Finite Group Databasebeta

Subgroups — #14512471

Order 1920.

250 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756726	C₂	1	2	yes	maximal
2	D₅×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#10604781	—	3	1	no	maximal, Hall
4	D₅×#41⋊C₃	C₂²	1	1	yes	characteristic
5	#91886	—	2	2	no	—
6	C₅×#131⋊C₃	C₄	1	1	yes	characteristic, complemented
7	#91222	—	4	1	no	maximal
8	#4094⋊C₃	C₄	1	1	yes	characteristic, complemented
9	#73125	—	5	1	no	maximal, Hall
10	D₅×#131	D₃	1	1	yes	characteristic
11	#65793	—	3	2	no	—
12	D₅⋊#108	—	3	1	no	—
13	C₂×#3993	—	3	1	no	—
14	#64899	—	3	2	no	—
15	C₅×#41⋊C₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊ΓL(1,𝔽₁₆)	—	4	2	no	—
17	D₅×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
18	C₅⋊(#41⋊C₃)	Q₈	1	1	yes	characteristic
19	C₃×C₈×D₅	—	4	1	no	—
20	C₂×#131⋊C₃	—	5	1	no	—
21	#5426	—	5	2	no	—
22	D₅×#41	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
24	D₅⋊M₁₆	—	3	2	no	—
25	C₅×#131	Dic₃	1	1	yes	characteristic, complemented
26	#4094	Dic₃	1	1	yes	characteristic, complemented
27	#4053	—	3	1	no	—
28	D₅×M₁₆	—	3	1	no	—
29	#4047	—	3	1	no	—
30	C₂×C₈×D₅	—	3	1	no	—
31	#4009	—	6	2	no	—
32	#4007	—	6	2	no	—
33	#3996	—	6	1	no	—
34	#3995	—	6	1	no	—
35	#3993	—	6	1	no	—
36	#3993	—	6	1	no	—
37	#1815	—	15	1	no	Sylow
38	C₂×ΓL(1,𝔽₁₆)	—	8	2	no	—
39	C₃×C₄×D₅	—	4	1	no	—
40	C₅×SL(2,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
41	C₃×C₅×C₈	—	4	1	no	—
42	C₃×#168	—	4	1	no	—
43	C₂×#41⋊C₃	—	5	1	no	—
44	#131⋊C₃	—	5	1	no	—
45	#131⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
46	#859	—	10	2	no	—
47	#817	—	20	1	no	—
48	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
49	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
50	#713	Dic₆	1	1	yes	characteristic
51	Q₈×D₅	D₁₂	1	1	yes	characteristic
52	#711	—	3	1	no	—
53	D₄×D₅	—	3	1	no	—
54	#709	—	3	1	no	—
55	C₂×C₄×D₅	—	3	1	no	—
56	#704	—	6	2	no	—
57	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
58	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
59	#700	—	6	2	no	—
60	D₅⋊C₈	—	6	2	no	—
61	C₅×M₁₆	—	3	1	no	—
62	C₂×C₅×C₈	—	3	1	no	—
63	#681	—	3	1	no	—
64	C₂×#168	—	3	1	no	—
65	#676	—	6	1	no	—
66	C₈×D₅	—	3	1	no	—
67	C₈×D₅	—	3	1	no	—
68	C₈×D₅	—	3	1	no	—
69	C₈×D₅	S₄	1	1	yes	characteristic
70	C₂×#131	—	5	1	no	—
71	C₂×#108	—	15	1	no	—
72	C₂×#106	—	15	1	no	—
73	#417	—	15	2	no	—
74	#343	—	15	2	no	—
75	C₂×C₃×D₅	—	4	1	no	—
76	ΓL(1,𝔽₁₆)	—	16	2	no	—
77	C₃×C₄×C₅	—	4	1	no	—
78	C₃×Dic₅	—	4	1	no	—
79	#41⋊C₃	—	5	1	no	—
80	#41⋊C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₂×SL(2,𝔽₃)	—	5	1	no	—
82	C₂×C₃×C₈	—	20	1	no	—
83	C₄⋊Dic₃	—	20	2	no	—
84	C₂²×D₅	—	3	1	no	—
85	C₂×AGL(1,𝔽₅)	—	12	2	no	—
86	C₂×AGL(1,𝔽₅)	—	24	2	no	—
87	C₅×Q₈	C₄⋊Dic₃	1	1	yes	characteristic
88	C₅×D₄	—	3	1	no	—
89	C₂×C₄×C₅	—	3	1	no	—
90	C₂²⋊D₅	—	6	1	no	—
91	C₂×Dic₅	—	3	1	no	—
92	D₂₀	—	3	1	no	—
93	C₄×D₅	—	3	1	no	—
94	C₄×D₅	—	3	1	no	—
95	C₄×D₅	—	3	1	no	—
96	C₄×D₅	C₂×S₄	1	1	yes	characteristic
97	Dic₁₀	—	3	1	no	—
98	#170	—	12	2	no	—
99	C₅×C₈	—	3	1	no	—
100	C₅×C₈	SL(2,ℤ/4)	1	1	yes	characteristic
101	#168	—	3	1	no	—
102	#168	SL(2,ℤ/4)	1	1	yes	characteristic
103	C₂×#41	—	5	1	no	—
104	#131	—	5	1	no	—
105	#131	—	15	1	no	—
106	#131	—	15	1	no	—
107	#131	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
108	C₂×M₁₆	—	15	1	no	—
109	C₂×M₁₆	—	15	2	no	—
110	C₂²×C₈	—	15	1	no	—
111	C₂×C₄⋊C₄	—	15	2	no	—
112	#108	—	30	1	no	—
113	#108	—	30	1	no	—
114	#106	—	30	1	no	—
115	#106	—	30	1	no	—
116	#103	—	30	2	no	—
117	#102	—	30	2	no	—
118	C₂×C₃×C₅	—	4	1	no	—
119	C₃×D₅	—	4	2	no	—
120	C₂×C₃×C₄	—	20	1	no	—
121	C₂×Dic₃	—	40	2	no	—
122	SL(2,𝔽₃)	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
123	C₃×C₈	—	4	1	no	—
124	C₃×C₈	—	20	1	no	—
125	C₂²×C₅	—	3	1	no	—
126	C₂×D₅	—	3	1	no	—
127	C₂×D₅	—	6	2	no	—
128	C₂×D₅	C₄⋊S₄	1	1	yes	characteristic
129	AGL(1,𝔽₅)	—	24	2	no	—
130	C₄×C₅	—	3	1	no	—
131	C₄×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
132	Dic₅	—	3	1	no	—
133	Dic₅	A₄⋊Q₈	1	1	yes	characteristic
134	#41	—	5	1	no	—
135	#41	—	15	1	no	—
136	#41	—	15	1	no	—
137	#41	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
138	C₂×Q₈	—	5	1	no	—
139	C₂×D₄	—	15	1	no	—
140	C₂²×C₄	—	15	1	no	—
141	C₂²×C₄	—	30	2	no	—
142	M₁₆	—	3	1	no	—
143	M₁₆	—	15	1	no	—
144	M₁₆	—	30	1	no	—
145	M₁₆	—	30	2	no	—
146	M₁₆	—	30	2	no	—
147	C₂×C₈	—	3	1	no	—
148	C₂×C₈	—	5	1	no	—
149	C₂×C₈	—	15	1	no	—
150	C₂×C₈	—	15	1	no	—
151	C₂×C₈	—	15	1	no	—
152	C₂×C₈	—	15	1	no	—
153	C₂×C₈	—	30	2	no	—
154	C₄⋊C₄	—	30	2	no	—
155	C₄⋊C₄	—	30	2	no	—
156	C₃×C₅	—	4	1	no	Hall
157	C₂²×C₃	—	20	1	no	—
158	C₃×C₄	—	4	1	no	—
159	C₃×C₄	—	20	1	no	—
160	Dic₃	—	80	2	no	—
161	C₂×C₅	—	6	1	no	—
162	C₂×C₅	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic
163	D₅	—	6	1	no	—
164	D₅	#5406	1	2	yes	—
165	C₂³	—	15	1	no	—
166	Q₈	—	15	1	no	—
167	Q₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
168	D₄	—	3	1	no	—
169	D₄	—	15	1	no	—
170	D₄	—	30	1	no	—
171	C₂×C₄	—	3	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	15	1	no	—
174	C₂×C₄	—	15	1	no	—
175	C₂×C₄	—	15	1	no	—
176	C₂×C₄	—	15	1	no	—
177	C₂×C₄	—	60	2	no	—
178	C₂×C₄	—	120	2	no	—
179	C₈	—	3	1	no	—
180	C₈	—	5	1	no	—
181	C₈	—	15	1	no	—
182	C₈	—	60	2	no	—
183	C₈	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₂×C₃	—	4	1	no	—
185	C₂×C₃	—	20	2	no	—
186	C₅	#73125	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	3	1	no	—
188	C₂²	—	5	1	no	—
189	C₂²	—	15	1	no	—
190	C₂²	—	30	2	no	—
191	C₄	—	3	1	no	—
192	C₄	—	5	1	no	—
193	C₄	—	15	1	no	—
194	C₄	—	120	2	no	—
195	C₄	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₃	—	4	1	no	Sylow
197	C₂	—	5	2	no	—
198	C₂	—	6	1	no	—
199	C₂	—	30	1	no	—
200	C₂	C₄⋊(A₄⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊(A₄⋊AGL(1,𝔽₅))
commutator	C₅×#41⋊C₃	C₂×C₄
Frattini	C₄	C₂×A₄⋊AGL(1,𝔽₅)
Fitting	C₅×#131	Dic₃
socle	C₂×C₅	C₄⋊SL(2,ℤ/4)
prime radical, cosocle	D₅×#41⋊C₃	C₂²
hypercenter	C₈	A₄⋊AGL(1,𝔽₅)
hypocenter	C₅×SL(2,𝔽₃)	C₄⋊C₄
LCS	G ▷ C₅×#41⋊C₃ ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
6. C₅×#131⋊C₃	C₄	semidirect
8. #4094⋊C₃	C₄	semidirect
25. C₅×#131	Dic₃	semidirect
26. #4094	Dic₃	semidirect
45. #131⋊C₃	AGL(1,𝔽₅)	semidirect
107. #131	ΓL(1,𝔽₁₆)	semidirect
186. C₅	#73125	semidirect

Maximal subgroups

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

subgroup	index
#11756726	2
D₅×#131⋊C₃	2
#10604781	3
#91222	4
#73125	5