Subgroups — #20246909

Order 6016.

500 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×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689059	C₂	1	2	yes	maximal
3	#15689053	C₂	1	2	yes	maximal, complemented
4	#15689030	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689019	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689011	C₂	1	2	yes	maximal
7	#15689004	C₂	1	2	yes	maximal, complemented
8	#15688992	C₂	1	1	yes	maximal, characteristic, complemented
9	#15688983	C₂	1	1	yes	maximal, characteristic
10	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₄₇	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
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	1	yes	characteristic
20	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₄₇⋊#95	C₂²	1	1	yes	characteristic
26	C₄₇⋊#95	C₂²	1	1	yes	characteristic
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²⋊Q₈×C₄₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
37	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₄₇	—	2	2	no	—
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	D₄	1	4	yes	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×Dic₄₇	C₂³	1	2	yes	—
49	C₂²×Dic₄₇	C₂³	1	2	yes	—
50	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₂²⋊C₄×C₄₇	—	2	4	no	—
57	C₂²⋊Dic₄₇	—	2	2	no	—
58	C₂²⋊Dic₄₇	—	4	2	no	—
59	C₂²⋊Dic₄₇	Q₈	1	4	yes	complemented
60	C₄⋊Dic₄₇	—	2	4	no	—
61	C₄⋊Dic₄₇	—	2	2	no	—
62	C₄⋊Dic₄₇	—	2	2	no	—
63	Dic₄₇⋊C₄	—	2	4	no	—
64	Dic₄₇⋊C₄	—	4	2	no	—
65	Dic₄₇⋊C₄	—	4	2	no	—
66	C₄×Dic₄₇	—	2	4	no	—
67	C₄×Dic₄₇	—	2	2	no	—
68	C₄×Dic₄₇	—	2	2	no	—
69	C₂³×C₄₇	—	2	1	no	—
70	C₂³×C₄₇	—	2	1	no	—
71	C₂³×C₄₇	—	2	1	no	—
72	C₂³×C₄₇	—	2	1	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
77	Q₈×C₄₇	—	2	8	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	2	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	—	4	4	no	—
87	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	#41	1	2	yes	—
90	C₂×C₄×C₄₇	#41	1	2	yes	—
91	C₂×Dic₄₇	—	2	1	no	—
92	C₂×Dic₄₇	—	2	1	no	—
93	C₂×Dic₄₇	—	2	1	no	—
94	C₂×Dic₄₇	—	2	1	no	—
95	C₂×Dic₄₇	—	2	1	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	1	no	—
102	C₂×Dic₄₇	—	2	1	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
107	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
108	C₂×Dic₄₇	#41	1	2	yes	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	4	1	no	—
113	C₂²×C₄₇	—	4	1	no	—
114	C₂²×C₄₇	—	4	1	no	—
115	C₂²×C₄₇	—	4	1	no	—
116	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₄×C₄₇	—	2	4	no	—
124	C₄×C₄₇	—	4	1	no	—
125	C₄×C₄₇	—	4	2	no	—
126	C₄×C₄₇	#123	1	4	yes	—
127	Dic₄₇	—	2	2	no	—
128	Dic₄₇	—	2	2	no	—
129	Dic₄₇	—	2	2	no	—
130	Dic₄₇	—	4	2	no	—
131	Dic₄₇	—	4	2	no	—
132	Dic₄₇	—	4	1	no	—
133	#2423	—	47	1	no	maximal, Sylow
134	C₂×C₄₇	—	4	2	no	—
135	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
136	C₂×C₄₇	#533	1	1	yes	characteristic
137	C₂×C₄₇	#542	1	1	yes	characteristic
138	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
139	C₂×C₄₇	#550	1	1	yes	characteristic
140	C₂×C₄₇	#552	1	1	yes	characteristic
141	C₂×C₄₇	#554	1	1	yes	characteristic
142	C₂×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
143	C₂×C₄×Q₈	—	47	1	no	—
144	#399	—	47	2	no	—
145	#398	—	47	2	no	—
146	#391	—	47	1	no	—
147	#389	—	47	1	no	—
148	#388	—	47	2	no	—
149	#387	—	47	2	no	—
150	#384	—	47	1	no	—
151	#380	—	47	1	no	—
152	C₄×C₂²⋊C₄	—	47	1	no	—
153	C₄₇	#2423	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
156	C₂²⋊Q₈	—	2	4	no	—
157	C₄×Q₈	—	94	4	no	—
158	C₂×C₄⋊C₄	—	47	1	no	—
159	C₂×C₄⋊C₄	—	47	2	no	—
160	C₂×C₄⋊C₄	—	47	2	no	—
161	C₂×C₄⋊C₄	—	47	1	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	—	47	2	no	—
164	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
166	C₂×C₂²⋊C₄	—	47	1	no	—
167	C₂×C₂²⋊C₄	—	47	1	no	—
168	C₂×C₂²⋊C₄	—	47	2	no	—
169	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
170	C₂×C₄²	—	47	1	no	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	—	47	1	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	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	C₂²⋊D₄₇	1	4	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	47	1	no	—
187	C₂²×C₄	—	47	1	no	—
188	C₂²×C₄	—	47	1	no	—
189	C₂²×C₄	—	47	2	no	—
190	C₂²×C₄	—	47	1	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
194	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	94	4	no	—
199	C₄⋊C₄	—	94	2	no	—
200	C₄⋊C₄	—	94	4	no	—
201	C₄⋊C₄	—	94	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₄⋊C₄	—	188	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	47	4	no	—
206	C₂²⋊C₄	—	94	2	no	—
207	C₂²⋊C₄	—	188	2	no	—
208	C₄²	—	94	2	no	—
209	C₄²	—	94	4	no	—
210	C₄²	—	94	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	Q₈×D₄₇	1	2	yes	—
218	C₂³	C₂³×D₄₇	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	47	2	no	—
230	C₂×C₄	—	47	2	no	—
231	C₂×C₄	—	47	2	no	—
232	C₂×C₄	—	94	1	no	—
233	C₂×C₄	—	94	1	no	—
234	C₂×C₄	—	94	1	no	—
235	C₂×C₄	—	94	1	no	—
236	C₂×C₄	—	94	1	no	—
237	C₂×C₄	—	94	1	no	—
238	C₂×C₄	—	94	1	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	1	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	1	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	#10625352	1	2	yes	—
248	C₂×C₄	#10625352	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
250	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
260	C₂²	#41×D₄₇	1	1	yes	characteristic
261	C₂²	C₂×#10625352	1	1	yes	characteristic
262	C₂²	C₂×#10625352	1	1	yes	characteristic
263	C₂²	#13094070	1	1	yes	characteristic
264	C₂²	#13094072	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	94	2	no	—
269	C₄	—	94	2	no	—
270	C₄	—	94	2	no	—
271	C₄	—	188	2	no	—
272	C₄	—	188	1	no	—
273	C₄	—	188	2	no	—
274	C₄	#13094005	1	4	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₂×#13094005	1	1	yes	characteristic
277	C₂	#15689675	1	1	yes	characteristic
278	C₂	#15689746	1	1	yes	characteristic
279	C₂	Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₂	#15689792	1	1	yes	characteristic
281	C₂	#15689807	1	1	yes	characteristic
282	C₂	#15689831	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₂²⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#2423
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₄₇	C₂	semidirect
3. #15689053	C₂	semidirect
4. #15689030	C₂	semidirect
5. #15689019	C₂	semidirect
7. #15689004	C₂	semidirect
8. #15688992	C₂	semidirect
59. C₂²⋊Dic₄₇	Q₈	semidirect
153. C₄₇	#2423	semidirect