Finite Group Databasebeta

Subgroups — #12019900

Order 1280.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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	Dic₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#64739	C₂	1	1	yes	maximal, characteristic
3	#10600110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1415	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598087	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598080	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598067	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
9	D₈×Dic₅	—	2	1	no	—
10	C₂×D₈₀	C₂²	1	1	yes	characteristic
11	#65161	—	2	1	no	—
12	C₂×C₅×C₃₂	C₂²	1	1	yes	characteristic
13	C₅×#366	C₂²	1	1	yes	characteristic
14	#64782	—	2	1	no	—
15	#64739	C₄	1	4	yes	complemented
16	#64724	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#3926	C₂²	1	1	yes	characteristic
19	#64711	—	2	1	no	—
20	#13654	—	5	1	no	maximal, Sylow
21	C₂×C₅×D₈	—	2	1	no	—
22	D₄×Dic₅	—	4	1	no	—
23	C₂×D₄₀	—	2	1	no	—
24	#4039	—	4	1	no	—
25	C₅×D₁₆	—	2	1	no	—
26	C₅×D₁₆	C₂×C₄	1	2	yes	complemented
27	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
28	C₅×#107	—	2	1	no	—
29	#3964	—	4	1	no	—
30	#3950	—	2	1	no	—
31	C₈×Dic₅	D₄	1	1	yes	characteristic
32	C₂×#672	D₄	1	1	yes	characteristic
33	#3941	—	4	1	no	—
34	D₈₀	—	2	1	no	—
35	D₈₀	C₂×C₄	1	2	yes	complemented
36	C₅×C₃₂	—	2	1	no	—
37	#3926	C₂×C₄	1	2	yes	complemented
38	C₂×D₃₂	—	5	1	no	—
39	C₄×D₁₆	—	5	1	no	—
40	C₄×D₁₆	—	5	1	no	—
41	#1424	—	5	1	no	—
42	#1415	—	5	1	no	—
43	#1415	D₅	1	1	yes	characteristic, complemented
44	#1397	—	5	1	no	—
45	C₂×C₅×D₄	—	4	1	no	—
46	C₂²×Dic₅	—	8	1	no	—
47	C₂×D₂₀	—	4	1	no	—
48	C₂×C₄×D₅	—	8	1	no	—
49	C₅×D₈	—	2	2	no	—
50	C₅×D₈	—	4	1	no	—
51	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
52	C₅×C₄⋊C₄	—	4	1	no	—
53	C₂²⋊Dic₅	—	8	1	no	—
54	#685	—	8	1	no	—
55	C₄⋊Dic₅	—	4	1	no	—
56	C₄×Dic₅	D₈	1	1	yes	characteristic
57	C₂×#168	D₈	1	1	yes	characteristic
58	D₄₀	—	2	2	no	—
59	D₄₀	—	4	1	no	—
60	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₅×C₁₆	#41	1	1	yes	characteristic
62	#672	—	2	1	no	—
63	C₂×D₁₆	—	5	1	no	—
64	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic
65	C₄×D₈	—	10	1	no	—
66	C₄×D₈	—	10	1	no	—
67	D₃₂	—	5	4	no	—
68	C₂×C₃₂	—	5	1	no	—
69	C₂×C₃₂	C₂×D₅	1	1	yes	characteristic
70	#366	—	5	1	no	—
71	#366	C₂×D₅	1	1	yes	characteristic
72	#357	—	10	1	no	—
73	#357	—	10	1	no	—
74	C₄×C₁₆	—	5	1	no	—
75	C₂³×C₅	—	8	1	no	—
76	C₂²×D₅	—	8	1	no	—
77	C₅×D₄	—	4	2	no	—
78	C₅×D₄	—	8	1	no	—
79	C₂×C₄×C₅	—	8	1	no	—
80	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
81	C₂×Dic₅	—	8	1	no	—
82	C₂×Dic₅	—	16	2	no	—
83	C₂×Dic₅	D₁₆	1	1	yes	characteristic
84	C₂×Dic₅	D₁₆	1	1	yes	characteristic
85	D₂₀	—	4	2	no	—
86	D₂₀	—	8	1	no	—
87	C₄×D₅	—	16	2	no	—
88	C₅×C₈	C₄×D₄	1	1	yes	characteristic
89	C₅×C₈	#135	1	1	yes	characteristic
90	#168	—	2	1	no	—
91	C₂×D₈	—	2	1	no	—
92	C₂×D₈	—	10	1	no	—
93	C₄×D₄	—	20	1	no	—
94	C₄×D₄	—	20	1	no	—
95	D₁₆	—	2	1	no	—
96	D₁₆	—	5	2	no	—
97	D₁₆	—	10	1	no	—
98	D₁₆	C₄×D₅	1	2	yes	complemented
99	C₂×C₁₆	—	5	1	no	—
100	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	10	1	no	—
103	#102	—	20	1	no	—
104	#102	—	20	1	no	—
105	C₄×C₈	—	5	1	no	—
106	C₃₂	—	2	1	no	—
107	C₃₂	—	5	2	no	—
108	C₂²×C₅	—	8	2	no	—
109	C₂²×C₅	—	16	1	no	—
110	C₂²×C₅	—	16	1	no	—
111	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
112	C₂×D₅	—	8	2	no	—
113	C₂×D₅	—	16	1	no	—
114	C₂×D₅	—	16	1	no	—
115	C₄×C₅	—	16	1	no	—
116	C₄×C₅	C₄×D₈	1	1	yes	characteristic
117	C₄×C₅	#508	1	1	yes	characteristic
118	Dic₅	—	2	1	no	—
119	Dic₅	—	2	1	no	—
120	Dic₅	—	16	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂×D₄	—	20	1	no	—
123	C₂²×C₄	—	40	1	no	—
124	C₂²×C₄	—	40	1	no	—
125	D₈	—	2	2	no	—
126	D₈	—	4	1	no	—
127	D₈	—	10	2	no	—
128	D₈	—	20	1	no	—
129	C₂×C₈	—	5	1	no	—
130	C₂×C₈	D₄×D₅	1	1	yes	characteristic
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	20	1	no	—
133	C₂²⋊C₄	—	40	1	no	—
134	C₂²⋊C₄	—	40	1	no	—
135	C₄²	—	5	1	no	—
136	C₁₆	—	10	1	no	—
137	C₁₆	C₂×C₄×D₅	1	1	yes	characteristic
138	C₁₆	#711	1	1	yes	characteristic
139	C₂×C₅	—	16	2	no	—
140	C₂×C₅	C₄×D₁₆	1	1	yes	characteristic
141	C₂×C₅	#2263	1	1	yes	characteristic
142	C₂×C₅	#2263	1	1	yes	characteristic
143	D₅	—	16	2	no	—
144	C₂³	—	8	1	no	—
145	C₂³	—	40	1	no	—
146	D₄	—	4	2	no	—
147	D₄	—	8	1	no	—
148	D₄	—	20	2	no	—
149	D₄	—	40	1	no	—
150	C₂×C₄	—	5	1	no	—
151	C₂×C₄	—	5	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	40	1	no	—
154	C₂×C₄	—	80	2	no	—
155	C₂×C₄	—	80	2	no	—
156	C₂×C₄	D₅×D₈	1	1	yes	characteristic
157	C₈	—	10	1	no	—
158	C₈	#4027	1	1	yes	characteristic
159	C₈	#4062	1	1	yes	characteristic
160	C₅	#13654	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	8	2	no	—
162	C₂²	—	16	1	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	40	2	no	—
165	C₂²	—	80	1	no	—
166	C₂²	—	80	1	no	—
167	C₂²	D₅×D₁₆	1	1	yes	characteristic
168	C₄	—	10	1	no	—
169	C₄	—	10	1	no	—
170	C₄	—	16	1	no	—
171	C₄	—	80	1	no	—
172	C₄	#65045	1	1	yes	characteristic
173	C₄	#65205	1	1	yes	characteristic
174	C₂	—	16	2	no	—
175	C₂	—	80	2	no	—
176	C₂	#10599967	1	1	yes	characteristic
177	C₂	#10600170	1	1	yes	characteristic
178	C₂	#10600173	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×D₁₆
commutator	C₅×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅
Fitting	C₅×#1415	C₂
socle	C₂²×C₅	C₂×D₁₆
prime radical	C₂×#64739	C₂
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	#1415	D₅
hypocenter	C₅	#13654
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
1. Dic₅×D₁₆	C₂	semidirect
3. #10600110	C₂	semidirect
4. C₅×#1415	C₂	semidirect
5. #10598087	C₂	semidirect
6. #10598080	C₂	semidirect
7. #10598067	C₂	semidirect
15. #64739	C₄	semidirect
26. C₅×D₁₆	C₂×C₄	semidirect
35. D₈₀	C₂×C₄	semidirect
37. #3926	C₂×C₄	semidirect
43. #1415	D₅	semidirect
98. D₁₆	C₄×D₅	semidirect
160. C₅	#13654	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×D₁₆	2
C₂×#64739	2
#10600110	2
C₅×#1415	2
#10598087	2
#10598080	2
#10598067	2
#13654	5