Subgroups — #26132197

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#20241465	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₈×Dic₄₇	—	2	4	no	—
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₁₆	—	2	4	no	—
15	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
18	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
19	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
20	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄₇⋊C₁₆	—	2	2	no	—
33	C₂×C₄₇⋊C₁₆	—	2	2	no	—
34	C₂×C₄₇⋊C₁₆	—	2	2	no	—
35	C₈×Dic₄₇	—	2	2	no	—
36	C₈×Dic₄₇	—	2	2	no	—
37	C₈×Dic₄₇	—	2	2	no	—
38	C₈×Dic₄₇	D₄	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	2	yes	—
44	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₇	—	2	2	no	—
47	C₂×C₁₆×C₄₇	—	2	2	no	—
48	C₂×C₁₆×C₄₇	—	2	2	no	—
49	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	2	no	—
59	C₂²×Dic₄₇	—	2	2	no	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂²×Dic₄₇	—	2	2	no	—
62	C₂²×Dic₄₇	—	2	2	no	—
63	C₂²×Dic₄₇	C₁₆	1	4	yes	complemented
64	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₄₇	#41	1	2	yes	—
77	C₄×Dic₄₇	—	2	1	no	—
78	C₄×Dic₄₇	—	2	1	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₄₇	M₁₆	1	2	yes	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
90	C₁₆×C₄₇	—	4	2	no	—
91	C₄₇⋊C₁₆	—	4	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
97	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
98	C₂³×C₄₇	M₃₂	1	2	yes	—
99	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	2	1	no	—
115	C₂×Dic₄₇	—	2	1	no	—
116	C₂×Dic₄₇	—	2	1	no	—
117	C₂×Dic₄₇	—	2	1	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	C₂×Dic₄₇	—	2	1	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
122	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
123	C₂×Dic₄₇	C₂×C₁₆	1	2	yes	—
124	C₂×Dic₄₇	C₂×C₁₆	1	2	yes	—
125	C₂×Dic₄₇	M₃₂	1	2	yes	—
126	C₂×Dic₄₇	M₃₂	1	2	yes	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	#121	1	2	yes	—
129	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₄₇⋊C₈	—	2	2	no	—
131	C₄₇⋊C₈	—	2	2	no	—
132	#12783	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	C₄⋊C₁₆	1	4	yes	complemented
140	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₂²×C₄₇	#431	1	1	yes	characteristic
143	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	#380	1	1	yes	characteristic
149	C₄×C₄₇	#407	1	1	yes	characteristic
150	C₄×C₄₇	#423	1	1	yes	characteristic
151	C₄×C₄₇	#432	1	1	yes	characteristic
152	Dic₄₇	—	2	2	no	—
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	2	2	no	—
155	Dic₄₇	C₂²⋊C₁₆	1	4	yes	complemented
156	C₂²×C₄×C₈	—	47	1	no	—
157	C₂×C₄⋊C₁₆	—	47	2	no	—
158	C₂×C₂²⋊C₁₆	—	47	1	no	—
159	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
160	#1374	—	47	2	no	—
161	C₂×C₄₇	—	2	4	no	—
162	C₂×C₄₇	#1797	1	1	yes	characteristic
163	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
164	C₂×C₄₇	#2112	1	1	yes	characteristic
165	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
166	C₂×C₄₇	#2150	1	1	yes	characteristic
167	C₂×C₄₇	#2162	1	1	yes	characteristic
168	C₂×C₄₇	#2163	1	1	yes	characteristic
169	C₂³×C₈	—	47	1	no	—
170	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₄²	—	47	1	no	—
172	C₂²×C₁₆	—	47	2	no	—
173	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
174	C₂×C₄×C₈	—	47	1	no	—
175	C₂×C₄×C₈	—	47	1	no	—
176	C₂×C₄×C₈	—	47	1	no	—
177	C₂×C₄×C₈	—	47	1	no	—
178	C₂×C₄×C₈	—	94	4	no	—
179	C₄⋊C₁₆	—	94	4	no	—
180	C₂²⋊C₁₆	—	2	2	no	—
181	C₂²⋊C₁₆	—	94	2	no	—
182	C₄₇	#12783	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	47	1	no	—
184	C₂³×C₄	—	47	1	no	—
185	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	47	1	no	—
189	C₂²×C₈	—	47	1	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	Dic₉₄	1	2	yes	—
194	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	—	47	1	no	—
198	C₂×C₄²	—	47	1	no	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	1	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	—	94	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	94	2	no	—
207	C₂×C₁₆	—	94	2	no	—
208	C₂×C₁₆	—	94	2	no	—
209	C₄×C₈	—	47	4	no	—
210	C₄×C₈	—	94	2	no	—
211	C₄×C₈	—	94	2	no	—
212	C₄×C₈	—	94	2	no	—
213	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	47	1	no	—
221	C₂²×C₄	—	47	1	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	#10625324	1	2	yes	—
230	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	47	2	no	—
238	C₂×C₈	—	94	8	no	—
239	C₂×C₈	—	94	1	no	—
240	C₂×C₈	—	94	1	no	—
241	C₂×C₈	—	94	1	no	—
242	C₂×C₈	—	94	1	no	—
243	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
244	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
245	C₂×C₈	D₄×D₄₇	1	2	yes	—
246	C₂×C₈	#10625352	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₄²	—	47	2	no	—
249	C₄²	—	47	2	no	—
250	C₄²	—	94	2	no	—
251	C₄²	—	94	2	no	—
252	C₄²	—	94	1	no	—
253	C₄²	—	94	1	no	—
254	C₁₆	—	4	2	no	—
255	C₁₆	—	188	2	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	Dic₄₇⋊C₈	1	2	yes	—
261	C₂³	C₁₆×D₄₇	1	2	yes	—
262	C₂³	#13093963	1	2	yes	—
263	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	47	2	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	47	2	no	—
274	C₂×C₄	—	47	2	no	—
275	C₂×C₄	—	47	2	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	94	8	no	—
278	C₂×C₄	—	94	1	no	—
279	C₂×C₄	—	94	1	no	—
280	C₂×C₄	—	94	1	no	—
281	C₂×C₄	—	94	1	no	—
282	C₂×C₄	—	94	1	no	—
283	C₂×C₄	—	94	1	no	—
284	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
285	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#13093980	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	94	2	no	—
293	C₈	—	94	2	no	—
294	C₈	#13093996	1	2	yes	—
295	C₈	#13094001	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	Dic₄₇⋊C₁₆	1	4	yes	complemented
303	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
304	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
305	C₂²	#15689173	1	1	yes	characteristic
306	C₂²	C₂×C₁₆×D₄₇	1	1	yes	characteristic
307	C₂²	M₃₂×D₄₇	1	1	yes	characteristic
308	C₂²	M₃₂×D₄₇	1	1	yes	characteristic
309	C₂²	C₂×#13093963	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	47	4	no	—
312	C₄	—	94	2	no	—
313	C₄	—	94	2	no	—
314	C₄	—	94	2	no	—
315	C₄	#15688981	1	1	yes	characteristic
316	C₄	#15689085	1	1	yes	characteristic
317	C₄	#15689146	1	1	yes	characteristic
318	C₄	#15689177	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	#20243366	1	1	yes	characteristic
321	C₂	C₂²⋊C₁₆×D₄₇	1	1	yes	characteristic
322	C₂	#20244997	1	1	yes	characteristic
323	C₂	C₂×Dic₄₇⋊C₁₆	1	1	yes	characteristic
324	C₂	#20245143	1	1	yes	characteristic
325	C₂	#20245184	1	1	yes	characteristic
326	C₂	#20245188	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#12783
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
4. #20241465	C₂	semidirect
63. C₂²×Dic₄₇	C₁₆	semidirect
139. C₂²×C₄₇	C₄⋊C₁₆	semidirect
155. Dic₄₇	C₂²⋊C₁₆	semidirect
182. C₄₇	#12783	semidirect
302. C₂²	Dic₄₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
C₂×Dic₄₇⋊C₁₆	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241465	2
C₂×C₄₇×C₂²⋊C₁₆	2
#12783	47