Subgroups — #26233999

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689160	C₂	1	1	yes	maximal, characteristic
2	#20246280	C₂	1	2	yes	maximal
3	#20244127	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243976	C₂	1	2	yes	maximal, complemented
5	#20243970	C₂	1	2	yes	maximal, complemented
6	#20243481	C₂	1	2	yes	maximal
7	#20243410	C₂	1	1	yes	maximal, characteristic
8	#20243288	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243285	C₂	1	1	yes	maximal, characteristic, complemented
10	#20243165	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1943	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×#13093982	C₂²	1	2	yes	—
14	#15689160	C₄	1	8	yes	complemented
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	1	yes	characteristic
17	C₂×#13093955	C₂²	1	2	yes	—
18	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#13093934	C₂²	1	1	yes	characteristic
20	#15689022	C₂²	1	2	yes	—
21	#15689015	—	2	2	no	—
22	#15689013	—	2	2	no	—
23	#15688990	—	2	2	no	—
24	#15688980	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
28	C₄×(C₂×D₄₇)⋊C₄	C₂²	1	1	yes	characteristic
29	#15688886	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	#15688885	C₂²	1	2	yes	—
32	#15688885	C₂²	1	2	yes	—
33	#15688878	C₂²	1	1	yes	characteristic
34	#15688877	C₂²	1	1	yes	characteristic
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#389	C₂²	1	2	yes	—
39	C₄₇×#340	C₂²	1	2	yes	—
40	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₄₇	—	2	2	no	—
44	C₂×C₈×D₄₇	—	2	2	no	—
45	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
46	#13093982	—	2	4	no	—
47	#13093982	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
51	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
52	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
53	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
54	C₂×C₄×Dic₄₇	—	2	2	no	—
55	C₂×C₄×Dic₄₇	—	2	2	no	—
56	C₂×C₄×Dic₄₇	—	2	2	no	—
57	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
58	#13093958	C₂×C₄	1	4	yes	—
59	#13093956	—	2	2	no	—
60	#13093955	C₂×C₄	1	8	yes	—
61	(C₂×D₄₇)⋊C₈	—	2	2	no	—
62	#13093934	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₄₇⋊#95	—	2	2	no	—
65	C₄₇⋊#95	—	2	2	no	—
66	C₄₇⋊#95	—	2	2	no	—
67	C₄₇⋊#95	D₄	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	#107×C₄₇	C₂×C₄	1	4	yes	—
75	#106×C₄₇	—	2	2	no	—
76	#97×C₄₇	—	2	2	no	—
77	#95×C₄₇	—	2	2	no	—
78	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×D₄₇	—	2	1	no	—
88	C₂×C₄×D₄₇	—	2	1	no	—
89	C₂×C₄×D₄₇	—	2	1	no	—
90	C₂×C₄×D₄₇	—	2	1	no	—
91	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₄₇	#41	1	2	yes	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	2	1	no	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₄₇	#41	1	2	yes	—
98	C₂×C₈×C₄₇	#41	1	2	yes	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	4	4	no	—
103	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₄₇	—	2	1	no	—
105	C₄²×C₄₇	—	2	1	no	—
106	(C₂×D₄₇)⋊C₄	—	2	8	no	—
107	(C₂×D₄₇)⋊C₄	—	4	2	no	—
108	(C₂×D₄₇)⋊C₄	—	8	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	4	4	no	—
113	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
114	C₄×Dic₄₇	—	2	1	no	—
115	C₄×Dic₄₇	—	2	1	no	—
116	C₄×Dic₄₇	—	2	8	no	—
117	C₄×Dic₄₇	—	8	2	no	—
118	C₄×Dic₄₇	—	8	2	no	—
119	C₂×C₄₇⋊C₈	—	2	1	no	—
120	C₂×C₄₇⋊C₈	—	2	1	no	—
121	C₂×C₄₇⋊C₈	—	2	1	no	—
122	C₂×C₄₇⋊C₈	—	2	1	no	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
125	C₈×D₄₇	—	2	8	no	—
126	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂²×D₄₇	—	2	1	no	—
128	C₂²×D₄₇	—	2	1	no	—
129	C₂²×D₄₇	—	2	2	no	—
130	C₂²×D₄₇	—	2	2	no	—
131	C₂²×D₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×Dic₄₇	—	2	1	no	—
148	C₂×Dic₄₇	—	2	1	no	—
149	C₂×Dic₄₇	—	2	4	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	C₄×D₄	1	2	yes	—
159	C₄×D₄₇	—	2	2	no	—
160	C₄×D₄₇	—	2	2	no	—
161	C₄×D₄₇	—	2	2	no	—
162	C₄×D₄₇	—	2	2	no	—
163	C₈×C₄₇	—	4	1	no	—
164	C₈×C₄₇	#117	1	4	yes	—
165	C₄₇⋊C₈	—	2	2	no	—
166	C₄₇⋊C₈	—	4	1	no	—
167	#22620	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₄₇	#547	1	1	yes	characteristic
170	C₂²×C₄₇	#547	1	1	yes	characteristic
171	C₂²×C₄₇	#575	1	2	yes	—
172	C₂²×C₄₇	#577	1	2	yes	—
173	C₂×D₄₇	—	2	2	no	—
174	C₂×D₄₇	—	2	2	no	—
175	C₂×D₄₇	—	2	2	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₄×C₄₇	—	4	2	no	—
179	C₄×C₄₇	—	4	4	no	—
180	C₄×C₄₇	—	8	2	no	—
181	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
182	C₄×C₄₇	#519	1	1	yes	characteristic
183	C₄×C₄₇	#552	1	1	yes	characteristic
184	C₄×C₄₇	#552	1	1	yes	characteristic
185	Dic₄₇	—	2	2	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	—	4	4	no	—
188	Dic₄₇	—	8	2	no	—
189	C₂×#427	—	47	1	no	—
190	#2347	—	47	2	no	—
191	#1943	—	47	1	no	—
192	#1943	D₄₇	1	1	yes	characteristic, complemented
193	#1920	—	47	2	no	—
194	#1920	—	47	2	no	—
195	#1825	—	47	2	no	—
196	#1807	—	47	1	no	—
197	#1776	—	47	1	no	—
198	#1776	—	47	1	no	—
199	#1752	—	47	1	no	—
200	C₂×C₄₇	#2344	1	1	yes	characteristic
201	C₂×C₄₇	#2943	1	1	yes	characteristic
202	C₂×C₄₇	#2952	1	1	yes	characteristic
203	C₂×C₄₇	#3350	1	2	yes	—
204	C₂×C₄₇	#3350	1	2	yes	—
205	D₄₇	—	4	2	no	—
206	C₂³×C₈	—	47	1	no	—
207	C₂×#117	—	47	2	no	—
208	#427	—	47	8	no	—
209	C₂×#107	—	47	1	no	—
210	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
211	C₂×#106	—	47	2	no	—
212	C₂×#106	—	47	1	no	—
213	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
214	C₂×C₂²⋊C₈	—	47	1	no	—
215	C₂×#97	—	47	1	no	—
216	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
217	#389	—	47	2	no	—
218	#389	C₂×D₄₇	1	2	yes	—
219	#388	—	94	2	no	—
220	#388	—	94	2	no	—
221	#383	—	94	2	no	—
222	#380	—	47	2	no	—
223	C₄×C₄⋊C₄	—	47	2	no	—
224	C₄×C₄⋊C₄	—	47	2	no	—
225	C₄×C₄⋊C₄	—	94	2	no	—
226	C₄×C₂²⋊C₄	—	47	1	no	—
227	#340	—	47	2	no	—
228	#340	—	47	2	no	—
229	#340	—	47	2	no	—
230	#340	—	47	2	no	—
231	#340	C₂×D₄₇	1	2	yes	—
232	#336	—	47	1	no	—
233	#336	—	47	1	no	—
234	C₄₇	#22620	1	1	yes	characteristic, complemented, Sylow
235	C₂³×C₄	—	47	1	no	—
236	C₂²×C₈	—	47	1	no	—
237	C₂²×C₈	—	47	4	no	—
238	C₂²×C₈	—	47	1	no	—
239	C₂²×C₈	—	94	2	no	—
240	C₂²×C₈	—	94	2	no	—
241	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	#117	—	47	8	no	—
244	#117	—	94	4	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	—	47	2	no	—
247	C₂×C₄⋊C₄	—	47	2	no	—
248	C₂×C₄⋊C₄	—	94	2	no	—
249	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
250	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
251	C₂×C₂²⋊C₄	—	47	2	no	—
252	C₂×C₂²⋊C₄	—	94	2	no	—
253	C₂×C₂²⋊C₄	—	94	2	no	—
254	C₂×C₄²	—	47	1	no	—
255	C₂×C₄²	—	94	2	no	—
256	C₂×C₄²	—	94	2	no	—
257	C₂×C₄²	—	94	2	no	—
258	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
259	#107	—	47	4	no	—
260	#107	C₄×D₄₇	1	4	yes	—
261	#106	—	2	2	no	—
262	#106	—	47	8	no	—
263	#106	—	94	2	no	—
264	C₂²⋊C₈	—	94	2	no	—
265	#97	—	2	2	no	—
266	#97	—	94	2	no	—
267	#95	—	2	2	no	—
268	#95	—	47	2	no	—
269	#95	—	94	2	no	—
270	#95	—	94	2	no	—
271	#95	—	94	2	no	—
272	#95	—	94	2	no	—
273	C₂⁴	—	47	1	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	47	1	no	—
277	C₂²×C₄	—	47	2	no	—
278	C₂²×C₄	—	47	2	no	—
279	C₂²×C₄	—	47	2	no	—
280	C₂²×C₄	—	94	1	no	—
281	C₂²×C₄	—	94	1	no	—
282	C₂²×C₄	—	94	1	no	—
283	C₂²×C₄	—	94	1	no	—
284	C₂²×C₄	—	94	2	no	—
285	C₂²×C₄	—	94	2	no	—
286	C₂²×C₄	D₄×D₄₇	1	2	yes	—
287	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	94	1	no	—
293	C₂×C₈	—	94	1	no	—
294	C₂×C₈	—	94	8	no	—
295	C₂×C₈	—	94	1	no	—
296	C₂×C₈	—	94	1	no	—
297	C₂×C₈	—	94	1	no	—
298	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
299	C₂×C₈	#10625352	1	2	yes	—
300	C₂×C₈	#10625354	1	2	yes	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	4	no	—
305	C₄⋊C₄	—	47	4	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	188	4	no	—
310	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
311	C₂²⋊C₄	—	94	8	no	—
312	C₂²⋊C₄	—	376	2	no	—
313	C₂²⋊C₄	—	188	2	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	2	1	no	—
316	C₄²	—	94	8	no	—
317	C₄²	—	94	1	no	—
318	C₄²	—	94	1	no	—
319	C₄²	—	376	2	no	—
320	C₄²	—	376	2	no	—
321	C₂³	—	47	2	no	—
322	C₂³	—	94	1	no	—
323	C₂³	—	94	1	no	—
324	C₂³	—	94	2	no	—
325	C₂³	—	94	2	no	—
326	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	47	2	no	—
337	C₂×C₄	—	94	1	no	—
338	C₂×C₄	—	94	1	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	—	94	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	—	94	2	no	—
343	C₂×C₄	—	94	2	no	—
344	C₂×C₄	—	94	4	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	188	2	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
353	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
356	C₂×C₄	#13094071	1	1	yes	characteristic
357	C₂×C₄	#13094072	1	1	yes	characteristic
358	C₈	—	4	1	no	—
359	C₈	—	94	2	no	—
360	C₈	—	188	1	no	—
361	C₈	#13093982	1	4	yes	—
362	C₂²	—	94	2	no	—
363	C₂²	—	94	2	no	—
364	C₂²	—	94	2	no	—
365	C₂²	—	188	2	no	—
366	C₂²	—	188	2	no	—
367	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
368	C₂²	#15689773	1	1	yes	characteristic
369	C₂²	#15689776	1	1	yes	characteristic
370	C₂²	#15689919	1	2	yes	—
371	C₂²	#15689932	1	2	yes	—
372	C₄	—	4	4	no	—
373	C₄	—	4	2	no	—
374	C₄	—	8	2	no	—
375	C₄	—	94	2	no	—
376	C₄	—	376	2	no	—
377	C₄	—	188	4	no	—
378	C₄	—	188	2	no	—
379	C₄	C₂×#13093982	1	1	yes	characteristic
380	C₄	#15689599	1	1	yes	characteristic
381	C₄	#15689807	1	1	yes	characteristic
382	C₄	#15689811	1	1	yes	characteristic
383	C₂	—	188	2	no	—
384	C₂	#20246256	1	1	yes	characteristic
385	C₂	#20252185	1	1	yes	characteristic
386	C₂	#20252268	1	1	yes	characteristic
387	C₂	#20255928	1	2	yes	—
388	C₂	#20255932	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1943	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1943	D₄₇
hypocenter	C₄₇	#22620
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1943

Complements

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

N	H ≅ G/N	type
3. #20244127	C₂	semidirect
4. #20243976	C₂	semidirect
5. #20243970	C₂	semidirect
8. #20243288	C₂	semidirect
9. #20243285	C₂	semidirect
11. C₄₇×#1943	C₂	semidirect
14. #15689160	C₄	semidirect
192. #1943	D₄₇	semidirect
234. C₄₇	#22620	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689160	2
#20246280	2
#20244127	2
#20243976	2
#20243970	2
#20243481	2
#20243410	2
#20243288	2
#20243285	2
#20243165	2
C₄₇×#1943	2
#22620	47