Finite Group Databasebeta

Subgroups — #12010932

Order 1280.

226 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
2	#10600525	C₂	1	1	yes	maximal, characteristic
3	#10600523	C₂	1	1	yes	maximal, characteristic
4	#10598004	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	#65215	—	2	1	no	—
8	#65213	—	2	1	no	—
9	C₅×#346	C₄	1	1	yes	characteristic
10	C₅×#346	C₄	1	1	yes	characteristic
11	C₅×#346	C₂²	1	2	yes	—
12	#64678	—	2	1	no	—
13	#64677	—	2	1	no	—
14	#64677	C₂²	1	2	yes	—
15	#64677	C₂²	1	2	yes	—
16	#12019	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×#681	—	4	1	no	—
21	C₂×C₅×C₁₆	—	2	4	no	—
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²⋊#168	—	4	1	no	—
29	C₄⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	4	1	no	—
32	#3935	—	4	1	no	—
33	C₂×#346	D₅	1	1	yes	characteristic
34	#1568	—	5	1	no	—
35	#1566	—	5	1	no	—
36	#1368	—	5	4	no	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
40	C₂×C₅×C₈	—	2	2	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	SD₁₆	1	1	yes	characteristic
48	C₄²×C₅	SD₁₆	1	1	yes	characteristic
49	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
50	#681	—	8	2	no	—
51	C₂×#168	—	4	2	no	—
52	C₂×#168	—	4	2	no	—
53	C₅×C₁₆	—	2	2	no	—
54	C₅×C₁₆	M₁₆	1	4	yes	complemented
55	C₂²×C₁₆	Dic₅	1	2	yes	—
56	#432	—	10	1	no	—
57	#423	—	10	1	no	—
58	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
59	#346	Dic₅	1	1	yes	characteristic
60	#346	Dic₅	1	1	yes	characteristic
61	#346	C₂×D₅	1	2	yes	—
62	#335	—	10	1	no	—
63	#334	—	5	2	no	—
64	#334	—	5	2	no	—
65	#334	—	10	1	no	—
66	C₂³×C₅	#106	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	4	1	no	—
71	C₂×C₄×C₅	#106	1	1	yes	characteristic
72	C₂×C₄×C₅	#106	1	1	yes	characteristic
73	C₂×C₄×C₅	#106	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	#108	1	2	yes	—
80	C₅×C₈	C₂×M₁₆	1	2	yes	—
81	#168	—	8	2	no	—
82	#168	—	8	2	no	—
83	C₂×M₁₆	—	20	1	no	—
84	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
85	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
86	C₂×C₄²	D₂₀	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
89	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
90	C₄⋊C₈	—	10	2	no	—
91	C₄⋊C₈	—	10	2	no	—
92	C₄⋊C₈	—	20	1	no	—
93	C₂²⋊C₈	—	20	1	no	—
94	#97	—	20	1	no	—
95	C₄×C₈	Dic₁₀	1	1	yes	characteristic
96	C₄×C₈	D₂₀	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
99	C₂²×C₅	—	2	1	no	—
100	C₂²×C₅	—	2	1	no	—
101	C₂²×C₅	#365	1	2	yes	—
102	C₂²×C₅	C₂×#106	1	1	yes	characteristic
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	#423	1	1	yes	characteristic
109	C₄×C₅	C₂×#108	1	1	yes	characteristic
110	C₂²×C₄	#677	1	1	yes	characteristic
111	C₂²×C₄	#677	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
113	M₁₆	—	40	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	20	2	no	—
117	C₂×C₈	—	20	2	no	—
118	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
122	C₄²	#677	1	1	yes	characteristic
123	C₄²	#677	1	1	yes	characteristic
124	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
125	C₄²	C₂×D₂₀	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	#681	1	4	yes	complemented
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	#1566	1	1	yes	characteristic
130	C₂×C₅	C₂×#365	1	1	yes	characteristic
131	C₂×C₅	#2155	1	1	yes	characteristic
132	C₂³	#3949	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	#3949	1	1	yes	characteristic
138	C₂×C₄	#3949	1	1	yes	characteristic
139	C₂×C₄	#3949	1	1	yes	characteristic
140	C₂×C₄	C₂×#677	1	1	yes	characteristic
141	C₂×C₄	C₂×#677	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	40	2	no	—
146	C₈	—	40	2	no	—
147	C₈	#3951	1	2	yes	—
148	C₈	C₂×#681	1	2	yes	—
149	C₅	#12019	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#64733	1	2	yes	—
153	C₂²	C₂×#3949	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#65213	1	1	yes	characteristic
160	C₄	C₂×#3951	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#10600523	1	1	yes	characteristic
163	C₂	C₂×#64733	1	1	yes	characteristic
164	C₂	#10602220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3949
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×#346	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂³×C₅	#106
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12019
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
4. #10598004	C₂	semidirect
54. C₅×C₁₆	M₁₆	semidirect
127. C₁₆	#681	semidirect
149. C₅	#12019	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#346	2
#10600525	2
#10600523	2
#10598004	2
#12019	5