Subgroups — #14296202

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91087	C₂	1	1	yes	maximal, characteristic
2	C₅×#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊#168	C₂	1	2	yes	maximal
5	#11745967	C₂	1	2	yes	maximal
6	#10600229	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	#91087	—	2	2	no	—
13	C₅×C₃⋊#95	C₄	1	4	yes	—
14	C₃×C₄×#168	—	2	2	no	—
15	Dic₃⋊#168	C₄	1	8	yes	complemented
16	#69755	—	5	1	no	maximal, Hall
17	C₅×#376	—	3	1	no	—
18	C₂×C₄⋊#168	—	3	2	no	—
19	C₂×#3935	—	3	1	no	—
20	C₂×C₄×#168	D₃	1	1	yes	characteristic
21	#64744	—	3	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
25	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	D₄	1	2	yes	—
27	C₃×C₄²×C₅	Q₈	1	2	yes	—
28	C₂×C₃⋊#168	—	2	2	no	—
29	C₂×C₃⋊#168	—	2	2	no	—
30	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
31	C₄×C₅×Dic₃	—	2	2	no	—
32	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄×C₈	—	5	1	no	—
37	C₂×Dic₃⋊C₈	—	5	2	no	—
38	#4932	D₅	1	1	yes	characteristic
39	C₂×#803	—	5	1	no	—
40	#4550	—	5	2	no	—
41	C₂×C₄²×C₅	—	3	1	no	—
42	C₂×C₄²×C₅	—	3	1	no	—
43	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
44	C₂²×#168	—	3	2	no	—
45	C₂²×#168	C₂×D₃	1	2	yes	—
46	C₅×#95	—	3	4	no	—
47	C₄⋊#168	—	3	8	no	—
48	#3935	—	6	2	no	—
49	C₄×#168	—	2	2	no	—
50	#1768	—	15	1	no	Sylow
51	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×C₅×Dic₃	—	2	4	no	—
59	C₂×C₅×Dic₃	—	2	2	no	—
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	C₄²	1	4	yes	—
62	C₃⋊#168	—	2	4	no	—
63	C₃×#168	—	2	4	no	—
64	C₂²×C₃×C₈	—	5	2	no	—
65	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
66	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
67	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
68	C₂²×C₃⋊C₈	—	5	2	no	—
69	C₃×C₄×C₈	—	10	2	no	—
70	C₃⋊#95	Dic₅	1	4	yes	—
71	Dic₃⋊C₈	—	5	8	no	—
72	#803	—	10	2	no	—
73	C₂²×C₄×C₅	—	3	2	no	—
74	C₂²×C₄×C₅	—	3	2	no	—
75	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
76	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₄²×C₅	—	3	4	no	—
78	C₄²×C₅	—	6	2	no	—
79	C₄²×C₅	Dic₆	1	2	yes	—
80	C₄²×C₅	#67	1	2	yes	—
81	C₂×#168	—	2	2	no	—
82	C₂×#168	—	2	2	no	—
83	C₂×#168	—	3	4	no	—
84	C₂×#168	—	6	2	no	—
85	C₂×#168	—	6	2	no	—
86	C₂×#168	C₄×D₃	1	4	yes	—
87	C₂×C₄⋊C₈	—	15	2	no	—
88	C₂×#97	—	15	1	no	—
89	C₂×C₄×C₈	—	5	1	no	—
90	#376	—	3	1	no	—
91	#336	—	15	2	no	—
92	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₃×C₅	#117	1	1	yes	characteristic
95	C₂²×C₃×C₅	#131	1	4	yes	—
96	C₃×C₄×C₅	—	2	4	no	—
97	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
98	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
99	C₅×Dic₃	—	2	4	no	—
100	C₅×Dic₃	—	4	2	no	—
101	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
103	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
104	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
105	C₂×C₃×C₈	—	5	4	no	—
106	C₂×C₃×C₈	—	10	2	no	—
107	C₂×C₃×C₈	—	10	2	no	—
108	C₃×C₄²	Dic₁₀	1	2	yes	—
109	C₃×C₄²	D₂₀	1	2	yes	—
110	C₄×Dic₃	—	2	2	no	—
111	C₄×Dic₃	C₄×D₅	1	4	yes	—
112	C₂×C₃⋊C₈	—	5	4	no	—
113	C₂×C₃⋊C₈	—	10	2	no	—
114	C₂×C₃⋊C₈	—	10	2	no	—
115	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	3	4	no	—
118	C₂×C₄×C₅	—	6	4	no	—
119	C₂×C₄×C₅	—	6	2	no	—
120	C₂×C₄×C₅	—	6	4	no	—
121	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
122	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
124	C₂×C₄×C₅	#235	1	2	yes	—
125	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
126	#168	—	2	4	no	—
127	#168	—	6	4	no	—
128	C₂²×C₈	—	5	2	no	—
129	C₂²×C₈	—	15	2	no	—
130	C₂×C₄²	—	3	1	no	—
131	C₂×C₄²	—	3	1	no	—
132	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
133	C₄⋊C₈	—	15	8	no	—
134	#97	—	30	2	no	—
135	C₄×C₈	—	10	2	no	—
136	#95	—	3	4	no	—
137	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₅	#405	1	2	yes	—
139	C₂×C₃×C₅	#424	1	2	yes	—
140	C₂×C₃×C₅	#433	1	2	yes	—
141	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
144	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
146	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
147	C₂×C₃×C₄	#709	1	2	yes	—
148	C₂×Dic₃	—	2	4	no	—
149	C₂×Dic₃	—	2	2	no	—
150	C₂×Dic₃	—	2	4	no	—
151	C₂×Dic₃	C₄×Dic₅	1	4	yes	—
152	C₃×C₈	—	10	4	no	—
153	C₃⋊C₈	—	10	4	no	—
154	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
155	C₂²×C₅	#872	1	1	yes	characteristic
156	C₂²×C₅	#901	1	4	yes	—
157	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
158	C₄×C₅	—	2	4	no	—
159	C₄×C₅	—	6	4	no	—
160	C₄×C₅	—	12	2	no	—
161	C₄×C₅	C₄×Dic₆	1	2	yes	—
162	C₄×C₅	C₄×#67	1	2	yes	—
163	C₂²×C₄	—	3	2	no	—
164	C₂²×C₄	—	3	2	no	—
165	C₂²×C₄	D₃×Dic₅	1	2	yes	—
166	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
167	C₂×C₈	—	5	4	no	—
168	C₂×C₈	—	10	2	no	—
169	C₂×C₈	—	10	2	no	—
170	C₂×C₈	—	15	4	no	—
171	C₂×C₈	—	30	2	no	—
172	C₂×C₈	—	30	2	no	—
173	C₄²	—	3	4	no	—
174	C₄²	—	6	2	no	—
175	C₄²	#1202	1	2	yes	—
176	C₄²	#1204	1	2	yes	—
177	C₃×C₅	#1768	1	1	yes	characteristic, complemented, Hall
178	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
180	C₂²×C₃	#4072	1	1	yes	characteristic
181	C₂²×C₃	#4094	1	4	yes	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
184	C₃×C₄	C₄×D₂₀	1	2	yes	—
185	Dic₃	—	2	4	no	—
186	Dic₃	—	4	2	no	—
187	C₂×C₅	#4689	1	2	yes	—
188	C₂×C₅	#4690	1	2	yes	—
189	C₂×C₅	C₄×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂×C₅	#5102	1	2	yes	—
191	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	3	4	no	—
194	C₂×C₄	—	6	4	no	—
195	C₂×C₄	—	6	4	no	—
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	Dic₃⋊Dic₅	1	4	yes	—
198	C₂×C₄	#6771	1	2	yes	—
199	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
200	C₂×C₄	C₂×#1202	1	1	yes	characteristic
201	C₂×C₄	C₂×#1204	1	1	yes	characteristic
202	C₈	—	10	4	no	—
203	C₈	—	30	4	no	—
204	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
205	C₂×C₃	#65286	1	2	yes	—
206	C₂×C₃	#65288	1	2	yes	—
207	C₂×C₃	#65289	1	2	yes	—
208	C₅	#69755	1	1	yes	characteristic, complemented, Sylow
209	C₂²	#91286	1	4	yes	—
210	C₂²	#91371	1	1	yes	characteristic
211	C₂²	C₄×D₃×Dic₅	1	1	yes	characteristic
212	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	6	4	no	—
215	C₄	—	12	2	no	—
216	C₄	C₄×#1202	1	2	yes	—
217	C₄	C₄×#1204	1	2	yes	—
218	C₃	#10600229	1	1	yes	characteristic, complemented, Sylow
219	C₂	#11746877	1	2	yes	—
220	C₂	#11746977	1	2	yes	—
221	C₂	#11748439	1	2	yes	—
222	C₂	C₄×Dic₃⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1768
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. Dic₃⋊#168	C₄	semidirect
177. C₃×C₅	#1768	semidirect
208. C₅	#69755	semidirect
218. C₃	#10600229	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91087	2
C₅×#4932	2
C₂×C₃×C₄×#168	2
C₂×Dic₃⋊#168	2
#11745967	2
#10600229	3
#69755	5