Subgroups — #26214531

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20243258	C₂	1	2	yes	maximal, complemented
4	#20243253	C₂	1	2	yes	maximal, complemented
5	#20243167	C₂	1	2	yes	maximal
6	#20243162	C₂	1	2	yes	maximal
7	#20243141	C₂	1	2	yes	maximal, complemented
8	#20243076	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13093934	C₂²	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
19	#15688991	C₄	1	2	yes	—
20	#15688989	C₄	1	2	yes	—
21	#15688986	C₄	1	2	yes	—
22	#15688983	C₄	1	2	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
25	#15688969	C₂²	1	2	yes	—
26	#15688877	C₂²	1	4	yes	—
27	#15688877	C₂²	1	4	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#432	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
32	C₂×#97×C₄₇	C₂²	1	2	yes	—
33	C₂²×C₄₇⋊C₈	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
44	#13093934	—	2	8	no	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
46	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₇	—	2	8	no	—
50	C₄₇⋊#95	C₂×C₄	1	4	yes	—
51	C₄₇⋊#95	C₂×C₄	1	4	yes	—
52	C₄₇⋊#95	C₂×C₄	1	4	yes	—
53	C₄₇⋊#95	C₂×C₄	1	4	yes	—
54	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
66	#97×C₄₇	C₂×C₄	1	8	yes	—
67	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
86	C₄²×C₄₇	C₄²	1	4	yes	—
87	C₄²×C₄₇	C₄²	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂²⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	8	no	—
99	C₂×C₄₇⋊C₈	—	2	8	no	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	#41	1	4	yes	—
105	C₂×C₄₇⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₇	#131	1	4	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	#131	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	2	8	no	—
133	C₈×C₄₇	—	2	8	no	—
134	C₄₇⋊C₈	—	2	2	no	—
135	C₄₇⋊C₈	—	2	2	no	—
136	C₄₇⋊C₈	#117	1	4	yes	—
137	C₄₇⋊C₈	#117	1	4	yes	—
138	#20476	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	8	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	#405	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₄₇	C₄×#41	1	2	yes	—
145	C₂²×C₄₇	C₂×#131	1	2	yes	—
146	C₂²×C₄₇	C₂×#131	1	2	yes	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	#405	1	4	yes	—
152	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₄×#41	1	2	yes	—
155	Dic₄₇	—	4	8	no	—
156	C₂×#432	D₄₇	1	1	yes	characteristic
157	C₂²×C₄×C₈	—	47	1	no	—
158	#2283	—	47	1	no	—
159	#1770	—	47	2	no	—
160	C₈×C₄⋊C₄	—	47	2	no	—
161	#1752	—	47	2	no	—
162	C₈×C₂²⋊C₄	—	47	2	no	—
163	#1746	—	47	2	no	—
164	#1726	—	47	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
169	C₂×C₄₇	#2966	1	4	yes	—
170	C₂³×C₈	—	47	1	no	—
171	C₂³×C₈	—	47	1	no	—
172	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
173	#432	Dic₄₇	1	8	yes	complemented
174	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
175	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
176	C₂×#97	—	47	4	no	—
177	C₂×#97	C₂×D₄₇	1	2	yes	—
178	C₂×C₄×C₈	—	47	8	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	C₂×C₄×C₈	—	47	1	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	C₂×C₄×C₈	—	47	4	no	—
184	#383	—	47	2	no	—
185	#382	—	47	2	no	—
186	#381	—	47	2	no	—
187	#380	—	47	2	no	—
188	C₄×C₄⋊C₄	—	47	2	no	—
189	C₄×C₂²⋊C₄	—	47	2	no	—
190	#376	—	47	2	no	—
191	#336	—	47	4	no	—
192	#336	—	47	4	no	—
193	C₄₇	#20476	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	47	1	no	—
198	C₂²×C₈	—	47	1	no	—
199	C₂²×C₈	—	47	4	no	—
200	C₂²×C₈	—	47	4	no	—
201	C₂²×C₈	—	47	1	no	—
202	C₂²×C₈	—	47	1	no	—
203	C₂²×C₈	—	94	4	no	—
204	C₂²×C₈	—	94	4	no	—
205	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₂²⋊C₄	—	47	4	no	—
208	C₂×C₄²	—	47	4	no	—
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₄₇	1	4	yes	—
212	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
216	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
217	#97	—	94	8	no	—
218	#97	C₂×Dic₄₇	1	8	yes	—
219	C₄×C₈	—	47	4	no	—
220	C₄×C₈	—	47	4	no	—
221	C₄×C₈	—	47	4	no	—
222	C₄×C₈	—	47	4	no	—
223	C₄×C₈	—	94	8	no	—
224	#95	—	47	4	no	—
225	#95	—	47	4	no	—
226	#95	—	47	4	no	—
227	#95	—	47	4	no	—
228	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
235	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	47	2	no	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	47	4	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	2	no	—
251	C₂×C₈	—	47	2	no	—
252	C₂×C₈	—	94	8	no	—
253	C₂×C₈	—	94	8	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
257	C₄⋊C₄	—	94	8	no	—
258	C₂²⋊C₄	—	94	8	no	—
259	C₄²	—	94	4	no	—
260	C₄²	—	94	4	no	—
261	C₄²	C₄×Dic₄₇	1	4	yes	—
262	C₄²	C₄×Dic₄₇	1	4	yes	—
263	C₄²	C₂×C₄×D₄₇	1	2	yes	—
264	C₄²	C₂×C₄×D₄₇	1	2	yes	—
265	C₄²	C₂²×Dic₄₇	1	2	yes	—
266	C₄²	C₂²×Dic₄₇	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
270	C₂³	#13094032	1	4	yes	—
271	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	94	8	no	—
282	C₂×C₄	—	94	8	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	#13094032	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	#41×D₄₇	1	2	yes	—
293	C₂×C₄	#41×D₄₇	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₈	—	47	4	no	—
296	C₈	—	47	4	no	—
297	C₈	—	94	2	no	—
298	C₈	—	94	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#15689081	1	4	yes	—
303	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
304	C₂²	#15689591	1	2	yes	—
305	C₂²	C₂×#13094032	1	2	yes	—
306	C₂²	C₂×#13094032	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	188	8	no	—
312	C₄	#15689081	1	4	yes	—
313	C₄	#117×D₄₇	1	1	yes	characteristic
314	C₄	#117×D₄₇	1	1	yes	characteristic
315	C₄	#41×Dic₄₇	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	#117×Dic₄₇	1	1	yes	characteristic
318	C₂	C₂×#15689081	1	1	yes	characteristic
319	C₂	C₂×#15689081	1	1	yes	characteristic
320	C₂	#20252357	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20476
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243258	C₂	semidirect
4. #20243253	C₂	semidirect
7. #20243141	C₂	semidirect
8. #20243076	C₂	semidirect
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
29. C₄₇×#432	C₄	semidirect
173. #432	Dic₄₇	semidirect
193. C₄₇	#20476	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
#20245786	2
#20243258	2
#20243253	2
#20243167	2
#20243162	2
#20243141	2
#20243076	2
C₂×C₄₇×#432	2
#20476	47