Subgroups — #14299030

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91233	C₂	1	4	yes	maximal
2	C₂×C₄⋊(C₃⋊#170)	C₂	1	2	yes	maximal
3	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10604191	—	3	1	no	maximal, Hall
5	C₂²×C₃⋊#170	C₂²	1	4	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
8	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
11	#91233	—	2	8	no	—
12	C₄⋊(C₃⋊#170)	—	2	4	no	—
13	#69394	—	5	1	no	maximal, Hall
14	C₂×#4004	—	3	4	no	—
15	C₂×C₄⋊#170	—	3	2	no	—
16	C₄²×Dic₅	D₃	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊#170	—	2	4	no	—
23	C₂×C₃⋊#170	—	2	8	no	—
24	C₃×C₄²×C₅	—	2	2	no	—
25	C₃×C₄×Dic₅	—	2	4	no	—
26	C₃×C₄×Dic₅	C₈	1	4	yes	complemented
27	C₃×C₄×Dic₅	D₄	1	2	yes	—
28	C₃×C₄×Dic₅	D₄	1	4	yes	—
29	C₃×C₄×Dic₅	Q₈	1	2	yes	—
30	C₃×C₄×Dic₅	Q₈	1	4	yes	—
31	C₃×C₄³	—	5	1	no	—
32	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
33	C₂×C₄⋊(C₃⋊C₈)	—	5	4	no	—
34	C₂²×#170	—	3	4	no	—
35	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
36	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
37	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
38	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
40	#4004	—	6	8	no	—
41	C₄⋊#170	—	6	4	no	—
42	#1842	—	15	1	no	Sylow
43	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	—	2	2	no	—
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
50	C₂×C₃×Dic₅	—	2	2	no	—
51	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
52	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
53	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
54	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
55	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
57	C₂×C₃×Dic₅	C₂×Q₈	1	2	yes	—
58	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
59	C₃⋊#170	—	4	4	no	—
60	C₂×C₃×C₄²	—	5	2	no	—
61	C₂×C₃×C₄²	—	5	1	no	—
62	C₂×C₃×C₄²	—	5	1	no	—
63	C₂×C₃×C₄²	—	5	2	no	—
64	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂²×C₃⋊C₈	—	5	4	no	—
66	C₄⋊(C₃⋊C₈)	—	10	4	no	—
67	C₄⋊(C₃⋊C₈)	—	10	8	no	—
68	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
69	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
70	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
71	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
72	C₂²×Dic₅	C₂²×D₃	1	1	yes	characteristic
73	C₂×#170	—	6	4	no	—
74	C₂×#170	—	6	8	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄×Dic₅	—	2	4	no	—
77	C₄×Dic₅	C₃⋊C₈	1	4	yes	complemented
78	C₄×Dic₅	Dic₆	1	2	yes	—
79	C₄×Dic₅	Dic₆	1	4	yes	—
80	C₄×Dic₅	D₁₂	1	2	yes	—
81	C₄×Dic₅	D₁₂	1	4	yes	—
82	C₂×C₄⋊C₈	—	15	2	no	—
83	C₂×C₄⋊C₈	—	15	4	no	—
84	C₄³	—	5	1	no	—
85	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
89	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₃×C₄×C₅	—	2	4	no	—
91	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	complemented
92	C₃×Dic₅	—	2	2	no	—
93	C₃×Dic₅	C₄⋊C₈	1	8	yes	complemented
94	C₃×Dic₅	#127	1	1	yes	characteristic
95	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
96	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
97	C₂²×C₃×C₄	—	5	2	no	—
98	C₂²×C₃×C₄	—	5	1	no	—
99	C₂²×C₃×C₄	—	5	1	no	—
100	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
101	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₃×C₄²	—	2	2	no	—
103	C₃×C₄²	—	5	4	no	—
104	C₃×C₄²	—	5	4	no	—
105	C₃×C₄²	—	5	2	no	—
106	C₃×C₄²	—	5	4	no	—
107	C₃×C₄²	—	5	2	no	—
108	C₃×C₄²	—	10	4	no	—
109	C₂×C₃⋊C₈	—	10	4	no	—
110	C₂×C₃⋊C₈	—	10	8	no	—
111	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
115	C₂×C₄×C₅	#208	1	2	yes	—
116	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
117	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
118	C₂×Dic₅	—	2	2	no	—
119	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
120	C₂×Dic₅	#208	1	4	yes	—
121	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
122	C₂×Dic₅	C₄⋊Dic₃	1	4	yes	—
123	C₂×Dic₅	C₂×Dic₆	1	2	yes	—
124	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
125	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
126	C₂×Dic₅	C₂×D₁₂	1	2	yes	—
127	#170	—	12	4	no	—
128	C₂²×C₈	—	15	4	no	—
129	C₂×C₄²	—	5	2	no	—
130	C₂×C₄²	—	5	1	no	—
131	C₂×C₄²	—	5	1	no	—
132	C₂×C₄²	—	5	2	no	—
133	C₂×C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
134	C₄⋊C₈	—	30	4	no	—
135	C₄⋊C₈	—	30	8	no	—
136	C₂×C₃×C₅	#384	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
138	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₃×C₅	#423	1	1	yes	characteristic
140	C₂×C₃×C₅	#423	1	2	yes	—
141	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	5	4	no	—
145	C₂×C₃×C₄	—	5	4	no	—
146	C₂×C₃×C₄	—	5	4	no	—
147	C₂×C₃×C₄	—	5	2	no	—
148	C₂×C₃×C₄	—	5	1	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	—	5	1	no	—
151	C₂×C₃×C₄	—	5	2	no	—
152	C₂×C₃×C₄	—	10	2	no	—
153	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
154	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
155	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
156	C₂×C₃×C₄	#704	1	2	yes	—
157	C₃⋊C₈	—	20	4	no	—
158	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
159	C₂²×C₅	C₂×#208	1	2	yes	—
160	C₂²×C₅	C₂×#208	1	1	yes	characteristic
161	C₂²×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
162	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
163	C₄×C₅	—	2	4	no	—
164	C₄×C₅	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
165	Dic₅	—	2	2	no	—
166	Dic₅	C₄⋊(C₃⋊C₈)	1	8	yes	complemented
167	Dic₅	#869	1	1	yes	characteristic
168	Dic₅	#869	1	2	yes	—
169	Dic₅	#874	1	1	yes	characteristic
170	C₂²×C₄	—	5	1	no	—
171	C₂²×C₄	—	5	2	no	—
172	C₂²×C₄	—	5	1	no	—
173	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
174	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
175	C₂×C₈	—	30	4	no	—
176	C₂×C₈	—	30	8	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	5	4	no	—
179	C₄²	—	5	4	no	—
180	C₄²	—	5	2	no	—
181	C₄²	—	5	4	no	—
182	C₄²	—	5	2	no	—
183	C₄²	—	10	4	no	—
184	C₃×C₅	#1842	1	1	yes	characteristic, complemented, Hall
185	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
186	C₂²×C₃	D₅⋊M₁₆	1	2	yes	—
187	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
188	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂²×C₃	C₂×#704	1	1	yes	characteristic
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	5	8	no	—
192	C₃×C₄	—	5	1	no	—
193	C₃×C₄	—	5	2	no	—
194	C₃×C₄	—	5	1	no	—
195	C₃×C₄	—	10	2	no	—
196	C₃×C₄	#4004	1	4	yes	complemented
197	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
198	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
199	C₂×C₅	#4924	1	1	yes	characteristic
200	C₂×C₅	#4924	1	2	yes	—
201	C₂×C₅	#4935	1	1	yes	characteristic
202	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	5	4	no	—
206	C₂×C₄	—	5	4	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	4	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	5	2	no	—
213	C₂×C₄	—	10	2	no	—
214	C₂×C₄	D₅⋊(C₃⋊C₈)	1	2	yes	—
215	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	—
216	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
217	C₂×C₄	#6762	1	2	yes	—
218	C₈	—	60	4	no	—
219	C₂×C₃	D₅⋊(C₄⋊C₈)	1	2	yes	—
220	C₂×C₃	#65682	1	1	yes	characteristic
221	C₂×C₃	#65688	1	1	yes	characteristic
222	C₂×C₃	C₂×#4004	1	1	yes	characteristic
223	C₂×C₃	#65758	1	2	yes	—
224	C₅	#69394	1	1	yes	characteristic, complemented, Sylow
225	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
226	C₂²	D₅⋊#208	1	2	yes	—
227	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
228	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
229	C₂²	C₂×#6762	1	1	yes	characteristic
230	C₄	—	2	4	no	—
231	C₄	—	5	8	no	—
232	C₄	—	5	1	no	—
233	C₄	—	5	2	no	—
234	C₄	—	5	1	no	—
235	C₄	—	10	2	no	—
236	C₄	#91233	1	4	yes	complemented
237	C₃	#10604191	1	1	yes	characteristic, complemented, Sylow
238	C₂	D₅⋊(C₄⋊(C₃⋊C₈))	1	2	yes	—
239	C₂	#11752341	1	1	yes	characteristic
240	C₂	#11752347	1	1	yes	characteristic
241	C₂	C₂×#91233	1	1	yes	characteristic
242	C₂	#11752417	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₄²×C₅	C₄
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×Dic₅	C₂³
hypercenter	C₂×C₄²	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1842
LCS	G ▷ C₂²×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
26. C₃×C₄×Dic₅	C₈	semidirect
77. C₄×Dic₅	C₃⋊C₈	semidirect
91. C₃×C₄×C₅	C₄⋊C₈	semidirect
93. C₃×Dic₅	C₄⋊C₈	semidirect
164. C₄×C₅	C₄⋊(C₃⋊C₈)	semidirect
166. Dic₅	C₄⋊(C₃⋊C₈)	semidirect
184. C₃×C₅	#1842	semidirect
196. C₃×C₄	#4004	semidirect
224. C₅	#69394	semidirect
236. C₄	#91233	semidirect
237. C₃	#10604191	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91233	2
C₂×C₄⋊(C₃⋊#170)	2
C₃×C₄²×Dic₅	2
#10604191	3
#69394	5