Finite Group Databasebeta

Subgroups — #14512467

Order 1920.

236 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 38 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	#11756724	C₂	1	2	yes	maximal
2	D₅×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#10604779	—	3	1	no	maximal, Hall
4	D₅×#41⋊C₃	C₂²	1	1	yes	characteristic
5	#91885	—	2	2	no	—
6	C₅×#131⋊C₃	C₄	1	1	yes	characteristic, complemented
7	#91224	—	4	1	no	maximal
8	#4094⋊C₃	C₄	1	1	yes	characteristic, complemented
9	#73120	—	5	1	no	maximal, Hall
10	D₅×#131	D₃	1	1	yes	characteristic
11	#65795	—	3	2	no	—
12	C₂×#3995	—	3	1	no	—
13	#65721	—	3	1	no	—
14	#64902	—	3	2	no	—
15	C₅×#41⋊C₃	C₂×C₄	1	1	yes	characteristic
16	#6758	—	4	2	no	—
17	D₅×SL(2,𝔽₃)	Q₈	1	1	yes	characteristic
18	C₅⋊(#41⋊C₃)	D₄	1	1	yes	characteristic
19	C₃×C₈×D₅	—	4	1	no	—
20	C₂×#131⋊C₃	—	5	1	no	—
21	#5423	—	5	2	no	—
22	D₅×#41	C₂×D₃	1	1	yes	characteristic
23	#4130	—	3	2	no	—
24	C₂×#700	—	3	2	no	—
25	C₅×#131	Dic₃	1	1	yes	characteristic
26	#4094	Dic₃	1	1	yes	characteristic
27	#4053	—	3	1	no	—
28	D₅×M₁₆	—	3	1	no	—
29	#4047	—	3	1	no	—
30	C₂×C₈×D₅	—	3	1	no	—
31	#4010	—	6	2	no	—
32	#4008	—	6	2	no	—
33	#3995	—	6	1	no	—
34	#3995	—	6	1	no	—
35	#3994	—	6	1	no	—
36	#3993	—	6	1	no	—
37	#1814	—	15	1	no	Sylow
38	C₃×C₄×D₅	—	4	1	no	—
39	C₅×SL(2,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
40	C₃⋊#170	—	8	2	no	—
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	GU(2,𝔽₃)	—	10	2	no	—
47	#819	—	20	1	no	—
48	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
49	#713	D₁₂	1	1	yes	characteristic
50	Q₈×D₅	Dic₆	1	1	yes	characteristic
51	#711	—	3	1	no	—
52	D₄×D₅	—	3	1	no	—
53	#709	—	3	1	no	—
54	C₂×C₄×D₅	—	3	1	no	—
55	C₂²⋊AGL(1,𝔽₅)	—	6	2	no	—
56	C₂×#170	—	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	#700	—	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	#427	—	15	1	no	—
73	#421	—	15	2	no	—
74	#344	—	15	2	no	—
75	C₂×C₃×D₅	—	4	1	no	—
76	C₃×C₄×C₅	—	4	1	no	—
77	C₃×Dic₅	—	4	1	no	—
78	#41⋊C₃	—	5	1	no	—
79	#41⋊C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₂×SL(2,𝔽₃)	—	5	1	no	—
81	C₂×C₃×C₈	—	20	1	no	—
82	#208	—	20	2	no	—
83	C₂²×D₅	—	3	1	no	—
84	C₂×AGL(1,𝔽₅)	—	12	2	no	—
85	C₅×Q₈	C₄⋊Dic₃	1	1	yes	characteristic
86	C₅×D₄	—	3	1	no	—
87	C₂×C₄×C₅	—	3	1	no	—
88	C₂²⋊D₅	—	6	1	no	—
89	C₂×Dic₅	—	3	1	no	—
90	D₂₀	—	3	1	no	—
91	C₄×D₅	—	3	1	no	—
92	C₄×D₅	—	3	1	no	—
93	C₄×D₅	—	3	1	no	—
94	C₄×D₅	C₂×S₄	1	1	yes	characteristic
95	Dic₁₀	—	3	1	no	—
96	#170	—	12	2	no	—
97	C₅×C₈	—	3	1	no	—
98	C₅×C₈	SL(2,ℤ/4)	1	1	yes	characteristic
99	#168	—	3	1	no	—
100	#168	SL(2,ℤ/4)	1	1	yes	characteristic
101	C₂×#41	—	5	1	no	—
102	#131	—	5	1	no	—
103	#131	—	15	1	no	—
104	#131	—	15	1	no	—
105	#131	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
106	C₂×M₁₆	—	15	1	no	—
107	C₂×M₁₆	—	15	2	no	—
108	C₂²×C₈	—	15	1	no	—
109	#117	—	15	2	no	—
110	#108	—	30	1	no	—
111	#108	—	30	1	no	—
112	#107	—	30	1	no	—
113	#106	—	30	1	no	—
114	#104	—	30	2	no	—
115	#104	—	30	2	no	—
116	C₂×C₃×C₅	—	4	1	no	—
117	C₃×D₅	—	8	1	no	—
118	C₂×C₃×C₄	—	20	1	no	—
119	SL(2,𝔽₃)	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₃×C₈	—	4	1	no	—
121	C₃×C₈	—	20	1	no	—
122	C₃⋊C₈	—	40	2	no	—
123	C₂²×C₅	—	3	1	no	—
124	C₂×D₅	—	3	1	no	—
125	C₂×D₅	—	12	1	no	—
126	C₂×D₅	A₄⋊Q₈	1	1	yes	characteristic
127	AGL(1,𝔽₅)	—	24	2	no	—
128	C₄×C₅	—	3	1	no	—
129	C₄×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
130	Dic₅	—	3	1	no	—
131	Dic₅	C₄⋊S₄	1	1	yes	characteristic
132	#41	—	5	1	no	—
133	#41	—	15	1	no	—
134	#41	—	15	1	no	—
135	#41	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
136	C₂×Q₈	—	5	1	no	—
137	C₂×D₄	—	15	1	no	—
138	C₂²×C₄	—	15	1	no	—
139	M₁₆	—	3	1	no	—
140	M₁₆	—	15	1	no	—
141	M₁₆	—	30	1	no	—
142	M₁₆	—	30	2	no	—
143	M₁₆	—	30	2	no	—
144	C₂×C₈	—	3	1	no	—
145	C₂×C₈	—	5	1	no	—
146	C₂×C₈	—	15	1	no	—
147	C₂×C₈	—	15	1	no	—
148	C₂×C₈	—	15	1	no	—
149	C₂×C₈	—	15	1	no	—
150	C₂×C₈	—	30	2	no	—
151	C₄⋊C₄	—	30	2	no	—
152	C₂²⋊C₄	—	30	2	no	—
153	C₄²	—	30	2	no	—
154	C₃×C₅	—	4	1	no	Hall
155	C₂²×C₃	—	20	1	no	—
156	C₃×C₄	—	4	1	no	—
157	C₃×C₄	—	20	1	no	—
158	C₂×C₅	—	6	1	no	—
159	C₂×C₅	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic
160	D₅	—	2	1	no	—
161	D₅	—	6	1	no	—
162	C₂³	—	15	1	no	—
163	Q₈	—	15	1	no	—
164	Q₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
165	D₄	—	3	1	no	—
166	D₄	—	15	1	no	—
167	D₄	—	30	1	no	—
168	C₂×C₄	—	3	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	15	1	no	—
171	C₂×C₄	—	15	1	no	—
172	C₂×C₄	—	15	1	no	—
173	C₂×C₄	—	15	1	no	—
174	C₂×C₄	—	60	2	no	—
175	C₈	—	3	1	no	—
176	C₈	—	5	1	no	—
177	C₈	—	15	1	no	—
178	C₈	—	60	2	no	—
179	C₈	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₂×C₃	—	4	1	no	—
181	C₂×C₃	—	40	1	no	—
182	C₅	#73120	1	1	yes	characteristic, complemented, Sylow
183	C₂²	—	3	1	no	—
184	C₂²	—	5	1	no	—
185	C₂²	—	15	1	no	—
186	C₂²	—	60	1	no	—
187	C₄	—	3	1	no	—
188	C₄	—	5	1	no	—
189	C₄	—	15	1	no	—
190	C₄	—	120	2	no	—
191	C₄	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₃	—	4	1	no	Sylow
193	C₂	—	6	1	no	—
194	C₂	—	10	1	no	—
195	C₂	—	30	1	no	—
196	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
45. #131⋊C₃	AGL(1,𝔽₅)	semidirect
182. C₅	#73120	semidirect

Maximal subgroups

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

subgroup	index
#11756724	2
D₅×#131⋊C₃	2
#10604779	3
#91224	4
#73120	5