Subgroups — #26133134

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	#20241466	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂²⋊(C₄₇⋊C₁₆)	—	2	4	no	—
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
15	C₈×D₄×C₄₇	—	2	2	no	—
16	C₈×D₄×C₄₇	—	2	2	no	—
17	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₄₇	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₄₇	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄₇⋊C₁₆	—	2	8	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
28	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	2	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	—	2	2	no	—
34	C₂²×C₈×C₄₇	—	2	1	no	—
35	C₂²×C₈×C₄₇	—	2	1	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	D₄	1	2	yes	—
41	C₂²×C₈×C₄₇	D₄	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₄×D₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₇	C₈	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₇	C₈	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	2	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	—	2	2	no	—
52	C₄×C₈×C₄₇	—	2	1	no	—
53	C₄×C₈×C₄₇	—	2	1	no	—
54	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
55	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	C₁₆	1	4	yes	complemented
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	—	2	1	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	#41	1	2	yes	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₂²⋊C₄×C₄₇	—	2	2	no	—
90	C₂²⋊C₄×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	1	no	—
92	C₄²×C₄₇	—	2	1	no	—
93	C₄₇⋊C₁₆	—	2	4	no	—
94	C₄₇⋊C₁₆	—	4	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
102	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
103	C₂³×C₄₇	C₂×C₁₆	1	2	yes	complemented
104	C₂³×C₄₇	M₃₂	1	2	yes	—
105	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
106	D₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	4	1	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	complemented
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#131	1	2	yes	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	—	4	1	no	—
128	C₈×C₄₇	—	4	1	no	—
129	C₈×C₄₇	#120	1	1	yes	characteristic
130	C₈×C₄₇	#121	1	2	yes	—
131	C₈×C₄₇	#123	1	1	yes	characteristic
132	#12976	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	2	no	—
142	C₂²×C₄₇	C₂²⋊C₁₆	1	4	yes	complemented
143	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₈×D₄	1	2	yes	—
145	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
147	C₂²×C₄₇	#504	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	4	1	no	—
151	C₄×C₄₇	—	4	1	no	—
152	C₄×C₄₇	#386	1	1	yes	characteristic
153	C₄×C₄₇	#408	1	1	yes	characteristic
154	C₄×C₄₇	#435	1	2	yes	—
155	C₂³×C₁₆	—	47	1	no	—
156	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₁₆	—	47	2	no	—
158	C₂×C₂²⋊C₁₆	—	47	1	no	—
159	#1374	—	47	2	no	—
160	C₂×C₄₇	—	2	4	no	—
161	C₂×C₄₇	—	4	2	no	—
162	C₂×C₄₇	#1869	1	1	yes	characteristic
163	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
164	C₂×C₄₇	#2113	1	1	yes	characteristic
165	C₂×C₄₇	D₄×C₁₆	1	2	yes	—
166	C₂×C₄₇	#2168	1	2	yes	—
167	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×D₄	Dic₄₇	1	1	yes	characteristic
170	C₂²×C₁₆	—	47	2	no	—
171	C₂²×C₁₆	—	47	2	no	—
172	C₂²×C₁₆	—	94	4	no	—
173	C₂²×C₁₆	—	94	2	no	—
174	C₈×D₄	—	2	2	no	—
175	C₈×D₄	—	2	2	no	—
176	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
177	C₂×C₂²⋊C₈	Dic₄₇	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	Dic₄₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	C₂²⋊C₁₆	—	94	4	no	—
181	C₂²⋊C₁₆	—	94	2	no	—
182	C₄₇	#12976	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂×C₁₆	—	47	4	no	—
205	C₂×C₁₆	—	94	8	no	—
206	C₂×C₁₆	—	94	2	no	—
207	C₂×C₁₆	—	94	2	no	—
208	C₂×C₁₆	—	94	2	no	—
209	C₂×C₁₆	—	94	2	no	—
210	C₂×C₁₆	—	94	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
217	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	C₄₇⋊C₁₆	1	4	yes	complemented
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
231	C₂²×C₄	#10625329	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
246	C₂×C₈	D₄×D₄₇	1	2	yes	—
247	C₂×C₈	#10625352	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₁₆	—	94	4	no	—
256	C₁₆	—	188	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
264	C₂³	C₂×C₄₇⋊C₁₆	1	2	yes	—
265	C₂³	C₂×C₄₇⋊C₁₆	1	2	yes	complemented
266	C₂³	#13093964	1	2	yes	—
267	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
268	D₄	—	2	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
283	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	complemented
284	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
287	C₂×C₄	#13094031	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	4	1	no	—
291	C₈	#13093992	1	1	yes	characteristic
292	C₈	#13093996	1	2	yes	—
293	C₈	#13094005	1	1	yes	characteristic
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	4	yes	complemented
304	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
305	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
306	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
307	C₂²	C₂×#13093964	1	1	yes	characteristic
308	C₂²	#15689540	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	#15689003	1	1	yes	characteristic
314	C₄	#15689090	1	1	yes	characteristic
315	C₄	#15689195	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	4	2	no	—
318	C₂	#20243681	1	1	yes	characteristic
319	C₂	C₂×C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
320	C₂	#20245002	1	1	yes	characteristic
321	C₂	D₄×C₄₇⋊C₁₆	1	2	yes	—
322	C₂	#20245214	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂²×D₄×C₄₇	C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#12976
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄₇⋊C₁₆	C₂	semidirect
2. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
3. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
4. #20241466	C₂	semidirect
27. C₂×C₄₇⋊C₁₆	D₄	semidirect
58. C₂×D₄×C₄₇	C₁₆	semidirect
103. C₂³×C₄₇	C₂×C₁₆	semidirect
121. C₂×C₄×C₄₇	C₂×C₁₆	semidirect
142. C₂²×C₄₇	C₂²⋊C₁₆	semidirect
182. C₄₇	#12976	semidirect
220. C₂×D₄	C₄₇⋊C₁₆	semidirect
265. C₂³	C₂×C₄₇⋊C₁₆	semidirect
283. C₂×C₄	C₂×C₄₇⋊C₁₆	semidirect
303. C₂²	C₂²⋊(C₄₇⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241466	2
C₂×C₈×D₄×C₄₇	2
#12976	47