Subgroups — #14307360

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#870	C₂	1	1	yes	maximal, characteristic
2	C₂×#91087	C₂	1	1	yes	maximal, characteristic
3	C₃×#65224	C₂	1	1	yes	maximal, characteristic
4	#11745979	C₂	1	4	yes	maximal
5	#10602534	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊#168	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
9	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
10	C₅×#870	—	2	2	no	—
11	C₅×#870	C₄	1	4	yes	complemented
12	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	#91087	—	2	2	no	—
15	#69583	—	5	1	no	maximal, Hall
16	C₂×C₅×#125	—	3	1	no	—
17	#65224	D₃	1	1	yes	characteristic
18	C₂×#3935	—	3	1	no	—
19	#64749	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	—	2	2	no	—
23	C₂²×C₃×Dic₅	—	2	2	no	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₃⋊#168	—	2	2	no	—
27	C₂×C₃⋊#168	—	2	2	no	—
28	C₂×C₃⋊#168	Q₈	1	4	yes	—
29	C₅×C₄⋊Dic₃	—	2	2	no	—
30	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
31	C₅×Dic₃⋊C₄	—	2	8	no	—
32	C₃×C₄⋊Dic₅	—	2	4	no	—
33	C₃×C₄⋊Dic₅	—	4	4	no	—
34	C₂×#870	D₅	1	1	yes	characteristic
35	C₃×#384	—	5	1	no	—
36	C₂×#803	—	5	1	no	—
37	#4529	—	5	4	no	—
38	C₅×#125	—	3	4	no	—
39	C₅×#125	—	6	2	no	—
40	C₂×C₅×C₄⋊C₄	—	3	2	no	—
41	C₂×C₅×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅	—	2	2	no	—
44	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
45	C₂²×#168	—	3	2	no	—
46	#3935	—	6	2	no	—
47	#1992	—	15	1	no	Sylow
48	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
55	C₂×C₃×C₄×C₅	#41	1	2	yes	—
56	C₂×C₅×Dic₃	—	2	4	no	—
57	C₂×C₅×Dic₃	—	4	2	no	—
58	C₂×C₅×Dic₃	—	4	2	no	—
59	C₂×C₃×Dic₅	—	4	2	no	—
60	C₂×C₃×Dic₅	—	4	4	no	—
61	C₃⋊#168	—	2	4	no	—
62	C₂×C₃×C₄⋊C₄	—	5	2	no	—
63	C₂×C₃×C₄⋊C₄	—	10	2	no	—
64	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
65	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
66	C₂×Dic₃⋊C₄	—	2	2	no	—
67	C₂²×C₃⋊C₈	—	5	2	no	—
68	#870	—	2	2	no	—
69	#870	Dic₅	1	4	yes	complemented
70	#803	—	10	2	no	—
71	C₂²×C₄×C₅	—	6	2	no	—
72	C₂²×C₄×C₅	#67	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
74	C₂²×Dic₅	—	2	2	no	—
75	C₅×C₄⋊C₄	—	3	4	no	—
76	C₅×C₄⋊C₄	—	6	8	no	—
77	C₅×C₄⋊C₄	—	6	2	no	—
78	C₄²×C₅	C₄×D₃	1	2	yes	—
79	C₄²×C₅	D₁₂	1	2	yes	—
80	C₄⋊Dic₅	—	2	4	no	—
81	C₄⋊Dic₅	—	4	4	no	—
82	C₂×#168	—	3	4	no	—
83	C₂×#168	—	6	2	no	—
84	C₂×#168	—	6	2	no	—
85	C₂×#125	—	3	1	no	—
86	C₂×#97	—	15	1	no	—
87	#384	—	5	1	no	—
88	#340	—	15	4	no	—
89	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃×C₅	#124	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²×C₃×C₅	#137	1	2	yes	—
94	C₃×C₄×C₅	—	2	4	no	—
95	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
96	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
97	C₃×Dic₅	—	8	2	no	—
98	C₅×Dic₃	—	4	4	no	—
99	C₂²×C₃×C₄	—	10	2	no	—
100	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
101	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
102	C₂²×Dic₃	—	2	2	no	—
103	C₃×C₄⋊C₄	—	10	4	no	—
104	C₃×C₄⋊C₄	—	20	4	no	—
105	C₃×C₄²	C₂×Dic₅	1	2	yes	—
106	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
107	C₄⋊Dic₃	—	2	2	no	—
108	C₄⋊Dic₃	C₂×Dic₅	1	4	yes	—
109	Dic₃⋊C₄	—	2	8	no	—
110	C₂×C₃⋊C₈	—	5	4	no	—
111	C₂×C₃⋊C₈	—	10	2	no	—
112	C₂×C₃⋊C₈	—	10	2	no	—
113	C₂³×C₅	C₂×#67	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	6	4	no	—
117	C₂×C₄×C₅	—	12	2	no	—
118	C₂×C₄×C₅	—	12	2	no	—
119	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
120	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
122	C₂×C₄×C₅	#237	1	2	yes	—
123	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
124	C₂×Dic₅	—	4	4	no	—
125	C₂×Dic₅	—	4	2	no	—
126	#168	—	6	4	no	—
127	C₂²×C₈	—	15	2	no	—
128	#125	—	3	4	no	—
129	#125	—	6	2	no	—
130	C₂×C₄⋊C₄	—	3	2	no	—
131	C₂×C₄⋊C₄	—	5	2	no	—
132	C₂×C₄⋊C₄	—	6	2	no	—
133	C₂×C₄⋊C₄	—	10	2	no	—
134	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
135	#97	—	30	2	no	—
136	C₂×C₃×C₅	#391	1	1	yes	characteristic
137	C₂×C₃×C₅	#417	1	2	yes	—
138	C₂×C₃×C₅	#490	1	1	yes	characteristic
139	C₂×C₃×C₅	#491	1	1	yes	characteristic
140	C₂×C₃×C₅	#501	1	2	yes	—
141	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	20	2	no	—
145	C₂×C₃×C₄	—	20	4	no	—
146	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
147	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
148	C₂×C₃×C₄	#713	1	2	yes	—
149	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×Dic₃	—	2	4	no	—
152	C₂×Dic₃	—	4	2	no	—
153	C₂×Dic₃	—	4	2	no	—
154	C₃⋊C₈	—	10	4	no	—
155	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
156	C₂²×C₅	#932	1	2	yes	—
157	C₂²×C₅	#936	1	1	yes	characteristic
158	C₂²×C₅	#942	1	2	yes	—
159	C₂²×C₅	#944	1	1	yes	characteristic
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	12	4	no	—
162	C₄×C₅	#887	1	2	yes	—
163	C₄×C₅	#897	1	2	yes	—
164	Dic₅	—	8	2	no	—
165	C₂²×C₄	—	6	2	no	—
166	C₂²×C₄	—	10	2	no	—
167	C₂²×C₄	#1201	1	2	yes	—
168	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
169	C₂×C₈	—	15	4	no	—
170	C₂×C₈	—	30	2	no	—
171	C₂×C₈	—	30	2	no	—
172	C₄⋊C₄	—	3	4	no	—
173	C₄⋊C₄	—	6	8	no	—
174	C₄⋊C₄	—	6	2	no	—
175	C₄⋊C₄	—	10	4	no	—
176	C₄⋊C₄	—	20	4	no	—
177	C₄²	D₃×Dic₅	1	2	yes	—
178	C₄²	#1203	1	2	yes	—
179	C₃×C₅	#1992	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	#4090	1	1	yes	characteristic
181	C₂²×C₃	#4093	1	1	yes	characteristic
182	C₂²×C₃	#4095	1	2	yes	—
183	C₂²×C₃	#4097	1	2	yes	—
184	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	40	2	no	—
187	C₃×C₄	Q₈×Dic₅	1	2	yes	—
188	C₃×C₄	#4092	1	2	yes	—
189	Dic₃	—	4	4	no	—
190	C₂×C₅	#4966	1	2	yes	—
191	C₂×C₅	#4973	1	1	yes	characteristic
192	C₂×C₅	#5071	1	1	yes	characteristic
193	C₂×C₅	#5081	1	2	yes	—
194	C₂×C₅	#5082	1	1	yes	characteristic
195	C₂³	C₂×#1201	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	6	4	no	—
199	C₂×C₄	—	12	2	no	—
200	C₂×C₄	—	12	2	no	—
201	C₂×C₄	—	20	4	no	—
202	C₂×C₄	—	20	2	no	—
203	C₂×C₄	(C₂×D₃)⋊Dic₅	1	4	yes	—
204	C₂×C₄	#6766	1	2	yes	—
205	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
206	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
207	C₂×C₄	C₂×#1203	1	1	yes	characteristic
208	C₈	—	30	4	no	—
209	C₂×C₃	#65354	1	2	yes	—
210	C₂×C₃	#65356	1	1	yes	characteristic
211	C₂×C₃	#65358	1	1	yes	characteristic
212	C₂×C₃	#65519	1	1	yes	characteristic
213	C₂×C₃	#65521	1	2	yes	—
214	C₅	#69583	1	1	yes	characteristic, complemented, Sylow
215	C₂²	#91298	1	2	yes	—
216	C₂²	#91309	1	2	yes	—
217	C₂²	#91368	1	1	yes	characteristic
218	C₂²	#91385	1	1	yes	characteristic
219	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	12	4	no	—
222	C₄	—	40	2	no	—
223	C₄	#91344	1	2	yes	—
224	C₄	#91383	1	2	yes	—
225	C₃	#10602534	1	1	yes	characteristic, complemented, Sylow
226	C₂	#11746897	1	1	yes	characteristic
227	C₂	#11746925	1	1	yes	characteristic
228	C₂	#11746969	1	2	yes	—
229	C₂	#11748384	1	2	yes	—
230	C₂	#11748607	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1992
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₅×#870	C₄	semidirect
69. #870	Dic₅	semidirect
179. C₃×C₅	#1992	semidirect
214. C₅	#69583	semidirect
225. C₃	#10602534	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#870	2
C₂×#91087	2
C₃×#65224	2
#11745979	2
#10602534	3
#69583	5