Finite Group Databasebeta

Subgroups — #12019925

Order 1280.

194 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₅×#2179	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602302	C₂	1	1	yes	maximal, characteristic
3	#10599871	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598319	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598074	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598073	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598010	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#508	C₂²	1	1	yes	characteristic
9	C₅×#443	—	2	1	no	—
10	C₂×D₈₀	C₂²	1	1	yes	characteristic
11	C₄×D₄₀	—	2	1	no	—
12	C₅×#368	C₂²	1	1	yes	characteristic
13	C₅×#359	—	2	1	no	—
14	C₅×#346	C₂²	1	1	yes	characteristic
15	#64741	C₄	1	2	yes	complemented
16	#64739	C₄	1	2	yes	complemented
17	#64729	—	2	1	no	—
18	#64725	C₂²	1	1	yes	characteristic
19	#64719	C₂²	1	1	yes	characteristic
20	#64719	C₂²	1	1	yes	characteristic
21	#13658	—	5	1	no	maximal, Sylow
22	C₅×#135	—	2	1	no	—
23	C₅×#131	—	4	1	no	—
24	C₂×D₄₀	—	2	1	no	—
25	C₄×D₂₀	—	4	1	no	—
26	C₅×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
27	C₅×SD₃₂	—	2	1	no	—
28	C₅×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
31	C₅×#108	—	2	1	no	—
32	C₅×#104	—	4	1	no	—
33	C₄×C₅×C₈	D₄	1	1	yes	characteristic
34	#3953	—	4	1	no	—
35	#3950	—	2	1	no	—
36	D₈₀	—	2	1	no	—
37	D₈₀	C₂×C₄	1	2	yes	—
38	#3926	—	2	1	no	—
39	#3926	C₂×C₄	1	2	yes	complemented
40	#2263	—	5	1	no	—
41	#2179	D₅	1	1	yes	characteristic, complemented
42	#2177	—	5	1	no	—
43	#1426	—	5	1	no	—
44	#1419	—	5	1	no	—
45	#1418	—	5	1	no	—
46	#1398	—	5	1	no	—
47	C₅×#41	—	4	1	no	—
48	C₂×D₂₀	—	4	1	no	—
49	C₂×C₄×D₅	—	8	1	no	—
50	C₅×Q₁₆	—	2	1	no	—
51	C₅×SD₁₆	—	4	1	no	—
52	C₅×D₈	—	2	1	no	—
53	C₅×M₁₆	—	4	1	no	—
54	C₅×M₁₆	—	8	1	no	—
55	C₂×C₅×C₈	—	8	1	no	—
56	C₂×C₅×C₈	D₈	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅	D₈	1	1	yes	characteristic
59	#685	—	8	1	no	—
60	C₄⋊Dic₅	—	4	1	no	—
61	D₄₀	—	2	2	no	—
62	D₄₀	—	4	1	no	—
63	C₅×C₁₆	—	2	1	no	—
64	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₅×C₁₆	#41	1	1	yes	characteristic
66	#508	C₂×D₅	1	1	yes	characteristic
67	C₂×D₁₆	—	5	1	no	—
68	#443	—	2	1	no	—
69	C₄×D₈	—	10	1	no	—
70	SD₆₄	—	5	2	no	—
71	D₃₂	—	5	2	no	—
72	M₆₄	—	5	1	no	—
73	M₆₄	—	5	1	no	—
74	#368	C₂×D₅	1	1	yes	characteristic
75	#367	—	5	1	no	—
76	#359	—	2	1	no	—
77	#357	—	10	1	no	—
78	#346	C₂×D₅	1	1	yes	characteristic
79	C₂²×D₅	—	8	1	no	—
80	C₅×Q₈	—	4	1	no	—
81	C₅×D₄	—	4	1	no	—
82	C₅×D₄	—	8	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	8	1	no	—
85	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₅	—	8	1	no	—
87	D₂₀	—	4	2	no	—
88	D₂₀	—	8	1	no	—
89	C₄×D₅	—	16	2	no	—
90	C₅×C₈	—	2	1	no	—
91	C₅×C₈	—	8	1	no	—
92	C₅×C₈	C₄×D₄	1	1	yes	characteristic
93	C₅×C₈	#135	1	1	yes	characteristic
94	#135	—	2	1	no	—
95	C₂×D₈	—	10	1	no	—
96	#131	—	4	1	no	—
97	C₄×D₄	—	20	1	no	—
98	Q₃₂	C₄×D₅	1	1	yes	characteristic, complemented
99	SD₃₂	—	2	1	no	—
100	D₁₆	—	5	2	no	—
101	D₁₆	—	10	1	no	—
102	D₁₆	C₄×D₅	1	1	yes	characteristic, complemented
103	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
105	#108	—	2	1	no	—
106	#107	—	10	1	no	—
107	#104	—	4	1	no	—
108	#102	—	20	1	no	—
109	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
110	C₃₂	—	5	2	no	—
111	C₃₂	—	10	1	no	—
112	C₂²×C₅	—	8	1	no	—
113	C₂²×C₅	#509	1	1	yes	characteristic
114	C₂×D₅	—	8	2	no	—
115	C₂×D₅	—	16	1	no	—
116	C₂×D₅	—	16	1	no	—
117	C₄×C₅	—	4	1	no	—
118	C₄×C₅	—	8	1	no	—
119	C₄×C₅	C₄×D₈	1	1	yes	characteristic
120	C₄×C₅	#509	1	1	yes	characteristic
121	Dic₅	—	16	1	no	—
122	#41	—	4	1	no	—
123	C₂×D₄	—	20	1	no	—
124	C₂²×C₄	—	40	1	no	—
125	Q₁₆	—	2	1	no	—
126	SD₁₆	—	4	1	no	—
127	D₈	—	2	1	no	—
128	D₈	—	10	2	no	—
129	D₈	—	20	1	no	—
130	M₁₆	—	4	1	no	—
131	M₁₆	—	8	1	no	—
132	C₂×C₈	—	8	1	no	—
133	C₂×C₈	#686	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
135	C₄⋊C₄	—	20	1	no	—
136	C₂²⋊C₄	—	40	1	no	—
137	C₄²	#686	1	1	yes	characteristic
138	C₁₆	—	2	1	no	—
139	C₁₆	C₂×C₄×D₅	1	1	yes	characteristic
140	C₁₆	#709	1	1	yes	characteristic
141	C₂×C₅	—	2	1	no	—
142	C₂×C₅	—	16	1	no	—
143	C₂×C₅	#2177	1	1	yes	characteristic
144	D₅	—	16	2	no	—
145	C₂³	—	40	1	no	—
146	Q₈	—	4	1	no	—
147	D₄	—	4	1	no	—
148	D₄	—	8	1	no	—
149	D₄	—	20	2	no	—
150	D₄	—	40	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	40	1	no	—
154	C₂×C₄	—	80	2	no	—
155	C₂×C₄	C₂×#686	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	8	1	no	—
158	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
159	C₈	#4096	1	1	yes	characteristic
160	C₅	#13658	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	8	1	no	—
162	C₂²	—	40	2	no	—
163	C₂²	—	80	1	no	—
164	C₂²	—	80	1	no	—
165	C₂²	#65482	1	1	yes	characteristic
166	C₄	—	4	1	no	—
167	C₄	—	8	1	no	—
168	C₄	—	80	1	no	—
169	C₄	C₄×#686	1	1	yes	characteristic
170	C₄	#65436	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	80	2	no	—
174	C₂	#10601678	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10601678
commutator	C₅×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅
Fitting	C₅×#2179	C₂
socle	C₂×C₅	#2177
prime radical	#10602302	C₂
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	#2179	D₅
hypocenter	C₅	#13658
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #2179

Complements

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

N	H ≅ G/N	type
1. C₅×#2179	C₂	semidirect
3. #10599871	C₂	semidirect
4. #10598319	C₂	semidirect
5. #10598074	C₂	semidirect
6. #10598073	C₂	semidirect
7. #10598010	C₂	semidirect
15. #64741	C₄	semidirect
16. #64739	C₄	semidirect
26. C₅×Q₃₂	C₂×C₄	semidirect
28. C₅×D₁₆	C₂×C₄	semidirect
39. #3926	C₂×C₄	semidirect
41. #2179	D₅	semidirect
98. Q₃₂	C₄×D₅	semidirect
102. D₁₆	C₄×D₅	semidirect
160. C₅	#13658	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2179	2
#10602302	2
#10599871	2
#10598319	2
#10598074	2
#10598073	2
#10598010	2
#13658	5