Finite Group Databasebeta

Subgroups — #13025317

Order 1280.

210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 64 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	#10604841	C₂	1	1	yes	maximal, characteristic
2	#10604060	C₂	1	1	yes	maximal, characteristic
3	#10600000	C₂	1	1	yes	maximal, characteristic
4	#10598809	C₂	1	1	yes	maximal, characteristic
5	#10598646	C₂	1	1	yes	maximal, characteristic
6	#170⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	#10598610	C₂	1	1	yes	maximal, characteristic
8	#65788	—	2	1	no	—
9	#65703	—	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	#64885	C₂²	1	1	yes	characteristic
15	#64882	C₂²	1	1	yes	characteristic
16	#64882	C₂²	1	1	yes	characteristic
17	C₈×#170	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	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
22	#64709	C₄	1	1	yes	characteristic
23	#12076	—	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	#4004	—	4	1	no	—
29	#4004	—	4	1	no	—
30	#4002	—	4	1	no	—
31	C₄⋊#170	—	2	1	no	—
32	C₄⋊#170	—	2	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	C₄×Q₃₂	—	5	1	no	—
46	#1591	—	5	1	no	—
47	C₈×Q₁₆	—	5	1	no	—
48	#1372	—	5	1	no	—
49	#1332	—	5	1	no	—
50	#1332	—	5	1	no	—
51	#1312	—	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	—	4	1	no	—
56	C₂×#170	—	4	1	no	—
57	C₂×#170	D₈	1	1	yes	characteristic
58	C₂×#170	D₈	1	1	yes	characteristic
59	C₅×Q₁₆	—	2	1	no	—
60	C₅×Q₁₆	—	2	1	no	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	2	1	no	—
63	C₄⋊Dic₅	—	2	1	no	—
64	C₄⋊Dic₅	—	4	1	no	—
65	#683	—	4	1	no	—
66	C₄×Dic₅	—	4	1	no	—
67	C₄×Dic₅	—	4	1	no	—
68	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
69	C₂×#168	#41	1	1	yes	characteristic
70	Dic₂₀	—	2	1	no	—
71	Dic₂₀	M₁₆	1	2	yes	—
72	#674	—	4	1	no	—
73	C₅×C₁₆	—	2	1	no	—
74	#672	—	2	1	no	—
75	C₂×Q₃₂	—	5	1	no	—
76	#446	—	10	1	no	—
77	C₈×Q₈	—	10	1	no	—
78	C₄×Q₁₆	—	5	1	no	—
79	C₄×Q₁₆	—	5	1	no	—
80	#366	—	5	1	no	—
81	#358	—	5	1	no	—
82	#358	AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₄×C₁₆	—	5	1	no	—
84	C₄×C₁₆	—	5	1	no	—
85	#335	—	5	1	no	—
86	#335	—	5	1	no	—
87	Q₈⋊C₈	—	10	1	no	—
88	Q₈⋊C₈	—	10	1	no	—
89	C₈²	—	5	1	no	—
90	C₅×Q₈	—	4	1	no	—
91	C₅×Q₈	—	4	1	no	—
92	C₂×C₄×C₅	—	4	1	no	—
93	C₂×C₄×C₅	—	4	1	no	—
94	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₂×Dic₅	—	4	1	no	—
96	C₂×Dic₅	—	4	1	no	—
97	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
98	C₂×Dic₅	#135	1	1	yes	characteristic
99	Dic₁₀	—	2	2	no	—
100	Dic₁₀	—	4	1	no	—
101	#170	—	2	2	no	—
102	#170	—	2	1	no	—
103	#170	—	2	1	no	—
104	#170	—	8	1	no	—
105	#170	—	8	1	no	—
106	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
107	C₅×C₈	#131	1	1	yes	characteristic
108	#168	—	2	1	no	—
109	C₂×Q₁₆	—	5	1	no	—
110	C₂×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₄×Q₈	—	10	1	no	—
112	C₄×Q₈	—	10	1	no	—
113	Q₃₂	—	10	2	no	—
114	C₂×C₁₆	—	5	1	no	—
115	C₂×C₁₆	—	10	1	no	—
116	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	#107	—	5	1	no	—
118	#107	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₄⋊C₈	—	10	1	no	—
120	C₄⋊C₈	—	10	1	no	—
121	C₄⋊C₈	—	20	1	no	—
122	C₄⋊C₈	—	20	1	no	—
123	#103	—	10	1	no	—
124	#103	—	10	1	no	—
125	#97	—	20	1	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	—	5	1	no	—
129	C₄×C₈	—	20	1	no	—
130	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
131	C₄×C₅	—	8	1	no	—
132	C₄×C₅	—	8	1	no	—
133	C₄×C₅	#435	1	1	yes	characteristic
134	C₄×C₅	#444	1	1	yes	characteristic
135	Dic₅	—	2	1	no	—
136	Dic₅	—	4	2	no	—
137	Dic₅	—	8	1	no	—
138	Dic₅	#510	1	1	yes	characteristic
139	Dic₅	#510	1	1	yes	characteristic
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	10	1	no	—
142	Q₁₆	—	2	1	no	—
143	Q₁₆	—	2	1	no	—
144	Q₁₆	—	5	2	no	—
145	Q₁₆	—	10	1	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	1	no	—
149	C₂×C₈	—	10	1	no	—
150	C₂×C₈	—	20	1	no	—
151	C₂×C₈	—	20	1	no	—
152	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	10	1	no	—
155	C₄⋊C₄	—	20	1	no	—
156	C₄⋊C₄	—	20	1	no	—
157	C₄²	—	5	1	no	—
158	C₄²	—	20	1	no	—
159	C₄²	—	20	1	no	—
160	C₁₆	—	2	1	no	—
161	C₁₆	—	10	1	no	—
162	C₁₆	—	20	1	no	—
163	C₂×C₅	#1581	1	1	yes	characteristic
164	C₂×C₅	#2176	1	1	yes	characteristic
165	C₂×C₅	#2178	1	1	yes	characteristic
166	Q₈	—	4	1	no	—
167	Q₈	—	4	1	no	—
168	Q₈	—	10	2	no	—
169	Q₈	—	20	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	20	1	no	—
175	C₂×C₄	—	20	1	no	—
176	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₈	—	10	1	no	—
178	C₈	—	10	1	no	—
179	C₈	—	10	2	no	—
180	C₈	—	10	1	no	—
181	C₈	—	40	1	no	—
182	C₈	—	40	1	no	—
183	C₈	D₅⋊M₁₆	1	1	yes	characteristic
184	C₈	#4131	1	1	yes	characteristic
185	C₅	#12076	1	1	yes	characteristic, complemented, Sylow
186	C₂²	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₄	—	5	1	no	—
188	C₄	—	5	1	no	—
189	C₄	—	8	1	no	—
190	C₄	—	8	1	no	—
191	C₄	—	10	1	no	—
192	C₄	—	20	2	no	—
193	C₄	—	40	1	no	—
194	C₄	#65694	1	1	yes	characteristic
195	C₄	#65737	1	1	yes	characteristic
196	C₂	#10603940	1	1	yes	characteristic
197	C₂	#10604128	1	1	yes	characteristic
198	C₂	#10604135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×AGL(1,𝔽₅)
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#358	C₄
socle, prime radical	C₂²×C₅	C₄×D₈
cosocle	C₈×Dic₅	C₂³
hypercenter	#358	AGL(1,𝔽₅)
hypocenter	C₅	#12076
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
185. C₅	#12076	semidirect

Maximal subgroups

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

subgroup	index
#10604841	2
#10604060	2
#10600000	2
#10598809	2
#10598646	2
#170⋊C₁₆	2
#10598610	2
#12076	5