Subgroups — #20252128

Order 6016.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094031	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094030	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689669	C₂	1	1	yes	maximal, characteristic
4	#15689198	C₂	1	2	yes	maximal, complemented
5	#15689197	C₂	1	2	yes	maximal, complemented
6	#15689191	C₂	1	2	yes	maximal, complemented
7	(C₂²⋊D₄₇)⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689091	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	1	yes	characteristic
13	#13094031	—	2	4	no	—
14	#13094030	—	2	4	no	—
15	C₂×#10625329	C₂²	1	2	yes	—
16	C₂×#10625329	C₂²	1	1	yes	characteristic
17	C₂×#10625324	C₂²	1	2	yes	—
18	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₈×D₄₇	C₂²	1	2	yes	—
21	#13094022	C₄	1	1	yes	characteristic
22	#13094020	C₄	1	1	yes	characteristic
23	#13094013	C₄	1	1	yes	characteristic
24	#13094011	C₄	1	1	yes	characteristic
25	#13093996	C₄	1	2	yes	—
26	#13093996	C₄	1	2	yes	—
27	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
29	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
30	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
36	#13093934	C₂²	1	2	yes	—
37	C₈×Dic₄₇	C₂²	1	2	yes	—
38	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
42	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
43	#41×C₄₇	—	2	2	no	—
44	#41×C₄₇	—	2	2	no	—
45	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
53	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
55	#10625350	—	2	2	no	—
56	#10625350	—	2	2	no	—
57	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₄₇	C₂³	1	2	yes	—
59	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂³	1	2	yes	—
65	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
67	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
68	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
69	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
70	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
71	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
74	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
75	#10625329	—	2	4	no	—
76	#10625329	D₄	1	4	yes	complemented
77	C₂×C₄₇⋊C₈	—	2	2	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
82	#10625324	—	2	4	no	—
83	C₈×D₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
87	Q₈×C₄₇	—	2	2	no	—
88	D₄×C₄₇	—	2	4	no	—
89	D₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	#41	1	2	yes	—
96	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₄₇	—	2	4	no	—
98	C₂²⋊D₄₇	—	2	4	no	—
99	C₂²⋊D₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	4	1	no	—
102	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
106	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
107	D₁₈₈	—	2	2	no	—
108	C₄×D₄₇	—	2	2	no	—
109	C₄×D₄₇	—	2	2	no	—
110	Dic₉₄	—	2	2	no	—
111	C₈×C₄₇	—	2	1	no	—
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	—	2	2	no	—
114	C₄₇⋊C₈	—	2	1	no	—
115	C₄₇⋊C₈	—	2	1	no	—
116	C₄₇⋊C₈	C₂×D₄	1	4	yes	—
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	C₄×D₄	1	2	yes	—
123	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₄₇	—	2	2	no	—
125	C₂×D₄₇	—	2	2	no	—
126	C₂×D₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	C₄×D₄	1	2	yes	—
129	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
131	Dic₄₇	—	2	2	no	—
132	Dic₄₇	—	2	1	no	—
133	Dic₄₇	—	2	1	no	—
134	Dic₄₇	—	2	1	no	—
135	Dic₄₇	#131	1	2	yes	complemented
136	#2935	—	47	1	no	maximal, Sylow
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	—	4	2	no	—
139	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
141	C₂×C₄₇	#568	1	1	yes	characteristic
142	D₄₇	—	4	2	no	—
143	C₂×#131	—	47	1	no	—
144	C₂×#131	—	47	1	no	—
145	#532	—	47	1	no	—
146	#436	—	47	2	no	—
147	#435	—	47	2	no	—
148	#435	—	47	2	no	—
149	C₈×D₄	—	47	2	no	—
150	C₂×C₄⋊C₈	—	47	1	no	—
151	#408	—	47	1	no	—
152	#408	D₄₇	1	1	yes	characteristic, complemented
153	C₄×M₁₆	—	47	1	no	—
154	C₄₇	#2935	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	47	1	no	—
156	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
157	#131	—	94	4	no	—
158	#131	—	94	4	no	—
159	C₂×M₁₆	—	47	1	no	—
160	C₂×M₁₆	—	47	2	no	—
161	C₂×M₁₆	—	47	2	no	—
162	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
163	C₂²×C₈	—	47	1	no	—
164	C₂²×C₈	—	47	2	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	47	1	no	—
168	#127	—	47	1	no	—
169	#124	—	47	1	no	—
170	#124	—	47	1	no	—
171	#121	—	47	2	no	—
172	#121	—	47	2	no	—
173	C₄×D₄	—	47	2	no	—
174	C₄×D₄	—	47	2	no	—
175	C₂×C₄²	—	47	1	no	—
176	C₄⋊C₈	—	47	1	no	—
177	C₄⋊C₈	—	47	1	no	—
178	C₄⋊C₈	—	47	1	no	—
179	C₄⋊C₈	—	47	1	no	—
180	C₂²⋊C₈	—	47	2	no	—
181	C₂²⋊C₈	—	47	2	no	—
182	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
183	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
184	#97	—	47	2	no	—
185	C₄×C₈	—	47	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	94	2	no	—
189	#41	—	94	2	no	—
190	C₂×Q₈	—	47	1	no	—
191	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
192	C₂×D₄	—	47	2	no	—
193	C₂×D₄	—	47	2	no	—
194	C₂×D₄	—	47	2	no	—
195	C₂×D₄	—	47	1	no	—
196	C₂×D₄	C₄×D₄₇	1	2	yes	—
197	C₂×D₄	C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	47	1	no	—
199	C₂²×C₄	—	47	1	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	47	4	no	—
205	M₁₆	—	94	4	no	—
206	M₁₆	—	94	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	47	2	no	—
209	C₂×C₈	—	47	1	no	—
210	C₂×C₈	—	47	1	no	—
211	C₂×C₈	—	94	4	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	94	2	no	—
214	C₂×C₈	C₂²×D₄₇	1	2	yes	—
215	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄⋊C₄	—	47	2	no	—
218	C₄⋊C₄	—	47	2	no	—
219	C₂²⋊C₄	—	47	2	no	—
220	C₂²⋊C₄	—	47	2	no	—
221	C₂²⋊C₄	—	47	2	no	—
222	C₂²⋊C₄	—	47	2	no	—
223	C₄²	—	47	1	no	—
224	C₄²	—	47	1	no	—
225	C₄²	—	47	1	no	—
226	C₄²	—	47	1	no	—
227	C₂³	—	47	2	no	—
228	C₂³	C₂×C₄×D₄₇	1	2	yes	—
229	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	94	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	94	4	no	—
235	D₄	—	94	4	no	—
236	D₄	—	94	2	no	—
237	D₄	—	188	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	47	1	no	—
241	C₂×C₄	—	47	1	no	—
242	C₂×C₄	—	47	1	no	—
243	C₂×C₄	—	47	1	no	—
244	C₂×C₄	—	47	2	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	—	188	1	no	—
249	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	D₄×D₄₇	1	2	yes	—
252	C₂×C₄	#10625352	1	2	yes	—
253	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	47	4	no	—
258	C₈	—	94	1	no	—
259	C₈	—	94	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	94	2	no	—
266	C₂²	—	94	2	no	—
267	C₂²	—	94	2	no	—
268	C₂²	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
269	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	47	2	no	—
272	C₄	—	94	2	no	—
273	C₄	—	94	1	no	—
274	C₄	—	94	1	no	—
275	C₄	—	94	1	no	—
276	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
277	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
278	C₄	C₂×#10625352	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	188	2	no	—
282	C₂	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
283	C₂	#131×D₄₇	1	1	yes	characteristic
284	C₂	#15689888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#408	C₂
hypercenter	#408	D₄₇
hypocenter	C₄₇	#2935
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#13094031	C₂	semidirect
2. C₂×#13094030	C₂	semidirect
4. #15689198	C₂	semidirect
5. #15689197	C₂	semidirect
6. #15689191	C₂	semidirect
7. (C₂²⋊D₄₇)⋊C₈	C₂	semidirect
8. C₂×Dic₄₇⋊C₈	C₂	semidirect
9. #15689091	C₂	semidirect
10. M₁₆×Dic₄₇	C₂	semidirect
11. C₄₇×#408	C₂	semidirect
76. #10625329	D₄	semidirect
135. Dic₄₇	#131	semidirect
152. #408	D₄₇	semidirect
154. C₄₇	#2935	semidirect