Subgroups — #14280401

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4685	C₂	1	1	yes	maximal, characteristic
2	C₄×#6650	C₂	1	1	yes	maximal, characteristic
3	C₄×D₃×#168	C₂	1	1	yes	maximal, characteristic
4	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601532	—	3	1	no	maximal, Hall
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂×#6650	C₄	1	2	yes	—
11	C₂×D₃×#168	C₄	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
16	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90953	C₄	1	2	yes	—
21	Dic₃×#168	C₄	1	2	yes	—
22	C₈×ΓL(1,𝔽₂₅)	—	5	1	no	maximal, Hall
23	C₅×#431	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
25	C₂×C₄×#168	—	3	1	no	—
26	#64674	—	3	1	no	—
27	#64674	—	3	1	no	—
28	C₈×#168	—	3	1	no	—
29	C₈×#168	D₃	1	1	yes	characteristic, complemented
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
37	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	#6650	C₈	1	8	yes	complemented
44	D₃×#168	—	2	4	no	—
45	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
46	#4685	D₅	1	1	yes	characteristic
47	C₄×D₃×C₈	—	5	1	no	—
48	C₃×C₈²	—	5	1	no	—
49	#4455	—	5	1	no	—
50	#4454	—	5	1	no	—
51	C₈×C₃⋊C₈	—	5	1	no	—
52	C₂×C₄²×C₅	—	3	1	no	—
53	C₂×#681	—	3	2	no	—
54	C₂²×#168	—	3	2	no	—
55	C₅×C₄⋊C₈	—	3	2	no	—
56	C₅×C₂²⋊C₈	—	3	2	no	—
57	C₄×C₅×C₈	—	3	1	no	—
58	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	#3935	—	3	2	no	—
60	C₄×#168	—	3	1	no	—
61	C₄×#168	—	3	1	no	—
62	C₄×#168	—	3	2	no	—
63	C₄×#168	C₂×D₃	1	1	yes	characteristic
64	C₄×#168	C₂×D₃	1	1	yes	characteristic
65	C₈×M₁₆	—	15	1	no	Sylow
66	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
72	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
73	C₄×C₅×D₃	—	2	2	no	—
74	C₄×C₅×D₃	—	2	2	no	—
75	C₄×C₅×D₃	C₂×C₈	1	4	yes	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃⋊#168	—	2	2	no	—
78	C₃⋊#168	C₂×C₈	1	4	yes	complemented
79	C₃×#168	C₂×C₈	1	4	yes	complemented
80	C₃×#168	M₁₆	1	4	yes	complemented
81	C₅×C₃⋊C₈	—	2	2	no	—
82	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
83	C₂×D₃×C₈	—	5	2	no	—
84	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
85	C₃×C₄×C₈	—	5	1	no	—
86	C₃×C₄×C₈	—	5	1	no	—
87	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	(C₂×D₃)⋊C₈	Dic₅	1	2	yes	—
89	#815	—	5	2	no	—
90	Dic₃⋊C₈	Dic₅	1	2	yes	—
91	C₈×Dic₃	—	5	2	no	—
92	C₄×C₃⋊C₈	—	5	1	no	—
93	C₄×C₃⋊C₈	—	5	1	no	—
94	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂²×C₄×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
100	C₄²×C₅	—	3	1	no	—
101	C₄²×C₅	—	3	1	no	—
102	C₄²×C₅	—	3	1	no	—
103	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
104	#681	—	3	8	no	—
105	C₂×#168	—	3	2	no	—
106	C₂×#168	—	3	2	no	—
107	C₂×#168	—	6	4	no	—
108	C₂×#168	C₄×D₃	1	2	yes	—
109	C₂×#168	C₄×D₃	1	2	yes	—
110	#431	—	3	1	no	—
111	C₄×M₁₆	—	15	1	no	—
112	C₂×C₄×C₈	—	15	1	no	—
113	#322	—	15	1	no	—
114	#322	—	15	1	no	—
115	C₈²	—	5	1	no	—
116	C₈²	—	15	1	no	—
117	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₅×D₃	—	2	1	no	—
119	C₂×C₅×D₃	—	2	1	no	—
120	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
121	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
122	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
123	C₃×C₄×C₅	#131	1	2	yes	—
124	C₅×Dic₃	—	2	1	no	—
125	C₅×Dic₃	—	2	1	no	—
126	C₅×Dic₃	C₄×C₈	1	2	yes	—
127	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
128	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
129	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₃×C₈	—	5	2	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
133	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
134	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
135	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
136	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
137	C₂×C₃⋊C₈	—	5	2	no	—
138	C₂×C₃⋊C₈	—	5	2	no	—
139	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
140	ΓL(1,𝔽₂₅)	—	5	8	no	—
141	D₃×C₈	—	10	4	no	—
142	C₂³×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	4	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	6	2	no	—
148	C₂×C₄×C₅	—	6	2	no	—
149	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	6	2	no	—
154	#168	—	3	4	no	—
155	#168	—	6	2	no	—
156	#168	D₃×C₈	1	4	yes	complemented
157	#168	ΓL(1,𝔽₂₅)	1	4	yes	complemented
158	C₂×M₁₆	—	15	2	no	—
159	C₂²×C₈	—	15	2	no	—
160	C₂×C₄²	—	3	1	no	—
161	C₄⋊C₈	—	3	2	no	—
162	C₂²⋊C₈	—	3	2	no	—
163	#97	—	15	2	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	5	1	no	—
166	C₄×C₈	—	5	1	no	—
167	C₄×C₈	—	15	2	no	—
168	C₄×C₈	—	15	1	no	—
169	C₄×C₈	—	15	1	no	—
170	C₄×C₈	D₃×D₅	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
172	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
173	C₂×C₃×C₅	#405	1	1	yes	characteristic
174	C₅×D₃	—	2	2	no	—
175	C₂²×D₃	C₄×Dic₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
179	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
180	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
181	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
182	C₄×D₃	—	2	2	no	—
183	C₄×D₃	—	2	2	no	—
184	C₄×D₃	C₈×D₅	1	4	yes	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	5	4	no	—
187	C₃×C₈	—	5	4	no	—
188	C₃⋊C₈	—	2	2	no	—
189	C₃⋊C₈	—	5	4	no	—
190	C₃⋊C₈	—	10	2	no	—
191	C₂²×C₅	—	3	2	no	—
192	C₂²×C₅	—	6	1	no	—
193	C₂²×C₅	—	6	1	no	—
194	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
195	C₄×C₅	—	3	2	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	—	6	1	no	—
198	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
199	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
200	C₄×C₅	#901	1	2	yes	—
201	C₂²×C₄	—	3	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	M₁₆	—	15	8	no	—
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	—	5	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	15	2	no	—
209	C₂×C₈	—	15	2	no	—
210	C₂×C₈	—	30	4	no	—
211	C₂×C₈	D₃×Dic₅	1	2	yes	—
212	C₄²	—	3	1	no	—
213	C₄²	—	3	1	no	—
214	C₄²	—	3	1	no	—
215	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
216	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
218	C₂×D₃	—	2	1	no	—
219	C₂×D₃	—	2	1	no	—
220	C₂×D₃	C₈×Dic₅	1	2	yes	—
221	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
222	C₃×C₄	D₅×M₁₆	1	2	yes	—
223	C₃×C₄	#4094	1	2	yes	—
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	1	no	—
226	Dic₃	C₈×Dic₅	1	2	yes	—
227	C₂×C₅	—	6	2	no	—
228	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
229	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
230	C₂×C₅	#4689	1	1	yes	characteristic
231	C₂³	—	3	1	no	—
232	C₂×C₄	—	3	1	no	—
233	C₂×C₄	—	3	4	no	—
234	C₂×C₄	—	3	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
239	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	5	4	no	—
243	C₈	—	5	4	no	—
244	C₈	—	6	2	no	—
245	C₈	—	15	4	no	—
246	C₈	—	30	2	no	—
247	C₂×C₃	#65286	1	1	yes	characteristic
248	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
249	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
250	D₃	—	2	2	no	—
251	C₅	C₈×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	3	2	no	—
253	C₂²	—	6	1	no	—
254	C₂²	—	6	1	no	—
255	C₂²	C₄×D₃×Dic₅	1	1	yes	characteristic
256	C₄	—	3	2	no	—
257	C₄	—	6	1	no	—
258	C₄	—	6	1	no	—
259	C₄	D₃×C₈×D₅	1	2	yes	—
260	C₄	D₅×ΓL(1,𝔽₂₅)	1	2	yes	—
261	C₄	#91286	1	2	yes	—
262	C₃	#10601532	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	6	2	no	—
264	C₂	#11746977	1	1	yes	characteristic
265	C₂	D₃×C₈×Dic₅	1	1	yes	characteristic
266	C₂	Dic₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₈×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11746275	C₂	semidirect
5. C₃×C₈×#168	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
29. C₈×#168	D₃	semidirect
43. #6650	C₈	semidirect
78. C₃⋊#168	C₂×C₈	semidirect
79. C₃×#168	C₂×C₈	semidirect
80. C₃×#168	M₁₆	semidirect
156. #168	D₃×C₈	semidirect
157. #168	ΓL(1,𝔽₂₅)	semidirect
216. C₃×C₅	C₈×M₁₆	semidirect
251. C₅	C₈×ΓL(1,𝔽₂₅)	semidirect
262. C₃	#10601532	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#4685	2
C₄×#6650	2
C₄×D₃×#168	2
#11746275	2
C₃×C₈×#168	2
#11745831	2
C₃⋊C₈×#168	2
#10601532	3
C₈×ΓL(1,𝔽₂₅)	5