Subgroups — #14307408

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756229	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
3	#11751025	C₂	1	1	yes	maximal, characteristic
4	#11746344	C₂	1	4	yes	maximal
5	#10600189	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	—	2	2	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91815	C₂²	1	2	yes	—
10	#91813	—	2	2	no	—
11	C₂×#6710	C₂²	1	2	yes	—
12	#91759	C₄	1	2	yes	complemented
13	#91758	C₄	1	2	yes	complemented
14	C₄×Dic₃₀	—	2	2	no	—
15	C₄⋊(C₃⋊#168)	—	2	2	no	—
16	#69396	—	5	1	no	maximal, Hall
17	#65804	—	3	1	no	—
18	C₄³×C₅	D₃	1	1	yes	characteristic
19	#65213	—	3	1	no	—
20	#64743	—	3	4	no	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₃₀	—	2	2	no	—
25	C₃×C₄²×C₅	—	2	8	no	—
26	C₃×C₄²×C₅	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
31	#6719	—	2	2	no	—
32	C₄⋊Dic₁₅	—	2	2	no	—
33	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
34	#6712	—	2	4	no	—
35	C₄×Dic₁₅	—	2	2	no	—
36	#6710	—	2	4	no	—
37	C₂×C₃⋊#168	—	2	2	no	—
38	#5470	—	5	1	no	—
39	C₃×C₄³	D₅	1	1	yes	characteristic
40	#4924	—	5	1	no	—
41	#4523	—	5	4	no	—
42	C₂×C₄²×C₅	—	2	2	no	—
43	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	#4072	—	3	2	no	—
46	#4070	—	6	2	no	—
47	C₂×#681	—	3	2	no	—
48	#4016	—	3	2	no	—
49	#4015	—	3	2	no	—
50	C₄×Dic₁₀	—	6	2	no	—
51	C₄⋊#168	—	6	2	no	—
52	#1994	—	15	1	no	Sylow
53	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	8	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×Dic₁₅	—	2	4	no	—
64	Dic₃₀	—	4	4	no	—
65	C₃⋊#168	—	4	2	no	—
66	C₂×C₃×C₄²	—	2	2	no	—
67	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
68	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
69	#926	—	5	2	no	—
70	#924	—	10	2	no	—
71	C₂×#208	—	5	2	no	—
72	#870	—	5	2	no	—
73	#869	—	5	2	no	—
74	C₄×Dic₆	—	10	2	no	—
75	C₄⋊(C₃⋊C₈)	—	10	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	#67	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₂×Dic₁₀	—	6	2	no	—
80	C₄²×C₅	—	2	8	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	Dic₆	1	4	yes	complemented
83	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
84	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
85	C₄²×C₅	D₁₂	1	2	yes	—
86	C₂²⋊Dic₅	—	6	2	no	—
87	C₄⋊Dic₅	—	3	4	no	—
88	C₄⋊Dic₅	—	6	2	no	—
89	#683	—	6	4	no	—
90	C₄×Dic₅	—	6	2	no	—
91	#681	—	6	4	no	—
92	C₂×#168	—	6	2	no	—
93	#533	—	15	1	no	—
94	#423	—	15	1	no	—
95	C₄³	D₁₅	1	1	yes	characteristic
96	#339	—	15	4	no	—
97	C₂²×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
100	C₃×C₄×C₅	—	2	8	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	#104	1	4	yes	—
103	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
104	C₃×C₄×C₅	#124	1	1	yes	characteristic
105	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
106	Dic₁₅	—	4	4	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
109	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₂×Dic₆	—	10	2	no	—
111	C₃×C₄²	—	2	8	no	—
112	C₃×C₄²	—	2	2	no	—
113	C₃×C₄²	Dic₁₀	1	4	yes	complemented
114	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
115	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
116	C₃×C₄²	D₂₀	1	2	yes	—
117	#217	—	10	2	no	—
118	C₄⋊Dic₃	—	5	4	no	—
119	C₄⋊Dic₃	—	10	2	no	—
120	Dic₃⋊C₄	—	10	4	no	—
121	C₄×Dic₃	—	10	2	no	—
122	#208	—	10	4	no	—
123	C₂×C₃⋊C₈	—	10	2	no	—
124	C₂³×C₅	C₂×#67	1	1	yes	characteristic
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₄×C₅	#235	1	2	yes	—
134	C₂×Dic₅	—	6	4	no	—
135	Dic₁₀	—	12	4	no	—
136	#168	—	12	2	no	—
137	C₂×M₁₆	—	15	2	no	—
138	C₄⋊Q₈	—	15	2	no	—
139	#125	—	15	2	no	—
140	C₂²⋊Q₈	—	30	2	no	—
141	C₄×Q₈	—	30	2	no	—
142	#117	—	15	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂×D₁₅	1	2	yes	—
145	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
146	C₄⋊C₈	—	30	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	#391	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#104	1	2	yes	—
150	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	#685	1	2	yes	—
155	C₂×C₃×C₄	#685	1	2	yes	—
156	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
159	C₂×C₃×C₄	#709	1	2	yes	—
160	C₂×Dic₃	—	10	4	no	—
161	Dic₆	—	20	4	no	—
162	C₃⋊C₈	—	20	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	#924	1	2	yes	—
165	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₄×C₅	—	2	8	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	#805	1	4	yes	—
169	C₄×C₅	C₄×Dic₆	1	2	yes	—
170	C₄×C₅	#869	1	1	yes	characteristic
171	C₄×C₅	#875	1	1	yes	characteristic
172	Dic₅	—	12	4	no	—
173	C₂×Q₈	—	30	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	#1220	1	2	yes	—
176	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
177	M₁₆	—	30	4	no	—
178	C₂×C₈	—	30	2	no	—
179	C₄⋊C₄	—	15	4	no	—
180	C₄⋊C₄	—	30	4	no	—
181	C₄⋊C₄	—	30	2	no	—
182	C₂²⋊C₄	—	30	2	no	—
183	C₄²	—	2	8	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	30	2	no	—
186	C₄²	Dic₃₀	1	4	yes	complemented
187	C₄²	C₄×D₁₅	1	1	yes	characteristic
188	C₄²	C₄×D₁₅	1	1	yes	characteristic
189	C₄²	D₆₀	1	2	yes	—
190	C₃×C₅	#1994	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	#4070	1	2	yes	—
193	C₂²×C₃	C₂×#685	1	1	yes	characteristic
194	C₃×C₄	—	2	8	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	#3937	1	4	yes	—
197	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
198	C₃×C₄	#4015	1	1	yes	characteristic
199	C₃×C₄	#4021	1	1	yes	characteristic
200	Dic₃	—	20	4	no	—
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	C₂×#805	1	2	yes	—
203	C₂×C₅	#4929	1	1	yes	characteristic
204	C₂³	C₂×#1220	1	1	yes	characteristic
205	Q₈	—	60	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	30	4	no	—
210	C₂×C₄	#6714	1	2	yes	—
211	C₂×C₄	#6714	1	2	yes	—
212	C₂×C₄	C₂×Dic₃₀	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
214	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
215	C₂×C₄	#6817	1	2	yes	—
216	C₈	—	60	2	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	C₂×#3937	1	2	yes	—
219	C₂×C₃	#65218	1	1	yes	characteristic
220	C₅	#69396	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	#91813	1	2	yes	—
223	C₂²	C₂×#6714	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	2	2	no	—
226	C₄	—	60	4	no	—
227	C₄	#91089	1	4	yes	—
228	C₄	C₄×Dic₃₀	1	2	yes	—
229	C₄	#91758	1	1	yes	characteristic
230	C₄	#91764	1	1	yes	characteristic
231	C₃	#10600189	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	2	no	—
233	C₂	C₂×#91089	1	2	yes	—
234	C₂	#11751030	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄³×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	#1994
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
12. #91759	C₄	semidirect
13. #91758	C₄	semidirect
30. C₃×C₄²×C₅	Q₈	semidirect
82. C₄²×C₅	Dic₆	semidirect
113. C₃×C₄²	Dic₁₀	semidirect
186. C₄²	Dic₃₀	semidirect
190. C₃×C₅	#1994	semidirect
220. C₅	#69396	semidirect
231. C₃	#10600189	semidirect

Maximal subgroups

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

subgroup	index
#11756229	2
C₃×C₄³×C₅	2
#11751025	2
#11746344	2
#10600189	3
#69396	5