Subgroups — #20251917

Order 6016.

726 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094071	C₂	1	1	yes	maximal, characteristic
2	C₂×#13094030	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689603	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689131	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689130	C₂	1	2	yes	maximal, complemented
6	#15689124	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689121	C₂	1	2	yes	maximal, complemented
8	#15689116	C₂	1	2	yes	maximal, complemented
9	C₂×#13093949	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₄₇	C₂	1	2	yes	maximal, complemented
11	C₄₇×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	#13094071	—	2	4	no	—
13	#13094071	C₄	1	8	yes	complemented
14	C₂×#10625350	—	2	1	no	—
15	C₂×#10625350	C₂²	1	1	yes	characteristic
16	C₂×#10625352	—	2	2	no	—
17	C₂×#10625352	C₂²	1	2	yes	—
18	C₂²×Dic₉₄	—	2	1	no	—
19	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₄₇	C₂²	1	2	yes	—
21	#13094030	—	2	4	no	—
22	C₂×#10625329	C₂²	1	1	yes	characteristic
23	C₂×#10625324	C₂²	1	2	yes	—
24	C₂×C₈×D₄₇	C₂²	1	2	yes	—
25	C₄×Dic₉₄	—	2	1	no	—
26	C₄×D₁₈₈	—	2	1	no	—
27	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
28	#13093982	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₄₇	C₂²	1	2	yes	—
31	#13093949	C₂²	1	2	yes	—
32	#13093949	C₂²	1	2	yes	—
33	#13093948	C₂²	1	2	yes	—
34	#13093947	C₂²	1	2	yes	—
35	#13093947	C₂²	1	2	yes	—
36	#13093945	C₂²	1	2	yes	—
37	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	#117×C₄₇	C₂²	1	1	yes	characteristic
40	#103×C₄₇	C₂²	1	1	yes	characteristic
41	#103×C₄₇	C₂²	1	1	yes	characteristic
42	#102×C₄₇	C₂²	1	1	yes	characteristic
43	#102×C₄₇	C₂²	1	1	yes	characteristic
44	#41×C₄₇	—	2	2	no	—
45	#41×C₄₇	C₂×C₄	1	4	yes	complemented
46	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₄₇	—	2	1	no	—
48	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₇	—	2	2	no	—
52	C₂×C₂²⋊D₄₇	D₄	1	2	yes	—
53	C₂²×Dic₄₇	—	2	1	no	—
54	C₂²×Dic₄₇	—	2	2	no	—
55	C₂²×Dic₄₇	—	2	1	no	—
56	Q₈×D₄₇	—	2	4	no	—
57	Q₈×D₄₇	C₂×C₄	1	8	yes	complemented
58	#10625352	—	2	16	no	—
59	#10625352	—	2	4	no	—
60	#10625352	C₂×C₄	1	8	yes	complemented
61	#10625350	—	2	8	no	—
62	#10625350	C₂×C₄	1	4	yes	—
63	#10625350	D₄	1	4	yes	—
64	C₂×D₁₈₈	D₄	1	1	yes	characteristic
65	C₂×C₄×D₄₇	—	2	2	no	—
66	C₂×C₄×D₄₇	—	2	2	no	—
67	C₂×C₄×D₄₇	C₂³	1	2	yes	—
68	C₂×Dic₉₄	—	2	1	no	—
69	C₂×Dic₉₄	—	2	2	no	—
70	C₂×Dic₉₄	—	2	2	no	—
71	C₂×Dic₉₄	—	2	8	no	—
72	C₂×Dic₉₄	C₂×C₄	1	4	yes	complemented
73	C₂×Dic₉₄	D₄	1	1	yes	characteristic
74	C₂×Dic₉₄	C₂³	1	2	yes	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₄₇	—	2	1	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	(C₂×D₄₇)⋊C₄	—	2	2	no	—
83	C₄⋊Dic₄₇	—	2	1	no	—
84	C₄⋊Dic₄₇	C₂³	1	2	yes	—
85	Dic₄₇⋊C₄	—	2	2	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	#10625329	—	2	2	no	—
88	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
89	#10625324	—	2	4	no	—
90	C₈×D₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×D₄₇	C₂×D₄	1	2	yes	—
94	Q₈×C₄₇	—	2	1	no	—
95	Q₈×C₄₇	C₂²×C₄	1	2	yes	—
96	D₄×C₄₇	—	2	4	no	—
97	D₄×C₄₇	—	2	1	no	—
98	D₄×C₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₄₇	—	2	8	no	—
108	C₂²⋊D₄₇	C₂²⋊C₄	1	8	yes	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	8	1	no	—
114	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
115	D₁₈₈	C₂²⋊C₄	1	4	yes	—
116	C₄×D₄₇	—	2	8	no	—
117	C₄×D₄₇	—	8	2	no	—
118	C₄×D₄₇	C₂²×C₄	1	4	yes	—
119	C₄×D₄₇	C₂×D₄	1	4	yes	—
120	Dic₉₄	—	2	4	no	—
121	Dic₉₄	—	2	2	no	—
122	Dic₉₄	—	2	8	no	—
123	Dic₉₄	C₂²⋊C₄	1	4	yes	—
124	Dic₉₄	C₂²×C₄	1	4	yes	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	—	2	1	no	—
127	C₄₇⋊C₈	—	2	2	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₄₇	—	2	2	no	—
136	C₂×D₄₇	—	2	2	no	—
137	C₂×D₄₇	C₂×C₂²⋊C₄	1	4	yes	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	4	1	no	—
140	C₄×C₄₇	—	4	1	no	—
141	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
142	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	4	2	no	—
146	Dic₄₇	C₂×C₂²⋊C₄	1	4	yes	—
147	#2898	—	47	1	no	maximal, Sylow
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	4	2	no	—
150	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄₇	#577	1	1	yes	characteristic
152	C₂×C₄₇	#578	1	1	yes	characteristic
153	D₄₇	—	2	4	no	—
154	C₂×2₋¹⁺⁴	—	47	1	no	—
155	C₂×#131	—	47	1	no	—
156	#520	—	47	1	no	—
157	#419	—	47	1	no	—
158	#419	—	47	2	no	—
159	#417	—	47	1	no	—
160	#417	—	47	2	no	—
161	#416	—	47	2	no	—
162	#416	D₄₇	1	1	yes	characteristic, complemented
163	C₂×#103	—	47	1	no	—
164	C₂×#103	—	47	2	no	—
165	C₄₇	#2898	1	1	yes	characteristic, complemented, Sylow
166	2₋¹⁺⁴	—	47	8	no	—
167	2₋¹⁺⁴	—	94	4	no	—
168	C₂×#41	—	47	1	no	—
169	C₂×#41	—	47	2	no	—
170	C₂×#41	—	94	1	no	—
171	C₂×#41	—	94	2	no	—
172	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×Q₈	—	47	2	no	—
174	C₂²×Q₈	—	47	1	no	—
175	C₂²×Q₈	—	94	1	no	—
176	#131	—	94	4	no	—
177	C₂×M₁₆	—	47	1	no	—
178	C₂×M₁₆	—	47	2	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₄×Q₈	—	94	1	no	—
182	C₄×D₄	—	94	1	no	—
183	C₄×D₄	—	94	2	no	—
184	#117	—	47	2	no	—
185	#117	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	47	1	no	—
187	C₂×C₄⋊C₄	—	47	2	no	—
188	#103	—	47	2	no	—
189	#103	—	47	2	no	—
190	#103	—	47	2	no	—
191	#103	—	47	2	no	—
192	#103	—	47	2	no	—
193	#103	C₂×D₄₇	1	1	yes	characteristic
194	#103	C₂×D₄₇	1	1	yes	characteristic
195	#102	—	47	2	no	—
196	#102	C₂×D₄₇	1	1	yes	characteristic
197	#102	C₂×D₄₇	1	1	yes	characteristic
198	#41	—	2	2	no	—
199	#41	—	47	4	no	—
200	#41	—	47	4	no	—
201	#41	—	47	8	no	—
202	#41	—	94	8	no	—
203	#41	—	94	4	no	—
204	#41	—	94	16	no	—
205	#41	C₄×D₄₇	1	4	yes	complemented
206	C₂×Q₈	—	47	1	no	—
207	C₂×Q₈	—	47	8	no	—
208	C₂×Q₈	—	47	4	no	—
209	C₂×Q₈	—	47	2	no	—
210	C₂×Q₈	—	94	2	no	—
211	C₂×Q₈	—	94	8	no	—
212	C₂×Q₈	—	94	4	no	—
213	C₂×Q₈	—	94	1	no	—
214	C₂×Q₈	—	94	2	no	—
215	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	47	2	no	—
218	C₂×D₄	—	47	1	no	—
219	C₂×D₄	—	94	2	no	—
220	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	47	2	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	1	no	—
227	C₂²×C₄	—	94	1	no	—
228	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
229	M₁₆	—	94	4	no	—
230	M₁₆	—	94	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	47	2	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊C₄	—	47	2	no	—
237	C₄⋊C₄	—	94	2	no	—
238	C₄⋊C₄	—	94	1	no	—
239	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
240	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	94	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	94	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	47	2	no	—
247	C₂³	D₄×D₄₇	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	—	47	4	no	—
250	Q₈	—	47	4	no	—
251	Q₈	—	94	8	no	—
252	Q₈	—	94	4	no	—
253	Q₈	—	94	2	no	—
254	Q₈	C₂×C₄×D₄₇	1	2	yes	—
255	D₄	—	2	4	no	—
256	D₄	—	2	1	no	—
257	D₄	—	47	8	no	—
258	D₄	—	47	4	no	—
259	D₄	—	94	8	no	—
260	D₄	C₂×C₄×D₄₇	1	2	yes	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	47	4	no	—
267	C₂×C₄	—	47	4	no	—
268	C₂×C₄	—	94	8	no	—
269	C₂×C₄	—	94	8	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	376	2	no	—
274	C₂×C₄	—	376	1	no	—
275	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
276	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	D₄×D₄₇	1	2	yes	—
278	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	94	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	47	4	no	—
288	C₂²	—	94	2	no	—
289	C₂²	—	94	2	no	—
290	C₂²	C₂²⋊C₄×D₄₇	1	2	yes	—
291	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	47	4	no	—
296	C₄	—	94	4	no	—
297	C₄	—	188	2	no	—
298	C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
299	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	—	94	4	no	—
304	C₂	C₂×C₂²⋊C₄×D₄₇	1	1	yes	characteristic
305	C₂	#15689930	1	1	yes	characteristic
306	C₂	#15689937	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#416	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#416	D₄₇
hypocenter	C₄₇	#2898
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. C₂×#13094030	C₂	semidirect
3. #15689603	C₂	semidirect
4. #15689131	C₂	semidirect
5. #15689130	C₂	semidirect
6. #15689124	C₂	semidirect
7. #15689121	C₂	semidirect
8. #15689116	C₂	semidirect
9. C₂×#13093949	C₂	semidirect
10. #103×D₄₇	C₂	semidirect
11. C₄₇×#416	C₂	semidirect
13. #13094071	C₄	semidirect
45. #41×C₄₇	C₂×C₄	semidirect
57. Q₈×D₄₇	C₂×C₄	semidirect
60. #10625352	C₂×C₄	semidirect
72. C₂×Dic₉₄	C₂×C₄	semidirect
162. #416	D₄₇	semidirect
165. C₄₇	#2898	semidirect
205. #41	C₄×D₄₇	semidirect