Subgroups — #14299092

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
2	C₅×#4935	C₂	1	1	yes	maximal, characteristic
3	C₃×#65224	C₂	1	1	yes	maximal, characteristic
4	C₂×#90989	C₂	1	4	yes	maximal
5	#10600370	—	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	—	2	2	no	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₄×C₃⋊#168	C₄	1	4	yes	complemented
13	#90989	—	2	8	no	—
14	#69577	—	5	1	no	maximal, Hall
15	C₅×#384	—	3	1	no	—
16	C₂×#3939	—	3	4	no	—
17	#65224	D₃	1	1	yes	characteristic
18	C₂×C₄×#168	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	—	2	2	no	—
22	C₂²×C₃×Dic₅	—	2	2	no	—
23	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₃×C₄²×C₅	Q₈	1	2	yes	—
25	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
26	C₂×C₃⋊#168	D₄	1	4	yes	—
27	C₂×C₃⋊#168	Q₈	1	4	yes	—
28	C₅×C₄⋊Dic₃	—	2	4	no	—
29	C₅×C₄⋊Dic₃	—	4	4	no	—
30	C₃×C₄⋊Dic₅	—	2	4	no	—
31	C₃×C₄⋊Dic₅	—	4	4	no	—
32	C₃×#384	—	5	1	no	—
33	C₂×#807	—	5	4	no	—
34	#4935	D₅	1	1	yes	characteristic
35	C₂×C₄×C₃⋊C₈	—	5	1	no	—
36	C₂×C₅×C₄⋊C₄	—	3	2	no	—
37	C₂×C₅×C₄⋊C₄	—	6	2	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
41	C₂²×#168	—	3	2	no	—
42	#3939	—	6	8	no	—
43	C₄×#168	—	3	4	no	—
44	#1845	—	15	1	no	Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
47	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
48	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₅×Dic₃	—	4	2	no	—
54	C₂×C₅×Dic₃	—	4	4	no	—
55	C₂×C₃×Dic₅	—	4	2	no	—
56	C₂×C₃×Dic₅	—	4	4	no	—
57	C₃⋊#168	C₄⋊C₄	1	8	yes	complemented
58	C₂×C₃×C₄⋊C₄	—	5	2	no	—
59	C₂×C₃×C₄⋊C₄	—	10	2	no	—
60	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	2	2	no	—
62	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
63	C₂²×C₃⋊C₈	—	5	2	no	—
64	#807	—	10	8	no	—
65	C₄×C₃⋊C₈	—	5	4	no	—
66	C₂²×C₄×C₅	—	6	2	no	—
67	C₂²×C₄×C₅	#67	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂²×Dic₅	—	2	2	no	—
70	C₅×C₄⋊C₄	—	6	4	no	—
71	C₅×C₄⋊C₄	—	12	4	no	—
72	C₄²×C₅	Dic₆	1	2	yes	—
73	C₄²×C₅	C₄×D₃	1	2	yes	—
74	C₄⋊Dic₅	—	2	4	no	—
75	C₄⋊Dic₅	—	4	4	no	—
76	C₂×#168	—	3	4	no	—
77	C₂×#168	—	3	4	no	—
78	C₂×#168	—	3	4	no	—
79	C₂×#107	—	15	4	no	—
80	C₂×C₄×C₈	—	15	1	no	—
81	#384	—	3	1	no	—
82	#384	—	5	1	no	—
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	#127	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
87	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
88	C₃×C₄×C₅	#107	1	8	yes	complemented
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
91	C₃×Dic₅	—	8	2	no	—
92	C₅×Dic₃	—	8	2	no	—
93	C₂²×C₃×C₄	—	10	2	no	—
94	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
95	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₃×C₄⋊C₄	—	10	4	no	—
98	C₃×C₄⋊C₄	—	20	4	no	—
99	C₃×C₄²	Dic₁₀	1	2	yes	—
100	C₃×C₄²	C₄×D₅	1	2	yes	—
101	C₄⋊Dic₃	—	2	4	no	—
102	C₄⋊Dic₃	—	4	4	no	—
103	C₂×C₃⋊C₈	—	5	4	no	—
104	C₂×C₃⋊C₈	—	5	4	no	—
105	C₂×C₃⋊C₈	—	5	4	no	—
106	C₂³×C₅	C₂×#67	1	1	yes	characteristic
107	C₂×C₄×C₅	—	12	4	no	—
108	C₂×C₄×C₅	—	12	2	no	—
109	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
110	C₂×C₄×C₅	#213	1	4	yes	—
111	C₂×C₄×C₅	#216	1	4	yes	—
112	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
114	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
115	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
116	C₂×Dic₅	—	4	4	no	—
117	C₂×Dic₅	—	4	2	no	—
118	#168	—	3	8	no	—
119	C₂²×C₈	—	15	2	no	—
120	C₂×C₄⋊C₄	—	3	2	no	—
121	C₂×C₄⋊C₄	—	5	2	no	—
122	C₂×C₄⋊C₄	—	6	2	no	—
123	C₂×C₄⋊C₄	—	10	2	no	—
124	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
125	#107	—	30	8	no	—
126	C₄×C₈	—	15	4	no	—
127	C₂×C₃×C₅	#384	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#107	1	2	yes	—
129	C₂×C₃×C₅	#493	1	1	yes	characteristic
130	C₂×C₃×C₅	#494	1	1	yes	characteristic
131	C₂×C₃×C₅	#500	1	2	yes	—
132	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
133	C₂×C₃×C₄	—	20	4	no	—
134	C₂×C₃×C₄	—	20	2	no	—
135	C₂×C₃×C₄	#683	1	4	yes	—
136	C₂×C₃×C₄	#686	1	4	yes	—
137	C₂×C₃×C₄	#689	1	4	yes	—
138	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
140	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
141	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
142	C₂×Dic₃	—	4	4	no	—
143	C₂×Dic₃	—	4	2	no	—
144	C₃⋊C₈	—	5	8	no	—
145	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
146	C₂²×C₅	C₂×#213	1	2	yes	—
147	C₂²×C₅	#940	1	1	yes	characteristic
148	C₂²×C₅	C₂×#216	1	2	yes	—
149	C₂²×C₅	#944	1	1	yes	characteristic
150	C₄×C₅	—	24	2	no	—
151	C₄×C₅	#807	1	8	yes	complemented
152	C₄×C₅	#888	1	2	yes	—
153	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
154	Dic₅	—	8	2	no	—
155	C₂²×C₄	—	6	2	no	—
156	C₂²×C₄	—	10	2	no	—
157	C₂²×C₄	#1201	1	2	yes	—
158	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
159	C₂×C₈	—	15	4	no	—
160	C₂×C₈	—	15	4	no	—
161	C₂×C₈	—	15	4	no	—
162	C₄⋊C₄	—	6	4	no	—
163	C₄⋊C₄	—	10	4	no	—
164	C₄⋊C₄	—	12	4	no	—
165	C₄⋊C₄	—	20	4	no	—
166	C₄²	#1200	1	2	yes	—
167	C₄²	#1204	1	2	yes	—
168	C₃×C₅	#1845	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂×#683	1	1	yes	characteristic
170	C₂²×C₃	C₂×#686	1	2	yes	—
171	C₂²×C₃	#4086	1	1	yes	characteristic
172	C₂²×C₃	C₂×#689	1	2	yes	—
173	C₂²×C₃	#4090	1	1	yes	characteristic
174	C₃×C₄	—	40	2	no	—
175	C₃×C₄	#3939	1	8	yes	complemented
176	C₃×C₄	#4034	1	2	yes	—
177	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
178	Dic₃	—	8	2	no	—
179	C₂×C₅	C₂×#807	1	2	yes	—
180	C₂×C₅	#4975	1	1	yes	characteristic
181	C₂×C₅	#5073	1	1	yes	characteristic
182	C₂×C₅	#5078	1	2	yes	—
183	C₂×C₅	#5092	1	1	yes	characteristic
184	C₂³	C₂×#1201	1	1	yes	characteristic
185	C₂×C₄	—	12	2	no	—
186	C₂×C₄	—	12	4	no	—
187	C₂×C₄	—	20	2	no	—
188	C₂×C₄	—	20	4	no	—
189	C₂×C₄	#6652	1	4	yes	—
190	C₂×C₄	#6661	1	4	yes	—
191	C₂×C₄	#6669	1	4	yes	—
192	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
193	C₂×C₄	#6777	1	2	yes	—
194	C₂×C₄	C₂×#1200	1	1	yes	characteristic
195	C₂×C₄	C₂×#1204	1	1	yes	characteristic
196	C₈	—	15	8	no	—
197	C₂×C₃	C₂×#3939	1	2	yes	—
198	C₂×C₃	#65265	1	1	yes	characteristic
199	C₂×C₃	#65362	1	1	yes	characteristic
200	C₂×C₃	#65367	1	2	yes	—
201	C₂×C₃	#65381	1	1	yes	characteristic
202	C₅	#69577	1	1	yes	characteristic, complemented, Sylow
203	C₂²	C₂×#6652	1	2	yes	—
204	C₂²	C₂×#6661	1	2	yes	—
205	C₂²	#91385	1	1	yes	characteristic
206	C₂²	#91463	1	1	yes	characteristic
207	C₂²	C₂×#6669	1	1	yes	characteristic
208	C₄	—	24	2	no	—
209	C₄	—	40	2	no	—
210	C₄	#90989	1	8	yes	complemented
211	C₄	#91437	1	2	yes	—
212	C₄	#91470	1	2	yes	—
213	C₃	#10600370	1	1	yes	characteristic, complemented, Sylow
214	C₂	#11746962	1	2	yes	—
215	C₂	#11747039	1	1	yes	characteristic
216	C₂	#11747103	1	1	yes	characteristic
217	C₂	C₂×#90989	1	2	yes	—
218	C₂	#11748866	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	#1845
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
12. C₄×C₃⋊#168	C₄	semidirect
57. C₃⋊#168	C₄⋊C₄	semidirect
88. C₃×C₄×C₅	#107	semidirect
151. C₄×C₅	#807	semidirect
168. C₃×C₅	#1845	semidirect
175. C₃×C₄	#3939	semidirect
202. C₅	#69577	semidirect
210. C₄	#90989	semidirect
213. C₃	#10600370	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊#168	2
C₅×#4935	2
C₃×#65224	2
C₂×#90989	2
#10600370	3
#69577	5