Subgroups — #12010913

Order 1280.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#10600537	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	C₄×#672	—	2	1	no	—
18	#12015	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	C₈	1	1	yes	characteristic
20	C₂×C₅×M₁₆	C₈	1	1	yes	characteristic
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×C₄⋊C₈	—	2	1	no	—
25	C₅×C₂²⋊C₈	—	2	1	no	—
26	C₅×#97	C₈	1	2	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	D₄	1	1	yes	characteristic
31	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	—	2	2	no	—
36	C₂×#672	—	4	2	no	—
37	#2162	—	5	1	no	—
38	C₂×C₄⋊C₁₆	—	5	1	no	—
39	C₈×M₁₆	D₅	1	1	yes	characteristic
40	#1370	—	5	2	no	—
41	#1367	—	5	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₅×M₁₆	—	2	2	no	—
46	C₅×M₁₆	C₁₆	1	4	yes	complemented
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	2	1	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	4	1	no	—
51	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅	M₁₆	1	1	yes	characteristic
60	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
61	#672	—	4	1	no	—
62	#672	—	4	2	no	—
63	#672	—	4	1	no	—
64	C₂²×C₁₆	—	10	1	no	—
65	#431	Dic₅	1	1	yes	characteristic
66	C₄×M₁₆	Dic₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	C₄⋊C₁₆	—	5	1	no	—
69	C₄⋊C₁₆	—	5	2	no	—
70	C₄⋊C₁₆	—	5	1	no	—
71	C₄⋊C₁₆	—	10	1	no	—
72	C₂²⋊C₁₆	—	10	1	no	—
73	C₄×C₁₆	—	10	1	no	—
74	#322	Dic₅	1	2	yes	—
75	C₈²	C₂×D₅	1	2	yes	—
76	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	4	1	no	—
80	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₅	M₃₂	1	2	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₅×C₈	—	2	1	no	—
89	C₅×C₈	—	2	1	no	—
90	C₅×C₈	—	2	1	no	—
91	C₅×C₈	—	2	2	no	—
92	C₅×C₈	—	4	2	no	—
93	C₅×C₈	C₂×C₁₆	1	4	yes	complemented
94	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₅×C₈	#137	1	1	yes	characteristic
96	C₂×M₁₆	#168	1	1	yes	characteristic
97	C₂×M₁₆	#168	1	1	yes	characteristic
98	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
99	C₂²×C₈	D₂₀	1	1	yes	characteristic
100	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
101	C₂×C₁₆	—	10	1	no	—
102	C₂×C₁₆	—	10	1	no	—
103	C₂×C₁₆	—	10	2	no	—
104	C₂×C₁₆	—	20	2	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₂²⋊C₈	—	2	1	no	—
107	#97	#168	1	2	yes	—
108	C₄×C₈	Dic₁₀	1	1	yes	characteristic
109	C₄×C₈	D₂₀	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₅	1	2	yes	—
111	C₄×C₈	C₂×Dic₅	1	2	yes	—
112	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₂²×C₅	—	2	1	no	—
115	C₂²×C₅	—	2	1	no	—
116	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
117	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₅	—	2	1	no	—
119	C₄×C₅	—	2	1	no	—
120	C₄×C₅	—	2	1	no	—
121	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
122	C₄×C₅	#428	1	1	yes	characteristic
123	C₄×C₅	#430	1	1	yes	characteristic
124	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
125	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
126	C₂²×C₄	C₂×#168	1	1	yes	characteristic
127	C₂²×C₄	#681	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	#672	1	4	yes	complemented
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	C₂×#168	1	2	yes	—
136	C₂×C₈	C₂×#168	1	2	yes	—
137	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
140	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
141	C₄²	C₂×#168	1	1	yes	characteristic
142	C₄²	#681	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
144	C₄²	C₂²×Dic₅	1	1	yes	characteristic
145	C₁₆	—	20	2	no	—
146	C₁₆	—	20	1	no	—
147	C₁₆	—	20	1	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	#1565	1	1	yes	characteristic
150	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
151	C₂×C₅	#2153	1	1	yes	characteristic
152	C₂³	C₄⋊#168	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
157	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
158	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
159	C₂×C₄	C₂×#672	1	2	yes	—
160	C₂×C₄	#3944	1	2	yes	—
161	C₂×C₄	C₂²×#168	1	1	yes	characteristic
162	C₂×C₄	C₂×#681	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	4	2	no	—
169	C₈	C₂×#672	1	4	yes	complemented
170	C₈	#4054	1	1	yes	characteristic
171	C₈	#4055	1	1	yes	characteristic
172	C₅	#12015	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#672	1	2	yes	—
176	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊#672	1	2	yes	—
181	C₄	C₂²×#672	1	1	yes	characteristic
182	C₄	C₂×#3944	1	1	yes	characteristic
183	C₄	#65410	1	1	yes	characteristic
184	C₄	#65414	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×C₄⋊#672	1	1	yes	characteristic
187	C₂	#10600643	1	1	yes	characteristic
188	C₂	#10600678	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₁₆
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	#12015
LCS	G ▷ 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
46. C₅×M₁₆	C₁₆	semidirect
93. C₅×C₈	C₂×C₁₆	semidirect
130. M₁₆	#672	semidirect
169. C₈	C₂×#672	semidirect
172. C₅	#12015	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
#10600537	2
C₂×C₄⋊#672	2
#10597931	2
#10597930	2
#12015	5