Finite Group Databasebeta

Subgroups — #13025196

Order 1280.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10604861	C₂	1	1	yes	maximal, characteristic, complemented
2	#10604083	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600031	C₂	1	1	yes	maximal, characteristic
4	#10598815	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598651	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598636	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598623	C₂	1	1	yes	maximal, characteristic, complemented
8	#65791	—	2	1	no	—
9	#65725	—	2	1	no	—
10	#65485	C₂²	1	1	yes	characteristic
11	#65483	C₄	1	1	yes	characteristic
12	#65184	C₂²	1	1	yes	characteristic
13	#64932	—	2	1	no	—
14	#64898	—	2	1	no	—
15	#64889	C₂²	1	1	yes	characteristic
16	#64889	C₂²	1	1	yes	characteristic
17	#64885	C₂²	1	1	yes	characteristic
18	C₈×#170	C₂²	1	1	yes	characteristic
19	C₅×#359	C₄	1	1	yes	characteristic
20	#64730	C₄	1	1	yes	characteristic
21	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
22	#64713	C₄	1	1	yes	characteristic
23	#12047	—	5	1	no	maximal, Sylow
24	C₅×#135	C₂×C₄	1	1	yes	characteristic
25	#4094	—	2	1	no	—
26	#4053	—	2	1	no	—
27	#4050	C₂×C₄	1	1	yes	characteristic
28	C₄×#170	—	2	1	no	—
29	#3998	—	2	1	no	—
30	#3998	—	2	1	no	—
31	#3998	—	4	1	no	—
32	C₂×#674	—	4	1	no	—
33	C₂×#674	D₄	1	1	yes	characteristic
34	C₂×#674	D₄	1	1	yes	characteristic
35	#3990	—	4	1	no	—
36	D₅⋊C₁₆	—	4	1	no	—
37	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₅×#108	C₂×C₄	1	1	yes	characteristic
39	#3969	—	2	1	no	—
40	#3961	C₈	1	1	yes	characteristic, complemented
41	#3960	C₈	1	1	yes	characteristic, complemented
42	#3959	C₈	1	1	yes	characteristic, complemented
43	#3958	C₈	1	1	yes	characteristic, complemented
44	#3957	—	2	1	no	—
45	#3951	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×#672	C₂×C₄	1	1	yes	characteristic
48	#2178	—	5	1	no	—
49	#2170	—	5	1	no	—
50	#2170	—	5	1	no	—
51	#1373	—	5	1	no	—
52	#1335	—	5	1	no	—
53	#1335	—	5	1	no	—
54	#1311	—	5	1	no	—
55	C₅×#41	—	2	1	no	—
56	#709	—	2	1	no	—
57	C₂×#170	—	2	2	no	—
58	C₅×Q₁₆	C₂×C₈	1	1	yes	characteristic
59	C₅×SD₁₆	—	2	1	no	—
60	C₅×D₈	C₂×C₈	1	1	yes	characteristic
61	C₅×M₁₆	—	2	1	no	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₅	#41	1	1	yes	characteristic
64	#681	—	2	1	no	—
65	#681	—	4	1	no	—
66	C₂×#168	—	4	1	no	—
67	C₂×#168	C₂×D₄	1	1	yes	characteristic
68	Dic₂₀	C₂×C₈	1	1	yes	characteristic
69	D₄₀	C₂×C₈	1	1	yes	characteristic
70	#677	—	2	1	no	—
71	#676	—	4	1	no	—
72	C₈×D₅	—	4	1	no	—
73	#674	—	2	1	no	—
74	#674	—	4	1	no	—
75	#674	—	4	1	no	—
76	#674	D₈	1	1	yes	characteristic
77	#674	D₈	1	1	yes	characteristic
78	C₅×C₁₆	—	2	1	no	—
79	#672	C₂×C₈	1	1	yes	characteristic
80	#672	C₂×C₈	1	1	yes	characteristic
81	#508	—	5	1	no	—
82	#504	—	10	1	no	—
83	#504	—	10	1	no	—
84	#443	—	5	1	no	—
85	#443	—	5	1	no	—
86	#368	—	5	1	no	—
87	#364	—	5	1	no	—
88	#364	—	5	1	no	—
89	#359	—	5	1	no	—
90	#359	AGL(1,𝔽₅)	1	1	yes	characteristic
91	#350	—	10	1	no	—
92	#350	—	10	1	no	—
93	C₄×C₁₆	—	5	1	no	—
94	C₄×C₁₆	—	5	1	no	—
95	C₈²	—	5	1	no	—
96	C₅×Q₈	—	2	1	no	—
97	C₅×D₄	—	2	1	no	—
98	C₅×D₄	—	4	1	no	—
99	C₂×C₄×C₅	—	4	1	no	—
100	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
101	C₂²⋊D₅	—	4	1	no	—
102	C₂×Dic₅	—	2	1	no	—
103	D₂₀	—	2	1	no	—
104	C₄×D₅	—	4	1	no	—
105	Dic₁₀	—	2	1	no	—
106	#170	—	4	2	no	—
107	C₅×C₈	—	4	1	no	—
108	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
109	C₅×C₈	#131	1	1	yes	characteristic
110	#168	—	4	1	no	—
111	#168	C₂×D₈	1	1	yes	characteristic
112	#168	#135	1	1	yes	characteristic
113	#135	—	5	1	no	—
114	#135	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	#131	—	10	1	no	—
116	#131	—	10	1	no	—
117	Q₃₂	—	5	1	no	—
118	SD₃₂	—	5	1	no	—
119	SD₃₂	—	5	1	no	—
120	D₁₆	—	5	1	no	—
121	M₃₂	—	10	1	no	—
122	M₃₂	—	10	1	no	—
123	M₃₂	—	20	1	no	—
124	M₃₂	—	20	1	no	—
125	C₂×C₁₆	—	5	1	no	—
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	—	5	1	no	—
128	C₂×C₁₆	—	20	1	no	—
129	C₂×C₁₆	—	20	1	no	—
130	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	#108	—	5	1	no	—
132	#108	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
133	#104	—	10	1	no	—
134	#104	—	10	1	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	—	10	1	no	—
137	C₂²×C₅	—	4	1	no	—
138	C₂²×C₅	#443	1	1	yes	characteristic
139	C₂×D₅	—	4	1	no	—
140	C₄×C₅	—	4	1	no	—
141	C₄×C₅	C₈×D₄	1	1	yes	characteristic
142	C₄×C₅	C₄×D₈	1	1	yes	characteristic
143	Dic₅	—	2	2	no	—
144	Dic₅	—	4	1	no	—
145	#41	—	2	1	no	—
146	#41	—	10	1	no	—
147	Q₁₆	—	5	1	no	—
148	Q₁₆	D₅⋊C₈	1	1	yes	characteristic
149	SD₁₆	—	2	1	no	—
150	SD₁₆	—	10	1	no	—
151	D₈	—	5	1	no	—
152	D₈	D₅⋊C₈	1	1	yes	characteristic
153	M₁₆	—	2	1	no	—
154	M₁₆	—	10	1	no	—
155	M₁₆	—	20	1	no	—
156	M₁₆	—	20	1	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	10	2	no	—
159	C₂×C₈	—	20	1	no	—
160	C₂×C₈	—	20	1	no	—
161	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₄²	—	5	1	no	—
163	C₁₆	—	2	1	no	—
164	C₁₆	—	5	1	no	—
165	C₁₆	—	5	1	no	—
166	C₁₆	—	5	1	no	—
167	C₁₆	—	5	1	no	—
168	C₁₆	—	10	1	no	—
169	C₁₆	—	20	1	no	—
170	C₁₆	—	20	1	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	—	8	1	no	—
173	C₂×C₅	C₈×D₈	1	1	yes	characteristic
174	D₅	—	8	1	no	—
175	Q₈	—	2	1	no	—
176	Q₈	—	10	1	no	—
177	D₄	—	2	1	no	—
178	D₄	—	4	1	no	—
179	D₄	—	10	1	no	—
180	D₄	—	20	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	10	1	no	—
183	C₂×C₄	—	20	1	no	—
184	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₈	—	4	1	no	—
186	C₈	—	5	1	no	—
187	C₈	—	5	1	no	—
188	C₈	—	20	2	no	—
189	C₈	—	20	1	no	—
190	C₈	C₂×D₅⋊C₈	1	1	yes	characteristic
191	C₈	#4131	1	1	yes	characteristic
192	C₅	#12047	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	4	1	no	—
194	C₂²	—	20	1	no	—
195	C₂²	#65736	1	1	yes	characteristic
196	C₄	—	4	1	no	—
197	C₄	—	10	2	no	—
198	C₄	—	20	1	no	—
199	C₄	#65692	1	1	yes	characteristic
200	C₄	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	8	1	no	—
203	C₂	—	40	1	no	—
204	C₂	#10603935	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×AGL(1,𝔽₅)
commutator	C₅×C₈	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#359	C₄
socle	C₂×C₅	C₈×D₈
prime radical	#65483	C₄
cosocle	C₈×Dic₅	C₂³
hypercenter	#359	AGL(1,𝔽₅)
hypocenter	C₅	#12047
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #10604861	C₂	semidirect
2. #10604083	C₂	semidirect
4. #10598815	C₂	semidirect
5. #10598651	C₂	semidirect
6. #10598636	C₂	semidirect
7. #10598623	C₂	semidirect
40. #3961	C₈	semidirect
41. #3960	C₈	semidirect
42. #3959	C₈	semidirect
43. #3958	C₈	semidirect
192. C₅	#12047	semidirect

Maximal subgroups

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

subgroup	index
#10604861	2
#10604083	2
#10600031	2
#10598815	2
#10598651	2
#10598636	2
#10598623	2
#12047	5