Subgroups — #11993064

Order 1280.

382 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	C₅×#1744	C₂	1	1	yes	maximal, characteristic
2	C₂×#64678	C₂	1	2	yes	maximal
3	C₂×#64677	C₂	1	2	yes	maximal
4	#10600187	C₂	1	1	yes	maximal, characteristic
5	#10600185	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#168	—	2	2	no	—
9	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#168	C₂²	1	2	yes	—
11	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
12	C₅×#336	C₄	1	2	yes	—
13	C₅×#336	C₄	1	2	yes	—
14	#64744	—	2	2	no	—
15	#64678	—	2	4	no	—
16	#64677	—	2	4	no	—
17	#8812	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	2	yes	—
22	C₂×C₄²×C₅	Q₈	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂²×#168	—	2	2	no	—
25	C₂²×#168	—	2	1	no	—
26	C₂²×#168	—	2	1	no	—
27	C₄×C₅×C₈	D₄	1	4	yes	—
28	C₄×C₅×C₈	Q₈	1	4	yes	—
29	C₄⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄⋊#168	—	4	4	no	—
34	#1842	—	5	1	no	—
35	#1831	—	5	1	no	—
36	#1744	D₅	1	1	yes	characteristic
37	C₂×#335	—	5	2	no	—
38	C₂×#334	—	5	2	no	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₅×C₈	—	2	4	no	—
45	C₂×C₅×C₈	—	2	4	no	—
46	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
47	C₄²×C₅	—	2	4	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	—	2	2	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	C₂×D₄	1	2	yes	—
53	C₄²×C₅	C₂×Q₈	1	2	yes	—
54	C₂×#168	—	4	2	no	—
55	C₂×#168	—	4	1	no	—
56	C₂×#168	—	4	1	no	—
57	C₂×#168	—	4	4	no	—
58	C₂×#168	—	4	2	no	—
59	C₂×#168	—	4	2	no	—
60	C₂×C₄⋊C₈	—	5	1	no	—
61	C₂×C₄⋊C₈	—	5	2	no	—
62	C₂×C₄⋊C₈	—	5	1	no	—
63	C₂×C₄⋊C₈	—	10	2	no	—
64	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
65	C₄³	C₂×D₅	1	1	yes	characteristic
66	#336	—	10	2	no	—
67	#336	Dic₅	1	2	yes	—
68	#336	Dic₅	1	2	yes	—
69	#335	—	10	4	no	—
70	#334	—	10	4	no	—
71	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	#108	1	4	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅	#127	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	#131	1	2	yes	—
84	C₂×C₄×C₅	#131	1	2	yes	—
85	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₅	#137	1	2	yes	—
87	C₅×C₈	—	2	8	no	—
88	#168	—	8	2	no	—
89	#168	—	8	1	no	—
90	#168	—	8	1	no	—
91	C₂²×C₈	—	10	1	no	—
92	C₂²×C₈	—	10	1	no	—
93	C₂²×C₈	—	10	2	no	—
94	C₂²×C₈	C₂×Dic₅	1	4	yes	—
95	C₂×C₄²	Dic₁₀	1	2	yes	—
96	C₂×C₄²	D₂₀	1	2	yes	—
97	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
99	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
100	C₄⋊C₈	—	10	2	no	—
101	C₄⋊C₈	—	10	2	no	—
102	C₄⋊C₈	—	10	2	no	—
103	C₄⋊C₈	—	10	2	no	—
104	C₄⋊C₈	—	20	4	no	—
105	C₄×C₈	Dic₁₀	1	4	yes	—
106	C₄×C₈	D₂₀	1	4	yes	—
107	C₂²×C₅	#384	1	1	yes	characteristic
108	C₂²×C₅	#424	1	1	yes	characteristic
109	C₂²×C₅	#424	1	1	yes	characteristic
110	C₂²×C₅	#428	1	2	yes	—
111	C₂²×C₅	C₂×#108	1	2	yes	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	4	2	no	—
115	C₄×C₅	#424	1	4	yes	—
116	C₄×C₅	#496	1	1	yes	characteristic
117	C₄×C₅	#497	1	1	yes	characteristic
118	C₄×C₅	#501	1	2	yes	—
119	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
120	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
122	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	20	2	no	—
127	C₂×C₈	—	20	1	no	—
128	C₂×C₈	—	20	1	no	—
129	C₂×C₈	—	20	4	no	—
130	C₂×C₈	—	20	2	no	—
131	C₂×C₈	—	20	2	no	—
132	C₂×C₈	C₄⋊Dic₅	1	8	yes	—
133	C₄²	—	2	4	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	C₂×Dic₁₀	1	2	yes	—
139	C₄²	C₂×D₂₀	1	2	yes	—
140	C₂×C₅	#1573	1	2	yes	—
141	C₂×C₅	#1574	1	2	yes	—
142	C₂×C₅	#1843	1	1	yes	characteristic
143	C₂×C₅	#1847	1	1	yes	characteristic
144	C₂×C₅	#1848	1	1	yes	characteristic
145	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	#3951	1	4	yes	—
153	C₂×C₄	#4015	1	2	yes	—
154	C₂×C₄	#4015	1	1	yes	characteristic
155	C₂×C₄	#4020	1	1	yes	characteristic
156	C₂×C₄	#4054	1	2	yes	—
157	C₂×C₄	#4055	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
159	C₂×C₄	#4094	1	2	yes	—
160	C₂×C₄	#4094	1	2	yes	—
161	C₈	—	2	8	no	—
162	C₈	—	40	2	no	—
163	C₈	—	40	1	no	—
164	C₈	—	40	1	no	—
165	C₅	#8812	1	1	yes	characteristic, complemented, Sylow
166	C₂²	#65224	1	1	yes	characteristic
167	C₂²	#65288	1	1	yes	characteristic
168	C₂²	#65288	1	1	yes	characteristic
169	C₂²	C₂×#3951	1	2	yes	—
170	C₂²	#65410	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#64991	1	2	yes	—
175	C₄	#65001	1	1	yes	characteristic
176	C₄	#65004	1	1	yes	characteristic
177	C₄	#65288	1	4	yes	—
178	C₂	#10600232	1	1	yes	characteristic
179	C₂	#10600585	1	1	yes	characteristic
180	C₂	#10600609	1	1	yes	characteristic
181	C₂	#10601545	1	2	yes	—
182	C₂	#10601546	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1744	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1744	D₅
hypocenter	C₅	#8812
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
165. C₅	#8812	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1744	2
C₂×#64678	2
C₂×#64677	2
#10600187	2
#10600185	2
#8812	5