Subgroups — #26369647

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093967	C₂	1	1	yes	maximal, characteristic
2	#20252241	C₂	1	1	yes	maximal, characteristic
3	#20245250	C₂	1	4	yes	maximal, complemented
4	C₄×#13093967	C₂	1	4	yes	maximal, complemented
5	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245108	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245103	C₂	1	1	yes	maximal, characteristic
8	#20244985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689597	—	2	1	no	—
12	C₂×#13093967	C₄	1	8	yes	complemented
13	C₂×#13093967	C₂²	1	2	yes	—
14	C₂×#13093967	C₂²	1	2	yes	—
15	C₂×#13093964	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#15689236	—	2	2	no	—
18	C₄×D₃₇₆	C₂²	1	4	yes	—
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689129	—	2	1	no	—
21	#15688950	C₂²	1	2	yes	complemented
22	#15688948	C₂²	1	2	yes	complemented
23	#15688927	C₂²	1	4	yes	complemented
24	#15688923	C₂²	1	4	yes	—
25	#15688899	C₂²	1	2	yes	complemented
26	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
27	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#519	—	2	1	no	—
29	C₄₇×#441	—	2	2	no	—
30	C₄×Q₁₆×C₄₇	C₂²	1	4	yes	complemented
31	C₄₇×#427	C₂²	1	1	yes	characteristic
32	C₄₇×#419	—	2	1	no	—
33	C₄₇×#405	C₂²	1	1	yes	characteristic
34	C₂²×D₁₈₈	—	2	1	no	—
35	C₂×D₃₇₆	—	2	1	no	—
36	C₂×D₃₇₆	—	2	2	no	—
37	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
38	C₂×D₃₇₆	C₂³	1	2	yes	—
39	C₄×D₁₈₈	—	2	4	no	—
40	#13093986	—	4	2	no	—
41	#13093983	—	2	1	no	—
42	C₂²⋊C₄×D₄₇	—	4	2	no	—
43	#13093967	C₂×C₄	1	8	yes	complemented
44	#13093967	C₂×C₄	1	8	yes	complemented
45	#13093964	—	2	2	no	—
46	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
50	#13093958	C₂³	1	2	yes	—
51	#13093956	—	2	1	no	—
52	#13093946	—	2	4	no	—
53	C₂²×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	1	no	—
55	C₂×Q₁₆×C₄₇	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₄₇	—	2	4	no	—
61	C₄×Q₈×C₄₇	—	4	2	no	—
62	#117×C₄₇	—	2	1	no	—
63	#117×C₄₇	—	4	2	no	—
64	#117×C₄₇	D₄	1	1	yes	characteristic
65	#107×C₄₇	C₂³	1	2	yes	—
66	#106×C₄₇	—	2	1	no	—
67	#103×C₄₇	—	2	4	no	—
68	#97×C₄₇	D₄	1	2	yes	—
69	C₄×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂³×D₄₇	—	4	1	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	4	1	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×D₁₈₈	—	2	4	no	—
78	C₂×D₁₈₈	—	2	2	no	—
79	C₂×D₁₈₈	—	4	1	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×C₄×D₄₇	—	4	4	no	—
82	Q₁₆×C₄₇	—	2	4	no	—
83	Q₁₆×C₄₇	—	2	2	no	—
84	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
85	M₁₆×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	#41	1	2	yes	—
90	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
94	C₂²⋊C₄×C₄₇	—	4	1	no	—
95	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
96	C₄²×C₄₇	—	4	1	no	—
97	C₄²×C₄₇	—	4	4	no	—
98	C₄²×C₄₇	C₂×D₄	1	2	yes	—
99	C₂²⋊Dic₄₇	—	4	1	no	—
100	(C₂×D₄₇)⋊C₄	—	4	4	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	4	1	no	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	—	2	2	no	—
105	D₃₇₆	C₂²×C₄	1	4	yes	—
106	C₄₇⋊C₁₆	—	2	2	no	—
107	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₄₇	—	4	2	no	—
111	C₂²×D₄₇	—	4	4	no	—
112	C₂²×D₄₇	—	16	2	no	—
113	Q₈×C₄₇	—	2	4	no	—
114	Q₈×C₄₇	—	4	2	no	—
115	Q₈×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#135	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	2	no	—
125	D₁₈₈	—	2	4	no	—
126	D₁₈₈	—	4	2	no	—
127	D₁₈₈	—	4	4	no	—
128	C₄×D₄₇	—	16	4	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	C₄×D₄	1	2	yes	—
131	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
133	#33457	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	C₄×D₈	1	2	yes	—
136	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	C₂×D₄₇	—	4	4	no	—
138	C₂×D₄₇	—	8	4	no	—
139	C₂×D₄₇	—	16	4	no	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	—	4	4	no	—
142	C₄×C₄₇	—	8	2	no	—
143	C₄×C₄₇	C₄×D₈	1	2	yes	—
144	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
146	Dic₄₇	—	8	2	no	—
147	C₂²×SD₃₂	—	47	1	no	—
148	#2950	D₄₇	1	1	yes	characteristic, complemented
149	#2948	—	47	1	no	—
150	#2175	—	47	4	no	—
151	C₄×SD₃₂	—	47	4	no	—
152	#2158	—	47	1	no	—
153	#2141	—	47	1	no	—
154	#2140	—	47	1	no	—
155	#2108	—	47	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₄₇	#3416	1	2	yes	—
159	D₄₇	—	8	4	no	—
160	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
161	C₂²×D₈	—	47	1	no	—
162	#519	—	2	1	no	—
163	#518	—	94	1	no	—
164	C₂×SD₃₂	—	47	8	no	—
165	C₂×SD₃₂	—	47	2	no	—
166	C₂×SD₃₂	—	47	2	no	—
167	C₂×M₃₂	—	47	1	no	—
168	C₂²×C₁₆	—	47	1	no	—
169	#442	—	94	2	no	—
170	#441	—	2	2	no	—
171	C₄×Q₁₆	C₂×D₄₇	1	4	yes	complemented
172	C₄×D₈	—	47	4	no	—
173	#427	—	47	1	no	—
174	#427	C₂×D₄₇	1	1	yes	characteristic
175	#419	—	2	1	no	—
176	#418	—	94	1	no	—
177	#405	C₂×D₄₇	1	1	yes	characteristic
178	#367	—	47	2	no	—
179	#366	—	47	2	no	—
180	#358	—	47	4	no	—
181	#357	—	47	4	no	—
182	#346	—	47	2	no	—
183	C₄×C₁₆	—	47	2	no	—
184	C₄₇	#33457	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	94	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₄×D₄₇	1	4	yes	complemented
190	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
191	C₂×D₈	—	47	4	no	—
192	C₂×D₈	—	47	2	no	—
193	C₂×D₈	—	94	2	no	—
194	C₂×D₈	—	94	1	no	—
195	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	4	2	no	—
199	C₄×D₄	—	94	4	no	—
200	C₄×D₄	—	188	2	no	—
201	#117	—	2	1	no	—
202	#117	—	4	2	no	—
203	#117	—	94	1	no	—
204	#117	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	188	2	no	—
206	SD₃₂	—	47	8	no	—
207	SD₃₂	—	47	8	no	—
208	M₃₂	—	94	2	no	—
209	C₂×C₁₆	—	47	4	no	—
210	C₂×C₁₆	—	47	1	no	—
211	C₂×C₁₆	—	47	1	no	—
212	C₂×C₁₆	—	47	2	no	—
213	#107	—	47	2	no	—
214	#107	C₂²×D₄₇	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	94	1	no	—
217	#103	—	2	4	no	—
218	#102	—	94	4	no	—
219	#97	C₂²⋊D₄₇	1	2	yes	—
220	C₄×C₈	C₂²×D₄₇	1	2	yes	—
221	C₂⁴	—	188	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	94	4	no	—
227	C₂×D₄	—	94	2	no	—
228	C₂×D₄	—	188	1	no	—
229	C₂×D₄	—	188	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂×C₄×D₄₇	1	4	yes	—
236	D₈	—	47	4	no	—
237	D₈	—	94	4	no	—
238	D₈	—	94	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
241	C₂×C₈	#10625350	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	#10625334	1	2	yes	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	188	1	no	—
251	C₂²⋊C₄	—	188	4	no	—
252	C₂²⋊C₄	#10625334	1	2	yes	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	4	no	—
255	C₄²	—	188	1	no	—
256	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
257	C₁₆	—	47	2	no	—
258	C₁₆	—	47	2	no	—
259	C₁₆	—	94	2	no	—
260	C₂³	—	188	2	no	—
261	C₂³	—	188	4	no	—
262	C₂³	—	752	2	no	—
263	C₂³	C₂×#10625334	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	94	4	no	—
268	D₄	—	188	2	no	—
269	D₄	—	188	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	752	4	no	—
275	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
276	C₂×C₄	C₂×#10625334	1	4	yes	—
277	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
278	C₂×C₄	#13094044	1	2	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
282	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₈	C₂×#10625350	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	376	4	no	—
286	C₂²	—	188	4	no	—
287	C₂²	—	752	4	no	—
288	C₂²	C₄×#10625334	1	2	yes	—
289	C₂²	C₂²×#10625334	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	2	no	—
293	C₄	—	376	2	no	—
294	C₄	C₄×#10625334	1	2	yes	—
295	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₄	C₂×#13094044	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	376	4	no	—
299	C₂	C₂×C₄×#10625334	1	1	yes	characteristic
300	C₂	#20256532	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2950	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2950	D₄₇
hypocenter	C₄₇	#33457
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #20245250	C₂	semidirect
4. C₄×#13093967	C₂	semidirect
5. #20245170	C₂	semidirect
6. #20245108	C₂	semidirect
8. #20244985	C₂	semidirect
9. C₄₇×#2950	C₂	semidirect
12. C₂×#13093967	C₄	semidirect
21. #15688950	C₂²	semidirect
22. #15688948	C₂²	semidirect
23. #15688927	C₂²	semidirect
25. #15688899	C₂²	semidirect
26. C₄×C₄₇⋊C₁₆	C₂²	semidirect
30. C₄×Q₁₆×C₄₇	C₂²	semidirect
43. #13093967	C₂×C₄	semidirect
44. #13093967	C₂×C₄	semidirect
46. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
56. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
148. #2950	D₄₇	semidirect
171. C₄×Q₁₆	C₂×D₄₇	semidirect
184. C₄₇	#33457	semidirect
189. C₂×Q₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093967	2
#20252241	2
#20245250	2
C₄×#13093967	2
#20245170	2
#20245108	2
#20245103	2
#20244985	2
C₄₇×#2950	2
#33457	47