Subgroups — #14299664

Order 1920.

504 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	#11751338	C₂	1	1	yes	maximal, characteristic
3	#11751335	C₂	1	1	yes	maximal, characteristic
4	#11751225	C₂	1	2	yes	maximal, complemented
5	#11746379	C₂	1	2	yes	maximal, complemented
6	#10602820	—	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	C₂×#6710	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	2	yes	—
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	#91120	C₄	1	2	yes	—
15	#91117	C₄	1	2	yes	—
16	#91114	—	2	2	no	—
17	#91106	—	2	4	no	—
18	#72027	—	5	1	no	maximal, Hall
19	C₅×#568	D₃	1	1	yes	characteristic
20	#65526	—	3	1	no	—
21	#65523	—	3	1	no	—
22	#65413	—	3	2	no	—
23	#64778	—	3	2	no	—
24	C₃×C₅×#41	—	2	2	no	—
25	C₃×C₅×#41	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×D₄	Q₈	1	2	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×M₁₆	—	2	4	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₃×C₅×M₁₆	D₄	1	4	yes	—
34	C₂×C₃×C₅×C₈	—	2	4	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	#6710	—	2	2	no	—
37	#6710	—	2	2	no	—
38	#6710	C₂×C₄	1	4	yes	—
39	C₂×C₃⋊#168	—	2	2	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	—	2	1	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₃×#568	D₅	1	1	yes	characteristic
44	#5237	—	5	1	no	—
45	#5234	—	5	1	no	—
46	#5124	—	5	2	no	—
47	#4558	—	5	2	no	—
48	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
49	C₅×#131	—	2	4	no	—
50	C₂×C₅×M₁₆	—	2	2	no	—
51	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
52	C₂×#681	—	3	2	no	—
53	C₂×#681	—	3	1	no	—
54	C₂²×#168	—	3	1	no	—
55	#3968	—	3	2	no	—
56	#3965	—	3	2	no	—
57	C₂²⋊#168	—	6	2	no	—
58	#3954	—	6	4	no	—
59	#1858	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
61	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×Q₈	—	2	2	no	—
63	C₃×C₅×D₄	—	2	4	no	—
64	C₃×C₅×D₄	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	2	yes	—
72	C₃×C₅×C₈	—	2	4	no	—
73	C₃⋊#168	—	2	4	no	—
74	C₃⋊#168	—	4	1	no	—
75	C₃⋊#168	—	4	1	no	—
76	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
77	C₃×#131	—	2	4	no	—
78	C₂×C₃×M₁₆	—	2	2	no	—
79	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
80	C₂×#208	—	5	2	no	—
81	C₂×#208	—	5	1	no	—
82	C₂²×C₃⋊C₈	—	5	1	no	—
83	#836	—	5	2	no	—
84	#833	—	5	2	no	—
85	#830	—	10	2	no	—
86	#822	—	10	4	no	—
87	C₅×#41	—	2	2	no	—
88	C₅×#41	—	2	2	no	—
89	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
90	C₂×C₅×D₄	Dic₆	1	2	yes	—
91	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	#67	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₅×M₁₆	—	2	4	no	—
95	C₅×M₁₆	—	2	2	no	—
96	C₅×M₁₆	#67	1	4	yes	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	#681	—	3	4	no	—
100	#681	—	6	2	no	—
101	#681	—	6	2	no	—
102	C₂×#168	—	6	2	no	—
103	C₂×#168	—	6	2	no	—
104	C₂×#168	—	6	1	no	—
105	C₂×#168	—	6	1	no	—
106	#568	D₁₅	1	1	yes	characteristic
107	#430	—	15	2	no	—
108	#413	—	15	1	no	—
109	#408	—	15	1	no	—
110	#341	—	15	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	4	1	no	—
113	C₂²×C₃×C₅	—	4	2	no	—
114	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#120	1	1	yes	characteristic
119	C₃×C₄×C₅	#123	1	1	yes	characteristic
120	C₃×#41	—	2	2	no	—
121	C₃×#41	—	2	2	no	—
122	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
123	C₂×C₃×D₄	Dic₁₀	1	2	yes	—
124	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₃×M₁₆	—	2	4	no	—
128	C₃×M₁₆	—	2	2	no	—
129	C₃×M₁₆	C₂²⋊D₅	1	4	yes	—
130	C₂×C₃×C₈	—	2	4	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	#208	—	5	4	no	—
133	#208	—	10	2	no	—
134	#208	—	10	2	no	—
135	C₂×C₃⋊C₈	—	10	2	no	—
136	C₂×C₃⋊C₈	—	10	2	no	—
137	C₂×C₃⋊C₈	—	10	1	no	—
138	C₂×C₃⋊C₈	—	10	1	no	—
139	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
140	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
141	C₅×Q₈	—	2	2	no	—
142	C₅×D₄	—	2	4	no	—
143	C₅×D₄	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	#235	1	2	yes	—
149	C₂×C₄×C₅	C₂×#67	1	2	yes	—
150	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
151	C₅×C₈	—	2	4	no	—
152	#168	—	6	4	no	—
153	#168	—	12	1	no	—
154	#168	—	12	1	no	—
155	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	15	2	no	—
159	C₂×M₁₆	—	15	1	no	—
160	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
161	C₂²×C₈	—	15	1	no	—
162	#108	—	30	4	no	—
163	#101	—	15	2	no	—
164	#100	—	15	2	no	—
165	C₂²⋊C₈	—	30	2	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	—	2	2	no	—
168	C₂×C₃×C₅	—	4	2	no	—
169	C₂×C₃×C₅	#385	1	1	yes	characteristic
170	C₂³×C₃	#683	1	2	yes	—
171	C₂³×C₃	C₂×Dic₁₀	1	1	yes	characteristic
172	C₃×Q₈	—	2	2	no	—
173	C₃×D₄	—	2	4	no	—
174	C₃×D₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	#683	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	#709	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₃×C₈	—	2	4	no	—
183	C₃⋊C₈	—	10	4	no	—
184	C₃⋊C₈	—	20	1	no	—
185	C₃⋊C₈	—	20	1	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	—	4	2	no	—
189	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	#924	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	C₄×#67	1	2	yes	—
193	C₄×C₅	#929	1	1	yes	characteristic
194	C₄×C₅	#953	1	1	yes	characteristic
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₄×D₁₅	1	1	yes	characteristic
198	C₂×D₄	Dic₃₀	1	2	yes	—
199	C₂×D₄	C₄×D₁₅	1	1	yes	characteristic
200	C₂²×C₄	#1220	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	—	15	4	no	—
205	M₁₆	—	30	2	no	—
206	M₁₆	—	30	2	no	—
207	M₁₆	#1220	1	4	yes	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	30	2	no	—
211	C₂×C₈	—	30	2	no	—
212	C₂×C₈	—	30	1	no	—
213	C₂×C₈	—	30	1	no	—
214	C₃×C₅	#1858	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	C₂×#683	1	1	yes	characteristic
219	C₂²×C₃	#4070	1	2	yes	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
222	C₃×C₄	#4075	1	1	yes	characteristic
223	C₃×C₄	#4099	1	1	yes	characteristic
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	#5210	1	1	yes	characteristic
228	C₂³	#6712	1	2	yes	—
229	C₂³	C₂×Dic₃₀	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	#6712	1	2	yes	—
236	C₂×C₄	C₂×C₄×D₁₅	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	—	30	4	no	—
242	C₈	—	60	1	no	—
243	C₈	—	60	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	#65499	1	1	yes	characteristic
248	C₅	#72027	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₂×#6712	1	1	yes	characteristic
253	C₂²	#91813	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	C₄×#1220	1	2	yes	—
256	C₄	#91818	1	1	yes	characteristic
257	C₄	#91842	1	1	yes	characteristic
258	C₃	#10602820	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	2	no	—
261	C₂	—	4	2	no	—
262	C₂	#11751311	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91842
commutator	C₂×C₃×C₄×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	#1858
LCS	G ▷ C₂×C₃×C₄×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. #11751225	C₂	semidirect
5. #11746379	C₂	semidirect
214. C₃×C₅	#1858	semidirect
248. C₅	#72027	semidirect
258. C₃	#10602820	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
#11751338	2
#11751335	2
#11751225	2
#11746379	2
#10602820	3
#72027	5