Subgroups — #13043653

Order 1280.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#713	C₂	1	1	yes	maximal, characteristic
2	Q₈×#674	C₂	1	2	yes	maximal, complemented
3	#10604021	C₂	1	6	yes	maximal, complemented
4	#10603829	C₂	1	3	yes	maximal, complemented
5	C₄×D₅⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₄×#713	C₄	1	1	yes	characteristic
7	C₂×D₅⋊C₁₆	C₂²	1	6	yes	—
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
9	Q₈×#168	C₂²	1	1	yes	characteristic
10	#65123	C₂²	1	3	yes	—
11	#65122	C₄	1	3	yes	—
12	C₈×D₂₀	C₄	1	3	yes	—
13	C₄×C₈×D₅	C₂²	1	3	yes	—
14	#64887	C₂²	1	6	yes	—
15	#64885	C₂²	1	1	yes	characteristic
16	#64885	C₂²	1	3	yes	—
17	C₄⋊#674	C₂²	1	6	yes	—
18	C₄×#674	C₂²	1	6	yes	—
19	C₁₆×#41	—	5	1	no	maximal, Sylow
20	C₂×#713	C₈	1	1	yes	characteristic
21	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
22	Q₈×Dic₅	C₈	1	1	yes	characteristic
23	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
24	C₂×C₈×D₅	C₂³	1	3	yes	—
25	#4039	C₈	1	3	yes	—
26	#4038	C₈	1	3	yes	—
27	C₄×D₂₀	C₂×C₄	1	3	yes	—
28	C₄²×D₅	C₂×C₄	1	3	yes	—
29	C₂×#674	C₂³	1	6	yes	—
30	C₂×#674	C₂³	1	2	yes	—
31	D₅⋊C₁₆	—	2	12	no	—
32	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
34	#3952	C₂×C₄	1	6	yes	—
35	C₈×Dic₅	C₂×C₄	1	3	yes	—
36	C₈×Dic₅	C₂³	1	1	yes	characteristic
37	C₄⋊#168	C₂×C₄	1	3	yes	—
38	C₄×#168	C₂³	1	3	yes	—
39	C₈×#41	—	5	1	no	—
40	Q₈×C₁₆	—	5	2	no	—
41	D₄×C₁₆	—	5	6	no	—
42	#2162	—	5	3	no	—
43	C₂×C₄×C₁₆	—	5	3	no	—
44	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
45	#713	C₁₆	1	8	yes	complemented
46	C₂×D₂₀	C₂×C₈	1	3	yes	—
47	C₂×C₄×D₅	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₅	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₅×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₄²×C₅	C₂²×C₄	1	3	yes	—
54	#685	C₂×C₈	1	6	yes	—
55	C₄⋊Dic₅	C₂×C₈	1	3	yes	—
56	C₄×Dic₅	C₂×C₈	1	3	yes	—
57	C₄×Dic₅	C₂×C₈	1	3	yes	—
58	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×#168	C₂²×C₄	1	3	yes	—
60	C₂×#168	C₂⁴	1	1	yes	characteristic
61	C₈×D₅	—	2	6	no	—
62	C₈×D₅	—	2	6	no	—
63	#674	—	2	6	no	—
64	#674	#41	1	4	yes	complemented
65	C₄×#41	—	5	1	no	—
66	C₂²×C₁₆	—	5	6	no	—
67	C₈×Q₈	—	5	1	no	—
68	C₈×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₈×D₄	—	5	3	no	—
70	C₈×D₄	—	5	3	no	—
71	#431	—	5	3	no	—
72	C₂×C₄×C₈	—	5	3	no	—
73	C₄⋊C₁₆	—	5	6	no	—
74	C₂²⋊C₁₆	—	5	6	no	—
75	C₄×C₁₆	—	5	1	no	—
76	C₄×C₁₆	—	5	6	no	—
77	C₄×C₁₆	—	5	3	no	—
78	C₂²×D₅	C₂²×C₈	1	3	yes	—
79	C₅×Q₈	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
82	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₅	C₂²×C₈	1	3	yes	—
84	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
86	D₂₀	C₂×C₁₆	1	12	yes	complemented
87	C₄×D₅	—	2	3	no	—
88	C₄×D₅	—	2	3	no	—
89	C₄×D₅	—	2	6	no	—
90	C₄×D₅	C₂×C₁₆	1	12	yes	complemented
91	C₅×C₈	—	2	3	no	—
92	C₅×C₈	#131	1	2	yes	—
93	#168	—	2	3	no	—
94	#168	C₂×#41	1	2	yes	—
95	C₂×#41	—	5	1	no	—
96	C₂²×C₈	—	5	3	no	—
97	C₂²×C₈	—	5	3	no	—
98	C₄×Q₈	—	5	1	no	—
99	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₄×D₄	—	5	3	no	—
101	C₄×D₄	—	5	3	no	—
102	#117	—	5	3	no	—
103	C₂×C₄²	—	5	3	no	—
104	C₂×C₁₆	—	5	2	no	—
105	C₂×C₁₆	—	5	6	no	—
106	C₂×C₁₆	—	10	12	no	—
107	C₄⋊C₈	—	5	3	no	—
108	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
109	C₂²⋊C₈	—	5	6	no	—
110	C₄×C₈	—	5	3	no	—
111	C₄×C₈	—	5	1	no	—
112	C₄×C₈	—	5	3	no	—
113	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
114	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
115	C₂×D₅	—	2	3	no	—
116	C₂×D₅	—	2	3	no	—
117	C₂×D₅	C₂²×C₁₆	1	6	yes	—
118	C₄×C₅	—	2	3	no	—
119	C₄×C₅	C₂²×C₁₆	1	6	yes	—
120	C₄×C₅	C₄×#41	1	1	yes	characteristic
121	C₄×C₅	C₂×#131	1	1	yes	characteristic
122	Dic₅	—	2	3	no	—
123	Dic₅	C₂²×C₁₆	1	2	yes	—
124	Dic₅	#504	1	2	yes	—
125	#41	—	5	8	no	—
126	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
127	C₂×D₄	—	5	3	no	—
128	C₂²×C₄	—	5	3	no	—
129	C₂²×C₄	—	5	3	no	—
130	C₂²×C₄	—	5	3	no	—
131	C₂×C₈	—	5	1	no	—
132	C₂×C₈	—	5	3	no	—
133	C₂×C₈	—	10	6	no	—
134	C₂×C₈	—	10	6	no	—
135	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
136	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₄⋊C₄	—	5	3	no	—
138	C₄⋊C₄	D₅⋊C₈	1	3	yes	—
139	C₂²⋊C₄	—	5	6	no	—
140	C₄²	—	5	3	no	—
141	C₄²	—	5	3	no	—
142	C₄²	—	5	1	no	—
143	C₄²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
144	C₁₆	—	5	4	no	—
145	C₁₆	—	10	6	no	—
146	C₂×C₅	C₈×#41	1	1	yes	characteristic
147	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
148	C₂×C₅	C₂×#504	1	1	yes	characteristic
149	D₅	—	2	6	no	—
150	C₂³	—	5	3	no	—
151	Q₈	D₅⋊C₁₆	1	4	yes	complemented
152	D₄	—	5	12	no	—
153	C₂×C₄	—	5	1	no	—
154	C₂×C₄	—	5	1	no	—
155	C₂×C₄	—	5	12	no	—
156	C₂×C₄	—	5	3	no	—
157	C₂×C₄	—	10	6	no	—
158	C₂×C₄	—	10	3	no	—
159	C₂×C₄	—	10	3	no	—
160	C₂×C₄	C₂×D₅⋊C₈	1	3	yes	—
161	C₂×C₄	C₂×D₅⋊C₈	1	3	yes	—
162	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₈	—	2	3	no	—
164	C₈	—	5	2	no	—
165	C₈	—	10	3	no	—
166	C₈	#4133	1	2	yes	—
167	C₅	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	5	6	no	—
169	C₂²	—	10	3	no	—
170	C₂²	—	10	3	no	—
171	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	5	2	no	—
174	C₄	—	5	2	no	—
175	C₄	—	10	3	no	—
176	C₄	C₂×D₅⋊C₁₆	1	6	yes	—
177	C₄	C₂×#4133	1	1	yes	characteristic
178	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
179	C₂	—	10	6	no	—
180	C₂	#10617195	1	1	yes	characteristic
181	C₂	C₂²×D₅⋊C₁₆	1	1	yes	characteristic
182	C₂	D₅⋊#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×C₈×Q₈	C₄
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×#713	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₈×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₁₆×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#674	C₂	semidirect
3. #10604021	C₂	semidirect
4. #10603829	C₂	semidirect
5. C₄×D₅⋊C₁₆	C₂	semidirect
45. #713	C₁₆	semidirect
64. #674	#41	semidirect
79. C₅×Q₈	C₂×C₁₆	semidirect
86. D₂₀	C₂×C₁₆	semidirect
90. C₄×D₅	C₂×C₁₆	semidirect
151. Q₈	D₅⋊C₁₆	semidirect
167. C₅	C₁₆×#41	semidirect

Maximal subgroups

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

subgroup	index
C₈×#713	2
Q₈×#674	2
#10604021	2
#10603829	2
C₄×D₅⋊C₁₆	2
C₁₆×#41	5