Subgroups — #20252203

Order 6016.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094055	C₂	1	1	yes	maximal, characteristic
2	#15689591	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689227	C₂	1	2	yes	maximal, complemented
5	#15689226	C₂	1	2	yes	maximal, complemented
6	#15689215	C₂	1	2	yes	maximal, complemented
7	#15689208	C₂	1	2	yes	maximal, complemented
8	#15689166	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689125	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#419	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	1	yes	characteristic
13	#13094055	C₄	1	8	yes	complemented
14	C₂×#10625337	C₂²	1	2	yes	—
15	C₂×#10625336	C₂²	1	2	yes	—
16	C₂×#10625329	C₂²	1	1	yes	characteristic
17	Q₈×Dic₄₇	—	2	1	no	—
18	Q₈×Dic₄₇	—	2	1	no	—
19	Q₈×Dic₄₇	C₂²	1	2	yes	—
20	Q₈×Dic₄₇	C₂²	1	2	yes	—
21	#13093989	C₂²	1	1	yes	characteristic
22	#13093989	C₂²	1	1	yes	characteristic
23	#13093986	C₂²	1	1	yes	characteristic
24	#13093986	C₂²	1	1	yes	characteristic
25	(C₂²⋊D₄₇)⋊C₄	—	2	1	no	—
26	(C₂²⋊D₄₇)⋊C₄	—	2	1	no	—
27	#13093981	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₄²×D₄₇	—	2	1	no	—
33	#13093957	C₂²	1	2	yes	—
34	#13093955	C₂²	1	2	yes	—
35	#13093949	C₂²	1	2	yes	—
36	#13093946	C₂²	1	2	yes	—
37	#13093934	C₂²	1	2	yes	—
38	C₈×Dic₄₇	C₂²	1	2	yes	—
39	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	#117×C₄₇	C₂²	1	1	yes	characteristic
42	#103×C₄₇	C₂²	1	2	yes	—
43	#103×C₄₇	C₂²	1	2	yes	—
44	C₂×Q₈×C₄₇	—	2	1	no	—
45	C₂×Q₈×C₄₇	—	2	2	no	—
46	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₄₇	—	2	1	no	—
51	C₂²×Dic₄₇	—	2	1	no	—
52	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
54	#10625350	—	2	2	no	—
55	#10625350	C₂×C₄	1	4	yes	—
56	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₄₇	—	2	1	no	—
58	C₂×C₄×D₄₇	—	2	1	no	—
59	C₂×C₄×D₄₇	—	2	1	no	—
60	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
61	M₁₆×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₄₇	—	2	1	no	—
66	C₄²×C₄₇	—	2	1	no	—
67	#10625337	C₂×C₄	1	8	yes	complemented
68	#10625336	C₂×C₄	1	8	yes	complemented
69	(C₂×D₄₇)⋊C₄	—	2	1	no	—
70	(C₂×D₄₇)⋊C₄	—	2	1	no	—
71	C₄⋊Dic₄₇	—	2	1	no	—
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	—	2	2	no	—
74	C₄⋊Dic₄₇	C₂³	1	2	yes	—
75	Dic₄₇⋊C₄	—	2	1	no	—
76	Dic₄₇⋊C₄	—	2	1	no	—
77	C₄×Dic₄₇	—	2	1	no	—
78	C₄×Dic₄₇	—	2	1	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	D₄	1	1	yes	characteristic
82	C₄×Dic₄₇	D₄	1	1	yes	characteristic
83	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
85	#10625329	C₂×C₄	1	4	yes	complemented
86	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
87	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₄₇	—	2	1	no	—
89	Q₈×C₄₇	—	2	4	no	—
90	Q₈×C₄₇	—	2	2	no	—
91	Q₈×C₄₇	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#41	1	2	yes	—
99	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₄₇	—	2	4	no	—
101	C₂×Dic₄₇	—	2	1	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	4	1	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
110	D₁₈₈	—	2	1	no	—
111	D₁₈₈	C₂²×C₄	1	2	yes	—
112	C₄×D₄₇	—	2	4	no	—
113	C₄×D₄₇	—	4	2	no	—
114	C₄×D₄₇	—	4	2	no	—
115	Dic₉₄	—	2	1	no	—
116	Dic₉₄	C₂²×C₄	1	2	yes	—
117	C₈×C₄₇	—	2	2	no	—
118	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	C₄×D₄	1	2	yes	—
122	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₄₇	—	2	2	no	—
124	C₂×D₄₇	—	4	1	no	—
125	C₂×D₄₇	—	4	1	no	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	4	1	no	—
128	C₄×C₄₇	—	4	1	no	—
129	C₄×C₄₇	C₄×D₄	1	2	yes	—
130	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	—	2	1	no	—
134	Dic₄₇	—	2	1	no	—
135	Dic₄₇	—	2	1	no	—
136	Dic₄₇	—	4	2	no	—
137	Dic₄₇	#137	1	2	yes	complemented
138	C₄×#137	—	47	1	no	maximal, Sylow
139	C₂×C₄₇	—	2	2	no	—
140	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
142	C₂×C₄₇	#575	1	1	yes	characteristic
143	D₄₇	—	4	2	no	—
144	C₂×#137	—	47	1	no	—
145	C₄×#41	—	47	1	no	—
146	C₂×C₄×Q₈	—	47	1	no	—
147	#441	—	47	2	no	—
148	#440	—	47	2	no	—
149	C₄×Q₁₆	—	47	2	no	—
150	C₄×SD₁₆	—	47	2	no	—
151	#428	—	47	1	no	—
152	#419	D₄₇	1	1	yes	characteristic, complemented
153	#417	—	47	1	no	—
154	C₄×M₁₆	—	47	1	no	—
155	C₄₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	47	1	no	—
157	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
158	#137	—	47	8	no	—
159	C₂×Q₁₆	—	47	2	no	—
160	C₂×SD₁₆	—	47	2	no	—
161	C₂×M₁₆	—	47	1	no	—
162	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
163	C₄×Q₈	—	47	2	no	—
164	C₄×Q₈	—	47	2	no	—
165	C₄×Q₈	—	47	1	no	—
166	C₄×Q₈	—	47	1	no	—
167	C₄×Q₈	—	94	1	no	—
168	C₄×Q₈	—	94	1	no	—
169	C₄×D₄	—	47	1	no	—
170	C₄×D₄	—	47	1	no	—
171	C₄×D₄	—	94	1	no	—
172	C₄×D₄	—	94	1	no	—
173	#117	—	94	1	no	—
174	#117	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	47	1	no	—
176	C₂×C₄⋊C₄	—	94	1	no	—
177	C₂×C₄²	—	47	1	no	—
178	C₂×C₄²	—	94	1	no	—
179	C₂×C₄²	—	94	1	no	—
180	#107	—	47	2	no	—
181	#106	—	47	2	no	—
182	#103	—	47	2	no	—
183	#103	C₂×D₄₇	1	2	yes	—
184	#103	C₂×D₄₇	1	2	yes	—
185	#102	—	47	2	no	—
186	#97	—	47	2	no	—
187	C₄×C₈	—	47	2	no	—
188	#41	—	47	4	no	—
189	#41	—	94	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	47	1	no	—
193	C₂×Q₈	C₄×D₄₇	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
195	C₂×D₄	—	47	1	no	—
196	C₂×D₄	—	94	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	47	1	no	—
199	C₂²×C₄	—	47	1	no	—
200	C₂²×C₄	—	94	1	no	—
201	C₂²×C₄	—	94	1	no	—
202	C₂²×C₄	—	94	1	no	—
203	C₂²×C₄	—	94	1	no	—
204	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	Q₁₆	—	47	8	no	—
206	SD₁₆	—	47	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	47	4	no	—
209	C₂×C₈	—	47	2	no	—
210	C₂×C₈	C₂²×D₄₇	1	2	yes	—
211	C₄⋊C₄	—	47	2	no	—
212	C₄⋊C₄	—	94	1	no	—
213	C₄⋊C₄	—	94	1	no	—
214	C₄⋊C₄	—	94	2	no	—
215	C₄⋊C₄	—	94	2	no	—
216	C₄⋊C₄	—	94	1	no	—
217	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	94	1	no	—
221	C₂²⋊C₄	—	94	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	47	1	no	—
224	C₄²	—	47	1	no	—
225	C₄²	—	47	1	no	—
226	C₄²	—	47	1	no	—
227	C₄²	—	94	1	no	—
228	C₄²	—	94	1	no	—
229	C₄²	—	94	2	no	—
230	C₄²	—	94	2	no	—
231	C₂³	—	94	1	no	—
232	C₂³	D₄×D₄₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	47	2	no	—
236	Q₈	—	94	1	no	—
237	Q₈	C₂×C₄×D₄₇	1	4	yes	—
238	D₄	—	47	2	no	—
239	D₄	—	94	4	no	—
240	D₄	—	94	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	47	1	no	—
246	C₂×C₄	—	47	1	no	—
247	C₂×C₄	—	47	1	no	—
248	C₂×C₄	—	47	1	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	—	94	1	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	—	188	2	no	—
255	C₂×C₄	—	188	1	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
258	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	#10625352	1	2	yes	—
260	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	47	4	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	94	2	no	—
266	C₂²	—	188	1	no	—
267	C₂²	—	188	1	no	—
268	C₂²	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
269	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	47	2	no	—
274	C₄	—	94	2	no	—
275	C₄	—	94	1	no	—
276	C₄	—	94	1	no	—
277	C₄	—	94	1	no	—
278	C₄	—	188	2	no	—
279	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
280	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
281	C₄	C₂×#10625352	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	188	2	no	—
284	C₂	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
285	C₂	#137×D₄₇	1	1	yes	characteristic
286	C₂	#15689922	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#419	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#419	D₄₇
hypocenter	C₄₇	C₄×#137
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×Dic₄₇	C₂	semidirect
4. #15689227	C₂	semidirect
5. #15689226	C₂	semidirect
6. #15689215	C₂	semidirect
7. #15689208	C₂	semidirect
8. #15689166	C₂	semidirect
10. M₁₆×Dic₄₇	C₂	semidirect
11. C₄₇×#419	C₂	semidirect
13. #13094055	C₄	semidirect
46. C₂×Q₈×C₄₇	C₂×C₄	semidirect
67. #10625337	C₂×C₄	semidirect
68. #10625336	C₂×C₄	semidirect
85. #10625329	C₂×C₄	semidirect
137. Dic₄₇	#137	semidirect
152. #419	D₄₇	semidirect
155. C₄₇	C₄×#137	semidirect
193. C₂×Q₈	C₄×D₄₇	semidirect