Subgroups — #14294875

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊#170	C₂	1	2	yes	maximal
2	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#11746789	C₂	1	4	yes	maximal
4	#10604174	—	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	1	yes	characteristic
8	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₄×C₃⋊#170	C₄	1	8	yes	complemented
12	#69388	—	5	1	no	maximal, Hall
13	C₂×C₄×#170	—	3	2	no	—
14	C₄²×Dic₅	D₃	1	1	yes	characteristic
15	#64918	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
19	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊#170	—	2	8	no	—
22	C₂×C₃⋊#170	C₂×C₄	1	8	yes	—
23	C₃×C₄²×C₅	—	2	2	no	—
24	C₃×C₄×Dic₅	—	2	2	no	—
25	C₃×C₄×Dic₅	—	2	4	no	—
26	C₃×C₄×Dic₅	C₈	1	8	yes	complemented
27	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	—
28	C₃×C₄³	—	5	1	no	—
29	C₂×C₄×C₃⋊C₈	—	5	2	no	—
30	#4524	—	5	4	no	—
31	C₂²×#170	—	3	4	no	—
32	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
33	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
34	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
35	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
36	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
37	C₄×#170	—	3	8	no	—
38	#1744	—	15	1	no	Sylow
39	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	—	2	2	no	—
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
44	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
46	C₂×C₃×Dic₅	—	2	2	no	—
47	C₂×C₃×Dic₅	—	2	2	no	—
48	C₂×C₃×Dic₅	C₄²	1	2	yes	—
49	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
50	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
51	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
52	C₂×C₃×Dic₅	#41	1	4	yes	—
53	C₃⋊#170	—	2	8	no	—
54	C₂×C₃×C₄²	—	5	1	no	—
55	C₂×C₃×C₄²	—	5	2	no	—
56	C₂×C₃×C₄²	—	5	2	no	—
57	C₂×C₃×C₄²	—	5	1	no	—
58	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂²×C₃⋊C₈	—	5	4	no	—
60	C₄×C₃⋊C₈	—	5	8	no	—
61	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
62	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
63	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
64	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
65	C₂²×Dic₅	C₂²×D₃	1	1	yes	characteristic
66	C₂×#170	—	3	8	no	—
67	C₂×#170	—	6	8	no	—
68	C₄²×C₅	—	2	2	no	—
69	C₄×Dic₅	—	2	2	no	—
70	C₄×Dic₅	—	2	4	no	—
71	C₄×Dic₅	C₃⋊C₈	1	8	yes	complemented
72	C₄×Dic₅	C₄×D₃	1	4	yes	—
73	C₂×C₄×C₈	—	15	2	no	—
74	C₄³	—	5	1	no	—
75	#336	—	15	4	no	—
76	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
77	C₂²×C₃×C₅	#117	1	2	yes	—
78	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
79	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
80	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
81	C₃×C₄×C₅	—	2	4	no	—
82	C₃×C₄×C₅	#97	1	4	yes	—
83	C₃×Dic₅	—	2	4	no	—
84	C₃×Dic₅	C₄×C₈	1	4	yes	—
85	C₃×Dic₅	#117	1	4	yes	—
86	C₂²×C₃×C₄	—	5	1	no	—
87	C₂²×C₃×C₄	—	5	1	no	—
88	C₂²×C₃×C₄	—	5	2	no	—
89	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
90	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₃×C₄²	—	2	2	no	—
92	C₃×C₄²	—	5	8	no	—
93	C₃×C₄²	—	5	4	no	—
94	C₃×C₄²	—	10	4	no	—
95	C₃×C₄²	—	10	2	no	—
96	C₂×C₃⋊C₈	—	5	8	no	—
97	C₂×C₃⋊C₈	—	10	8	no	—
98	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
102	C₂×C₄×C₅	#208	1	2	yes	—
103	C₂×C₄×C₅	#208	1	2	yes	—
104	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
105	C₂×Dic₅	—	2	2	no	—
106	C₂×Dic₅	—	2	2	no	—
107	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
108	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
109	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
110	C₂×Dic₅	C₂×C₄×D₃	1	2	yes	—
111	C₂×Dic₅	#235	1	4	yes	—
112	#170	—	6	8	no	—
113	C₂²×C₈	—	15	4	no	—
114	C₂×C₄²	—	5	2	no	—
115	C₂×C₄²	—	5	1	no	—
116	C₂×C₄²	—	5	2	no	—
117	C₂×C₄²	—	5	1	no	—
118	C₂×C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
119	C₄×C₈	—	15	8	no	—
120	C₂×C₃×C₅	#376	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
123	C₂×C₃×C₅	#431	1	2	yes	—
124	C₂×C₃×C₅	#431	1	2	yes	—
125	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	5	4	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	5	4	no	—
132	C₂×C₃×C₄	—	5	4	no	—
133	C₂×C₃×C₄	—	10	2	no	—
134	C₂×C₃×C₄	—	10	2	no	—
135	C₂×C₃×C₄	D₅⋊C₈	1	4	yes	—
136	C₂×C₃×C₄	#700	1	2	yes	—
137	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
138	C₂×C₃×C₄	#704	1	2	yes	—
139	C₃⋊C₈	—	10	8	no	—
140	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
141	C₂²×C₅	C₂×#208	1	1	yes	characteristic
142	C₂²×C₅	C₂×#208	1	1	yes	characteristic
143	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
144	C₂²×C₅	#926	1	2	yes	—
145	C₄×C₅	—	2	4	no	—
146	C₄×C₅	#803	1	4	yes	—
147	Dic₅	—	2	4	no	—
148	Dic₅	C₄×C₃⋊C₈	1	4	yes	—
149	Dic₅	#872	1	4	yes	—
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	5	1	no	—
152	C₂²×C₄	—	5	2	no	—
153	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
154	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
155	C₂×C₈	—	15	8	no	—
156	C₂×C₈	—	30	8	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	5	8	no	—
159	C₄²	—	5	4	no	—
160	C₄²	—	10	4	no	—
161	C₄²	—	10	2	no	—
162	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
163	C₂²×C₃	C₂×D₅⋊C₈	1	2	yes	—
164	C₂²×C₃	C₂×#700	1	1	yes	characteristic
165	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂²×C₃	#4130	1	2	yes	—
167	C₂²×C₃	C₂×#704	1	1	yes	characteristic
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	—	5	4	no	—
170	C₃×C₄	—	5	4	no	—
171	C₃×C₄	—	10	4	no	—
172	C₃×C₄	#4002	1	4	yes	—
173	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
174	C₂×C₅	C₂×#803	1	1	yes	characteristic
175	C₂×C₅	#4925	1	2	yes	—
176	C₂×C₅	#4925	1	2	yes	—
177	C₂×C₅	#4932	1	1	yes	characteristic
178	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	5	4	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	5	4	no	—
185	C₂×C₄	—	5	4	no	—
186	C₂×C₄	—	10	2	no	—
187	C₂×C₄	—	10	2	no	—
188	C₂×C₄	D₅⋊(C₃⋊C₈)	1	4	yes	—
189	C₂×C₄	#6758	1	2	yes	—
190	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
191	C₂×C₄	#6762	1	2	yes	—
192	C₈	—	30	8	no	—
193	C₂×C₃	C₄×D₅⋊C₈	1	1	yes	characteristic
194	C₂×C₃	#65680	1	2	yes	—
195	C₂×C₃	#65686	1	1	yes	characteristic
196	C₂×C₃	C₂×#4002	1	1	yes	characteristic
197	C₂×C₃	#65754	1	2	yes	—
198	C₅	#69388	1	1	yes	characteristic, complemented, Sylow
199	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	2	yes	—
200	C₂²	C₂×#6758	1	1	yes	characteristic
201	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
202	C₂²	#91989	1	2	yes	—
203	C₂²	C₂×#6762	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	5	4	no	—
206	C₄	—	5	4	no	—
207	C₄	—	10	4	no	—
208	C₄	#91231	1	4	yes	—
209	C₃	#10604174	1	1	yes	characteristic, complemented, Sylow
210	C₂	C₄×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
211	C₂	#11752339	1	2	yes	—
212	C₂	#11752345	1	1	yes	characteristic
213	C₂	C₂×#91231	1	1	yes	characteristic
214	C₂	#11752413	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂³	C₂²×ΓL(1,𝔽₁₆)
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	#1744
LCS	G ▷ C₂×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
11. C₄×C₃⋊#170	C₄	semidirect
26. C₃×C₄×Dic₅	C₈	semidirect
71. C₄×Dic₅	C₃⋊C₈	semidirect
162. C₃×C₅	#1744	semidirect
198. C₅	#69388	semidirect
209. C₃	#10604174	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⋊#170	2
C₃×C₄²×Dic₅	2
#11746789	2
#10604174	3
#69388	5