Subgroups — #12010918

Order 1280.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10600538	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
4	#10597931	C₂	1	2	yes	maximal, complemented
5	#10597930	C₂	1	2	yes	maximal, complemented
6	C₅×#431	C₄	1	1	yes	characteristic
7	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×#672	—	2	1	no	—
10	C₅×#322	C₄	1	2	yes	—
11	C₅×C₈²	C₂²	1	2	yes	—
12	C₂²⋊#672	—	2	1	no	—
13	C₄⋊#672	—	2	1	no	—
14	C₄⋊#672	C₂²	1	1	yes	characteristic
15	C₄⋊#672	C₂²	1	2	yes	—
16	C₄⋊#672	C₂²	1	1	yes	characteristic
17	#64681	—	2	1	no	—
18	#12016	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₅×C₄⋊C₈	C₈	1	2	yes	—
24	C₅×C₂²⋊C₈	C₈	1	2	yes	—
25	C₅×#97	—	2	1	no	—
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×#672	—	2	1	no	—
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	2	no	—
35	C₂×#672	—	4	2	no	—
36	#2163	—	5	1	no	—
37	C₂×C₄⋊C₁₆	—	5	1	no	—
38	C₈×M₁₆	D₅	1	1	yes	characteristic
39	#1370	—	5	2	no	—
40	#1367	—	5	2	no	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₅×M₁₆	—	2	4	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	4	1	no	—
49	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
52	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₅	M₁₆	1	1	yes	characteristic
57	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
58	#672	—	4	1	no	—
59	#672	—	4	2	no	—
60	#672	—	4	1	no	—
61	C₂²×C₁₆	—	10	1	no	—
62	#431	Dic₅	1	1	yes	characteristic
63	C₄×M₁₆	Dic₅	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
65	C₄⋊C₁₆	—	5	1	no	—
66	C₄⋊C₁₆	—	5	2	no	—
67	C₄⋊C₁₆	—	5	1	no	—
68	C₄⋊C₁₆	—	10	1	no	—
69	C₂²⋊C₁₆	—	10	1	no	—
70	#346	—	10	1	no	—
71	#322	Dic₅	1	2	yes	—
72	C₈²	C₂×D₅	1	2	yes	—
73	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₅×C₈	—	2	4	no	—
85	C₅×C₈	—	2	1	no	—
86	C₅×C₈	—	2	1	no	—
87	C₅×C₈	—	2	1	no	—
88	C₅×C₈	—	2	4	no	—
89	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₅×C₈	#137	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
93	C₂²×C₈	D₂₀	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
95	C₂×C₁₆	—	10	1	no	—
96	C₂×C₁₆	—	10	1	no	—
97	C₂×C₁₆	—	10	2	no	—
98	C₂×C₁₆	—	20	2	no	—
99	C₄⋊C₈	#168	1	2	yes	—
100	C₂²⋊C₈	#168	1	2	yes	—
101	#97	—	2	1	no	—
102	C₄×C₈	Dic₁₀	1	1	yes	characteristic
103	C₄×C₈	D₂₀	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₅	1	2	yes	—
105	C₄×C₈	C₂×Dic₅	1	2	yes	—
106	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	—	2	1	no	—
110	C₂²×C₅	#364	1	2	yes	—
111	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	#428	1	1	yes	characteristic
117	C₄×C₅	#430	1	1	yes	characteristic
118	C₄×C₅	#504	1	2	yes	—
119	C₂²×C₄	C₂×#168	1	1	yes	characteristic
120	C₂²×C₄	#681	1	1	yes	characteristic
121	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂×#168	1	4	yes	—
128	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
132	C₄²	C₂×#168	1	1	yes	characteristic
133	C₄²	#681	1	1	yes	characteristic
134	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
135	C₄²	C₂²×Dic₅	1	1	yes	characteristic
136	C₁₆	—	20	2	no	—
137	C₁₆	—	20	1	no	—
138	C₁₆	—	20	1	no	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	#1565	1	1	yes	characteristic
141	C₂×C₅	#2151	1	1	yes	characteristic
142	C₂×C₅	C₂×#364	1	1	yes	characteristic
143	C₂³	C₄⋊#168	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
149	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
151	C₂×C₄	C₂²×#168	1	1	yes	characteristic
152	C₂×C₄	C₂×#681	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
154	C₈	—	2	4	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	4	no	—
159	C₈	#4054	1	1	yes	characteristic
160	C₈	#4055	1	1	yes	characteristic
161	C₅	#12016	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#64683	1	2	yes	—
165	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#65410	1	1	yes	characteristic
171	C₄	#65414	1	1	yes	characteristic
172	C₄	#65429	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	C₂×#64683	1	1	yes	characteristic
175	C₂	#10600643	1	1	yes	characteristic
176	C₂	#10600675	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#12016
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10597931	C₂	semidirect
5. #10597930	C₂	semidirect
161. C₅	#12016	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×M₁₆	2
#10600538	2
C₂×C₄⋊#672	2
#10597931	2
#10597930	2
#12016	5