Finite Group Databasebeta

Subgroups — #12019926

Order 1280.

194 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 67 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
2	#10602305	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602254	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598319	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598316	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598092	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598091	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	D₈×Dic₅	—	2	1	no	—
12	C₅×SD₆₄	C₄	1	2	yes	complemented
13	C₅×D₃₂	C₄	1	2	yes	complemented
14	C₅×M₆₄	C₂²	1	1	yes	characteristic
15	#64787	—	2	1	no	—
16	#64782	—	2	1	no	—
17	#64726	C₂²	1	1	yes	characteristic
18	#64725	C₂²	1	1	yes	characteristic
19	#64722	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	C₂×C₅×D₈	—	2	1	no	—
24	#4094	—	4	1	no	—
25	D₄×Dic₅	—	4	1	no	—
26	C₅×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
27	C₅×SD₃₂	—	2	1	no	—
28	C₅×D₁₆	—	2	1	no	—
29	C₅×D₁₆	C₂×C₄	1	2	yes	—
30	C₅×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
32	#3969	—	4	1	no	—
33	#3964	—	4	1	no	—
34	#3951	—	2	1	no	—
35	#3950	—	2	1	no	—
36	C₈×Dic₅	D₄	1	1	yes	characteristic
37	C₂×#672	D₄	1	1	yes	characteristic
38	C₅×C₃₂	C₂×C₄	1	2	yes	complemented
39	#3926	—	2	1	no	—
40	#2263	D₅	1	1	yes	characteristic
41	#2179	—	5	1	no	—
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×C₅×D₄	—	4	1	no	—
49	C₂²×Dic₅	—	8	1	no	—
50	C₅×Q₁₆	—	2	1	no	—
51	C₅×SD₁₆	—	4	1	no	—
52	C₅×D₈	—	2	2	no	—
53	C₅×D₈	—	2	1	no	—
54	C₅×D₈	—	4	1	no	—
55	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊Dic₅	—	8	1	no	—
57	C₄⋊Dic₅	—	4	1	no	—
58	C₄×Dic₅	D₈	1	1	yes	characteristic
59	#681	—	4	1	no	—
60	#681	—	8	1	no	—
61	C₂×#168	—	8	1	no	—
62	C₂×#168	D₈	1	1	yes	characteristic
63	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
64	C₅×C₁₆	#41	1	1	yes	characteristic
65	#672	—	2	1	no	—
66	#508	C₂×D₅	1	1	yes	characteristic
67	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic
68	#443	—	10	1	no	—
69	C₄×D₈	—	10	1	no	—
70	SD₆₄	Dic₅	1	2	yes	complemented
71	D₃₂	Dic₅	1	2	yes	complemented
72	M₆₄	—	5	1	no	—
73	M₆₄	C₂×D₅	1	1	yes	characteristic
74	#368	—	5	1	no	—
75	#367	—	5	1	no	—
76	#359	—	10	1	no	—
77	#357	—	10	1	no	—
78	#346	—	5	1	no	—
79	C₂³×C₅	—	8	1	no	—
80	C₅×Q₈	—	4	1	no	—
81	C₅×D₄	—	4	2	no	—
82	C₅×D₄	—	4	1	no	—
83	C₅×D₄	—	8	1	no	—
84	C₅×D₄	—	8	1	no	—
85	C₂×C₄×C₅	—	8	1	no	—
86	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
87	C₂×Dic₅	—	2	1	no	—
88	C₂×Dic₅	—	8	1	no	—
89	C₂×Dic₅	—	16	2	no	—
90	C₅×C₈	C₄×D₄	1	1	yes	characteristic
91	C₅×C₈	#135	1	1	yes	characteristic
92	#168	—	2	1	no	—
93	#168	—	8	1	no	—
94	#135	—	2	1	no	—
95	C₂×D₈	—	2	1	no	—
96	#131	—	20	1	no	—
97	C₄×D₄	—	20	1	no	—
98	Q₃₂	C₂×Dic₅	1	1	yes	characteristic, complemented
99	SD₃₂	—	2	1	no	—
100	D₁₆	—	2	1	no	—
101	D₁₆	C₂×Dic₅	1	2	yes	—
102	D₁₆	C₂×Dic₅	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	5	1	no	—
104	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
105	#108	—	10	1	no	—
106	#107	—	10	1	no	—
107	#104	—	20	1	no	—
108	#102	—	20	1	no	—
109	C₄×C₈	—	5	1	no	—
110	C₃₂	—	10	1	no	—
111	C₃₂	C₂×Dic₅	1	2	yes	complemented
112	C₂²×C₅	—	8	1	no	—
113	C₂²×C₅	—	8	2	no	—
114	C₂²×C₅	—	16	1	no	—
115	C₂²×C₅	—	16	1	no	—
116	C₂²×C₅	#509	1	1	yes	characteristic
117	C₄×C₅	—	8	1	no	—
118	C₄×C₅	C₄×D₈	1	1	yes	characteristic
119	C₄×C₅	#509	1	1	yes	characteristic
120	Dic₅	—	4	1	no	—
121	Dic₅	—	16	1	no	—
122	#41	—	4	1	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂²×C₄	—	40	1	no	—
125	Q₁₆	—	2	1	no	—
126	SD₁₆	—	4	1	no	—
127	D₈	—	2	2	no	—
128	D₈	—	2	1	no	—
129	D₈	—	4	1	no	—
130	M₁₆	—	20	1	no	—
131	M₁₆	—	40	1	no	—
132	C₂×C₈	—	5	1	no	—
133	C₂×C₈	—	40	1	no	—
134	C₂×C₈	D₄×D₅	1	1	yes	characteristic
135	C₄⋊C₄	—	20	1	no	—
136	C₂²⋊C₄	—	40	1	no	—
137	C₄²	—	5	1	no	—
138	C₁₆	—	10	1	no	—
139	C₁₆	#711	1	1	yes	characteristic
140	C₁₆	C₂²×Dic₅	1	1	yes	characteristic
141	C₂×C₅	—	2	1	no	—
142	C₂×C₅	—	16	1	no	—
143	C₂×C₅	—	16	2	no	—
144	C₂×C₅	#2177	1	1	yes	characteristic
145	C₂³	—	8	1	no	—
146	Q₈	—	4	1	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	10	1	no	—
153	C₂×C₄	—	40	1	no	—
154	C₂×C₄	—	80	2	no	—
155	C₂×C₄	D₅×D₈	1	1	yes	characteristic
156	C₈	—	10	1	no	—
157	C₈	—	40	1	no	—
158	C₈	#4058	1	1	yes	characteristic
159	C₈	D₄×Dic₅	1	1	yes	characteristic
160	C₅	#13658	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	8	1	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	#65200	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	20	1	no	—
168	C₄	—	80	1	no	—
169	C₄	#65200	1	1	yes	characteristic
170	C₄	D₈×Dic₅	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	2	no	—
173	C₂	—	16	1	no	—
174	C₂	#10602254	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10602254
commutator	C₅×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅
Fitting, prime radical	C₅×#2263	C₂
socle	C₂×C₅	#2177
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	#2263	D₅
hypocenter	C₅	#13658
LCS	G ▷ 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
2. #10602305	C₂	semidirect
3. #10602254	C₂	semidirect
4. #10598319	C₂	semidirect
5. #10598316	C₂	semidirect
6. #10598092	C₂	semidirect
7. #10598091	C₂	semidirect
12. C₅×SD₆₄	C₄	semidirect
13. C₅×D₃₂	C₄	semidirect
26. C₅×Q₃₂	C₂×C₄	semidirect
30. C₅×D₁₆	C₂×C₄	semidirect
38. C₅×C₃₂	C₂×C₄	semidirect
70. SD₆₄	Dic₅	semidirect
71. D₃₂	Dic₅	semidirect
98. Q₃₂	C₂×Dic₅	semidirect
102. D₁₆	C₂×Dic₅	semidirect
111. C₃₂	C₂×Dic₅	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×#2263	2
#10602305	2
#10602254	2
#10598319	2
#10598316	2
#10598092	2
#10598091	2
#13658	5