Subgroups — #26369663

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689556	C₂	1	1	yes	maximal, characteristic
2	#20252263	C₂	1	2	yes	maximal, complemented
3	#20245255	C₂	1	2	yes	maximal, complemented
4	#20245250	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245234	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245099	C₂	1	2	yes	maximal, complemented
10	#20244983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094047	C₂²	1	2	yes	—
13	#15689605	—	2	2	no	—
14	#15689556	C₄	1	8	yes	complemented
15	C₂×#13093967	C₂²	1	2	yes	—
16	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
17	C₂×#13093963	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
19	#15689226	—	2	2	no	—
20	#15689221	—	2	2	no	—
21	Q₁₆×Dic₄₇	C₂²	1	2	yes	—
22	C₄×D₃₇₆	C₂²	1	2	yes	—
23	#15689205	C₂²	1	2	yes	—
24	#107×D₄₇	C₂²	1	2	yes	—
25	#15689123	—	2	2	no	—
26	#15689077	C₂²	1	1	yes	characteristic
27	#15688949	C₂²	1	1	yes	characteristic
28	#15688948	C₂²	1	2	yes	—
29	#15688927	C₂²	1	2	yes	—
30	#15688925	C₂²	1	2	yes	—
31	#15688923	C₂²	1	2	yes	—
32	#15688899	C₂²	1	1	yes	characteristic
33	#15688898	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#366	C₂²	1	1	yes	characteristic
38	C₄₇×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#10625354	—	2	2	no	—
41	#13094047	—	2	4	no	—
42	#13094047	C₂×C₄	1	8	yes	complemented
43	C₂×#10625336	—	2	2	no	—
44	C₂×D₃₇₆	C₂³	1	2	yes	—
45	C₂×#10625324	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
47	Q₈×Dic₄₇	—	2	2	no	—
48	C₄×D₁₈₈	—	2	2	no	—
49	C₄×D₁₈₈	—	4	2	no	—
50	#13093986	—	2	2	no	—
51	#13093986	—	4	2	no	—
52	#13093981	—	4	2	no	—
53	#13093981	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₄₇	—	2	2	no	—
55	C₄⋊C₄×D₄₇	—	4	2	no	—
56	#13093967	C₂×C₄	1	8	yes	complemented
57	D₇₅₂	C₂×C₄	1	4	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	#13093957	—	2	2	no	—
64	#13093948	—	2	2	no	—
65	#13093946	—	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×Q₁₆×C₄₇	C₂³	1	2	yes	—
71	C₄×Q₈×C₄₇	—	2	2	no	—
72	Q₃₂×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	#103×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₄₇	—	2	2	no	—
79	#10625354	—	2	8	no	—
80	#10625354	—	4	4	no	—
81	C₂×D₁₈₈	—	2	2	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×C₄×D₄₇	—	4	2	no	—
84	C₂×C₄×D₄₇	—	4	2	no	—
85	C₂×C₄×D₄₇	—	4	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₄₇	—	2	2	no	—
89	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	4	2	no	—
94	C₄²×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	#10625336	—	2	8	no	—
97	(C₂×D₄₇)⋊C₄	—	4	2	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
100	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	4	2	no	—
103	Dic₄₇⋊C₄	—	4	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	C₂²×D₄₇	—	4	2	no	—
120	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₄₇	—	2	4	no	—
122	Q₈×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
132	D₁₈₈	—	2	4	no	—
133	D₁₈₈	—	4	8	no	—
134	D₁₈₈	—	4	2	no	—
135	C₄×D₄₇	—	4	8	no	—
136	C₄×D₄₇	—	16	2	no	—
137	C₄×D₄₇	—	16	2	no	—
138	C₄×D₄₇	—	16	2	no	—
139	C₄×D₄₇	C₄×D₄	1	2	yes	—
140	C₄×D₄₇	#135	1	2	yes	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₄₇⋊C₈	—	2	1	no	—
145	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
146	#33459	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₄₇	—	2	1	no	—
149	C₂×D₄₇	—	2	1	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₂×D₄₇	—	8	2	no	—
152	C₂×D₄₇	—	8	2	no	—
153	C₂×D₄₇	C₄×D₈	1	2	yes	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
160	Dic₄₇	—	2	1	no	—
161	Dic₄₇	—	2	1	no	—
162	Dic₄₇	—	8	2	no	—
163	Dic₄₇	C₄×D₈	1	2	yes	—
164	C₂×#508	—	47	1	no	—
165	#2951	—	47	2	no	—
166	#2175	—	47	2	no	—
167	#2175	—	47	2	no	—
168	C₄×Q₃₂	—	47	1	no	—
169	C₄×Q₃₂	D₄₇	1	1	yes	characteristic, complemented
170	C₄×D₁₆	—	47	1	no	—
171	C₄×D₁₆	—	47	1	no	—
172	C₂×#366	—	47	1	no	—
173	#2139	—	47	2	no	—
174	#2108	—	47	1	no	—
175	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
176	C₂×C₄₇	#3415	1	1	yes	characteristic
177	C₂×C₄₇	#3417	1	1	yes	characteristic
178	D₄₇	—	2	2	no	—
179	D₄₇	—	8	4	no	—
180	C₂×#135	—	47	2	no	—
181	#520	—	94	2	no	—
182	#508	—	47	8	no	—
183	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
184	C₂×SD₃₂	—	47	2	no	—
185	C₂×D₁₆	—	47	1	no	—
186	C₂×M₃₂	—	47	1	no	—
187	C₂²×C₁₆	—	47	1	no	—
188	#440	—	94	2	no	—
189	#440	—	94	2	no	—
190	C₄×Q₁₆	—	47	2	no	—
191	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
192	C₄×D₈	—	47	2	no	—
193	C₄×D₈	—	47	2	no	—
194	C₂×#107	—	47	2	no	—
195	#417	—	94	2	no	—
196	#405	—	47	1	no	—
197	#366	—	47	1	no	—
198	#366	—	47	2	no	—
199	#366	C₂×D₄₇	1	1	yes	characteristic
200	#358	—	47	2	no	—
201	#358	C₂×D₄₇	1	2	yes	—
202	#357	—	47	2	no	—
203	#357	—	47	2	no	—
204	#346	—	47	1	no	—
205	#346	—	47	1	no	—
206	C₄×C₁₆	—	47	1	no	—
207	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
208	C₄₇	#33459	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	94	2	no	—
210	#135	—	47	8	no	—
211	#135	—	94	4	no	—
212	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
213	C₂×SD₁₆	—	94	2	no	—
214	C₂×D₈	—	47	2	no	—
215	C₂×M₁₆	—	47	1	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	94	2	no	—
219	C₄×D₄	—	94	2	no	—
220	C₄×D₄	—	94	2	no	—
221	C₄×D₄	—	188	2	no	—
222	C₄×D₄	—	188	2	no	—
223	#117	—	47	1	no	—
224	#117	—	188	2	no	—
225	C₂×C₄⋊C₄	—	94	2	no	—
226	C₂×C₄⋊C₄	—	188	2	no	—
227	Q₃₂	C₄×D₄₇	1	4	yes	complemented
228	SD₃₂	—	47	8	no	—
229	D₁₆	—	47	4	no	—
230	M₃₂	—	94	2	no	—
231	C₂×C₁₆	—	47	1	no	—
232	C₂×C₁₆	—	47	4	no	—
233	C₂×C₁₆	—	47	1	no	—
234	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
236	#107	—	47	2	no	—
237	#107	—	94	2	no	—
238	#107	C₂²×D₄₇	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	—	94	2	no	—
241	#102	—	94	2	no	—
242	#102	—	94	2	no	—
243	#97	—	47	1	no	—
244	#97	—	47	1	no	—
245	C₄×C₈	—	47	1	no	—
246	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	#41	—	94	8	no	—
248	#41	—	188	4	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	94	2	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	C₂×C₄×D₄₇	1	4	yes	—
259	SD₁₆	—	94	8	no	—
260	D₈	—	47	4	no	—
261	D₈	—	94	2	no	—
262	M₁₆	—	94	2	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	1	no	—
267	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	47	1	no	—
272	C₄⋊C₄	—	47	1	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₄⋊C₄	—	188	2	no	—
276	C₂²⋊C₄	—	47	1	no	—
277	C₂²⋊C₄	—	47	1	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₂²⋊C₄	—	188	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	188	2	no	—
283	C₄²	D₄×D₄₇	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	47	2	no	—
286	C₁₆	—	94	1	no	—
287	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
288	C₂³	—	47	1	no	—
289	C₂³	—	188	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	94	4	no	—
293	D₄	—	188	8	no	—
294	D₄	—	188	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	47	1	no	—
298	C₂×C₄	—	47	2	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	1	no	—
301	C₂×C₄	—	47	1	no	—
302	C₂×C₄	—	188	8	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	752	2	no	—
305	C₂×C₄	—	752	2	no	—
306	C₂×C₄	—	752	2	no	—
307	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
308	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	47	2	no	—
312	C₈	—	94	1	no	—
313	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₈	#41×D₄₇	1	1	yes	characteristic
315	C₂²	—	47	2	no	—
316	C₂²	—	94	1	no	—
317	C₂²	—	94	1	no	—
318	C₂²	—	376	2	no	—
319	C₂²	—	376	2	no	—
320	C₂²	—	188	4	no	—
321	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	47	2	no	—
327	C₄	—	94	1	no	—
328	C₄	—	94	1	no	—
329	C₄	—	376	2	no	—
330	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
331	C₄	#135×D₄₇	1	1	yes	characteristic
332	C₂	—	94	2	no	—
333	C₂	—	376	4	no	—
334	C₂	C₄×D₈×D₄₇	1	1	yes	characteristic
335	C₂	#20256526	1	1	yes	characteristic
336	C₂	#20256537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄×Q₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689556	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×Q₃₂	D₄₇
hypocenter	C₄₇	#33459
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20252263	C₂	semidirect
3. #20245255	C₂	semidirect
4. #20245250	C₂	semidirect
5. Q₃₂×Dic₄₇	C₂	semidirect
6. C₄×D₇₅₂	C₂	semidirect
7. #20245234	C₂	semidirect
8. #366×D₄₇	C₂	semidirect
9. #20245099	C₂	semidirect
10. #20244983	C₂	semidirect
11. C₄×Q₃₂×C₄₇	C₂	semidirect
14. #15689556	C₄	semidirect
42. #13094047	C₂×C₄	semidirect
56. #13093967	C₂×C₄	semidirect
57. D₇₅₂	C₂×C₄	semidirect
61. C₁₆×D₄₇	C₂×C₄	semidirect
72. Q₃₂×C₄₇	C₂×C₄	semidirect
169. C₄×Q₃₂	D₄₇	semidirect
208. C₄₇	#33459	semidirect
227. Q₃₂	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×#15689556	2
#20252263	2
#20245255	2
#20245250	2
Q₃₂×Dic₄₇	2
C₄×D₇₅₂	2
#20245234	2
#366×D₄₇	2
#20245099	2
#20244983	2
C₄×Q₃₂×C₄₇	2
#33459	47