Subgroups — #20248564

Order 6016.

584 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689062	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689059	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689043	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689035	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689007	C₂	1	2	yes	maximal, complemented
7	#15689003	C₂	1	2	yes	maximal
8	#15688996	C₂	1	2	yes	maximal, complemented
9	#15688981	C₂	1	2	yes	maximal, complemented
10	C₂×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂³×Dic₄₇	C₂²	1	2	yes	—
14	#13094003	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄₇⋊#95	C₂²	1	2	yes	—
23	C₄₇⋊#95	C₂²	1	1	yes	characteristic
24	C₄₇⋊#95	C₂²	1	1	yes	characteristic
25	C₄₇⋊#95	C₂²	1	2	yes	—
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₄₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
37	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
38	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₄₇	—	2	2	no	—
40	C₂×D₄×C₄₇	—	2	2	no	—
41	C₂×D₄×C₄₇	Q₈	1	4	yes	complemented
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	D₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	4	no	—
52	C₂²×Dic₄₇	—	2	4	no	—
53	C₂²×Dic₄₇	—	4	2	no	—
54	C₂²×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×Dic₄₇	C₂³	1	2	yes	—
56	C₂²×Dic₄₇	C₂³	1	2	yes	—
57	C₂²×Dic₄₇	C₂³	1	2	yes	—
58	C₂×Dic₉₄	—	2	2	no	—
59	C₂×Dic₉₄	—	4	2	no	—
60	C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂²⋊C₄×C₄₇	—	2	2	no	—
62	C₂²⋊C₄×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₂²⋊Dic₄₇	—	2	4	no	—
65	C₂²⋊Dic₄₇	—	2	4	no	—
66	C₂²⋊Dic₄₇	—	4	2	no	—
67	C₄⋊Dic₄₇	—	2	2	no	—
68	Dic₄₇⋊C₄	—	2	4	no	—
69	Dic₄₇⋊C₄	—	2	4	no	—
70	Dic₄₇⋊C₄	—	4	2	no	—
71	C₂³×C₄₇	—	2	2	no	—
72	C₂³×C₄₇	—	2	2	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	C₂×D₄	1	2	yes	—
78	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
80	C₂³×C₄₇	#41	1	2	yes	—
81	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
82	D₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	2	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	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂×Dic₄₇	—	2	2	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	4	8	no	—
108	C₂×Dic₄₇	#41	1	4	yes	—
109	Dic₉₄	—	4	4	no	—
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	—	2	4	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	4	1	no	—
117	C₂²×C₄₇	—	4	1	no	—
118	C₂²×C₄₇	—	4	2	no	—
119	C₂²×C₄₇	C₂²⋊Q₈	1	4	yes	—
120	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#41	1	2	yes	—
123	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	4	1	no	—
129	C₄×C₄₇	—	4	1	no	—
130	Dic₄₇	—	2	4	no	—
131	Dic₄₇	—	4	2	no	—
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	4	2	no	—
134	#2564	—	47	1	no	maximal, Sylow
135	C₂×C₄₇	—	2	4	no	—
136	C₂×C₄₇	—	4	2	no	—
137	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄₇	#532	1	1	yes	characteristic
139	C₂×C₄₇	#534	1	1	yes	characteristic
140	C₂×C₄₇	#551	1	2	yes	—
141	C₂×C₄₇	#554	1	2	yes	—
142	C₂×C₂²⋊Q₈	—	47	1	no	—
143	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
144	#400	—	47	1	no	—
145	#399	—	47	1	no	—
146	#395	—	47	1	no	—
147	#393	—	47	1	no	—
148	#387	—	47	2	no	—
149	#386	—	47	2	no	—
150	#385	—	47	2	no	—
151	#380	—	47	2	no	—
152	C₂×#95	—	47	1	no	—
153	C₄₇	#2564	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	47	1	no	—
155	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
156	C₂³×C₄	—	47	2	no	—
157	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
158	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
159	C₂²⋊Q₈	—	94	4	no	—
160	C₄×D₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	47	1	no	—
163	C₂×C₄⋊C₄	—	47	2	no	—
164	C₂×C₄⋊C₄	—	47	2	no	—
165	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	47	2	no	—
167	C₂×C₂²⋊C₄	—	47	2	no	—
168	C₂×C₂²⋊C₄	—	47	2	no	—
169	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#95	—	47	2	no	—
173	#95	—	47	1	no	—
174	#95	—	47	1	no	—
175	#95	—	47	2	no	—
176	#95	—	47	2	no	—
177	#95	—	47	2	no	—
178	#95	—	47	2	no	—
179	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
180	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×Q₈	—	94	2	no	—
182	C₂×Q₈	—	188	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	Dic₉₄	1	4	yes	complemented
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	47	2	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	188	2	no	—
196	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
197	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	94	4	no	—
204	C₄⋊C₄	—	94	4	no	—
205	C₄⋊C₄	—	94	2	no	—
206	C₄⋊C₄	—	188	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	94	4	no	—
210	C₂²⋊C₄	—	94	4	no	—
211	C₂²⋊C₄	—	188	2	no	—
212	C₄²	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂×Dic₉₄	1	2	yes	—
220	C₂³	C₂×Dic₉₄	1	2	yes	—
221	C₂³	#10625350	1	2	yes	—
222	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₂³	C₂³×D₄₇	1	1	yes	characteristic
224	Q₈	—	188	4	no	—
225	D₄	—	2	8	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	47	4	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	94	2	no	—
242	C₂×C₄	—	94	2	no	—
243	C₂×C₄	—	94	2	no	—
244	C₂×C₄	—	94	2	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	—	188	8	no	—
249	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	#13094003	1	4	yes	—
262	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
264	C₂²	#41×D₄₇	1	2	yes	—
265	C₂²	C₂×#10625350	1	1	yes	characteristic
266	C₂²	#13094069	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	94	4	no	—
273	C₄	—	188	2	no	—
274	C₄	—	188	2	no	—
275	C₄	—	188	2	no	—
276	C₂	—	2	4	no	—
277	C₂	—	4	2	no	—
278	C₂	C₂×#13094003	1	1	yes	characteristic
279	C₂	#15689669	1	1	yes	characteristic
280	C₂	#15689682	1	1	yes	characteristic
281	C₂	#15689802	1	2	yes	—
282	C₂	#15689832	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×D₄×C₄₇	C₂
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#2564
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094003	C₂	semidirect
2. #15689062	C₂	semidirect
3. #15689059	C₂	semidirect
4. #15689043	C₂	semidirect
5. #15689035	C₂	semidirect
6. #15689007	C₂	semidirect
8. #15688996	C₂	semidirect
9. #15688981	C₂	semidirect
10. C₂×C₄₇⋊#95	C₂	semidirect
41. C₂×D₄×C₄₇	Q₈	semidirect
153. C₄₇	#2564	semidirect
185. C₂×D₄	Dic₉₄	semidirect