Finite Group Databasebeta

Subgroups — #13025342

Order 1280.

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

normal subgroups only

This group has 218 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, complemented
2	#10604070	C₂	1	1	yes	maximal, characteristic
3	#10600001	C₂	1	1	yes	maximal, characteristic
4	#10598807	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598642	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598640	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598607	C₂	1	1	yes	maximal, characteristic
8	Q₈×#170	—	2	1	no	—
9	#65705	—	2	1	no	—
10	Q₁₆×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×#3960	C₄	1	1	yes	characteristic
12	#65161	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	#64871	—	2	1	no	—
19	C₅×#358	C₄	1	1	yes	characteristic
20	#64729	C₄	1	1	yes	characteristic
21	#64722	C₂²	1	1	yes	characteristic
22	#64709	C₄	1	1	yes	characteristic
23	#12080	—	5	1	no	maximal, Sylow
24	C₂×#700	—	4	1	no	—
25	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
26	Q₈×Dic₅	—	2	1	no	—
27	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
28	#4039	—	2	1	no	—
29	#4003	—	4	1	no	—
30	C₄⋊#170	—	2	1	no	—
31	C₄⋊#170	—	2	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	#3960	—	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	#3941	—	2	1	no	—
45	#2175	—	5	1	no	—
46	#1594	—	5	1	no	—
47	#1579	—	5	1	no	—
48	#1368	—	5	1	no	—
49	#1334	—	5	1	no	—
50	#1333	—	5	1	no	—
51	#1313	—	5	1	no	—
52	C₂×C₅×Q₈	—	2	1	no	—
53	C₂×D₂₀	—	2	1	no	—
54	C₂×C₄×D₅	—	4	1	no	—
55	C₂×#170	—	2	1	no	—
56	C₂×#170	—	2	1	no	—
57	C₂×#170	—	4	1	no	—
58	C₂×#170	—	4	1	no	—
59	#700	—	8	2	no	—
60	C₅×Q₁₆	—	2	1	no	—
61	C₅×Q₁₆	M₁₆	1	2	yes	complemented
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₅×C₄⋊C₄	—	2	1	no	—
64	#685	—	4	1	no	—
65	C₄⋊Dic₅	—	2	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	D₄₀	—	2	1	no	—
71	D₄₀	—	2	1	no	—
72	#674	—	2	2	no	—
73	C₅×C₁₆	—	2	1	no	—
74	#672	—	2	1	no	—
75	C₂×SD₃₂	—	5	1	no	—
76	C₈×Q₈	—	10	1	no	—
77	C₄×Q₁₆	—	5	1	no	—
78	C₄×D₈	—	5	1	no	—
79	#436	—	10	1	no	—
80	#366	—	5	1	no	—
81	#358	AGL(1,𝔽₅)	1	1	yes	characteristic
82	#357	—	5	1	no	—
83	#346	—	5	1	no	—
84	#346	—	5	1	no	—
85	#334	—	5	1	no	—
86	#334	—	5	1	no	—
87	Q₈⋊C₈	—	10	1	no	—
88	D₄⋊C₈	—	10	1	no	—
89	#322	—	5	1	no	—
90	C₂²×D₅	—	4	1	no	—
91	C₅×Q₈	—	2	2	no	—
92	C₅×Q₈	—	4	1	no	—
93	C₂×C₄×C₅	—	4	1	no	—
94	C₂×C₄×C₅	—	4	1	no	—
95	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×Dic₅	—	4	1	no	—
97	C₂×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
99	D₂₀	—	4	1	no	—
100	D₂₀	—	4	1	no	—
101	C₄×D₅	—	8	1	no	—
102	C₄×D₅	—	8	1	no	—
103	#170	—	4	1	no	—
104	#170	—	4	1	no	—
105	#170	—	8	1	no	—
106	#170	—	8	1	no	—
107	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
108	C₅×C₈	#131	1	1	yes	characteristic
109	#168	—	2	1	no	—
110	C₂×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₂×D₈	—	5	1	no	—
112	C₂×M₁₆	—	20	1	no	—
113	C₄×Q₈	—	10	1	no	—
114	C₄×D₄	—	10	1	no	—
115	SD₃₂	—	10	2	no	—
116	C₂×C₁₆	—	5	1	no	—
117	C₂×C₁₆	—	10	1	no	—
118	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	#107	—	5	1	no	—
120	#107	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₄⋊C₈	—	10	1	no	—
122	C₄⋊C₈	—	10	1	no	—
123	C₄⋊C₈	—	20	1	no	—
124	#103	—	10	1	no	—
125	#102	—	10	1	no	—
126	C₂²⋊C₈	—	20	1	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	—	5	1	no	—
129	C₄×C₈	—	5	1	no	—
130	C₄×C₈	—	20	1	no	—
131	C₂²×C₅	#442	1	1	yes	characteristic
132	C₂×D₅	—	8	1	no	—
133	C₂×D₅	—	8	1	no	—
134	C₂×D₅	—	8	1	no	—
135	C₄×C₅	—	4	2	no	—
136	C₄×C₅	—	8	1	no	—
137	C₄×C₅	#435	1	1	yes	characteristic
138	C₄×C₅	#443	1	1	yes	characteristic
139	Dic₅	—	2	1	no	—
140	Dic₅	—	2	1	no	—
141	Dic₅	—	8	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×D₄	—	10	1	no	—
144	C₂²×C₄	—	20	1	no	—
145	Q₁₆	—	2	1	no	—
146	Q₁₆	#700	1	2	yes	complemented
147	D₈	—	10	1	no	—
148	D₈	—	10	1	no	—
149	M₁₆	—	40	2	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	10	1	no	—
152	C₂×C₈	—	10	1	no	—
153	C₂×C₈	—	20	1	no	—
154	C₂×C₈	—	20	1	no	—
155	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	10	1	no	—
158	C₄⋊C₄	—	20	1	no	—
159	C₂²⋊C₄	—	20	1	no	—
160	C₄²	—	5	1	no	—
161	C₄²	—	20	1	no	—
162	C₁₆	—	2	1	no	—
163	C₁₆	—	10	2	no	—
164	C₁₆	—	10	1	no	—
165	C₂×C₅	#1586	1	1	yes	characteristic
166	C₂×C₅	#2180	1	1	yes	characteristic
167	C₂×C₅	AGL(1,ℤ/16)	1	1	yes	characteristic
168	D₅	—	16	1	no	—
169	C₂³	—	20	1	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	4	1	no	—
172	D₄	—	20	1	no	—
173	D₄	—	20	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	5	1	no	—
177	C₂×C₄	—	5	1	no	—
178	C₂×C₄	—	20	1	no	—
179	C₂×C₄	—	40	1	no	—
180	C₂×C₄	—	40	1	no	—
181	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₈	—	10	1	no	—
183	C₈	—	20	1	no	—
184	C₈	—	20	1	no	—
185	C₈	—	40	1	no	—
186	C₈	—	40	1	no	—
187	C₈	C₂×#700	1	1	yes	characteristic
188	C₈	#4131	1	1	yes	characteristic
189	C₅	#12080	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	40	1	no	—
191	C₂²	—	40	1	no	—
192	C₂²	—	40	1	no	—
193	C₂²	#65731	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	10	1	no	—
197	C₄	—	10	1	no	—
198	C₄	—	40	1	no	—
199	C₄	#65693	1	1	yes	characteristic
200	C₄	#65736	1	1	yes	characteristic
201	C₂	—	80	1	no	—
202	C₂	#10603944	1	1	yes	characteristic
203	C₂	#10604133	1	1	yes	characteristic
204	C₂	#10604138	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	C₂²×C₅	#442
prime radical	C₂×D₄₀	C₂×C₄
cosocle	C₈×Dic₅	C₂³
hypercenter	#358	AGL(1,𝔽₅)
hypocenter	C₅	#12080
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
1. #10604838	C₂	semidirect
4. #10598807	C₂	semidirect
5. #10598642	C₂	semidirect
6. #10598640	C₂	semidirect
61. C₅×Q₁₆	M₁₆	semidirect
146. Q₁₆	#700	semidirect
189. C₅	#12080	semidirect

Maximal subgroups

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

subgroup	index
#10604838	2
#10604070	2
#10600001	2
#10598807	2
#10598642	2
#10598640	2
#10598607	2
#12080	5