Subgroups — #14298713

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊#168)	C₂	1	2	yes	maximal
3	#11746345	C₂	1	4	yes	maximal
4	#10600185	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊#168)	—	2	4	no	—
12	#69392	—	5	1	no	maximal, Hall
13	C₄³×C₅	D₃	1	1	yes	characteristic
14	C₂×C₄⋊#168	—	3	2	no	—
15	#64744	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₃×C₄²×C₅	—	2	4	no	—
22	C₃×C₄²×C₅	—	2	2	no	—
23	C₃×C₄²×C₅	—	2	4	no	—
24	C₃×C₄²×C₅	C₈	1	4	yes	complemented
25	C₃×C₄²×C₅	D₄	1	2	yes	—
26	C₃×C₄²×C₅	Q₈	1	2	yes	—
27	C₂×C₃⋊#168	—	2	4	no	—
28	C₂×C₃⋊#168	—	2	8	no	—
29	C₃×C₄³	D₅	1	1	yes	characteristic
30	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
31	#4524	—	5	4	no	—
32	C₂×C₄²×C₅	Dic₃	1	2	yes	—
33	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
34	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
36	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
37	C₂²×#168	—	3	4	no	—
38	C₄⋊#168	—	6	4	no	—
39	#1831	—	15	1	no	Sylow
40	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	—	2	2	no	—
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
49	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
50	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
51	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	#41	1	4	yes	—
55	C₃⋊#168	—	4	4	no	—
56	C₂×C₃×C₄²	Dic₅	1	2	yes	—
57	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
58	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
60	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	—	5	4	no	—
62	C₄⋊(C₃⋊C₈)	—	10	4	no	—
63	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
64	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
65	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
68	C₄²×C₅	—	2	4	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	4	no	—
71	C₄²×C₅	C₃⋊C₈	1	4	yes	complemented
72	C₄²×C₅	Dic₆	1	2	yes	—
73	C₄²×C₅	D₁₂	1	2	yes	—
74	C₂×#168	—	6	4	no	—
75	C₂×#168	—	6	8	no	—
76	C₂×C₄⋊C₈	—	15	2	no	—
77	C₄³	D₁₅	1	1	yes	characteristic
78	#336	—	15	4	no	—
79	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₃×C₅	#117	1	2	yes	—
81	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	2	4	no	—
86	C₃×C₄×C₅	—	2	4	no	—
87	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	—
88	C₃×C₄×C₅	#124	1	2	yes	—
89	C₃×C₄×C₅	#125	1	2	yes	—
90	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
91	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
95	C₃×C₄²	—	2	4	no	—
96	C₃×C₄²	—	2	2	no	—
97	C₃×C₄²	—	2	4	no	—
98	C₃×C₄²	#168	1	4	yes	complemented
99	C₃×C₄²	Dic₁₀	1	2	yes	—
100	C₃×C₄²	D₂₀	1	2	yes	—
101	C₂×C₃⋊C₈	—	10	4	no	—
102	C₂×C₃⋊C₈	—	10	8	no	—
103	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
110	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
111	C₂×C₄×C₅	#208	1	4	yes	—
112	C₂×C₄×C₅	#208	1	2	yes	—
113	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
114	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
115	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
117	C₂×C₄×C₅	#235	1	4	yes	—
118	#168	—	12	4	no	—
119	C₂²×C₈	—	15	4	no	—
120	C₂×C₄²	Dic₁₅	1	2	yes	—
121	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
122	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
123	C₂×C₄²	C₂×D₁₅	1	2	yes	—
124	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
125	C₄⋊C₈	—	30	4	no	—
126	C₂×C₃×C₅	#382	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₅	#423	1	1	yes	characteristic
129	C₂×C₃×C₅	#431	1	2	yes	—
130	C₂×C₃×C₅	#432	1	2	yes	—
131	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	C₂×#168	1	4	yes	—
138	C₂×C₃×C₄	C₂×#168	1	2	yes	—
139	C₂×C₃×C₄	#681	1	4	yes	—
140	C₂×C₃×C₄	#681	1	2	yes	—
141	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
142	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
143	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
145	C₂×C₃×C₄	#709	1	4	yes	—
146	C₃⋊C₈	—	20	4	no	—
147	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×#208	1	1	yes	characteristic
149	C₂²×C₅	C₂×#208	1	2	yes	—
150	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
151	C₂²×C₅	#926	1	2	yes	—
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	C₄⋊(C₃⋊C₈)	1	4	yes	—
156	C₄×C₅	#870	1	2	yes	—
157	C₄×C₅	#875	1	2	yes	—
158	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
159	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
160	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
163	C₂×C₈	—	30	4	no	—
164	C₂×C₈	—	30	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	C₃⋊#168	1	4	yes	complemented
169	C₄²	Dic₃₀	1	2	yes	—
170	C₄²	D₆₀	1	2	yes	—
171	C₃×C₅	#1831	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
173	C₂²×C₃	C₂×#681	1	1	yes	characteristic
174	C₂²×C₃	C₂×#681	1	2	yes	—
175	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
176	C₂²×C₃	#4072	1	2	yes	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	4	no	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	C₄⋊#168	1	4	yes	—
181	C₃×C₄	#4016	1	2	yes	—
182	C₃×C₄	#4021	1	2	yes	—
183	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
184	C₂×C₅	#4924	1	1	yes	characteristic
185	C₂×C₅	#4925	1	2	yes	—
186	C₂×C₅	#4926	1	2	yes	—
187	C₂×C₅	#4936	1	1	yes	characteristic
188	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
195	C₂×C₄	C₂×C₃⋊#168	1	2	yes	—
196	C₂×C₄	#6710	1	4	yes	—
197	C₂×C₄	#6710	1	2	yes	—
198	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
199	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
200	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
201	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
202	C₂×C₄	#6817	1	4	yes	—
203	C₈	—	60	4	no	—
204	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
205	C₂×C₃	#65213	1	1	yes	characteristic
206	C₂×C₃	#65214	1	2	yes	—
207	C₂×C₃	#65215	1	2	yes	—
208	C₂×C₃	#65225	1	1	yes	characteristic
209	C₅	#69392	1	1	yes	characteristic, complemented, Sylow
210	C₂²	C₂²×C₃⋊#168	1	1	yes	characteristic
211	C₂²	C₂×#6710	1	1	yes	characteristic
212	C₂²	C₂×#6710	1	2	yes	—
213	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
214	C₂²	#91815	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	4	no	—
218	C₄	C₄⋊(C₃⋊#168)	1	4	yes	—
219	C₄	#91759	1	2	yes	—
220	C₄	#91764	1	2	yes	—
221	C₃	#10600185	1	1	yes	characteristic, complemented, Sylow
222	C₂	C₂×C₄⋊(C₃⋊#168)	1	1	yes	characteristic
223	C₂	#11751025	1	1	yes	characteristic
224	C₂	#11751026	1	2	yes	—
225	C₂	#11751027	1	2	yes	—
226	C₂	#11751037	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	#1831
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
24. C₃×C₄²×C₅	C₈	semidirect
71. C₄²×C₅	C₃⋊C₈	semidirect
98. C₃×C₄²	#168	semidirect
168. C₄²	C₃⋊#168	semidirect
171. C₃×C₅	#1831	semidirect
209. C₅	#69392	semidirect
221. C₃	#10600185	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₄⋊(C₃⋊#168)	2
#11746345	2
#10600185	3
#69392	5