Finite Group Databasebeta

← #10617418⋊C₃ #14512552

Subgroups — #10617418⋊C₃ #14512552

Order 1920.

240 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 44 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	#66264⋊C₃	C₂	1	2	yes	maximal
2	D₅×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#10617418	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	D₅×#41⋊C₃	C₂²	1	1	yes	characteristic
5	#4133⋊C₃	—	2	2	no	—
6	C₅×#131⋊C₃	C₄	1	1	yes	characteristic
7	C₃×#3995	—	4	1	no	maximal
8	#4094⋊C₃	C₄	1	1	yes	characteristic
9	#2916⋊C₃	—	5	1	no	maximal, Hall
10	#66264	C₂×C₃	1	2	yes	—
11	D₅×#131	C₂×C₃	1	1	yes	characteristic
12	#65729	—	3	2	no	—
13	D₅⋊#108	—	3	1	no	—
14	C₂×#3995	—	3	1	no	—
15	C₅×#41⋊C₃	C₂×C₄	1	1	yes	characteristic
16	C₃×#700	—	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	#568⋊C₃	—	5	2	no	—
21	C₂×#131⋊C₃	—	5	1	no	—
22	D₅×#41	C₂²×C₃	1	1	yes	characteristic
23	#4133	—	2	2	no	—
24	#4131	—	6	2	no	—
25	D₅⋊M₁₆	—	3	2	no	—
26	C₂×#700	—	3	2	no	—
27	C₅×#131	C₃×C₄	1	1	yes	characteristic
28	#4094	C₃×C₄	1	1	yes	characteristic
29	#4053	—	3	1	no	—
30	D₅×M₁₆	—	3	1	no	—
31	#4047	—	3	1	no	—
32	C₂×C₈×D₅	—	3	1	no	—
33	#3996	—	3	2	no	—
34	#3995	—	3	2	no	—
35	#3995	—	3	1	no	—
36	#3995	A₄	1	1	yes	characteristic
37	#2916	—	5	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	—	10	2	no	—
46	#131⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₃×#108	—	20	1	no	—
48	C₅×#41	C₂×C₃×C₄	1	1	yes	characteristic
49	#713	C₃×D₄	1	1	yes	characteristic
50	Q₈×D₅	C₃×Q₈	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	#704	—	6	2	no	—
56	C₂×#170	—	6	2	no	—
57	#700	—	3	2	no	—
58	#700	—	6	2	no	—
59	#700	C₂×A₄	1	2	yes	—
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	—	3	2	no	—
66	C₈×D₅	—	3	1	no	—
67	C₈×D₅	—	3	1	no	—
68	C₈×D₅	—	3	1	no	—
69	C₈×D₅	C₂×A₄	1	1	yes	characteristic
70	#568	—	5	2	no	—
71	C₂×#131	—	5	1	no	—
72	#430	—	15	2	no	—
73	C₂×#108	—	15	1	no	—
74	C₂×#108	—	15	1	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	C₃×M₁₆	—	20	2	no	—
82	C₂×C₃×C₈	—	20	1	no	—
83	C₂²×D₅	—	3	1	no	—
84	C₅×Q₈	C₃×C₄⋊C₄	1	1	yes	characteristic
85	C₅×D₄	—	3	1	no	—
86	C₂×C₄×C₅	—	3	1	no	—
87	C₂²⋊D₅	—	3	2	no	—
88	C₂×Dic₅	—	3	1	no	—
89	D₂₀	—	3	1	no	—
90	C₄×D₅	—	3	1	no	—
91	C₄×D₅	—	3	1	no	—
92	C₄×D₅	—	3	1	no	—
93	C₄×D₅	C₂²×A₄	1	1	yes	characteristic
94	Dic₁₀	—	3	1	no	—
95	#170	—	2	2	no	—
96	#170	—	6	2	no	—
97	C₅×C₈	—	3	1	no	—
98	C₅×C₈	C₄×A₄	1	1	yes	characteristic
99	#168	—	3	1	no	—
100	#168	C₄×A₄	1	1	yes	characteristic
101	C₂×#41	—	5	1	no	—
102	#131	—	5	1	no	—
103	#131	—	10	2	no	—
104	#131	—	15	1	no	—
105	#131	—	15	1	no	—
106	#131	—	30	2	no	—
107	#131	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₂×M₁₆	—	15	1	no	—
109	C₂×M₁₆	—	15	2	no	—
110	C₂×M₁₆	—	15	2	no	—
111	C₂²×C₈	—	15	1	no	—
112	#108	—	5	1	no	—
113	#108	—	15	1	no	—
114	#108	—	15	2	no	—
115	#108	—	15	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	Q₈×A₄	1	1	yes	characteristic
127	C₄×C₅	—	3	1	no	—
128	C₄×C₅	C₂×C₄×A₄	1	1	yes	characteristic
129	Dic₅	—	3	1	no	—
130	Dic₅	D₄×A₄	1	1	yes	characteristic
131	#41	—	5	1	no	—
132	#41	—	15	1	no	—
133	#41	—	15	1	no	—
134	#41	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×Q₈	—	5	1	no	—
136	C₂×D₄	—	15	1	no	—
137	C₂²×C₄	—	15	1	no	—
138	M₁₆	—	3	1	no	—
139	M₁₆	—	5	2	no	—
140	M₁₆	—	15	2	no	—
141	M₁₆	—	15	1	no	—
142	M₁₆	—	15	2	no	—
143	M₁₆	—	30	2	no	—
144	M₁₆	—	30	2	no	—
145	C₂×C₈	—	3	1	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	15	1	no	—
148	C₂×C₈	—	15	1	no	—
149	C₂×C₈	—	15	1	no	—
150	C₂×C₈	—	15	1	no	—
151	C₂×C₈	—	30	2	no	—
152	C₂×C₈	—	30	2	no	—
153	C₃×C₅	—	4	1	no	Hall
154	C₂²×C₃	—	20	1	no	—
155	C₃×C₄	—	4	1	no	—
156	C₃×C₄	—	20	1	no	—
157	C₂×C₅	—	6	1	no	—
158	C₂×C₅	A₄×C₄⋊C₄	1	1	yes	characteristic
159	D₅	—	2	1	no	—
160	D₅	—	6	1	no	—
161	C₂³	—	15	1	no	—
162	Q₈	—	15	1	no	—
163	Q₈	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
164	D₄	—	3	1	no	—
165	D₄	—	15	1	no	—
166	D₄	—	15	2	no	—
167	C₂×C₄	—	3	1	no	—
168	C₂×C₄	—	5	1	no	—
169	C₂×C₄	—	15	1	no	—
170	C₂×C₄	—	15	1	no	—
171	C₂×C₄	—	15	1	no	—
172	C₂×C₄	—	15	1	no	—
173	C₈	—	3	1	no	—
174	C₈	—	5	1	no	—
175	C₈	—	10	2	no	—
176	C₈	—	15	1	no	—
177	C₈	—	30	2	no	—
178	C₈	A₄×AGL(1,𝔽₅)	1	1	yes	characteristic
179	C₂×C₃	—	4	1	no	—
180	C₂×C₃	—	40	1	no	—
181	C₅	#2916⋊C₃	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	3	1	no	—
183	C₂²	—	5	1	no	—
184	C₂²	—	15	1	no	—
185	C₂²	—	60	1	no	—
186	C₄	—	3	1	no	—
187	C₄	—	5	1	no	—
188	C₄	—	15	1	no	—
189	C₄	C₂×A₄×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₃	—	4	1	no	Sylow
191	C₂	—	6	1	no	—
192	C₂	—	10	1	no	—
193	C₂	—	30	1	no	—
194	C₂	A₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	A₄×C₄⋊AGL(1,𝔽₅)
commutator	C₅×#41	C₂×C₃×C₄
Frattini	C₄	C₂×A₄×AGL(1,𝔽₅)
Fitting	C₅×#131	C₃×C₄
socle	C₂×C₅	A₄×C₄⋊C₄
prime radical	D₅×#41⋊C₃	C₂²
cosocle	D₅×#41	C₂²×C₃
hypercenter	C₈	A₄×AGL(1,𝔽₅)
hypocenter	C₅×Q₈	C₃×C₄⋊C₄
LCS	G ▷ C₅×#41 ▷ C₅×Q₈
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
3. #10617418	C₃	semidirect
181. C₅	#2916⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#66264⋊C₃	2
D₅×#131⋊C₃	2
#10617418	3
C₃×#3995	4
#2916⋊C₃	5