Subgroups — #14280456

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11750786	C₂	1	1	yes	maximal, characteristic
2	C₄×#6648	C₂	1	1	yes	maximal, characteristic
3	#11746937	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64674	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	#10598840	—	3	1	no	maximal, Hall
9	C₄²×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×#6648	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91104	C₄	1	2	yes	—
13	#91098	C₄	1	2	yes	—
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×#168	C₂²	1	1	yes	characteristic
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	#90983	C₄	1	2	yes	—
20	#90957	C₄	1	2	yes	—
21	#90951	C₄	1	2	yes	—
22	#68068	—	5	1	no	maximal, Hall
23	#64974	—	3	1	no	—
24	#64974	—	3	1	no	—
25	C₄×C₈×D₅	—	3	1	no	—
26	C₅×C₈²	—	3	1	no	—
27	#64674	—	3	1	no	—
28	#64674	D₃	1	1	yes	characteristic, complemented
29	C₈×#168	—	3	1	no	—
30	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₅	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	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
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	#6648	C₈	1	8	yes	complemented
44	#4685	—	5	1	no	—
45	#4685	—	5	1	no	—
46	C₄×D₃×C₈	—	5	1	no	—
47	C₃×#322	—	5	1	no	—
48	#4455	—	5	1	no	—
49	C₈×C₃⋊C₈	—	5	1	no	—
50	C₈×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
51	C₂×C₈×D₅	—	3	2	no	—
52	C₄²×D₅	—	3	1	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×C₅×C₈	—	3	1	no	—
55	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	#3952	—	3	2	no	—
57	#3952	—	3	2	no	—
58	#3946	—	3	2	no	—
59	#3946	—	3	2	no	—
60	C₈×Dic₅	—	3	2	no	—
61	C₄×#168	—	3	1	no	—
62	C₄×#168	C₂×D₃	1	1	yes	characteristic
63	C₄×#168	C₂×D₃	1	1	yes	characteristic
64	#1450	—	15	1	no	Sylow
65	C₂²×D₁₅	C₄²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	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×Dic₁₅	C₄²	1	1	yes	characteristic
70	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
71	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
72	C₄×D₁₅	—	2	2	no	—
73	C₄×D₁₅	—	2	2	no	—
74	C₄×D₁₅	C₂×C₈	1	4	yes	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃⋊#168	—	2	2	no	—
77	C₃×#168	—	2	2	no	—
78	C₃×#168	C₂×C₈	1	4	yes	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₅×C₃⋊C₈	C₂×C₈	1	4	yes	—
81	C₂×D₃×C₈	—	5	2	no	—
82	C₄²×D₃	—	5	1	no	—
83	C₃×C₄×C₈	—	5	1	no	—
84	C₃×C₄×C₈	—	5	1	no	—
85	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
86	(C₂×D₃)⋊C₈	—	5	2	no	—
87	(C₂×D₃)⋊C₈	—	5	2	no	—
88	Dic₃⋊C₈	—	5	2	no	—
89	Dic₃⋊C₈	—	5	2	no	—
90	C₈×Dic₃	—	5	2	no	—
91	C₄×C₃⋊C₈	—	5	1	no	—
92	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×D₅	—	3	1	no	—
95	C₂×C₄×D₅	—	3	1	no	—
96	C₂×C₄×D₅	—	3	1	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
100	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₄×Dic₅	—	3	1	no	—
102	C₄×Dic₅	—	3	1	no	—
103	C₄×Dic₅	—	3	1	no	—
104	C₂×#168	—	3	2	no	—
105	C₂×#168	C₄×D₃	1	2	yes	—
106	C₂×#168	C₄×D₃	1	2	yes	—
107	C₈×D₅	—	3	8	no	—
108	#431	—	15	1	no	—
109	#431	—	15	1	no	—
110	C₂×C₄×C₈	—	15	1	no	—
111	#322	—	5	1	no	—
112	#322	—	15	1	no	—
113	C₈²	—	3	1	no	—
114	C₈²	—	15	1	no	—
115	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
116	C₂×D₁₅	—	2	1	no	—
117	C₂×D₁₅	—	2	1	no	—
118	C₂×D₁₅	C₄×C₈	1	2	yes	—
119	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	Dic₁₅	—	2	1	no	—
123	Dic₁₅	—	2	1	no	—
124	Dic₁₅	C₄×C₈	1	2	yes	—
125	C₂×C₄×D₃	—	5	1	no	—
126	C₂×C₄×D₃	—	5	1	no	—
127	C₂×C₄×D₃	—	5	1	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	—	5	2	no	—
130	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
131	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
132	C₄×Dic₃	—	5	1	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	1	no	—
135	C₂×C₃⋊C₈	—	5	2	no	—
136	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
137	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
138	D₃×C₈	—	5	8	no	—
139	C₂²×D₅	—	3	1	no	—
140	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₄×D₅	—	3	4	no	—
147	C₄×D₅	—	6	2	no	—
148	C₄×D₅	—	6	2	no	—
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	3	4	no	—
151	C₅×C₈	—	6	2	no	—
152	#168	—	2	2	no	—
153	#168	—	6	2	no	—
154	#168	D₃×C₈	1	4	yes	—
155	C₂²×C₈	—	15	2	no	—
156	C₂×C₄²	—	15	1	no	—
157	C₄⋊C₈	—	15	2	no	—
158	C₄⋊C₈	—	15	2	no	—
159	C₂²⋊C₈	—	15	2	no	—
160	C₂²⋊C₈	—	15	2	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	5	1	no	—
164	C₄×C₈	—	5	1	no	—
165	C₄×C₈	—	15	2	no	—
166	C₄×C₈	—	15	1	no	—
167	C₄×C₈	D₃×D₅	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂×C₃×C₅	#405	1	1	yes	characteristic
170	C₂×C₃×C₅	#405	1	1	yes	characteristic
171	D₁₅	—	2	2	no	—
172	C₂²×D₃	—	5	1	no	—
173	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	1	no	—
177	C₂×Dic₃	—	5	1	no	—
178	C₂×Dic₃	—	5	1	no	—
179	C₄×D₃	—	5	4	no	—
180	C₄×D₃	—	10	2	no	—
181	C₄×D₃	—	10	2	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	5	4	no	—
184	C₃×C₈	—	10	2	no	—
185	C₃⋊C₈	—	2	2	no	—
186	C₃⋊C₈	—	10	2	no	—
187	C₃⋊C₈	C₈×D₅	1	4	yes	—
188	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
189	C₂×D₅	—	3	2	no	—
190	C₂×D₅	—	6	1	no	—
191	C₂×D₅	—	6	1	no	—
192	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
193	C₄×C₅	#907	1	2	yes	—
194	C₄×C₅	#907	1	2	yes	—
195	Dic₅	—	3	2	no	—
196	Dic₅	—	6	1	no	—
197	Dic₅	—	6	1	no	—
198	C₂²×C₄	—	15	1	no	—
199	C₂²×C₄	—	15	1	no	—
200	C₂²×C₄	—	15	1	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	15	8	no	—
206	C₂×C₈	—	15	2	no	—
207	C₂×C₈	#1200	1	2	yes	—
208	C₄²	—	15	1	no	—
209	C₄²	—	15	1	no	—
210	C₄²	—	15	1	no	—
211	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
212	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
214	C₂×D₃	—	5	2	no	—
215	C₂×D₃	—	10	1	no	—
216	C₂×D₃	—	10	1	no	—
217	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
218	C₃×C₄	#4047	1	2	yes	—
219	C₃×C₄	#4047	1	2	yes	—
220	Dic₃	—	5	2	no	—
221	Dic₃	—	10	1	no	—
222	Dic₃	—	10	1	no	—
223	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
224	C₂×C₅	#4708	1	1	yes	characteristic
225	C₂×C₅	#4708	1	1	yes	characteristic
226	D₅	—	6	2	no	—
227	C₂³	—	15	1	no	—
228	C₂×C₄	—	15	1	no	—
229	C₂×C₄	—	15	4	no	—
230	C₂×C₄	—	15	1	no	—
231	C₂×C₄	—	15	1	no	—
232	C₂×C₄	—	30	2	no	—
233	C₂×C₄	—	30	2	no	—
234	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
235	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×#1200	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	3	4	no	—
239	C₈	—	5	4	no	—
240	C₈	—	6	2	no	—
241	C₈	—	10	2	no	—
242	C₈	—	30	2	no	—
243	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
244	C₂×C₃	#64978	1	1	yes	characteristic
245	C₂×C₃	#64978	1	1	yes	characteristic
246	D₃	—	10	2	no	—
247	C₅	#68068	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	15	2	no	—
249	C₂²	—	30	1	no	—
250	C₂²	—	30	1	no	—
251	C₂²	C₄×#1200	1	1	yes	characteristic
252	C₄	—	15	2	no	—
253	C₄	—	30	1	no	—
254	C₄	—	30	1	no	—
255	C₄	D₃×C₈×D₅	1	2	yes	—
256	C₄	#91267	1	2	yes	—
257	C₄	#91291	1	2	yes	—
258	C₃	#10598840	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	30	2	no	—
260	C₂	#11746982	1	1	yes	characteristic
261	C₂	C₈×#1200	1	1	yes	characteristic
262	C₂	#11748054	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×D₁₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1450
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. C₅×C₈×C₃⋊C₈	C₂	semidirect
5. C₃×#64674	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
28. #64674	D₃	semidirect
43. #6648	C₈	semidirect
50. C₈×C₃⋊C₈	D₅	semidirect
212. C₃×C₅	#1450	semidirect
247. C₅	#68068	semidirect
258. C₃	#10598840	semidirect

Maximal subgroups

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

subgroup	index
#11750786	2
C₄×#6648	2
#11746937	2
C₅×C₈×C₃⋊C₈	2
C₃×#64674	2
#11745831	2
C₃⋊C₈×#168	2
#10598840	3
#68068	5