Subgroups — #12008329

Order 1280.

306 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	2	yes	maximal
3	#10597929	C₂	1	2	yes	maximal, complemented
4	C₈×#672	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×C₈²	C₄	1	1	yes	characteristic
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	C₅×C₈²	C₂²	1	1	yes	characteristic
12	C₂²⋊#672	C₄	1	4	yes	—
13	C₄⋊#672	C₄	1	4	yes	—
14	C₄×#672	C₂²	1	4	yes	—
15	#11410	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₈	1	2	yes	—
17	C₂²×C₅×C₈	C₈	1	2	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅×C₈	C₈	1	2	yes	—
21	C₄×C₅×C₈	C₈	1	2	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×#672	C₂×C₄	1	8	yes	—
30	#2162	—	5	2	no	—
31	C₂×C₈²	D₅	1	1	yes	characteristic
32	#1312	—	5	2	no	—
33	C₈×C₁₆	—	5	2	no	—
34	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
37	C₂×C₅×C₈	—	2	4	no	—
38	C₂×C₅×C₈	—	2	4	no	—
39	C₂×C₅×C₈	C₁₆	1	8	yes	complemented
40	C₂×C₅×C₈	C₄²	1	2	yes	—
41	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
44	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
45	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
46	C₄²×C₅	C₄²	1	1	yes	characteristic
47	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
50	#672	—	2	8	no	—
51	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
52	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
54	C₄⋊C₁₆	—	5	4	no	—
55	C₂²⋊C₁₆	—	5	4	no	—
56	C₄×C₁₆	—	5	4	no	—
57	C₈²	Dic₅	1	1	yes	characteristic
58	C₈²	Dic₅	1	1	yes	characteristic
59	C₈²	C₂×D₅	1	1	yes	characteristic
60	C₈²	C₂×D₅	1	1	yes	characteristic
61	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
68	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
71	C₅×C₈	—	2	2	no	—
72	C₅×C₈	—	2	2	no	—
73	C₅×C₈	C₂×C₁₆	1	4	yes	—
74	C₅×C₈	C₂×C₁₆	1	4	yes	—
75	C₅×C₈	M₃₂	1	4	yes	—
76	C₅×C₈	#131	1	4	yes	—
77	C₂²×C₈	#168	1	2	yes	—
78	C₂²×C₈	#168	1	2	yes	—
79	C₂²×C₈	C₄×D₅	1	2	yes	—
80	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
81	C₂×C₁₆	—	5	8	no	—
82	C₄×C₈	#168	1	2	yes	—
83	C₄×C₈	#168	1	2	yes	—
84	C₄×C₈	C₄×D₅	1	2	yes	—
85	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
89	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
90	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
91	C₂²×C₅	—	2	1	no	—
92	C₂²×C₅	—	2	1	no	—
93	C₂²×C₅	C₄×C₁₆	1	2	yes	—
94	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₅	—	2	1	no	—
96	C₄×C₅	—	2	1	no	—
97	C₄×C₅	C₄×C₁₆	1	2	yes	—
98	C₄×C₅	#405	1	2	yes	—
99	C₄×C₅	C₂²×C₁₆	1	2	yes	—
100	C₄×C₅	C₂×M₃₂	1	2	yes	—
101	C₂²×C₄	C₂×#168	1	1	yes	characteristic
102	C₂²×C₄	C₂×#168	1	1	yes	characteristic
103	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	#672	1	8	yes	complemented
107	C₂×C₈	C₂×#168	1	2	yes	—
108	C₂×C₈	C₂×#168	1	2	yes	—
109	C₂×C₈	C₂×#168	1	2	yes	—
110	C₂×C₈	C₂×#168	1	2	yes	—
111	C₂×C₈	C₄×Dic₅	1	2	yes	—
112	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
113	C₄²	C₂×#168	1	1	yes	characteristic
114	C₄²	C₂×#168	1	1	yes	characteristic
115	C₄²	C₄×Dic₅	1	1	yes	characteristic
116	C₄²	C₂²×Dic₅	1	1	yes	characteristic
117	C₁₆	—	10	8	no	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	#1450	1	1	yes	characteristic
120	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
121	C₂×C₅	C₄×M₃₂	1	1	yes	characteristic
122	C₂³	C₄×#168	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₄×#168	1	1	yes	characteristic
126	C₂×C₄	C₄×#168	1	1	yes	characteristic
127	C₂×C₄	C₄×#168	1	1	yes	characteristic
128	C₂×C₄	C₂×#672	1	4	yes	—
129	C₂×C₄	C₂²×#168	1	1	yes	characteristic
130	C₂×C₄	C₂²×#168	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	2	no	—
134	C₈	C₂×#672	1	4	yes	—
135	C₈	C₂×#672	1	4	yes	—
136	C₈	#3944	1	4	yes	—
137	C₈	#4047	1	4	yes	—
138	C₅	#11410	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×#672	1	2	yes	—
142	C₂²	C₂×C₄×#168	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×#672	1	2	yes	—
146	C₄	C₂²×#672	1	2	yes	—
147	C₄	C₂×#3944	1	2	yes	—
148	C₄	#65390	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#10600517	1	1	yes	characteristic
151	C₂	C₂×C₄×#672	1	1	yes	characteristic
152	C₂	C₄×#3944	1	1	yes	characteristic

	subgroup	G/H
center	C₈²	C₂×D₅
commutator	C₂×C₅	C₂×C₄×C₁₆
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×C₈²	C₂
socle	C₂²×C₅	C₂×C₄×C₈
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#11410
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #10597929	C₂	semidirect
4. C₈×#672	C₂	semidirect
39. C₂×C₅×C₈	C₁₆	semidirect
106. C₂×C₈	#672	semidirect
138. C₅	#11410	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈²	2
#10600537	2
#10597929	2
C₈×#672	2
#11410	5