Subgroups — #20252231

Order 6016.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#15689603	C₂	1	1	yes	maximal, characteristic
3	#15689228	C₂	1	2	yes	maximal, complemented
4	#15689221	C₂	1	2	yes	maximal, complemented
5	C₄×#10625337	C₂	1	2	yes	maximal, complemented
6	C₄×#10625336	C₂	1	2	yes	maximal, complemented
7	#15689166	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689117	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093948	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689079	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#519	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	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₉₄	C₂²	1	2	yes	—
19	C₄×D₁₈₈	C₂²	1	2	yes	—
20	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
21	#13093982	—	2	1	no	—
22	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₄₇	—	2	1	no	—
24	#13093957	C₂²	1	2	yes	—
25	#13093955	C₂²	1	2	yes	—
26	#13093948	C₂²	1	2	yes	—
27	#13093948	C₂²	1	2	yes	—
28	#13093947	C₂²	1	1	yes	characteristic
29	#13093947	C₂²	1	1	yes	characteristic
30	#13093944	C₂²	1	1	yes	characteristic
31	#13093944	C₂²	1	1	yes	characteristic
32	#13093935	C₂²	1	2	yes	—
33	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
34	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₄₇	—	2	1	no	—
36	C₄×Q₈×C₄₇	—	2	1	no	—
37	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
38	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	C₂²	1	1	yes	characteristic
41	#117×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×Q₈×C₄₇	—	2	1	no	—
43	C₂×Q₈×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₄₇	—	2	1	no	—
49	C₂²×Dic₄₇	—	2	1	no	—
50	#10625350	—	2	2	no	—
51	#10625350	C₂×C₄	1	4	yes	—
52	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₄₇	—	2	1	no	—
54	C₂×C₄×D₄₇	—	2	2	no	—
55	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₄₇	—	2	1	no	—
63	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	1	no	—
67	C₄²×C₄₇	C₂³	1	2	yes	—
68	#10625337	C₂×C₄	1	8	yes	complemented
69	#10625336	C₂×C₄	1	8	yes	complemented
70	(C₂×D₄₇)⋊C₄	—	2	2	no	—
71	C₄⋊Dic₄₇	—	2	1	no	—
72	C₄⋊Dic₄₇	C₂³	1	2	yes	—
73	Dic₄₇⋊C₄	—	2	2	no	—
74	C₄×Dic₄₇	—	2	2	no	—
75	#10625329	C₂×C₄	1	4	yes	complemented
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₄₇	—	2	1	no	—
82	Q₈×C₄₇	—	2	4	no	—
83	Q₈×C₄₇	—	2	2	no	—
84	Q₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#41	1	2	yes	—
94	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₄₇	—	2	4	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	8	1	no	—
99	D₁₈₈	—	2	1	no	—
100	D₁₈₈	C₂²×C₄	1	2	yes	—
101	C₄×D₄₇	—	2	4	no	—
102	C₄×D₄₇	—	8	2	no	—
103	Dic₉₄	—	2	1	no	—
104	Dic₉₄	C₂²×C₄	1	2	yes	—
105	C₄₇⋊C₈	—	2	1	no	—
106	C₄₇⋊C₈	—	2	1	no	—
107	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	C₄×D₄	1	2	yes	—
111	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₄₇	—	2	2	no	—
113	C₂×D₄₇	—	4	1	no	—
114	C₂×D₄₇	—	4	1	no	—
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	4	1	no	—
119	C₄×C₄₇	—	4	1	no	—
120	C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	Dic₄₇	—	2	2	no	—
124	Dic₄₇	—	4	2	no	—
125	#2947	—	47	1	no	maximal, Sylow
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₄₇	#577	1	1	yes	characteristic
129	C₂×C₄₇	#578	1	1	yes	characteristic
130	D₄₇	—	4	2	no	—
131	C₂×#137	—	47	1	no	—
132	#520	—	47	1	no	—
133	#519	D₄₇	1	1	yes	characteristic, complemented
134	#441	—	47	2	no	—
135	#440	—	47	2	no	—
136	C₄×Q₁₆	—	47	2	no	—
137	C₄×SD₁₆	—	47	2	no	—
138	#428	—	47	1	no	—
139	#416	—	47	1	no	—
140	C₂×#103	—	47	1	no	—
141	#405	—	47	1	no	—
142	C₄₇	#2947	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	47	1	no	—
144	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
145	#137	—	47	8	no	—
146	C₂×Q₁₆	—	47	2	no	—
147	C₂×SD₁₆	—	47	2	no	—
148	C₂×M₁₆	—	47	1	no	—
149	C₂²×C₈	—	47	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	47	2	no	—
153	C₄×Q₈	C₂×D₄₇	1	2	yes	—
154	C₄×Q₈	C₂×D₄₇	1	2	yes	—
155	C₄×D₄	—	47	2	no	—
156	C₄×D₄	—	94	2	no	—
157	#117	—	2	2	no	—
158	#117	—	94	1	no	—
159	#117	C₂×D₄₇	1	1	yes	characteristic
160	#117	C₂×D₄₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	47	1	no	—
162	C₂×C₄⋊C₄	—	94	1	no	—
163	#107	—	47	2	no	—
164	#106	—	47	2	no	—
165	#103	—	47	2	no	—
166	#103	—	47	2	no	—
167	#103	—	47	1	no	—
168	#103	—	47	1	no	—
169	#102	—	47	1	no	—
170	#102	—	47	1	no	—
171	#97	—	47	2	no	—
172	C₄×C₈	—	47	2	no	—
173	#41	—	47	4	no	—
174	#41	—	94	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	47	1	no	—
178	C₂×Q₈	C₄×D₄₇	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
180	C₂×D₄	—	47	1	no	—
181	C₂×D₄	—	94	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	94	1	no	—
184	C₂²×C₄	—	94	1	no	—
185	C₂²×C₄	—	94	2	no	—
186	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	Q₁₆	—	47	8	no	—
188	SD₁₆	—	47	8	no	—
189	M₁₆	—	47	4	no	—
190	C₂×C₈	—	47	2	no	—
191	C₂×C₈	—	47	1	no	—
192	C₂×C₈	—	47	1	no	—
193	C₂×C₈	—	94	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	47	2	no	—
197	C₄⋊C₄	—	94	2	no	—
198	C₄⋊C₄	—	94	1	no	—
199	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	94	2	no	—
205	C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	94	2	no	—
210	C₄²	C₂²×D₄₇	1	2	yes	—
211	C₂³	—	94	1	no	—
212	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	47	2	no	—
216	Q₈	—	94	1	no	—
217	Q₈	C₂×C₄×D₄₇	1	4	yes	—
218	D₄	—	47	2	no	—
219	D₄	—	94	4	no	—
220	D₄	—	94	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	94	1	no	—
226	C₂×C₄	—	94	4	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	376	2	no	—
229	C₂×C₄	—	376	1	no	—
230	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂×C₄	#10625350	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₈	—	47	4	no	—
237	C₈	—	94	1	no	—
238	C₈	—	94	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	94	2	no	—
242	C₂²	—	188	1	no	—
243	C₂²	—	188	1	no	—
244	C₂²	C₄×C₂²⋊D₄₇	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	94	2	no	—
252	C₄	—	188	2	no	—
253	C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
254	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
255	C₄	C₂×#10625350	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	188	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂	#15689931	1	1	yes	characteristic
260	C₂	#15689938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#519	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#519	D₄₇
hypocenter	C₄₇	#2947
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #15689228	C₂	semidirect
4. #15689221	C₂	semidirect
5. C₄×#10625337	C₂	semidirect
6. C₄×#10625336	C₂	semidirect
7. #15689166	C₂	semidirect
9. C₂×#13093948	C₂	semidirect
10. #15689079	C₂	semidirect
11. C₄₇×#519	C₂	semidirect
13. #13094055	C₄	semidirect
44. C₂×Q₈×C₄₇	C₂×C₄	semidirect
68. #10625337	C₂×C₄	semidirect
69. #10625336	C₂×C₄	semidirect
75. #10625329	C₂×C₄	semidirect
133. #519	D₄₇	semidirect
142. C₄₇	#2947	semidirect
178. C₂×Q₈	C₄×D₄₇	semidirect