Subgroups — #26369635

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20252235	C₂	1	2	yes	maximal, complemented
3	#20245262	C₂	1	2	yes	maximal, complemented
4	#20245261	C₂	1	2	yes	maximal, complemented
5	C₄×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245234	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245169	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245102	C₂	1	2	yes	maximal, complemented
10	#20244983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₇	C₂²	1	2	yes	—
13	#15689595	—	2	2	no	—
14	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
15	C₂×#13093965	C₂²	1	2	yes	—
16	D₁₆×D₄₇	C₄	1	8	yes	complemented
17	C₂×#13093963	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
19	#15689233	—	2	2	no	—
20	#15689232	—	2	2	no	—
21	C₄×D₃₇₆	C₂²	1	2	yes	complemented
22	D₈×Dic₄₇	C₂²	1	2	yes	complemented
23	#15689205	C₂²	1	2	yes	complemented
24	#15689160	C₂²	1	2	yes	—
25	#15689126	—	2	2	no	—
26	#15689077	C₂²	1	1	yes	characteristic
27	#15688950	C₂²	1	2	yes	complemented
28	#15688949	C₂²	1	1	yes	characteristic, complemented
29	#15688925	C₂²	1	2	yes	complemented
30	#15688924	C₂²	1	2	yes	complemented
31	#15688923	C₂²	1	2	yes	complemented
32	#15688899	C₂²	1	1	yes	characteristic, complemented
33	#15688898	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	2	yes	complemented
37	C₄₇×#366	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#357	C₂²	1	2	yes	complemented
39	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₄₇	—	2	2	no	—
41	C₂×D₃₇₆	C₂³	1	2	yes	—
42	C₂×#10625334	—	2	2	no	—
43	D₈×D₄₇	—	2	4	no	—
44	D₈×D₄₇	C₂×C₄	1	8	yes	complemented
45	C₂×#10625324	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
47	C₄×D₁₈₈	—	2	2	no	—
48	D₄×Dic₄₇	—	2	2	no	—
49	#13093986	—	2	2	no	—
50	C₄×C₂²⋊D₄₇	—	4	2	no	—
51	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
52	#13093982	—	2	2	no	—
53	#13093981	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
55	C₂²⋊C₄×D₄₇	—	4	2	no	—
56	D₇₅₂	C₂×C₄	1	4	yes	complemented
57	#13093965	C₂×C₄	1	8	yes	complemented
58	#13093963	—	2	2	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₄₇	C₂×C₄	1	4	yes	complemented
62	#13093958	C₂³	1	2	yes	—
63	#13093955	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	#13093945	—	2	2	no	—
66	#13093944	—	2	2	no	—
67	#13093935	D₄	1	1	yes	characteristic
68	#13093934	D₄	1	1	yes	characteristic
69	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₄₇	C₂³	1	2	yes	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	D₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	C₂³	1	2	yes	—
76	#102×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂³×D₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	4	2	no	—
81	C₂×C₂²⋊D₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	D₄×D₄₇	—	2	8	no	—
84	D₄×D₄₇	—	4	4	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
89	D₈×C₄₇	—	2	2	no	—
90	D₈×C₄₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₄₇	—	4	2	no	—
97	#10625334	—	2	8	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	—	4	2	no	—
100	(C₂×D₄₇)⋊C₄	—	4	2	no	—
101	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
102	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
103	C₄⋊Dic₄₇	—	2	2	no	—
104	Dic₄₇⋊C₄	D₈	1	1	yes	characteristic
105	Dic₄₇⋊C₄	D₈	1	1	yes	characteristic
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
110	D₃₇₆	—	2	2	no	—
111	D₃₇₆	C₂²×C₄	1	4	yes	—
112	#10625324	—	2	2	no	—
113	C₈×D₄₇	C₂²×C₄	1	2	yes	—
114	C₈×D₄₇	#41	1	2	yes	—
115	C₁₆×C₄₇	—	2	1	no	—
116	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
117	C₄₇⋊C₁₆	—	2	1	no	—
118	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂²×D₄₇	—	4	2	no	—
121	C₂²×D₄₇	—	4	8	no	—
122	C₂²×D₄₇	—	16	2	no	—
123	C₂²×D₄₇	—	16	2	no	—
124	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
125	D₄×C₄₇	—	2	4	no	—
126	D₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	16	2	no	—
129	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₄₇	—	4	8	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	16	2	no	—
135	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
138	D₁₈₈	—	2	4	no	—
139	D₁₈₈	—	4	2	no	—
140	C₄×D₄₇	—	16	2	no	—
141	C₄×D₄₇	—	16	2	no	—
142	C₄×D₄₇	C₄×D₄	1	2	yes	—
143	C₄×D₄₇	#135	1	2	yes	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₄₇⋊C₈	—	2	1	no	—
148	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
149	#33456	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	4	4	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	—	8	2	no	—
153	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₄₇	—	2	1	no	—
155	C₂×D₄₇	—	2	1	no	—
156	C₂×D₄₇	—	4	4	no	—
157	C₂×D₄₇	—	8	2	no	—
158	C₂×D₄₇	—	8	2	no	—
159	C₂×D₄₇	—	16	8	no	—
160	C₂×D₄₇	C₄×D₈	1	2	yes	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	8	2	no	—
164	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
166	Dic₄₇	—	2	1	no	—
167	Dic₄₇	—	2	1	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	C₄×D₈	1	2	yes	—
170	C₂²×D₁₆	—	47	1	no	—
171	#2948	—	47	2	no	—
172	#2177	—	47	2	no	—
173	#2177	—	47	2	no	—
174	C₄×D₁₆	—	47	1	no	—
175	C₄×D₁₆	—	47	1	no	—
176	C₄×D₁₆	—	47	1	no	—
177	C₄×D₁₆	D₄₇	1	1	yes	characteristic, complemented
178	#2158	—	47	1	no	—
179	#2140	—	47	2	no	—
180	#2108	—	47	1	no	—
181	C₂×C₄₇	—	8	4	no	—
182	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₄₇	#3415	1	1	yes	characteristic
184	C₂×C₄₇	#3415	1	1	yes	characteristic
185	D₄₇	—	2	2	no	—
186	D₄₇	—	8	4	no	—
187	C₂²×D₈	—	47	2	no	—
188	#518	—	94	2	no	—
189	C₂×D₁₆	—	47	2	no	—
190	C₂×D₁₆	—	47	8	no	—
191	C₂×D₁₆	—	47	1	no	—
192	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
193	C₂×M₃₂	—	47	1	no	—
194	C₂²×C₁₆	—	47	1	no	—
195	#442	—	94	2	no	—
196	#442	—	94	2	no	—
197	C₄×D₈	—	47	2	no	—
198	C₄×D₈	—	47	2	no	—
199	C₄×D₈	—	47	2	no	—
200	C₄×D₈	C₂×D₄₇	1	2	yes	complemented
201	#427	—	47	2	no	—
202	#418	—	94	2	no	—
203	#405	—	47	1	no	—
204	#367	—	47	2	no	—
205	#366	—	47	1	no	—
206	#366	C₂×D₄₇	1	1	yes	characteristic, complemented
207	#357	—	47	2	no	—
208	#357	—	47	2	no	—
209	#357	—	47	2	no	—
210	#357	C₂×D₄₇	1	2	yes	complemented
211	#346	—	47	1	no	—
212	#346	—	47	1	no	—
213	C₄×C₁₆	—	47	1	no	—
214	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
215	C₄₇	#33456	1	1	yes	characteristic, complemented, Sylow
216	C₂²×D₄	—	94	2	no	—
217	C₂×D₈	—	47	8	no	—
218	C₂×D₈	—	47	2	no	—
219	C₂×D₈	—	94	4	no	—
220	C₂×D₈	—	94	2	no	—
221	C₂×D₈	C₂²×D₄₇	1	2	yes	—
222	C₂×M₁₆	—	47	1	no	—
223	C₂²×C₈	—	47	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	94	2	no	—
226	C₄×D₄	—	94	2	no	—
227	C₄×D₄	—	94	2	no	—
228	C₄×D₄	—	188	2	no	—
229	C₄×D₄	—	188	2	no	—
230	#117	—	47	1	no	—
231	#117	—	94	2	no	—
232	C₂×C₂²⋊C₄	—	188	2	no	—
233	C₂×C₂²⋊C₄	—	188	2	no	—
234	D₁₆	—	47	8	no	—
235	D₁₆	—	47	4	no	—
236	D₁₆	C₄×D₄₇	1	4	yes	complemented
237	M₃₂	—	94	2	no	—
238	C₂×C₁₆	—	47	1	no	—
239	C₂×C₁₆	—	47	4	no	—
240	C₂×C₁₆	—	47	1	no	—
241	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
243	#107	—	47	2	no	—
244	#107	C₂²×D₄₇	1	2	yes	—
245	#106	—	94	2	no	—
246	#102	—	2	2	no	—
247	#102	—	94	2	no	—
248	#102	—	94	2	no	—
249	#102	—	94	2	no	—
250	#97	—	47	1	no	—
251	#97	—	47	1	no	—
252	C₄×C₈	—	47	1	no	—
253	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
254	C₂⁴	—	188	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	94	8	no	—
257	C₂×D₄	—	94	2	no	—
258	C₂×D₄	—	188	4	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	47	1	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	—	188	2	no	—
264	C₂²×C₄	—	188	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	47	4	no	—
267	D₈	—	94	8	no	—
268	D₈	—	94	2	no	—
269	D₈	C₂×C₄×D₄₇	1	4	yes	—
270	M₁₆	—	94	2	no	—
271	C₂×C₈	—	47	1	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	—	47	2	no	—
274	C₂×C₈	—	47	1	no	—
275	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	47	1	no	—
279	C₄⋊C₄	—	47	1	no	—
280	C₄⋊C₄	—	94	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	47	1	no	—
283	C₂²⋊C₄	—	47	1	no	—
284	C₂²⋊C₄	—	188	2	no	—
285	C₂²⋊C₄	—	188	2	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₂²⋊C₄	—	188	2	no	—
288	C₄²	—	47	1	no	—
289	C₄²	—	188	2	no	—
290	C₄²	D₄×D₄₇	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	47	2	no	—
293	C₁₆	—	94	1	no	—
294	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	47	1	no	—
297	C₂³	—	188	8	no	—
298	C₂³	—	188	2	no	—
299	C₂³	—	752	2	no	—
300	C₂³	—	752	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	94	4	no	—
304	D₄	—	188	8	no	—
305	D₄	—	188	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	47	1	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	47	1	no	—
312	C₂×C₄	—	47	1	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	752	2	no	—
315	C₂×C₄	—	752	2	no	—
316	C₂×C₄	—	752	2	no	—
317	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
318	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	47	2	no	—
322	C₈	—	94	1	no	—
323	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
324	C₈	#41×D₄₇	1	1	yes	characteristic
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	47	2	no	—
329	C₂²	—	94	1	no	—
330	C₂²	—	94	1	no	—
331	C₂²	—	376	2	no	—
332	C₂²	—	376	2	no	—
333	C₂²	—	188	4	no	—
334	C₂²	—	752	8	no	—
335	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	47	2	no	—
340	C₄	—	94	1	no	—
341	C₄	—	94	1	no	—
342	C₄	—	376	2	no	—
343	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
344	C₄	#135×D₄₇	1	1	yes	characteristic
345	C₂	—	8	4	no	—
346	C₂	—	94	2	no	—
347	C₂	—	376	4	no	—
348	C₂	C₄×D₈×D₄₇	1	1	yes	characteristic
349	C₂	#20256525	1	1	yes	characteristic
350	C₂	#20256526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄×D₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×D₁₆×D₄₇	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×D₁₆	D₄₇
hypocenter	C₄₇	#33456
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #20252235	C₂	semidirect
3. #20245262	C₂	semidirect
4. #20245261	C₂	semidirect
5. C₄×D₇₅₂	C₂	semidirect
6. D₁₆×Dic₄₇	C₂	semidirect
7. #20245234	C₂	semidirect
8. #20245169	C₂	semidirect
9. #20245102	C₂	semidirect
10. #20244983	C₂	semidirect
11. C₄×D₁₆×C₄₇	C₂	semidirect
16. D₁₆×D₄₇	C₄	semidirect
21. C₄×D₃₇₆	C₂²	semidirect
22. D₈×Dic₄₇	C₂²	semidirect
23. #15689205	C₂²	semidirect
27. #15688950	C₂²	semidirect
28. #15688949	C₂²	semidirect
29. #15688925	C₂²	semidirect
30. #15688924	C₂²	semidirect
31. #15688923	C₂²	semidirect
32. #15688899	C₂²	semidirect
33. #15688898	C₂²	semidirect
34. C₁₆×Dic₄₇	C₂²	semidirect
36. C₄×D₈×C₄₇	C₂²	semidirect
37. C₄₇×#366	C₂²	semidirect
38. C₄₇×#357	C₂²	semidirect
39. C₄×C₁₆×C₄₇	C₂²	semidirect
44. D₈×D₄₇	C₂×C₄	semidirect
56. D₇₅₂	C₂×C₄	semidirect
57. #13093965	C₂×C₄	semidirect
61. C₁₆×D₄₇	C₂×C₄	semidirect
72. D₁₆×C₄₇	C₂×C₄	semidirect
177. C₄×D₁₆	D₄₇	semidirect
200. C₄×D₈	C₂×D₄₇	semidirect
206. #366	C₂×D₄₇	semidirect
210. #357	C₂×D₄₇	semidirect
214. C₄×C₁₆	C₂×D₄₇	semidirect
215. C₄₇	#33456	semidirect
236. D₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₄₇	2
#20252235	2
#20245262	2
#20245261	2
C₄×D₇₅₂	2
D₁₆×Dic₄₇	2
#20245234	2
#20245169	2
#20245102	2
#20244983	2
C₄×D₁₆×C₄₇	2
#33456	47