Subgroups — #20247084

Order 6016.

608 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094023	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094003	C₂	1	2	yes	maximal
3	#15689056	C₂	1	2	yes	maximal, complemented
4	#15689043	C₂	1	2	yes	maximal, complemented
5	#15689022	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689019	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688996	C₂	1	2	yes	maximal
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	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₉₄	C₂²	1	2	yes	—
13	#13094023	—	2	4	no	—
14	#13094003	—	2	8	no	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₄₇⋊#95	C₂²	1	1	yes	characteristic
28	C₄₇⋊#95	C₂²	1	1	yes	characteristic
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₄₇⋊#95	C₂²	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	Q₈	1	4	yes	complemented
45	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₂³	1	2	yes	—
50	C₂²×Dic₄₇	C₂³	1	2	yes	—
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	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₉₄	—	2	4	no	—
56	C₂×Dic₉₄	—	2	4	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	D₄	1	4	yes	—
61	C₂²⋊Dic₄₇	—	2	4	no	—
62	C₂²⋊Dic₄₇	D₄	1	4	yes	—
63	C₂²⋊Dic₄₇	Q₈	1	4	yes	complemented
64	C₄⋊Dic₄₇	—	2	2	no	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	Dic₄₇⋊C₄	—	2	4	no	—
68	Dic₄₇⋊C₄	—	2	4	no	—
69	Dic₄₇⋊C₄	—	2	4	no	—
70	Dic₄₇⋊C₄	—	4	2	no	—
71	C₄×Dic₄₇	—	2	2	no	—
72	C₄×Dic₄₇	—	2	2	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	1	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	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₄₇	#41	1	2	yes	—
95	C₂×Dic₄₇	—	2	1	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	1	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	1	no	—
100	C₂×Dic₄₇	—	2	1	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	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
110	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
111	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
112	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
113	Dic₉₄	—	4	8	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	C₄⋊Q₈	1	4	yes	complemented
121	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	complemented
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	—	2	2	no	—
134	Dic₄₇	—	2	2	no	—
135	Dic₄₇	—	2	2	no	—
136	Dic₄₇	—	4	2	no	—
137	Dic₄₇	—	4	2	no	—
138	#2440	—	47	1	no	maximal, Sylow
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₄₇	#544	1	1	yes	characteristic
143	C₂×C₄₇	D₄²	1	1	yes	characteristic
144	C₂×C₄₇	#547	1	1	yes	characteristic
145	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
146	C₂×C₄₇	#554	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	—	47	1	no	—
148	C₂×C₂²⋊Q₈	—	47	2	no	—
149	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
150	#398	—	47	2	no	—
151	#395	—	47	2	no	—
152	#389	—	47	1	no	—
153	#389	—	47	1	no	—
154	#385	—	47	2	no	—
155	#384	—	47	1	no	—
156	#380	—	47	1	no	—
157	C₄×C₂²⋊C₄	—	47	1	no	—
158	C₄₇	#2440	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	—	47	2	no	—
160	C₂³×C₄	C₂×D₄₇	1	2	yes	—
161	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	94	4	no	—
163	C₂²⋊Q₈	—	94	8	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	47	2	no	—
166	C₂×C₄⋊C₄	—	47	2	no	—
167	C₂×C₄⋊C₄	—	47	2	no	—
168	C₂×C₄⋊C₄	—	47	2	no	—
169	C₂×C₄⋊C₄	—	47	1	no	—
170	C₂×C₄⋊C₄	—	47	1	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	47	1	no	—
176	C₂×C₂²⋊C₄	—	47	1	no	—
177	C₂×C₂²⋊C₄	—	47	2	no	—
178	C₂×C₄²	—	47	1	no	—
179	C₂×C₄²	—	47	1	no	—
180	#95	—	47	1	no	—
181	#95	—	47	1	no	—
182	#95	—	47	2	no	—
183	#95	—	47	2	no	—
184	#95	—	47	2	no	—
185	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×Q₈	—	94	4	no	—
187	C₂×Q₈	—	94	4	no	—
188	C₂×Q₈	—	188	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	47	1	no	—
195	C₂²×C₄	—	47	1	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	1	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	2	no	—
200	C₂²×C₄	Dic₉₄	1	4	yes	complemented
201	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
204	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	94	4	no	—
208	C₄⋊C₄	—	94	4	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₄⋊C₄	—	94	2	no	—
211	C₄⋊C₄	—	94	4	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	—	188	2	no	—
214	C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
215	C₂²⋊C₄	—	47	4	no	—
216	C₂²⋊C₄	—	47	4	no	—
217	C₂²⋊C₄	—	94	4	no	—
218	C₄²	—	94	2	no	—
219	C₄²	—	94	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	C₂×Dic₉₄	1	2	yes	—
225	C₂³	D₄×D₄₇	1	2	yes	—
226	C₂³	Q₈×D₄₇	1	2	yes	—
227	C₂³	C₂³×D₄₇	1	1	yes	characteristic
228	Q₈	—	188	8	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	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	47	2	no	—
239	C₂×C₄	—	47	2	no	—
240	C₂×C₄	—	47	2	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	94	1	no	—
243	C₂×C₄	—	94	1	no	—
244	C₂×C₄	—	94	1	no	—
245	C₂×C₄	—	94	1	no	—
246	C₂×C₄	—	94	1	no	—
247	C₂×C₄	—	94	1	no	—
248	C₂×C₄	—	94	1	no	—
249	C₂×C₄	—	94	1	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
257	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
258	C₂×C₄	#10625350	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#13094023	1	4	yes	complemented
268	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
271	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
272	C₂²	C₂×#10625350	1	1	yes	characteristic
273	C₂²	#13094069	1	1	yes	characteristic
274	C₂²	#13094072	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	94	2	no	—
279	C₄	—	94	2	no	—
280	C₄	—	94	2	no	—
281	C₄	—	94	2	no	—
282	C₄	—	188	2	no	—
283	C₄	—	188	2	no	—
284	C₄	#13094003	1	4	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	C₂×#13094003	1	1	yes	characteristic
287	C₂	C₂×#13094023	1	1	yes	characteristic
288	C₂	#15689754	1	1	yes	characteristic
289	C₂	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₂	#15689774	1	1	yes	characteristic
291	C₂	Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₂	#15689828	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2440
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094023	C₂	semidirect
3. #15689056	C₂	semidirect
4. #15689043	C₂	semidirect
5. #15689022	C₂	semidirect
6. #15689019	C₂	semidirect
8. #15688992	C₂	semidirect
44. C₂²×C₄×C₄₇	Q₈	semidirect
63. C₂²⋊Dic₄₇	Q₈	semidirect
120. C₂²×C₄₇	C₄⋊Q₈	semidirect
131. C₄×C₄₇	C₂²⋊Q₈	semidirect
158. C₄₇	#2440	semidirect
200. C₂²×C₄	Dic₉₄	semidirect
267. C₂²	#13094023	semidirect
284. C₄	#13094003	semidirect