Finite Group Databasebeta

Subgroups — #12020119

Order 1280.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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×#2263	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602304	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602302	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602257	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598316	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598095	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598090	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#508	C₂²	1	1	yes	characteristic
9	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
10	#65485	—	2	1	no	—
11	#65440	—	2	1	no	—
12	C₂×#3958	—	2	1	no	—
13	C₂×D₈₀	C₂²	1	1	yes	characteristic
14	C₅×SD₆₄	—	2	1	no	—
15	C₅×D₃₂	—	2	1	no	—
16	C₅×M₆₄	C₂²	1	1	yes	characteristic
17	#64787	—	2	1	no	—
18	#64741	—	2	1	no	—
19	#64739	—	2	1	no	—
20	#64726	C₂²	1	1	yes	characteristic
21	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
22	#64719	C₂²	1	1	yes	characteristic
23	#13692	—	5	1	no	maximal, Sylow
24	C₅×#135	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	#4094	—	4	1	no	—
27	#4086	—	4	1	no	—
28	C₂×#686	—	4	1	no	—
29	C₂×D₄₀	—	2	1	no	—
30	C₅×Q₃₂	D₄	1	1	yes	characteristic, complemented
31	C₅×SD₃₂	—	2	1	no	—
32	C₅×D₁₆	—	2	1	no	—
33	C₅×D₁₆	—	2	1	no	—
34	C₅×D₁₆	D₄	1	1	yes	characteristic, complemented
35	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
36	#3969	—	4	1	no	—
37	#3958	—	4	2	no	—
38	#3951	—	2	1	no	—
39	C₈×Dic₅	D₄	1	1	yes	characteristic
40	C₂×#672	D₄	1	1	yes	characteristic
41	D₈₀	—	2	1	no	—
42	D₈₀	—	2	1	no	—
43	C₅×C₃₂	—	2	1	no	—
44	#3926	—	2	1	no	—
45	#2263	—	5	1	no	—
46	#2263	D₅	1	1	yes	characteristic, complemented
47	#2246	—	5	1	no	—
48	#2178	—	5	1	no	—
49	#1422	—	5	1	no	—
50	#1419	—	5	1	no	—
51	#1419	—	5	1	no	—
52	C₅×#41	—	4	1	no	—
53	C₂×C₅×D₄	—	4	1	no	—
54	C₂×C₂²⋊D₅	—	8	2	no	—
55	C₂×D₂₀	—	4	1	no	—
56	C₅×Q₁₆	—	2	1	no	—
57	C₅×SD₁₆	—	4	1	no	—
58	C₅×D₈	—	2	1	no	—
59	C₅×D₈	—	4	1	no	—
60	C₅×D₈	—	4	1	no	—
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	#686	—	8	1	no	—
63	#686	—	8	1	no	—
64	C₄×Dic₅	D₈	1	1	yes	characteristic
65	#681	—	4	1	no	—
66	#681	—	8	1	no	—
67	C₂×#168	—	8	1	no	—
68	C₂×#168	D₈	1	1	yes	characteristic
69	D₄₀	—	4	1	no	—
70	D₄₀	—	4	1	no	—
71	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
72	C₅×C₁₆	#41	1	1	yes	characteristic
73	#672	—	2	1	no	—
74	#508	C₂×D₅	1	1	yes	characteristic
75	C₂×D₁₆	—	5	1	no	—
76	C₂×D₁₆	—	10	1	no	—
77	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic
78	#493	—	10	1	no	—
79	#443	—	10	1	no	—
80	SD₆₄	—	2	1	no	—
81	SD₆₄	—	10	1	no	—
82	D₃₂	—	2	1	no	—
83	D₃₂	—	10	1	no	—
84	M₆₄	—	5	1	no	—
85	M₆₄	C₂×D₅	1	1	yes	characteristic
86	#368	—	5	1	no	—
87	#359	—	10	1	no	—
88	C₄×C₁₆	—	5	1	no	—
89	C₂³×C₅	—	8	1	no	—
90	C₂²×D₅	—	8	1	no	—
91	C₅×Q₈	—	4	1	no	—
92	C₅×D₄	—	4	1	no	—
93	C₅×D₄	—	8	1	no	—
94	C₅×D₄	—	8	1	no	—
95	C₅×D₄	—	8	1	no	—
96	C₂×C₄×C₅	—	8	1	no	—
97	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
98	C₂²⋊D₅	—	16	4	no	—
99	C₂×Dic₅	—	2	1	no	—
100	D₂₀	—	8	1	no	—
101	D₂₀	—	8	1	no	—
102	C₅×C₈	#121	1	1	yes	characteristic
103	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
104	#168	—	8	1	no	—
105	#168	D₁₆	1	1	yes	characteristic
106	#168	D₁₆	1	1	yes	characteristic
107	#135	—	2	1	no	—
108	C₂×D₈	—	2	1	no	—
109	C₂×D₈	—	10	1	no	—
110	C₂×D₈	—	20	1	no	—
111	#131	—	20	1	no	—
112	#127	—	20	1	no	—
113	Q₃₂	C₂²⋊D₅	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	2	1	no	—
116	D₁₆	—	2	1	no	—
117	D₁₆	—	10	1	no	—
118	D₁₆	—	10	1	no	—
119	D₁₆	—	20	2	no	—
120	D₁₆	C₂²⋊D₅	1	1	yes	characteristic, complemented
121	C₂×C₁₆	—	5	1	no	—
122	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
123	#108	—	10	1	no	—
124	#104	—	20	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₃₂	—	2	1	no	—
127	C₃₂	—	10	1	no	—
128	C₂²×C₅	—	8	1	no	—
129	C₂²×C₅	—	16	2	no	—
130	C₂²×C₅	—	16	1	no	—
131	C₂²×C₅	#509	1	1	yes	characteristic
132	C₂×D₅	—	16	2	no	—
133	C₂×D₅	—	16	1	no	—
134	C₄×C₅	—	8	1	no	—
135	C₄×C₅	#459	1	1	yes	characteristic
136	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
137	Dic₅	—	2	2	no	—
138	#41	—	4	1	no	—
139	C₂×D₄	—	4	1	no	—
140	C₂×D₄	—	20	1	no	—
141	C₂×D₄	—	40	2	no	—
142	Q₁₆	—	2	1	no	—
143	SD₁₆	—	4	1	no	—
144	D₈	—	2	1	no	—
145	D₈	—	4	1	no	—
146	D₈	—	4	1	no	—
147	D₈	—	20	1	no	—
148	D₈	—	20	1	no	—
149	D₈	—	40	1	no	—
150	D₈	—	40	1	no	—
151	M₁₆	—	20	1	no	—
152	M₁₆	—	40	1	no	—
153	C₂×C₈	—	5	1	no	—
154	C₂×C₈	—	40	1	no	—
155	C₂×C₈	D₄×D₅	1	1	yes	characteristic
156	C₄²	—	5	1	no	—
157	C₁₆	—	10	1	no	—
158	C₁₆	#711	1	1	yes	characteristic
159	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₅	—	2	1	no	—
161	C₂×C₅	—	16	1	no	—
162	C₂×C₅	—	32	1	no	—
163	C₂×C₅	#2206	1	1	yes	characteristic
164	D₅	—	32	1	no	—
165	C₂³	—	8	1	no	—
166	C₂³	—	40	1	no	—
167	Q₈	—	4	1	no	—
168	D₄	—	4	1	no	—
169	D₄	—	8	1	no	—
170	D₄	—	8	1	no	—
171	D₄	—	8	1	no	—
172	D₄	—	40	1	no	—
173	D₄	—	40	1	no	—
174	D₄	—	80	4	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	10	1	no	—
177	C₂×C₄	D₅×D₈	1	1	yes	characteristic
178	C₈	—	5	1	no	—
179	C₈	—	5	1	no	—
180	C₈	—	40	1	no	—
181	C₈	#4057	1	1	yes	characteristic
182	C₈	#4085	1	1	yes	characteristic
183	C₅	#13692	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	8	1	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	16	1	no	—
187	C₂²	—	80	2	no	—
188	C₂²	—	80	1	no	—
189	C₂²	#65200	1	1	yes	characteristic
190	C₄	—	8	1	no	—
191	C₄	—	10	2	no	—
192	C₄	D₅×D₁₆	1	1	yes	characteristic
193	C₄	#65439	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	16	1	no	—
196	C₂	—	32	1	no	—
197	C₂	—	160	1	no	—
198	C₂	#10602255	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10602255
commutator, cosocle	C₂×C₅×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₅
Fitting	C₅×#2263	C₂
socle	C₂×C₅	#2206
hypercenter	#2263	D₅
hypocenter	C₅	#13692
LCS	G ▷ C₂×C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2263

Complements

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

N	H ≅ G/N	type
1. C₅×#2263	C₂	semidirect
2. #10602304	C₂	semidirect
3. #10602302	C₂	semidirect
4. #10602257	C₂	semidirect
5. #10598316	C₂	semidirect
6. #10598095	C₂	semidirect
7. #10598090	C₂	semidirect
30. C₅×Q₃₂	D₄	semidirect
34. C₅×D₁₆	D₄	semidirect
46. #2263	D₅	semidirect
113. Q₃₂	C₂²⋊D₅	semidirect
120. D₁₆	C₂²⋊D₅	semidirect
183. C₅	#13692	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2263	2
#10602304	2
#10602302	2
#10602257	2
#10598316	2
#10598095	2
#10598090	2
#13692	5