Finite Group Databasebeta

Subgroups — #13025334

Order 1280.

204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10604838	C₂	1	1	yes	maximal, characteristic
2	#10604069	C₂	1	1	yes	maximal, characteristic
3	#10600002	C₂	1	1	yes	maximal, characteristic
4	#10598807	C₂	1	1	yes	maximal, characteristic
5	#10598641	C₂	1	1	yes	maximal, characteristic
6	#10598640	C₂	1	1	yes	maximal, characteristic
7	#10598608	C₂	1	1	yes	maximal, characteristic
8	Q₈×#170	—	2	1	no	—
9	#65709	—	2	1	no	—
10	Q₁₆×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×#3961	C₄	1	1	yes	characteristic
12	#65162	C₂²	1	1	yes	characteristic
13	Q₈⋊#170	—	2	1	no	—
14	#64886	C₂²	1	1	yes	characteristic
15	#64881	C₂²	1	1	yes	characteristic
16	#64881	C₂²	1	1	yes	characteristic
17	#64879	C₂²	1	1	yes	characteristic
18	#64872	—	2	1	no	—
19	C₅×#358	C₄	1	1	yes	characteristic
20	#64723	C₄	1	1	yes	characteristic
21	#64722	C₂²	1	1	yes	characteristic
22	#64710	C₄	1	1	yes	characteristic
23	#12079	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	Q₈×Dic₅	—	2	1	no	—
26	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
27	#4033	—	2	1	no	—
28	#4002	—	4	1	no	—
29	C₄⋊#170	—	2	1	no	—
30	C₄⋊#170	—	2	1	no	—
31	C₄⋊#170	—	4	1	no	—
32	C₄⋊#170	—	4	1	no	—
33	C₄×#170	—	4	1	no	—
34	C₄×#170	D₄	1	1	yes	characteristic
35	C₄×#170	D₄	1	1	yes	characteristic
36	C₂×#674	—	2	1	no	—
37	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₅×#107	C₂×C₄	1	1	yes	characteristic
39	#3967	—	2	1	no	—
40	#3961	—	2	2	no	—
41	#3950	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅	C₂³	1	1	yes	characteristic
43	C₂×#672	C₂×C₄	1	1	yes	characteristic
44	#3942	—	2	1	no	—
45	#2176	—	5	1	no	—
46	#1593	—	5	1	no	—
47	#1579	—	5	1	no	—
48	#1368	—	5	1	no	—
49	#1334	—	5	1	no	—
50	#1334	—	5	1	no	—
51	#1313	—	5	1	no	—
52	C₂×C₅×Q₈	—	2	1	no	—
53	C₂×Dic₁₀	—	2	1	no	—
54	C₂×#170	—	2	1	no	—
55	C₂×#170	—	2	1	no	—
56	C₂×#170	—	4	1	no	—
57	C₂×#170	—	4	1	no	—
58	C₅×Q₁₆	—	2	1	no	—
59	C₅×Q₁₆	M₁₆	1	2	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	2	1	no	—
62	C₄⋊Dic₅	—	2	1	no	—
63	C₄⋊Dic₅	—	4	1	no	—
64	#683	—	4	1	no	—
65	C₄×Dic₅	—	4	1	no	—
66	C₄×Dic₅	—	4	1	no	—
67	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
68	C₂×#168	#41	1	1	yes	characteristic
69	Dic₂₀	—	2	1	no	—
70	Dic₂₀	—	2	1	no	—
71	#674	—	2	2	no	—
72	C₅×C₁₆	—	2	1	no	—
73	#672	—	2	1	no	—
74	C₂×Q₃₂	—	5	1	no	—
75	#446	—	10	1	no	—
76	C₈×Q₈	—	10	1	no	—
77	C₄×Q₁₆	—	5	1	no	—
78	C₄×Q₁₆	—	5	1	no	—
79	#367	—	5	1	no	—
80	#358	—	5	1	no	—
81	#358	AGL(1,𝔽₅)	1	1	yes	characteristic
82	#346	—	5	1	no	—
83	#346	—	5	1	no	—
84	#334	—	5	1	no	—
85	#334	—	5	1	no	—
86	Q₈⋊C₈	—	10	1	no	—
87	Q₈⋊C₈	—	10	1	no	—
88	#322	—	5	1	no	—
89	C₅×Q₈	—	2	2	no	—
90	C₅×Q₈	—	4	1	no	—
91	C₂×C₄×C₅	—	4	1	no	—
92	C₂×C₄×C₅	—	4	1	no	—
93	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
94	C₂×Dic₅	—	4	1	no	—
95	C₂×Dic₅	—	4	1	no	—
96	C₂×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
98	Dic₁₀	—	4	1	no	—
99	Dic₁₀	—	4	1	no	—
100	#170	—	4	1	no	—
101	#170	—	4	1	no	—
102	#170	—	8	1	no	—
103	#170	—	8	1	no	—
104	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
105	C₅×C₈	#131	1	1	yes	characteristic
106	#168	—	2	1	no	—
107	C₂×Q₁₆	—	5	1	no	—
108	C₂×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₄×Q₈	—	10	1	no	—
110	C₄×Q₈	—	10	1	no	—
111	Q₃₂	—	10	2	no	—
112	C₂×C₁₆	—	5	1	no	—
113	C₂×C₁₆	—	10	1	no	—
114	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	#107	—	5	1	no	—
116	#107	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₄⋊C₈	—	10	1	no	—
118	C₄⋊C₈	—	10	1	no	—
119	C₄⋊C₈	—	20	1	no	—
120	C₄⋊C₈	—	20	1	no	—
121	#103	—	10	1	no	—
122	#103	—	10	1	no	—
123	#97	—	20	1	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	—	20	1	no	—
128	C₂²×C₅	#442	1	1	yes	characteristic
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	—	8	1	no	—
131	C₄×C₅	#435	1	1	yes	characteristic
132	C₄×C₅	#443	1	1	yes	characteristic
133	Dic₅	—	2	1	no	—
134	Dic₅	—	2	1	no	—
135	Dic₅	—	8	1	no	—
136	Dic₅	—	8	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	10	1	no	—
139	Q₁₆	—	2	1	no	—
140	Q₁₆	—	10	1	no	—
141	Q₁₆	—	10	1	no	—
142	Q₁₆	#700	1	2	yes	—
143	C₂×C₈	—	5	1	no	—
144	C₂×C₈	—	10	1	no	—
145	C₂×C₈	—	10	1	no	—
146	C₂×C₈	—	20	1	no	—
147	C₂×C₈	—	20	1	no	—
148	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	10	1	no	—
151	C₄⋊C₄	—	20	1	no	—
152	C₄⋊C₄	—	20	1	no	—
153	C₄²	—	5	1	no	—
154	C₄²	—	20	1	no	—
155	C₄²	—	20	1	no	—
156	C₁₆	—	2	1	no	—
157	C₁₆	—	10	2	no	—
158	C₁₆	—	10	1	no	—
159	C₂×C₅	#1586	1	1	yes	characteristic
160	C₂×C₅	#2180	1	1	yes	characteristic
161	C₂×C₅	#2180	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	20	1	no	—
165	Q₈	—	20	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	5	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	20	1	no	—
171	C₂×C₄	—	20	1	no	—
172	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₈	—	10	1	no	—
174	C₈	—	20	1	no	—
175	C₈	—	20	1	no	—
176	C₈	—	40	1	no	—
177	C₈	—	40	1	no	—
178	C₈	C₂×#700	1	1	yes	characteristic
179	C₈	#4131	1	1	yes	characteristic
180	C₅	#12079	1	1	yes	characteristic, complemented, Sylow
181	C₂²	#65731	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	10	1	no	—
185	C₄	—	10	1	no	—
186	C₄	—	40	1	no	—
187	C₄	—	40	1	no	—
188	C₄	#65693	1	1	yes	characteristic
189	C₄	#65736	1	1	yes	characteristic
190	C₂	#10603944	1	1	yes	characteristic
191	C₂	#10604130	1	1	yes	characteristic
192	C₂	#10604139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65731
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#358	C₄
socle, prime radical	C₂²×C₅	#442
cosocle	C₈×Dic₅	C₂³
hypercenter	#358	AGL(1,𝔽₅)
hypocenter	C₅	#12079
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
180. C₅	#12079	semidirect

Maximal subgroups

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

subgroup	index
#10604838	2
#10604069	2
#10600002	2
#10598807	2
#10598641	2
#10598640	2
#10598608	2
#12079	5