Subgroups — #12252304

Order 1280.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#10615511	C₂	1	2	yes	maximal
3	#10600678	C₂	1	6	yes	maximal, complemented
4	C₂×#64683	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#131	C₄	1	1	yes	characteristic
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₄×C₅×#41	C₄	1	1	yes	characteristic
8	C₅×#405	C₄	1	3	yes	—
9	C₅×#405	C₂²	1	3	yes	—
10	C₄×C₅×M₁₆	C₄	1	3	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
12	#65429	—	2	8	no	—
13	C₂×#3944	C₂²	1	12	yes	—
14	#64683	C₂²	1	4	yes	—
15	#64683	C₂²	1	12	yes	—
16	#33401	—	5	1	no	maximal, Sylow
17	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
18	C₅×#131	C₈	1	8	yes	—
19	C₅×#131	D₄	1	4	yes	—
20	C₅×#131	Q₈	1	4	yes	—
21	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
22	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
24	C₂²×C₅×C₈	C₂³	1	3	yes	—
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×D₄	—	2	3	no	—
27	C₅×#117	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
29	C₅×#97	C₂×C₄	1	6	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
31	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
32	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×C₈	C₂³	1	3	yes	—
34	#3944	—	2	12	no	—
35	#3944	C₂³	1	8	yes	—
36	C₂×#672	—	2	12	no	—
37	#3407	—	5	2	no	—
38	C₄×#131	D₅	1	1	yes	characteristic
39	#2153	—	5	6	no	—
40	C₂×#364	—	5	6	no	—
41	C₅×#41	C₄⋊C₄	1	4	yes	—
42	C₅×#41	C₂×C₈	1	4	yes	—
43	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
44	C₂×C₅×D₄	M₁₆	1	3	yes	—
45	C₂²×C₄×C₅	—	2	3	no	—
46	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
47	C₅×M₁₆	C₄⋊C₄	1	12	yes	—
48	C₅×M₁₆	C₂×C₈	1	12	yes	—
49	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
53	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
54	C₂×C₅×C₈	C₂×Q₈	1	6	yes	—
55	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
56	C₅×C₄⋊C₄	—	2	3	no	—
57	C₅×C₂²⋊C₄	—	2	3	no	—
58	C₄²×C₅	—	2	3	no	—
59	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅	C₂²×C₄	1	3	yes	—
61	#672	—	2	8	no	—
62	C₂×#131	Dic₅	1	1	yes	characteristic
63	C₂×#131	C₂×D₅	1	1	yes	characteristic
64	C₄×#41	Dic₅	1	1	yes	characteristic
65	#504	—	10	8	no	—
66	C₂×M₃₂	—	5	12	no	—
67	#405	Dic₅	1	3	yes	—
68	#405	C₂×D₅	1	3	yes	—
69	C₄×M₁₆	Dic₅	1	3	yes	—
70	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
71	#364	—	5	4	no	—
72	#364	—	5	12	no	—
73	C₂³×C₅	C₂×M₁₆	1	3	yes	—
74	C₅×Q₈	C₄⋊C₈	1	4	yes	—
75	C₅×D₄	C₄⋊C₈	1	12	yes	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	3	no	—
78	C₂×C₄×C₅	—	4	6	no	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
80	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
81	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
82	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
83	C₅×C₈	C₂×C₄⋊C₄	1	6	yes	—
84	C₅×C₈	C₂²×C₈	1	2	yes	—
85	C₅×C₈	C₂²×C₈	1	6	yes	—
86	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
87	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
88	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
89	#131	#168	1	8	yes	—
90	#131	Dic₁₀	1	4	yes	—
91	#131	D₂₀	1	4	yes	—
92	C₂×M₁₆	C₂×Dic₅	1	3	yes	—
93	C₂×M₁₆	C₂×Dic₅	1	3	yes	—
94	C₂²×C₈	C₂×Dic₅	1	3	yes	—
95	C₂²×C₈	C₂²×D₅	1	3	yes	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×D₄	—	2	3	no	—
98	#117	C₂×Dic₅	1	3	yes	—
99	C₂×C₄²	C₂×Dic₅	1	3	yes	—
100	M₃₂	—	5	8	no	—
101	M₃₂	—	10	12	no	—
102	C₂×C₁₆	—	10	12	no	—
103	#97	C₂×Dic₅	1	6	yes	—
104	C₄×C₈	C₂×Dic₅	1	3	yes	—
105	C₄×C₈	C₂×Dic₅	1	3	yes	—
106	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₅	1	3	yes	—
108	C₂²×C₅	—	4	3	no	—
109	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
110	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	4	3	no	—
113	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
114	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
116	#41	C₂×#168	1	4	yes	—
117	#41	C₄⋊Dic₅	1	4	yes	—
118	C₂×Q₈	#681	1	1	yes	characteristic
119	C₂×D₄	#681	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
122	M₁₆	C₂×#168	1	12	yes	—
123	M₁₆	C₄⋊Dic₅	1	12	yes	—
124	C₂×C₈	C₂×#168	1	12	yes	—
125	C₂×C₈	C₂×Dic₁₀	1	6	yes	—
126	C₂×C₈	C₂×D₂₀	1	6	yes	—
127	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₅	1	3	yes	—
129	C₂×C₈	C₂²×Dic₅	1	3	yes	—
130	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
131	C₄⋊C₄	—	2	3	no	—
132	C₂²⋊C₄	—	2	3	no	—
133	C₄²	—	2	3	no	—
134	C₄²	C₂²×Dic₅	1	1	yes	characteristic
135	C₄²	C₂²×Dic₅	1	3	yes	—
136	C₁₆	—	10	8	no	—
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	—	2	6	no	—
139	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	C₂³	C₂×#681	1	3	yes	—
141	Q₈	C₄⋊#168	1	4	yes	—
142	D₄	C₄⋊#168	1	12	yes	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	3	no	—
145	C₂×C₄	—	4	6	no	—
146	C₂×C₄	C₂²×#168	1	6	yes	—
147	C₂×C₄	C₂×#681	1	3	yes	—
148	C₂×C₄	C₂×C₄⋊Dic₅	1	6	yes	—
149	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
150	C₈	C₂²×#168	1	2	yes	—
151	C₈	C₂²×#168	1	6	yes	—
152	C₈	C₂×C₄⋊Dic₅	1	6	yes	—
153	C₈	C₂²×Dic₁₀	1	1	yes	characteristic
154	C₈	C₂²×D₂₀	1	1	yes	characteristic
155	C₅	#33401	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊#168	1	6	yes	—
158	C₂²	C₂²×#681	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊#168	1	6	yes	—
162	C₄	C₂³×#168	1	1	yes	characteristic
163	C₄	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊#168	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₀
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₄×C₅×#131	C₂
socle	C₂×C₅	C₂²×C₄⋊C₈
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#33401
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600678	C₂	semidirect
4. C₂×#64683	C₂	semidirect
155. C₅	#33401	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#10615511	2
#10600678	2
C₂×#64683	2
#33401	5