Finite Group Databasebeta

Subgroups — #13016679

Order 1280.

354 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 132 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	#10603838	C₂	1	1	yes	maximal, characteristic
2	#10603809	C₂	1	1	yes	maximal, characteristic
3	#10598861	C₂	1	1	yes	maximal, characteristic
4	#10598529	C₂	1	2	yes	maximal
5	#10598526	C₂	1	2	yes	maximal
6	C₂×#3992	C₄	1	2	yes	—
7	C₂×C₈×AGL(1,𝔽₅)	C₄	1	2	yes	—
8	#65686	C₂²	1	1	yes	characteristic
9	#65678	C₂²	1	2	yes	—
10	#65676	C₂²	1	1	yes	characteristic
11	#64974	C₂²	1	2	yes	—
12	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
13	#64879	C₄	1	2	yes	complemented
14	C₈×#170	C₄	1	2	yes	complemented
15	C₅×#322	C₄	1	1	yes	characteristic, complemented
16	#64675	C₄	1	1	yes	characteristic, complemented
17	C₈×#168	C₄	1	2	yes	complemented
18	#8138	—	5	1	no	maximal, Sylow
19	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
20	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₅	C₂³	1	1	yes	characteristic
24	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
25	#4004	—	2	2	no	—
26	#4003	—	2	2	no	—
27	#4002	—	2	2	no	—
28	C₄×#170	C₂×C₄	1	4	yes	—
29	#3992	C₈	1	8	yes	complemented
30	C₈×AGL(1,𝔽₅)	—	2	4	no	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	#3952	—	2	2	no	—
34	#3946	—	2	2	no	—
35	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₄×#168	C₂×C₄	1	2	yes	—
38	C₄×#168	C₂×C₄	1	1	yes	characteristic
39	#1726	—	5	1	no	—
40	#1450	—	5	1	no	—
41	#1450	—	5	1	no	—
42	#1275	—	5	2	no	—
43	#1274	—	5	2	no	—
44	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
45	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅	#41	1	2	yes	—
47	C₂×#170	—	2	2	no	—
48	C₂×#170	—	2	4	no	—
49	C₄×AGL(1,𝔽₅)	—	2	2	no	—
50	C₄×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
51	D₅⋊C₈	—	2	2	no	—
52	D₅⋊C₈	C₂×C₈	1	4	yes	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
55	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₅	#41	1	2	yes	—
59	C₂×#168	—	2	2	no	—
60	C₂×#168	C₄²	1	1	yes	characteristic
61	C₂×#168	C₄²	1	1	yes	characteristic
62	C₈×D₅	C₂×C₈	1	4	yes	—
63	C₈×D₅	M₁₆	1	4	yes	—
64	#431	—	5	2	no	—
65	#431	—	5	2	no	—
66	C₂×#97	—	5	2	no	—
67	C₂×#97	—	5	1	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	C₂×C₄×C₈	—	5	2	no	—
70	#376	—	5	1	no	—
71	#322	—	5	2	no	—
72	#322	—	5	1	no	—
73	#322	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
74	C₈²	—	5	2	no	—
75	C₈²	—	5	2	no	—
76	C₂²×D₅	#117	1	1	yes	characteristic
77	C₂×AGL(1,𝔽₅)	—	2	4	no	—
78	C₂×AGL(1,𝔽₅)	—	4	4	no	—
79	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅	#117	1	2	yes	—
81	C₂×Dic₅	#117	1	2	yes	—
82	C₂×Dic₅	#117	1	1	yes	characteristic
83	C₄×D₅	—	4	2	no	—
84	C₄×D₅	C₂²×C₈	1	2	yes	—
85	C₄×D₅	C₂×M₁₆	1	2	yes	—
86	#170	—	2	4	no	—
87	#170	—	4	4	no	—
88	C₅×C₈	—	4	2	no	—
89	C₅×C₈	C₄×C₈	1	2	yes	—
90	C₅×C₈	#97	1	2	yes	—
91	#168	—	4	2	no	—
92	#168	C₄×C₈	1	2	yes	—
93	#168	#97	1	2	yes	—
94	C₂²×C₈	—	5	1	no	—
95	C₂²×C₈	—	5	1	no	—
96	C₂²×C₈	—	5	2	no	—
97	C₂×C₄²	—	5	1	no	—
98	C₂×C₄²	—	5	2	no	—
99	C₄⋊C₈	—	10	2	no	—
100	C₄⋊C₈	—	10	2	no	—
101	C₂²⋊C₈	—	10	2	no	—
102	C₂²⋊C₈	—	10	2	no	—
103	#97	—	5	8	no	—
104	#97	—	10	2	no	—
105	C₄×C₈	—	5	1	no	—
106	C₄×C₈	—	5	1	no	—
107	C₄×C₈	—	5	4	no	—
108	C₄×C₈	—	5	1	no	—
109	C₄×C₈	—	5	2	no	—
110	C₄×C₈	—	10	4	no	—
111	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
112	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
113	#95	—	10	2	no	—
114	C₂²×C₅	#376	1	1	yes	characteristic
115	C₂×D₅	—	2	2	no	—
116	C₂×D₅	#431	1	2	yes	—
117	AGL(1,𝔽₅)	—	4	4	no	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
120	C₄×C₅	C₂×#97	1	1	yes	characteristic
121	Dic₅	—	2	2	no	—
122	Dic₅	#431	1	2	yes	—
123	C₂²×C₄	—	5	1	no	—
124	C₂²×C₄	—	5	2	no	—
125	C₂²×C₄	—	10	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	5	1	no	—
128	C₂×C₈	—	5	1	no	—
129	C₂×C₈	—	5	4	no	—
130	C₂×C₈	—	5	4	no	—
131	C₂×C₈	—	5	4	no	—
132	C₂×C₈	—	10	2	no	—
133	C₂×C₈	—	10	2	no	—
134	C₂×C₈	—	10	4	no	—
135	C₂×C₈	—	10	2	no	—
136	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₄²	—	5	1	no	—
139	C₄²	—	5	2	no	—
140	C₄²	—	5	4	no	—
141	C₄²	—	10	2	no	—
142	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₂×C₅	#1744	1	1	yes	characteristic
144	C₂×C₅	#1747	1	2	yes	—
145	D₅	—	2	2	no	—
146	C₂³	—	5	1	no	—
147	C₂×C₄	—	5	1	no	—
148	C₂×C₄	—	5	2	no	—
149	C₂×C₄	—	5	2	no	—
150	C₂×C₄	—	5	2	no	—
151	C₂×C₄	—	10	4	no	—
152	C₂×C₄	—	20	2	no	—
153	C₂×C₄	—	20	4	no	—
154	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
155	C₂×C₄	#4130	1	2	yes	—
156	C₈	—	4	2	no	—
157	C₈	—	5	2	no	—
158	C₈	—	5	2	no	—
159	C₈	—	10	4	no	—
160	C₈	—	20	2	no	—
161	C₈	—	20	4	no	—
162	C₈	C₈×AGL(1,𝔽₅)	1	2	yes	—
163	C₈	#3992	1	2	yes	—
164	C₅	#8138	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	5	2	no	—
166	C₂²	—	10	2	no	—
167	C₂²	#65686	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	5	2	no	—
170	C₄	—	10	2	no	—
171	C₄	—	20	4	no	—
172	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₄	C₂×#3992	1	1	yes	characteristic
174	C₂	—	10	2	no	—
175	C₂	#10603839	1	1	yes	characteristic
176	C₂	#10603884	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂²×C₅	#376
prime radical	C₂²×D₅	#117
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#8138
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #322

Complements

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

N	H ≅ G/N	type
13. #64879	C₄	semidirect
14. C₈×#170	C₄	semidirect
15. C₅×#322	C₄	semidirect
16. #64675	C₄	semidirect
17. C₈×#168	C₄	semidirect
29. #3992	C₈	semidirect
73. #322	AGL(1,𝔽₅)	semidirect
164. C₅	#8138	semidirect

Maximal subgroups

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

subgroup	index
#10603838	2
#10603809	2
#10598861	2
#10598529	2
#10598526	2
#8138	5