Subgroups — #14300360

Order 1920.

520 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	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	#11751341	C₂	1	1	yes	maximal, characteristic
3	#11751335	C₂	1	1	yes	maximal, characteristic
4	#11751220	C₂	1	2	yes	maximal, complemented
5	#11746377	C₂	1	2	yes	maximal, complemented
6	#10602819	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#131	—	2	4	no	—
9	C₂×C₃×C₅×M₁₆	—	2	2	no	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	#91815	C₂²	1	2	yes	—
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	#91118	C₄	1	4	yes	—
15	#91114	—	2	2	no	—
16	#91098	—	2	4	no	—
17	#72026	—	5	1	no	maximal, Hall
18	C₅×#568	D₃	1	1	yes	characteristic
19	#65529	—	3	1	no	—
20	#65523	—	3	1	no	—
21	#65408	—	3	2	no	—
22	#64776	—	3	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₃×C₅×#41	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
26	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	4	no	—
34	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
35	#6719	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₁₅	—	2	2	no	—
37	C₄×Dic₁₅	—	2	2	no	—
38	#6710	—	2	2	no	—
39	C₂×C₃⋊#168	—	2	2	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	—	2	1	no	—
42	C₃×#568	D₅	1	1	yes	characteristic
43	#5240	—	5	1	no	—
44	#5234	—	5	1	no	—
45	#5119	—	5	2	no	—
46	#4556	—	5	2	no	—
47	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
48	C₅×#131	—	2	4	no	—
49	C₂×C₅×M₁₆	—	2	2	no	—
50	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
51	#4072	—	3	2	no	—
52	C₂×#681	—	3	1	no	—
53	C₂²×#168	—	3	1	no	—
54	#3966	—	3	4	no	—
55	C₂²⋊#168	—	6	2	no	—
56	#3946	—	6	4	no	—
57	#1868	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
59	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₅×Q₈	—	2	2	no	—
61	C₃×C₅×D₄	—	2	4	no	—
62	C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	2	yes	—
70	C₂×Dic₁₅	—	2	4	no	—
71	C₃×C₅×C₈	—	2	4	no	—
72	C₃⋊#168	—	4	1	no	—
73	C₃⋊#168	—	4	1	no	—
74	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
75	C₃×#131	—	2	4	no	—
76	C₂×C₃×M₁₆	—	2	2	no	—
77	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
78	#926	—	5	2	no	—
79	C₂×#208	—	5	1	no	—
80	C₂²×C₃⋊C₈	—	5	1	no	—
81	#834	—	5	4	no	—
82	#830	—	10	2	no	—
83	Dic₃⋊C₈	—	10	4	no	—
84	C₅×#41	—	2	2	no	—
85	C₅×#41	—	2	2	no	—
86	C₂×C₅×Q₈	Dic₆	1	1	yes	characteristic
87	C₂×C₅×D₄	Dic₆	1	1	yes	characteristic
88	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
89	C₂²×C₄×C₅	#67	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	#67	1	4	yes	—
96	C₂²⋊Dic₅	—	3	4	no	—
97	C₄⋊Dic₅	—	6	2	no	—
98	C₄×Dic₅	—	6	2	no	—
99	#681	—	6	2	no	—
100	C₂×#168	—	6	2	no	—
101	C₂×#168	—	6	1	no	—
102	C₂×#168	—	6	1	no	—
103	#568	D₁₅	1	1	yes	characteristic
104	#424	—	15	2	no	—
105	#410	—	15	1	no	—
106	#408	—	15	1	no	—
107	#344	—	15	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	2	no	—
111	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	#120	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
116	C₃×C₄×C₅	#123	1	1	yes	characteristic
117	Dic₁₅	—	4	4	no	—
118	C₃×#41	—	2	2	no	—
119	C₃×#41	—	2	2	no	—
120	C₂×C₃×Q₈	Dic₁₀	1	1	yes	characteristic
121	C₂×C₃×D₄	Dic₁₀	1	1	yes	characteristic
122	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
123	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
124	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
125	C₃×M₁₆	—	2	4	no	—
126	C₃×M₁₆	—	2	2	no	—
127	C₃×M₁₆	—	2	2	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
130	#217	—	5	4	no	—
131	C₄⋊Dic₃	—	10	2	no	—
132	C₄×Dic₃	—	10	2	no	—
133	#208	—	10	2	no	—
134	C₂×C₃⋊C₈	—	10	2	no	—
135	C₂×C₃⋊C₈	—	10	1	no	—
136	C₂×C₃⋊C₈	—	10	1	no	—
137	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
138	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
139	C₅×Q₈	—	2	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
145	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
146	C₂×C₄×C₅	#235	1	2	yes	—
147	C₂×C₄×C₅	C₂×#67	1	2	yes	—
148	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
149	C₂×Dic₅	—	6	4	no	—
150	C₅×C₈	—	2	4	no	—
151	#168	—	12	1	no	—
152	#168	—	12	1	no	—
153	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
154	#131	—	2	4	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	15	1	no	—
157	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
158	C₂²×C₈	—	15	1	no	—
159	#117	—	15	2	no	—
160	C₄⋊C₈	—	30	4	no	—
161	#99	—	15	4	no	—
162	C₂²⋊C₈	—	30	2	no	—
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	—	2	1	no	—
165	C₂×C₃×C₅	—	4	2	no	—
166	C₂×C₃×C₅	#385	1	1	yes	characteristic
167	C₂³×C₃	#683	1	2	yes	—
168	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
169	C₃×Q₈	—	2	2	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	#683	1	2	yes	—
175	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
176	C₂×C₃×C₄	#709	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
179	C₂×Dic₃	—	10	4	no	—
180	C₃×C₈	—	2	4	no	—
181	C₃⋊C₈	—	20	1	no	—
182	C₃⋊C₈	—	20	1	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
187	C₂²×C₅	C₄×#67	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	#924	1	2	yes	—
190	C₄×C₅	#929	1	1	yes	characteristic
191	C₄×C₅	#953	1	1	yes	characteristic
192	Dic₅	—	12	4	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	Dic₃₀	1	1	yes	characteristic
196	C₂×D₄	Dic₃₀	1	1	yes	characteristic
197	C₂×D₄	C₄×D₁₅	1	2	yes	—
198	C₂²×C₄	#1220	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	30	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	30	1	no	—
206	C₂×C₈	—	30	1	no	—
207	C₂×C₈	—	30	2	no	—
208	C₂×C₈	#1220	1	4	yes	—
209	C₄⋊C₄	—	30	2	no	—
210	C₂²⋊C₄	—	15	4	no	—
211	C₄²	—	30	2	no	—
212	C₃×C₅	#1868	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	C₂×#683	1	1	yes	characteristic
217	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	#4070	1	2	yes	—
220	C₃×C₄	#4075	1	1	yes	characteristic
221	C₃×C₄	#4099	1	1	yes	characteristic
222	Dic₃	—	20	4	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	4	2	no	—
226	C₂×C₅	#5210	1	1	yes	characteristic
227	C₂³	#6712	1	2	yes	—
228	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	30	4	no	—
235	C₂×C₄	#6712	1	2	yes	—
236	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
237	C₂×C₄	#6817	1	2	yes	—
238	C₂×C₄	C₂×#1220	1	2	yes	—
239	C₂×C₄	C₂×#1220	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	60	1	no	—
242	C₈	—	60	1	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	#65499	1	1	yes	characteristic
247	C₅	#72026	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×#6712	1	1	yes	characteristic
252	C₂²	C₄×#1220	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	60	4	no	—
255	C₄	#91813	1	2	yes	—
256	C₄	#91818	1	1	yes	characteristic
257	C₄	#91842	1	1	yes	characteristic
258	C₃	#10602819	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	2	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	#11751311	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91842
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#568	D₁₅
hypocenter	C₃×C₅	#1868
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
4. #11751220	C₂	semidirect
5. #11746377	C₂	semidirect
212. C₃×C₅	#1868	semidirect
247. C₅	#72026	semidirect
258. C₃	#10602819	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#568	2
#11751341	2
#11751335	2
#11751220	2
#11746377	2
#10602819	3
#72026	5