Subgroups — C₅×#12022 #11934742

Order 1280.

242 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#1567	C₂	1	2	yes	maximal
3	C₅×#1371	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×C₁₆	C₄	1	2	yes	—
5	C₅×#423	—	2	2	no	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₅×C₁₆	C₂²	1	2	yes	—
10	C₅×#335	—	2	2	no	—
11	C₅×#335	C₂²	1	4	yes	—
12	#12022	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₅×M₁₆	—	4	2	no	—
14	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
17	C₂×C₅×C₁₆	—	2	4	no	—
18	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
20	C₅×C₄⋊C₈	—	2	4	no	—
21	C₅×C₄⋊C₈	—	4	2	no	—
22	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
27	#1567	C₂×C₅	1	2	yes	—
28	#1371	C₂×C₅	1	4	yes	complemented
29	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
32	C₅×M₁₆	—	8	4	no	—
33	C₂×C₅×C₈	—	2	2	no	—
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	—	4	4	no	—
36	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₅	D₈	1	1	yes	characteristic
42	C₄²×C₅	Q₁₆	1	1	yes	characteristic
43	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₅×C₁₆	—	2	2	no	—
45	C₅×C₁₆	M₁₆	1	4	yes	complemented
46	C₂²×C₁₆	C₄×C₅	1	2	yes	—
47	#423	—	2	2	no	—
48	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
49	C₄×C₁₆	C₄×C₅	1	1	yes	characteristic
50	C₄×C₁₆	C₄×C₅	1	1	yes	characteristic
51	C₄×C₁₆	C₂²×C₅	1	2	yes	—
52	#335	—	2	2	no	—
53	#335	C₂²×C₅	1	4	yes	—
54	C₂³×C₅	#107	1	1	yes	characteristic
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	1	no	—
57	C₂×C₄×C₅	—	2	1	no	—
58	C₂×C₄×C₅	#107	1	1	yes	characteristic
59	C₂×C₄×C₅	#107	1	1	yes	characteristic
60	C₂×C₄×C₅	#107	1	1	yes	characteristic
61	C₂×C₄×C₅	D₁₆	1	2	yes	—
62	C₂×C₄×C₅	Q₃₂	1	2	yes	—
63	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
66	C₅×C₈	—	2	1	no	—
67	C₅×C₈	—	2	1	no	—
68	C₅×C₈	—	8	4	no	—
69	C₅×C₈	#108	1	2	yes	—
70	C₅×C₈	C₂×M₁₆	1	2	yes	—
71	C₂×M₁₆	—	4	2	no	—
72	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
73	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
74	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
77	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
78	C₄⋊C₈	—	2	4	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
81	C₄×C₈	C₅×D₄	1	1	yes	characteristic
82	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
83	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
84	C₂²×C₅	—	2	1	no	—
85	C₂²×C₅	—	2	1	no	—
86	C₂²×C₅	#366	1	2	yes	—
87	C₂²×C₅	C₂×#107	1	1	yes	characteristic
88	C₄×C₅	—	2	1	no	—
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	#366	1	2	yes	—
91	C₄×C₅	#368	1	2	yes	—
92	C₄×C₅	#423	1	1	yes	characteristic
93	C₄×C₅	C₂×#108	1	1	yes	characteristic
94	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
95	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
96	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
98	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
99	M₁₆	—	8	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
106	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
107	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
108	C₄²	C₅×D₈	1	1	yes	characteristic
109	C₄²	C₅×Q₁₆	1	1	yes	characteristic
110	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	C₅×M₁₆	1	4	yes	complemented
113	C₂×C₅	—	2	2	no	—
114	C₂×C₅	#1567	1	1	yes	characteristic
115	C₂×C₅	C₂×#366	1	1	yes	characteristic
116	C₂×C₅	C₂×#368	1	1	yes	characteristic
117	C₂³	C₅×#107	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₅×#107	1	1	yes	characteristic
122	C₂×C₄	C₅×#107	1	1	yes	characteristic
123	C₂×C₄	C₅×#107	1	1	yes	characteristic
124	C₂×C₄	C₅×D₁₆	1	2	yes	—
125	C₂×C₄	C₅×Q₃₂	1	2	yes	—
126	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
128	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	2	1	no	—
131	C₈	—	8	4	no	—
132	C₈	C₅×#108	1	2	yes	—
133	C₈	C₂×C₅×M₁₆	1	2	yes	—
134	C₅	#12022	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	C₅×#366	1	2	yes	—
138	C₂²	C₂×C₅×#107	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	C₅×#366	1	2	yes	—
142	C₄	C₅×#368	1	2	yes	—
143	C₄	C₅×#423	1	1	yes	characteristic
144	C₄	C₂×C₅×#108	1	1	yes	characteristic
145	C₄	C₂×C₅×D₁₆	1	1	yes	characteristic
146	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	C₅×#1567	1	1	yes	characteristic
149	C₂	C₂×C₅×#366	1	1	yes	characteristic
150	C₂	C₂×C₅×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1371	C₂	semidirect
12. #12022	C₅	direct (N × H)
28. #1371	C₂×C₅	semidirect
45. C₅×C₁₆	M₁₆	semidirect
112. C₁₆	C₅×M₁₆	semidirect
134. C₅	#12022	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₅×#1567	2
C₅×#1371	2
#12022	5

Subgroups — C5×#12022 #11934742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12022 #11934742