Subgroups — #20252156

Order 6016.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094044	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689591	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689215	C₂	1	2	yes	maximal, complemented
5	#15689214	C₂	1	2	yes	maximal, complemented
6	#15689208	C₂	1	2	yes	maximal, complemented
7	#15689203	C₂	1	2	yes	maximal, complemented
8	#15689159	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689120	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	1	yes	characteristic
13	#13094044	C₄	1	8	yes	complemented
14	C₂×#10625337	C₂²	1	1	yes	characteristic
15	C₂×#10625335	C₂²	1	1	yes	characteristic
16	C₂×#10625336	C₂²	1	1	yes	characteristic
17	C₂×#10625334	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₄₇	C₂²	1	2	yes	—
20	#13093989	C₂²	1	2	yes	—
21	D₄×Dic₄₇	—	2	2	no	—
22	D₄×Dic₄₇	C₂²	1	2	yes	—
23	#13093986	C₂²	1	2	yes	—
24	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
25	#13093983	—	2	1	no	—
26	#13093983	C₂²	1	1	yes	characteristic
27	#13093981	—	2	1	no	—
28	C₂×C₄×Dic₄₇	—	2	1	no	—
29	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
30	C₄²×D₄₇	—	2	1	no	—
31	#13093957	C₂²	1	2	yes	—
32	#13093955	C₂²	1	2	yes	—
33	#13093949	C₂²	1	2	yes	—
34	#13093946	C₂²	1	2	yes	—
35	C₈×Dic₄₇	C₂²	1	2	yes	—
36	C₈×Dic₄₇	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	2	yes	—
41	#102×C₄₇	C₂²	1	2	yes	—
42	#41×C₄₇	—	2	2	no	—
43	#41×C₄₇	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₇	—	2	1	no	—
46	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₄₇	—	2	1	no	—
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂²×Dic₄₇	D₄	1	2	yes	—
52	#10625350	—	2	2	no	—
53	#10625350	C₂×C₄	1	4	yes	complemented
54	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₄₇	—	2	1	no	—
56	C₂×C₄×D₄₇	—	2	2	no	—
57	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	C₂³	1	2	yes	—
60	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₄₇	—	2	1	no	—
62	C₄²×C₄₇	—	2	1	no	—
63	C₂²⋊Dic₄₇	—	2	1	no	—
64	C₂²⋊Dic₄₇	—	2	2	no	—
65	#10625337	C₂×C₄	1	4	yes	complemented
66	#10625336	C₂×C₄	1	4	yes	complemented
67	#10625335	C₂×C₄	1	4	yes	complemented
68	#10625334	C₂×C₄	1	4	yes	complemented
69	(C₂×D₄₇)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₄₇	—	2	2	no	—
71	C₄⋊Dic₄₇	C₂³	1	2	yes	—
72	Dic₄₇⋊C₄	—	2	2	no	—
73	C₄×Dic₄₇	—	2	1	no	—
74	C₄×Dic₄₇	—	2	2	no	—
75	C₄×Dic₄₇	—	2	2	no	—
76	C₄×Dic₄₇	D₄	1	1	yes	characteristic
77	C₄×Dic₄₇	D₄	1	1	yes	characteristic
78	C₄×Dic₄₇	C₂³	1	2	yes	—
79	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
80	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₄₇	—	2	1	no	—
85	Q₈×C₄₇	—	2	1	no	—
86	Q₈×C₄₇	C₂²×C₄	1	2	yes	—
87	D₄×C₄₇	—	2	4	no	—
88	D₄×C₄₇	—	2	1	no	—
89	D₄×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#41	1	2	yes	—
96	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₄₇	—	2	4	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	4	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	4	4	no	—
102	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
103	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
104	D₁₈₈	—	2	1	no	—
105	D₁₈₈	C₂²×C₄	1	2	yes	—
106	C₄×D₄₇	—	2	4	no	—
107	C₄×D₄₇	—	4	4	no	—
108	Dic₉₄	—	2	1	no	—
109	Dic₉₄	C₂²×C₄	1	2	yes	—
110	C₈×C₄₇	—	2	2	no	—
111	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
112	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	C₄×D₄	1	2	yes	—
117	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₄₇	—	2	2	no	—
119	C₂×D₄₇	—	4	2	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	4	2	no	—
122	C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	Dic₄₇	—	2	2	no	—
126	Dic₄₇	—	2	2	no	—
127	Dic₄₇	—	4	2	no	—
128	Dic₄₇	#135	1	4	yes	complemented
129	C₄×#135	—	47	1	no	maximal, Sylow
130	C₂×C₄₇	—	2	2	no	—
131	C₂×C₄₇	—	4	2	no	—
132	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#135	1	2	yes	—
134	D₄₇	—	4	2	no	—
135	C₂×#135	—	47	1	no	—
136	C₄×#41	—	47	1	no	—
137	C₄×#41	—	47	1	no	—
138	C₄×Q₁₆	—	47	2	no	—
139	C₄×SD₁₆	—	47	2	no	—
140	C₄×SD₁₆	—	47	2	no	—
141	C₄×D₈	—	47	2	no	—
142	#427	—	47	1	no	—
143	#416	—	47	1	no	—
144	#416	D₄₇	1	1	yes	characteristic, complemented
145	C₂×C₄×C₈	—	47	1	no	—
146	C₄₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	47	1	no	—
148	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
149	#135	—	47	8	no	—
150	C₂×Q₁₆	—	47	1	no	—
151	C₂×SD₁₆	—	47	1	no	—
152	C₂×SD₁₆	—	47	1	no	—
153	C₂×D₈	—	47	1	no	—
154	C₂²×C₈	—	47	1	no	—
155	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₄×Q₈	—	47	2	no	—
157	C₄×Q₈	—	47	2	no	—
158	C₄×D₄	—	47	2	no	—
159	C₄×D₄	—	47	2	no	—
160	C₄×D₄	—	94	2	no	—
161	C₄×D₄	—	94	2	no	—
162	#117	—	47	1	no	—
163	#117	—	94	1	no	—
164	#117	—	94	1	no	—
165	#117	C₂×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	—	47	1	no	—
167	C₂×C₄²	—	94	1	no	—
168	C₂×C₄²	—	94	1	no	—
169	#107	—	47	2	no	—
170	#106	—	47	2	no	—
171	#103	—	47	2	no	—
172	#103	C₂×D₄₇	1	2	yes	—
173	#102	—	47	2	no	—
174	#102	C₂×D₄₇	1	2	yes	—
175	C₄×C₈	—	47	2	no	—
176	C₄×C₈	—	47	2	no	—
177	#41	—	2	2	no	—
178	#41	—	47	4	no	—
179	#41	—	94	2	no	—
180	#41	C₄×D₄₇	1	4	yes	complemented
181	C₂×Q₈	—	47	1	no	—
182	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	47	1	no	—
185	C₂×D₄	—	94	1	no	—
186	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	94	2	no	—
190	C₂²×C₄	—	94	2	no	—
191	C₂²×C₄	—	94	1	no	—
192	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	Q₁₆	—	47	4	no	—
194	SD₁₆	—	47	4	no	—
195	SD₁₆	—	47	4	no	—
196	D₈	—	47	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	47	4	no	—
199	C₂×C₈	—	47	1	no	—
200	C₂×C₈	—	47	1	no	—
201	C₂×C₈	C₂²×D₄₇	1	2	yes	—
202	C₄⋊C₄	—	47	2	no	—
203	C₄⋊C₄	—	94	2	no	—
204	C₄⋊C₄	—	94	2	no	—
205	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	94	2	no	—
208	C₂²⋊C₄	—	94	2	no	—
209	C₂²⋊C₄	—	94	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	47	1	no	—
212	C₄²	—	47	1	no	—
213	C₄²	—	47	2	no	—
214	C₄²	—	94	2	no	—
215	C₄²	—	94	2	no	—
216	C₄²	—	94	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	94	1	no	—
219	C₂³	D₄×D₄₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	47	2	no	—
222	Q₈	—	94	1	no	—
223	Q₈	C₂×C₄×D₄₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	47	2	no	—
227	D₄	—	94	4	no	—
228	D₄	—	94	1	no	—
229	D₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	47	2	no	—
234	C₂×C₄	—	47	2	no	—
235	C₂×C₄	—	94	4	no	—
236	C₂×C₄	—	94	1	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	188	4	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
242	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄	#10625352	1	2	yes	—
244	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	47	2	no	—
247	C₈	—	47	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	94	2	no	—
252	C₂²	—	188	2	no	—
253	C₂²	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	47	4	no	—
258	C₄	—	94	2	no	—
259	C₄	—	94	2	no	—
260	C₄	—	188	2	no	—
261	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
263	C₄	C₂×#10625352	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	188	2	no	—
267	C₂	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂	#135×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	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	C₄×#135
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₄₇	C₂	semidirect
3. #15689591	C₂	semidirect
4. #15689215	C₂	semidirect
5. #15689214	C₂	semidirect
6. #15689208	C₂	semidirect
7. #15689203	C₂	semidirect
8. #15689159	C₂	semidirect
9. #15689120	C₂	semidirect
10. C₂×C₈×Dic₄₇	C₂	semidirect
11. C₄₇×#416	C₂	semidirect
13. #13094044	C₄	semidirect
43. #41×C₄₇	C₂×C₄	semidirect
53. #10625350	C₂×C₄	semidirect
65. #10625337	C₂×C₄	semidirect
66. #10625336	C₂×C₄	semidirect
67. #10625335	C₂×C₄	semidirect
68. #10625334	C₂×C₄	semidirect
79. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
128. Dic₄₇	#135	semidirect
144. #416	D₄₇	semidirect
146. C₄₇	C₄×#135	semidirect
180. #41	C₄×D₄₇	semidirect