Subgroups — C₅×#8824 #11931544

Order 1280.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1837	C₂	1	2	yes	maximal
2	C₅×#1567	C₂	1	2	yes	maximal
3	C₅×#1566	C₂	1	2	yes	maximal
4	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₅×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#341	—	2	4	no	—
10	C₅×#335	—	2	4	no	—
11	C₅×#334	—	2	4	no	—
12	C₅×C₈²	C₄	1	1	yes	characteristic
13	C₅×C₈²	C₄	1	1	yes	characteristic
14	C₅×C₈²	C₄	1	1	yes	characteristic
15	C₅×C₈²	C₄	1	1	yes	characteristic
16	#8824	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×M₁₆	—	2	4	no	—
18	C₂²×C₅×C₈	D₄	1	2	yes	—
19	C₂²×C₅×C₈	Q₈	1	2	yes	—
20	C₂²×C₅×C₈	Q₈	1	2	yes	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₅×C₄⋊C₈	—	2	4	no	—
23	C₅×C₄⋊C₈	—	2	4	no	—
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×C₈	D₄	1	2	yes	—
31	C₄×C₅×C₈	D₄	1	2	yes	—
32	C₄×C₅×C₈	Q₈	1	2	yes	—
33	#1837	C₂×C₅	1	2	yes	—
34	#1567	C₂×C₅	1	2	yes	—
35	#1566	C₂×C₅	1	2	yes	—
36	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
40	C₅×M₁₆	—	4	8	no	—
41	C₂×C₅×C₈	—	2	4	no	—
42	C₂×C₅×C₈	—	2	4	no	—
43	C₂×C₅×C₈	—	2	4	no	—
44	C₂×C₅×C₈	—	4	4	no	—
45	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
51	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
55	#423	C₂²×C₅	1	4	yes	—
56	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
59	#341	—	2	4	no	—
60	#335	—	2	4	no	—
61	#334	—	2	4	no	—
62	C₈²	C₄×C₅	1	1	yes	characteristic
63	C₈²	C₄×C₅	1	1	yes	characteristic
64	C₈²	C₄×C₅	1	1	yes	characteristic
65	C₈²	C₄×C₅	1	1	yes	characteristic
66	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	#127	1	1	yes	characteristic
73	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅	#135	1	4	yes	—
76	C₅×C₈	—	2	2	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	—	2	2	no	—
80	C₅×C₈	—	8	4	no	—
81	C₅×C₈	#108	1	4	yes	—
82	C₅×C₈	#108	1	4	yes	—
83	C₂×M₁₆	—	2	4	no	—
84	C₂²×C₈	C₅×D₄	1	2	yes	—
85	C₂²×C₈	C₅×Q₈	1	2	yes	—
86	C₂²×C₈	C₅×Q₈	1	2	yes	—
87	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	2	4	no	—
90	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
91	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
92	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
96	C₄×C₈	C₅×D₄	1	2	yes	—
97	C₄×C₈	C₅×D₄	1	2	yes	—
98	C₄×C₈	C₅×Q₈	1	2	yes	—
99	C₂²×C₅	—	2	1	no	—
100	C₂²×C₅	—	2	1	no	—
101	C₂²×C₅	#384	1	1	yes	characteristic
102	C₂²×C₅	#499	1	2	yes	—
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	#427	1	2	yes	—
106	C₄×C₅	C₂×#108	1	2	yes	—
107	C₄×C₅	C₂×#108	1	2	yes	—
108	C₄×C₅	#495	1	2	yes	—
109	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
112	M₁₆	—	4	8	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	4	4	no	—
117	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
118	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
119	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
120	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
121	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
122	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
123	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
124	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
125	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
126	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
127	C₂×C₅	—	2	2	no	—
128	C₂×C₅	#1846	1	1	yes	characteristic
129	C₂×C₅	#1848	1	1	yes	characteristic
130	C₂×C₅	#1848	1	1	yes	characteristic
131	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₅×#127	1	1	yes	characteristic
138	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₄	C₅×#135	1	4	yes	—
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	8	4	no	—
146	C₈	C₅×#108	1	4	yes	—
147	C₈	C₅×#108	1	4	yes	—
148	C₅	#8824	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	C₅×#384	1	1	yes	characteristic
152	C₂²	C₅×#499	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	C₅×#427	1	2	yes	—
156	C₄	C₂×C₅×#108	1	2	yes	—
157	C₄	C₂×C₅×#108	1	2	yes	—
158	C₄	C₅×#495	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	C₅×#1846	1	1	yes	characteristic
161	C₂	C₅×#1848	1	1	yes	characteristic
162	C₂	C₅×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#384
prime radical	C₂³×C₅	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
16. #8824	C₅	direct (N × H)
148. C₅	#8824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1837	2
C₅×#1567	2
C₅×#1566	2
C₂×C₅×C₈²	2
#8824	5

Subgroups — C5×#8824 #11931544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8824 #11931544