Subgroups — #11988677

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
2	C₄×#3947	C₂	1	1	yes	maximal, characteristic
3	C₄×#3947	C₂	1	2	yes	maximal
4	C₂×#64674	C₂	1	2	yes	maximal
5	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₂×#3947	C₄	1	8	yes	—
9	C₂×#3947	C₄	1	4	yes	—
10	C₄²×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×#168	C₂²	1	2	yes	—
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₅×#322	C₄	1	4	yes	complemented
14	#64674	C₄	1	8	yes	complemented
15	C₈×#168	C₄	1	4	yes	complemented
16	#7846	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
22	C₂²×#168	C₂×C₄	1	2	yes	—
23	C₂²×#168	C₂×C₄	1	4	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	#3947	—	2	8	no	—
29	#3947	—	2	16	no	—
30	C₄×#168	C₂×C₄	1	4	yes	—
31	C₄×#168	C₂×C₄	1	4	yes	—
32	C₄×#168	C₂×C₄	1	4	yes	—
33	C₄×#97	—	5	2	no	—
34	C₄×#97	—	5	1	no	—
35	C₂×#322	—	5	2	no	—
36	C₂×#322	D₅	1	1	yes	characteristic
37	C₂×C₈²	—	5	1	no	—
38	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
40	C₂²×Dic₅	C₄²	1	4	yes	—
41	C₂×C₅×C₈	—	2	8	no	—
42	C₂×C₅×C₈	C₄²	1	8	yes	—
43	C₂×C₅×C₈	C₄²	1	4	yes	—
44	C₂×C₅×C₈	M₁₆	1	8	yes	—
45	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅	C₂²×C₄	1	2	yes	—
47	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₄×Dic₅	—	2	4	no	—
49	C₄×Dic₅	—	2	8	no	—
50	C₂×#168	—	2	4	no	—
51	C₂×#168	C₄²	1	4	yes	—
52	C₂×#168	C₄²	1	8	yes	—
53	C₂×#168	M₁₆	1	16	yes	—
54	C₂×#97	—	5	8	no	—
55	C₂×#97	—	5	4	no	—
56	C₂×C₄×C₈	—	5	2	no	—
57	C₂×C₄×C₈	—	5	1	no	—
58	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
60	C₄³	—	5	1	no	—
61	#322	—	5	8	no	—
62	#322	Dic₅	1	4	yes	complemented
63	C₈²	—	5	4	no	—
64	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
66	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
69	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
70	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
71	C₂×Dic₅	—	2	4	no	—
72	C₂×Dic₅	—	2	8	no	—
73	C₅×C₈	—	2	8	no	—
74	C₅×C₈	#97	1	8	yes	complemented
75	#168	—	2	4	no	—
76	#168	#97	1	16	yes	complemented
77	C₂²×C₈	—	5	4	no	—
78	C₂²×C₈	—	5	2	no	—
79	C₂²×C₈	C₄×D₅	1	4	yes	—
80	C₂²×C₈	C₄×D₅	1	2	yes	—
81	C₂×C₄²	—	5	2	no	—
82	C₂×C₄²	—	5	4	no	—
83	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
84	#97	—	10	16	no	—
85	#97	—	10	8	no	—
86	C₄×C₈	—	5	4	no	—
87	C₄×C₈	—	5	4	no	—
88	C₄×C₈	—	5	4	no	—
89	C₄×C₈	C₄×D₅	1	2	yes	—
90	C₄×C₈	C₄×D₅	1	4	yes	—
91	C₄×C₈	C₂×Dic₅	1	4	yes	—
92	C₄×C₈	C₂×Dic₅	1	2	yes	—
93	C₂²×C₅	C₄³	1	1	yes	characteristic
94	C₂²×C₅	C₄×M₁₆	1	2	yes	—
95	C₂²×C₅	C₄×M₁₆	1	4	yes	—
96	C₄×C₅	C₂×#97	1	2	yes	—
97	C₄×C₅	C₂×#97	1	4	yes	—
98	C₄×C₅	C₄×M₁₆	1	4	yes	—
99	C₄×C₅	C₄×M₁₆	1	2	yes	—
100	Dic₅	—	4	4	no	—
101	C₂²×C₄	—	5	4	no	—
102	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
104	C₂×C₈	—	2	8	no	—
105	C₂×C₈	—	5	8	no	—
106	C₂×C₈	—	5	16	no	—
107	C₂×C₈	—	5	4	no	—
108	C₂×C₈	—	10	4	no	—
109	C₂×C₈	#676	1	8	yes	—
110	C₂×C₈	C₄×Dic₅	1	8	yes	—
111	C₂×C₈	C₄×Dic₅	1	4	yes	—
112	C₄²	—	10	4	no	—
113	C₄²	—	10	8	no	—
114	C₄²	C₂×C₄×D₅	1	2	yes	—
115	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
116	C₄²	C₂²×Dic₅	1	1	yes	characteristic
117	C₂×C₅	#1455	1	4	yes	—
118	C₂×C₅	C₄×#97	1	1	yes	characteristic
119	C₂×C₅	C₄×#97	1	2	yes	—
120	C₂³	C₄²×D₅	1	1	yes	characteristic
121	C₂×C₄	—	10	4	no	—
122	C₂×C₄	—	10	8	no	—
123	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
124	C₂×C₄	C₄²×D₅	1	2	yes	—
125	C₂×C₄	C₂×#676	1	4	yes	—
126	C₂×C₄	D₅×M₁₆	1	8	yes	—
127	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
128	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
129	C₈	—	2	8	no	—
130	C₈	—	5	16	no	—
131	C₈	—	10	4	no	—
132	C₈	#3947	1	8	yes	complemented
133	C₅	#7846	1	1	yes	characteristic, complemented, Sylow
134	C₂²	C₄×#676	1	2	yes	—
135	C₂²	#64996	1	4	yes	—
136	C₂²	C₄²×Dic₅	1	1	yes	characteristic
137	C₄	—	20	4	no	—
138	C₄	C₄×#676	1	2	yes	—
139	C₄	D₅×#97	1	4	yes	—
140	C₄	C₂×#3947	1	2	yes	—
141	C₄	M₁₆×Dic₅	1	4	yes	—
142	C₂	#10598868	1	4	yes	—
143	C₂	C₄×#3947	1	1	yes	characteristic
144	C₂	Dic₅×#97	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₄³
Fitting	C₂×C₅×#322	C₂
socle, prime radical	C₂³×C₅	C₂×C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#322	D₅
hypocenter	C₅	#7846
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
13. C₅×#322	C₄	semidirect
14. #64674	C₄	semidirect
15. C₈×#168	C₄	semidirect
62. #322	Dic₅	semidirect
74. C₅×C₈	#97	semidirect
76. #168	#97	semidirect
132. C₈	#3947	semidirect
133. C₅	#7846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#322	2
C₄×#3947	2
C₂×#64674	2
C₂×C₈×#168	2
#7846	5