Subgroups — #12014308

Order 1280.

338 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10600542	C₂	1	1	yes	maximal, characteristic
3	C₄×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598302	C₂	1	4	yes	maximal
5	C₂×C₅×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3947	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅	C₄	1	2	yes	—
9	C₂×#3944	C₂²	1	2	yes	—
10	#65221	C₄	1	1	yes	characteristic
11	C₂×#3935	C₄	1	1	yes	characteristic
12	C₅×#346	C₄	1	2	yes	complemented
13	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
14	#64681	C₄	1	2	yes	complemented
15	C₄×#672	C₄	1	2	yes	complemented
16	#12593	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×#168	C₂×C₄	1	2	yes	—
22	C₅×M₃₂	—	2	4	no	—
23	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
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	#3963	—	2	2	no	—
29	#3947	—	2	4	no	—
30	C₈×Dic₅	C₈	1	8	yes	—
31	#3944	—	2	4	no	—
32	C₂×#672	C₂×C₄	1	4	yes	—
33	#3935	—	2	2	no	—
34	C₄×M₃₂	—	5	1	no	—
35	C₄×M₃₂	D₅	1	1	yes	characteristic
36	#1726	—	5	1	no	—
37	#1375	—	5	4	no	—
38	C₂²×C₄×C₅	M₁₆	1	2	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅	—	2	2	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
44	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	#41	1	4	yes	—
47	C₄²×C₅	C₄²	1	1	yes	characteristic
48	C₄²×C₅	C₄²	1	1	yes	characteristic
49	C₄²×C₅	M₁₆	1	2	yes	—
50	C₄×Dic₅	—	2	2	no	—
51	C₄×Dic₅	C₂×C₈	1	4	yes	—
52	C₂×#168	—	2	2	no	—
53	C₂×#168	—	2	2	no	—
54	C₂×#168	C₂×C₈	1	4	yes	—
55	C₅×C₁₆	—	2	4	no	—
56	#672	—	2	4	no	—
57	C₂×M₃₂	—	5	2	no	—
58	C₂×M₃₂	C₂×D₅	1	2	yes	—
59	C₂×#97	—	5	2	no	—
60	C₂×#97	—	5	1	no	—
61	C₂×C₄×C₈	—	5	2	no	—
62	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
63	#376	—	5	1	no	—
64	#346	—	5	2	no	—
65	#346	Dic₅	1	2	yes	complemented
66	C₄×C₁₆	—	5	2	no	—
67	C₄×C₁₆	Dic₅	1	2	yes	complemented
68	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅	—	4	2	no	—
70	C₂×C₄×C₅	#97	1	2	yes	—
71	C₂×C₄×C₅	#97	1	2	yes	—
72	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅	#117	1	2	yes	—
74	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₅	—	2	4	no	—
77	C₂×Dic₅	—	4	4	no	—
78	C₅×C₈	C₄×C₈	1	2	yes	—
79	C₅×C₈	C₄×C₈	1	2	yes	—
80	C₅×C₈	#117	1	2	yes	—
81	C₅×C₈	#117	1	2	yes	—
82	#168	—	2	4	no	—
83	C₂²×C₈	—	5	2	no	—
84	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
86	C₂×C₄²	—	5	2	no	—
87	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
88	M₃₂	—	2	4	no	—
89	M₃₂	—	10	4	no	—
90	C₂×C₁₆	—	5	4	no	—
91	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
92	#97	—	10	4	no	—
93	#97	—	10	2	no	—
94	C₄×C₈	—	5	8	no	—
95	C₄×C₈	C₄×D₅	1	1	yes	characteristic
96	C₄×C₈	C₄×D₅	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
99	#95	—	10	2	no	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	C₂×#97	1	1	yes	characteristic
102	C₂²×C₅	#431	1	2	yes	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	#376	1	1	yes	characteristic
106	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₅	#431	1	2	yes	—
108	Dic₅	—	4	4	no	—
109	C₂²×C₄	—	10	2	no	—
110	C₂²×C₄	#676	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
112	C₂×C₈	—	5	4	no	—
113	C₂×C₈	—	10	2	no	—
114	C₂×C₈	—	10	2	no	—
115	C₂×C₈	C₈×D₅	1	4	yes	—
116	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
117	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
119	C₂×C₈	#709	1	4	yes	—
120	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
121	C₄²	—	5	4	no	—
122	C₄²	—	10	2	no	—
123	C₄²	#676	1	2	yes	—
124	C₄²	C₄×Dic₅	1	1	yes	characteristic
125	C₄²	C₄×Dic₅	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₁₆	—	10	4	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	#1744	1	1	yes	characteristic
130	C₂×C₅	#2110	1	2	yes	—
131	C₂³	C₂×#676	1	1	yes	characteristic
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	10	4	no	—
134	C₂×C₄	—	20	4	no	—
135	C₂×C₄	#3947	1	2	yes	—
136	C₂×C₄	#3947	1	2	yes	—
137	C₂×C₄	#4018	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
139	C₂×C₄	C₂×#676	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
141	C₈	—	10	4	no	—
142	C₈	C₈×Dic₅	1	2	yes	—
143	C₈	C₈×Dic₅	1	2	yes	—
144	C₈	#4018	1	2	yes	—
145	C₈	#4072	1	2	yes	—
146	C₅	#12593	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	2	2	no	—
148	C₂²	#64974	1	2	yes	—
149	C₂²	C₂×#3947	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	20	4	no	—
153	C₄	#64974	1	2	yes	—
154	C₄	#65221	1	1	yes	characteristic
155	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#10600543	1	1	yes	characteristic
158	C₂	#10602218	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#12593
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
12. C₅×#346	C₄	semidirect
13. C₄×C₅×C₁₆	C₄	semidirect
14. #64681	C₄	semidirect
15. C₄×#672	C₄	semidirect
65. #346	Dic₅	semidirect
67. C₄×C₁₆	Dic₅	semidirect
146. C₅	#12593	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
#10600542	2
C₄×#3944	2
#10598302	2
#12593	5