Subgroups — #26235212

Order 12032.

712 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689252	C₂	1	2	yes	maximal
2	#20245865	C₂	1	1	yes	maximal, characteristic
3	#20244110	C₂	1	2	yes	maximal
4	#20244109	C₂	1	2	yes	maximal
5	#20244105	C₂	1	2	yes	maximal
6	#20243506	C₂	1	2	yes	maximal
7	#20243244	C₂	1	2	yes	maximal
8	#20243241	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	#15689252	—	2	8	no	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	#15689029	C₄	1	1	yes	characteristic
21	#15689029	C₄	1	1	yes	characteristic
22	#15689021	C₄	1	1	yes	characteristic
23	#15689021	C₄	1	1	yes	characteristic
24	#15689007	C₄	1	2	yes	—
25	#15689007	C₄	1	2	yes	—
26	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
28	#15688969	C₂²	1	2	yes	—
29	#15688878	C₂²	1	1	yes	characteristic
30	#15688878	C₂²	1	1	yes	characteristic
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
34	C₄³×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#336	C₂²	1	2	yes	—
36	C₄₇×#336	C₂²	1	1	yes	characteristic
37	C₄₇×#336	C₂²	1	1	yes	characteristic
38	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₄×Dic₉₄	D₄	1	4	yes	—
42	C₄×Dic₉₄	Q₈	1	4	yes	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
52	Dic₄₇⋊C₈	—	2	4	no	—
53	Dic₄₇⋊C₈	—	2	4	no	—
54	Dic₄₇⋊C₈	—	2	4	no	—
55	Dic₄₇⋊C₈	—	2	4	no	—
56	#13093935	—	2	4	no	—
57	#13093934	—	2	4	no	—
58	C₈×Dic₄₇	—	2	4	no	—
59	C₄₇⋊#95	C₂×C₄	1	2	yes	—
60	C₄₇⋊#95	C₂×C₄	1	2	yes	—
61	C₄₇⋊#95	C₂×C₄	1	2	yes	—
62	C₄₇⋊#95	C₂×C₄	1	2	yes	—
63	C₄₇⋊#95	C₂×C₄	1	2	yes	—
64	C₄₇⋊#95	C₂×C₄	1	2	yes	—
65	C₄₇⋊#95	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₉₄	—	4	1	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	—	4	1	no	—
101	C₄²×C₄₇	—	4	1	no	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
104	C₄⋊Dic₄₇	—	4	1	no	—
105	C₄⋊Dic₄₇	—	4	1	no	—
106	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
107	Dic₄₇⋊C₄	—	2	4	no	—
108	Dic₄₇⋊C₄	—	4	2	no	—
109	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
113	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	#41	1	4	yes	—
120	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₄₇	#131	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
155	Dic₉₄	—	4	4	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₄₇⋊C₈	—	2	4	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	#22737	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
163	C₂²×C₄₇	#519	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
165	C₂²×C₄₇	#568	1	1	yes	characteristic
166	C₂²×C₄₇	#568	1	1	yes	characteristic
167	C₂²×C₄₇	#568	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	—	4	1	no	—
173	C₄×C₄₇	—	4	1	no	—
174	C₄×C₄₇	#548	1	2	yes	—
175	C₄×C₄₇	#557	1	2	yes	—
176	Dic₄₇	—	2	4	no	—
177	Dic₄₇	—	2	4	no	—
178	Dic₄₇	—	4	2	no	—
179	Dic₄₇	—	4	2	no	—
180	C₂×#446	—	47	2	no	—
181	#2295	—	47	1	no	—
182	#1940	—	47	2	no	—
183	#1940	—	47	2	no	—
184	#1939	—	47	2	no	—
185	#1832	—	47	2	no	—
186	#1831	D₄₇	1	1	yes	characteristic
187	#1768	—	47	2	no	—
188	#1767	—	47	1	no	—
189	C₂×C₄₇	#2351	1	1	yes	characteristic
190	C₂×C₄₇	#2905	1	1	yes	characteristic
191	C₂×C₄₇	#2906	1	1	yes	characteristic
192	C₂×C₄₇	#2967	1	1	yes	characteristic
193	C₂×C₄₇	#2968	1	1	yes	characteristic
194	C₂×C₄₇	#2985	1	1	yes	characteristic
195	C₂×C₄₇	#2987	1	1	yes	characteristic
196	C₂×C₄×Q₈	—	47	1	no	—
197	#446	—	94	8	no	—
198	C₂×C₄⋊C₈	—	47	2	no	—
199	C₂×C₄⋊C₈	—	47	2	no	—
200	C₂×C₄⋊C₈	—	47	2	no	—
201	C₂×C₄⋊C₈	—	47	2	no	—
202	C₂×C₄⋊C₈	—	47	2	no	—
203	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
204	C₂×#97	—	47	2	no	—
205	C₂×#97	—	47	2	no	—
206	C₂×C₄×C₈	—	47	2	no	—
207	#391	—	47	1	no	—
208	#391	—	47	1	no	—
209	#389	—	47	1	no	—
210	#389	—	47	1	no	—
211	#387	—	47	2	no	—
212	#387	—	47	2	no	—
213	C₄×C₄⋊C₄	—	47	1	no	—
214	C₄×C₄⋊C₄	—	47	2	no	—
215	#376	—	47	2	no	—
216	C₄³	C₂×D₄₇	1	1	yes	characteristic
217	#336	—	47	2	no	—
218	#336	—	47	2	no	—
219	#336	—	47	1	no	—
220	#336	—	47	1	no	—
221	#336	C₂×D₄₇	1	2	yes	—
222	#336	C₂×D₄₇	1	1	yes	characteristic
223	#336	C₂×D₄₇	1	1	yes	characteristic
224	C₄₇	#22737	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	47	1	no	—
226	C₂²×C₈	—	47	2	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
229	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
230	C₄×Q₈	—	47	4	no	—
231	C₄×Q₈	—	47	4	no	—
232	C₂×C₄⋊C₄	—	47	2	no	—
233	C₂×C₄⋊C₄	—	47	1	no	—
234	C₂×C₄⋊C₄	—	47	2	no	—
235	C₂×C₄⋊C₄	—	47	2	no	—
236	C₂×C₄⋊C₄	—	47	1	no	—
237	C₂×C₄⋊C₄	—	47	1	no	—
238	C₂×C₄²	—	47	2	no	—
239	C₂×C₄²	—	47	2	no	—
240	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
247	C₄⋊C₈	—	2	4	no	—
248	C₄⋊C₈	—	94	4	no	—
249	C₄⋊C₈	—	94	4	no	—
250	C₄⋊C₈	—	94	4	no	—
251	C₄⋊C₈	—	94	4	no	—
252	C₄⋊C₈	—	94	4	no	—
253	#97	—	94	4	no	—
254	#97	—	94	4	no	—
255	C₄×C₈	—	94	4	no	—
256	#95	—	47	2	no	—
257	#95	—	47	2	no	—
258	#95	—	47	2	no	—
259	#95	—	47	2	no	—
260	#95	—	47	2	no	—
261	#95	—	47	2	no	—
262	#95	—	47	2	no	—
263	C₂×Q₈	—	47	4	no	—
264	C₂×Q₈	—	188	1	no	—
265	C₂×Q₈	—	188	1	no	—
266	C₂²×C₄	—	47	2	no	—
267	C₂²×C₄	—	47	2	no	—
268	C₂²×C₄	—	47	2	no	—
269	C₂²×C₄	—	47	2	no	—
270	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
276	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	47	4	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₄⋊C₄	—	47	8	no	—
290	C₄⋊C₄	—	47	4	no	—
291	C₄⋊C₄	—	94	4	no	—
292	C₄⋊C₄	—	188	2	no	—
293	C₄⋊C₄	—	188	1	no	—
294	C₄⋊C₄	—	188	1	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	47	4	no	—
302	C₄²	—	47	4	no	—
303	C₄²	—	94	2	no	—
304	C₄²	—	94	2	no	—
305	C₄²	D₄×D₄₇	1	2	yes	—
306	C₄²	Q₈×D₄₇	1	2	yes	—
307	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
308	Q₈	—	188	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	47	4	no	—
325	C₂×C₄	—	47	4	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	94	2	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
337	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
338	C₂×C₄	#13094030	1	2	yes	—
339	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
341	C₂×C₄	#41×D₄₇	1	2	yes	—
342	C₂×C₄	#13094071	1	2	yes	—
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	94	4	no	—
346	C₈	—	188	2	no	—
347	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
348	C₂²	#15689591	1	1	yes	characteristic
349	C₂²	#15689600	1	1	yes	characteristic
350	C₂²	C₂×#13094030	1	1	yes	characteristic
351	C₂²	#15689889	1	1	yes	characteristic
352	C₂²	#15689889	1	1	yes	characteristic
353	C₂²	#15689889	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	94	4	no	—
361	C₄	—	94	4	no	—
362	C₄	—	188	2	no	—
363	C₄	—	188	2	no	—
364	C₄	#15689781	1	2	yes	—
365	C₄	#15689850	1	2	yes	—
366	C₂	#20246305	1	1	yes	characteristic
367	C₂	#20251950	1	1	yes	characteristic
368	C₂	#20251953	1	1	yes	characteristic
369	C₂	#20252368	1	1	yes	characteristic
370	C₂	#20252377	1	1	yes	characteristic
371	C₂	#20252482	1	1	yes	characteristic
372	C₂	#20252493	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1831	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#22737
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
224. C₄₇	#22737	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689252	2
#20245865	2
#20244110	2
#20244109	2
#20244105	2
#20243506	2
#20243244	2
#20243241	2
C₄₇×#1831	2
#22737	47