1	no	—
280	C₂²×C₄	—	47	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	94	4	no	—
283	Q₁₆	—	94	2	no	—
284	M₁₆	—	188	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	47	1	no	—
288	C₂×C₈	—	47	1	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	—	94	4	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	94	1	no	—
295	C₂×C₈	—	188	2	no	—
296	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	D₄×D₄₇	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	47	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
308	C₂²⋊C₄	—	94	2	no	—
309	C₂²⋊C₄	—	94	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	47	1	no	—
312	C₄²	—	47	1	no	—
313	C₄²	—	47	1	no	—
314	C₄²	—	94	2	no	—
315	C₄²	C₂³×D₄₇	1	1	yes	characteristic
316	C₂³	—	47	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	94	2	no	—
320	Q₈	—	94	2	no	—
321	Q₈	—	188	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	47	1	no	—
325	C₂×C₄	—	47	1	no	—
326	C₂×C₄	—	47	4	no	—
327	C₂×C₄	—	47	1	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	1	no	—
330	C₂×C₄	—	94	1	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	47	2	no	—
341	C₈	—	94	1	no	—
342	C₈	—	94	2	no	—
343	C₈	—	188	1	no	—
344	C₈	—	188	2	no	—
345	C₈	#13094030	1	2	yes	—
346	C₂²	—	47	2	no	—
347	C₂²	—	94	1	no	—
348	C₂²	—	94	1	no	—
349	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	47	2	no	—
354	C₄	—	47	2	no	—
355	C₄	—	94	1	no	—
356	C₄	—	188	2	no	—
357	C₄	—	188	2	no	—
358	C₄	#131×D₄₇	1	1	yes	characteristic
359	C₄	C₂×#13094030	1	1	yes	characteristic
360	C₄	#137×D₄₇	1	2	yes	—
361	C₂	—	94	2	no	—
362	C₂	#20252095	1	1	yes	characteristic
363	C₂	#441×D₄₇	1	1	yes	characteristic
364	C₂	#444×D₄₇	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×#1579	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1579	D₄₇
hypocenter	C₄₇	#17961
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #20242297	C₂	semidirect
4. #20242296	C₂	semidirect
5. #20242294	C₂	semidirect
6. #20242208	C₂	semidirect
7. #20242205	C₂	semidirect
11. C₄₇×#1579	C₂	semidirect
186. #1579	D₄₇	semidirect
227. C₄₇	#17961	semidirect

Maximal subgroups

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

subgroup	index
#20253971	2
#20252317	2
#20242297	2
#20242296	2
#20242294	2
#20242208	2
#20242205	2
#20242142	2
#20241841	2
#322×D₄₇	2
C₄₇×#1579	2
#17961	47