Subgroups — #20255131

Order 6016.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094046	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689782	C₂	1	2	yes	maximal, complemented
4	#15689507	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689364	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689359	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689346	C₂	1	2	yes	maximal, complemented
8	#15689287	C₂	1	2	yes	maximal, complemented
9	#15689264	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	2	yes	—
13	C₂²×Dic₉₄	C₂²	1	2	yes	—
14	#13094046	—	2	4	no	—
15	C₂×Dic₁₈₈	—	2	4	no	—
16	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	C₂²	1	2	yes	—
18	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
19	C₂×#10625325	C₂²	1	2	yes	—
20	C₂×#10625325	C₂²	1	2	yes	—
21	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
22	#13094012	C₂²	1	2	yes	—
23	#13094010	—	2	2	no	—
24	#13094003	C₂²	1	2	yes	—
25	#13094002	—	2	2	no	—
26	#13093998	C₂²	1	2	yes	—
27	#13093996	—	2	2	no	—
28	C₄×Dic₉₄	C₂²	1	2	yes	—
29	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
30	#13093958	C₂²	1	1	yes	characteristic
31	#13093949	C₂²	1	2	yes	—
32	#13093949	C₂²	1	2	yes	—
33	#13093946	C₂²	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₄₇	—	2	2	no	—
45	C₂×C₂²⋊D₄₇	—	2	2	no	—
46	C₂²×Dic₄₇	—	2	2	no	—
47	#10625350	—	2	4	no	—
48	#10625350	—	4	2	no	—
49	C₂×D₁₈₈	C₂³	1	2	yes	—
50	C₂×C₄×D₄₇	—	2	2	no	—
51	C₂×Dic₉₄	—	2	4	no	—
52	C₂×Dic₉₄	—	2	2	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	C₂³	1	2	yes	—
55	C₂×Dic₉₄	C₂³	1	2	yes	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₄₇	—	2	2	no	—
67	(C₂×D₄₇)⋊C₄	—	2	2	no	—
68	(C₂×D₄₇)⋊C₄	—	2	2	no	—
69	(C₂×D₄₇)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₄₇	C₂³	1	2	yes	—
71	Dic₄₇⋊C₄	—	2	2	no	—
72	Dic₄₇⋊C₄	—	2	2	no	—
73	C₄×Dic₄₇	—	2	2	no	—
74	Dic₁₈₈	—	2	4	no	—
75	Dic₁₈₈	—	2	2	no	—
76	Dic₁₈₈	D₄	1	4	yes	complemented
77	D₃₇₆	—	2	2	no	—
78	#10625325	—	2	4	no	—
79	#10625325	—	4	2	no	—
80	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₄₇	—	2	2	no	—
83	D₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₄₇	—	4	4	no	—
94	C₂²⋊D₄₇	—	4	4	no	—
95	C₂×Dic₄₇	—	2	2	no	—
96	C₂×Dic₄₇	—	2	2	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	4	4	no	—
99	D₁₈₈	—	2	2	no	—
100	D₁₈₈	—	2	2	no	—
101	C₄×D₄₇	—	4	4	no	—
102	Dic₉₄	—	2	2	no	—
103	Dic₉₄	—	2	2	no	—
104	Dic₉₄	—	2	2	no	—
105	Dic₉₄	—	4	4	no	—
106	Dic₉₄	C₂×D₄	1	4	yes	—
107	C₈×C₄₇	—	2	1	no	—
108	C₈×C₄₇	—	2	1	no	—
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	2	1	no	—
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	#135	1	2	yes	complemented
117	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₄₇	—	4	2	no	—
119	C₂×D₄₇	—	4	2	no	—
120	C₂×D₄₇	—	4	2	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₄₇	—	2	4	no	—
128	Dic₄₇	—	4	2	no	—
129	Dic₄₇	—	4	2	no	—
130	#3285	—	47	1	no	maximal, Sylow
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	—	4	1	no	—
133	C₂×C₄₇	D₄²	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
135	C₂×C₄₇	#578	1	1	yes	characteristic
136	D₄₇	—	8	2	no	—
137	C₂×#135	—	47	1	no	—
138	C₂²×Q₁₆	—	47	1	no	—
139	#548	—	47	2	no	—
140	#495	—	47	1	no	—
141	#467	—	47	1	no	—
142	#466	—	47	1	no	—
143	#464	—	47	2	no	—
144	#452	—	47	2	no	—
145	#448	—	47	2	no	—
146	C₄×Q₁₆	—	47	1	no	—
147	C₈×D₄	D₄₇	1	1	yes	characteristic, complemented
148	C₄₇	#3285	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	47	2	no	—
150	C₂²×Q₈	—	47	2	no	—
151	#135	—	94	4	no	—
152	C₂×Q₁₆	—	47	1	no	—
153	C₂×Q₁₆	—	47	2	no	—
154	C₂×Q₁₆	—	47	1	no	—
155	C₂×Q₁₆	—	94	4	no	—
156	C₂×SD₁₆	—	47	2	no	—
157	C₂×SD₁₆	—	47	2	no	—
158	C₂×D₈	—	47	1	no	—
159	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
161	#124	—	47	2	no	—
162	#124	—	94	2	no	—
163	C₂²⋊Q₈	—	47	2	no	—
164	C₂²⋊Q₈	—	94	2	no	—
165	#121	—	47	2	no	—
166	#121	—	94	2	no	—
167	C₄×Q₈	—	47	2	no	—
168	C₄×D₄	—	94	2	no	—
169	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
170	#107	—	47	1	no	—
171	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
172	#103	—	47	2	no	—
173	#103	—	47	2	no	—
174	#102	—	47	2	no	—
175	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#41	—	94	4	no	—
179	#41	—	188	2	no	—
180	C₂×Q₈	—	47	2	no	—
181	C₂×Q₈	—	47	2	no	—
182	C₂×Q₈	—	94	4	no	—
183	C₂×Q₈	—	94	2	no	—
184	C₂×Q₈	—	188	2	no	—
185	C₂×D₄	—	47	2	no	—
186	C₂×D₄	—	94	2	no	—
187	C₂×D₄	—	94	2	no	—
188	C₂×D₄	D₁₈₈	1	1	yes	characteristic
189	C₂²×C₄	—	94	2	no	—
190	C₂²×C₄	—	94	2	no	—
191	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	Q₁₆	—	47	4	no	—
194	Q₁₆	—	94	4	no	—
195	Q₁₆	—	94	2	no	—
196	SD₁₆	—	94	4	no	—
197	SD₁₆	—	188	2	no	—
198	D₈	—	94	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊C₄	—	47	2	no	—
206	C₄⋊C₄	—	94	2	no	—
207	C₄⋊C₄	—	94	2	no	—
208	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	94	2	no	—
210	C₂²⋊C₄	—	94	2	no	—
211	C₂²⋊C₄	—	94	2	no	—
212	C₂²⋊C₄	—	94	2	no	—
213	C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
214	C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
215	C₄²	—	94	2	no	—
216	C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₂³	—	94	2	no	—
218	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
220	Q₈	—	47	4	no	—
221	Q₈	—	94	2	no	—
222	Q₈	—	94	2	no	—
223	Q₈	—	94	2	no	—
224	Q₈	—	188	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	94	2	no	—
227	D₄	—	94	2	no	—
228	D₄	—	188	4	no	—
229	D₄	—	188	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	94	2	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	188	4	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₄₇	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	188	2	no	—
254	C₂²	—	188	2	no	—
255	C₂²	—	188	2	no	—
256	C₂²	#13094046	1	2	yes	complemented
257	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	94	4	no	—
263	C₄	—	188	2	no	—
264	C₄	—	188	2	no	—
265	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
266	C₄	#13094070	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	376	2	no	—
270	C₂	D₄×D₁₈₈	1	1	yes	characteristic
271	C₂	C₂×#13094046	1	1	yes	characteristic
272	C₂	#15689940	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₈×D₄	D₄₇
hypocenter	C₄₇	#3285
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094046	C₂	semidirect
2. C₂²×Dic₁₈₈	C₂	semidirect
3. #15689782	C₂	semidirect
4. #15689507	C₂	semidirect
5. #15689364	C₂	semidirect
6. #15689359	C₂	semidirect
7. #15689346	C₂	semidirect
8. #15689287	C₂	semidirect
9. #15689264	C₂	semidirect
10. C₄×Dic₁₈₈	C₂	semidirect
11. C₈×D₄×C₄₇	C₂	semidirect
76. Dic₁₈₈	D₄	semidirect
116. C₂²×C₄₇	#135	semidirect
147. C₈×D₄	D₄₇	semidirect
148. C₄₇	#3285	semidirect
256. C₂²	#13094046	semidirect