Subgroups — #11986136

Order 1280.

246 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1450	C₂	1	1	yes	maximal, characteristic
2	C₄×#3944	C₂	1	1	yes	maximal, characteristic
3	C₂×#64681	C₂	1	1	yes	maximal, characteristic
4	C₅×#431	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#3944	C₄	1	2	yes	—
7	C₂²×#672	C₄	1	2	yes	—
8	C₅×#322	—	2	1	no	—
9	C₅×C₈²	—	2	1	no	—
10	#64681	—	2	1	no	—
11	#64681	—	2	1	no	—
12	#64681	—	2	1	no	—
13	C₄×#672	—	2	1	no	—
14	#7140	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₅×C₄⋊C₈	—	2	2	no	—
19	C₅×C₂²⋊C₈	C₈	1	4	yes	—
20	C₄×C₅×C₈	—	2	2	no	—
21	C₄×C₅×C₈	D₄	1	1	yes	characteristic
22	C₄×C₅×C₈	D₄	1	1	yes	characteristic
23	C₄×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
25	#3944	—	2	2	no	—
26	#3944	C₈	1	4	yes	complemented
27	C₂×#672	—	2	2	no	—
28	C₂×#672	—	2	2	no	—
29	C₂×#672	—	2	2	no	—
30	C₂×#672	C₈	1	4	yes	complemented
31	C₄×M₃₂	—	5	1	no	—
32	C₂×#346	—	5	1	no	—
33	#1450	D₅	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
36	C₂×C₅×C₈	—	2	4	no	—
37	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₅×C₈	M₁₆	1	2	yes	—
44	C₂×C₅×C₈	M₁₆	1	2	yes	—
45	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	M₁₆	1	2	yes	—
48	#672	—	2	4	no	—
49	#672	—	2	4	no	—
50	C₂×M₃₂	—	5	2	no	—
51	C₂²×C₁₆	—	5	2	no	—
52	#431	Dic₅	1	2	yes	—
53	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
54	#346	—	10	1	no	—
55	#346	—	10	1	no	—
56	#346	—	10	1	no	—
57	C₄×C₁₆	—	10	1	no	—
58	#322	—	2	1	no	—
59	C₈²	—	2	1	no	—
60	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅	—	4	2	no	—
62	C₂×C₄×C₅	#95	1	1	yes	characteristic
63	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅	#97	1	1	yes	characteristic
66	C₂×C₄×C₅	#97	1	1	yes	characteristic
67	C₂×C₄×C₅	#97	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
70	C₅×C₈	—	4	4	no	—
71	C₅×C₈	C₂²⋊C₈	1	2	yes	—
72	C₅×C₈	C₂²⋊C₈	1	2	yes	—
73	C₅×C₈	C₄⋊C₈	1	2	yes	—
74	C₅×C₈	C₄⋊C₈	1	2	yes	—
75	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
76	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
77	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
78	M₃₂	—	5	4	no	—
79	M₃₂	—	10	2	no	—
80	C₂×C₁₆	—	5	4	no	—
81	C₂×C₁₆	—	10	2	no	—
82	C₂×C₁₆	—	10	2	no	—
83	C₂×C₁₆	—	10	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	#168	1	4	yes	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	Dic₁₀	1	1	yes	characteristic
88	C₄×C₈	D₂₀	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
91	C₂²×C₅	—	2	2	no	—
92	C₂²×C₅	C₈²	1	1	yes	characteristic
93	C₂²×C₅	#322	1	1	yes	characteristic
94	C₂²×C₅	#336	1	1	yes	characteristic
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	#322	1	1	yes	characteristic
98	C₄×C₅	#322	1	1	yes	characteristic
99	C₄×C₅	#336	1	1	yes	characteristic
100	C₄×C₅	#336	1	1	yes	characteristic
101	C₂²×C₄	C₂×#168	1	2	yes	—
102	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	C₈×D₅	1	2	yes	—
105	C₂×C₈	C₈×D₅	1	2	yes	—
106	C₂×C₈	#676	1	2	yes	—
107	C₂×C₈	#676	1	2	yes	—
108	C₂×C₈	#683	1	1	yes	characteristic
109	C₂×C₈	#683	1	1	yes	characteristic
110	C₂×C₈	#685	1	1	yes	characteristic
111	C₂×C₈	#685	1	1	yes	characteristic
112	C₄²	#681	1	2	yes	—
113	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
114	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
115	C₁₆	—	10	4	no	—
116	C₁₆	—	10	4	no	—
117	C₂×C₅	—	2	1	no	—
118	C₂×C₅	—	2	1	no	—
119	C₂×C₅	#1273	1	1	yes	characteristic
120	C₂×C₅	#1378	1	2	yes	—
121	C₂³	C₄×#168	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	#3935	1	1	yes	characteristic
124	C₂×C₄	C₄⋊#168	1	2	yes	—
125	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
126	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
127	C₂×C₄	#3947	1	1	yes	characteristic
128	C₂×C₄	#3947	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#168	1	2	yes	—
130	C₂×C₄	#3963	1	1	yes	characteristic
131	C₈	—	4	4	no	—
132	C₈	#3946	1	2	yes	—
133	C₈	#3946	1	2	yes	—
134	C₈	#3952	1	2	yes	—
135	C₈	#3952	1	2	yes	—
136	C₅	#7140	1	1	yes	characteristic, complemented, Sylow
137	C₂²	—	2	2	no	—
138	C₂²	C₈×#168	1	1	yes	characteristic
139	C₂²	#64675	1	1	yes	characteristic
140	C₂²	#64744	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	#64674	1	1	yes	characteristic
144	C₄	#64674	1	1	yes	characteristic
145	C₄	#64770	1	1	yes	characteristic
146	C₄	#64770	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#10598144	1	1	yes	characteristic
150	C₂	#10598163	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂²×C₅	C₈²
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	#1450	D₅
hypocenter	C₅	#7140
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
26. #3944	C₈	semidirect
30. C₂×#672	C₈	semidirect
136. C₅	#7140	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1450	2
C₄×#3944	2
C₂×#64681	2
#7140	5