Subgroups — C₅×#12556 #11935276

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	2	yes	maximal
3	C₅×#1380	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₅×#117	—	2	2	no	—
6	C₂²×C₅×C₁₆	—	2	4	no	—
7	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₅×#427	—	2	4	no	—
10	C₅×#427	C₄	1	8	yes	complemented
11	C₂×C₅×#107	C₂²	1	4	yes	—
12	C₂×C₅×#106	—	2	2	no	—
13	#12556	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	D₄	1	4	yes	—
19	C₂²×C₅×C₈	Q₈	1	2	yes	—
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₅×#117	—	2	8	no	—
22	C₅×#117	—	4	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	4	no	—
24	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
25	C₂×C₄²×C₅	—	4	2	no	—
26	C₂×C₅×C₁₆	—	2	8	no	—
27	C₂×C₅×C₁₆	—	2	4	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
29	C₅×#107	—	2	4	no	—
30	C₅×#107	C₂×C₄	1	8	yes	—
31	C₅×#106	—	2	8	no	—
32	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
33	C₂×#427	C₂×C₅	1	2	yes	—
34	#1380	C₂×C₅	1	4	yes	complemented
35	C₂⁴×C₅	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₅	—	4	4	no	—
37	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄²	1	4	yes	—
44	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
45	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
46	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₅×C₄⋊C₄	—	2	8	no	—
53	C₅×C₄⋊C₄	—	4	4	no	—
54	C₅×C₂²⋊C₄	—	4	8	no	—
55	C₄²×C₅	—	4	8	no	—
56	C₅×C₁₆	—	2	4	no	—
57	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
58	C₂×#117	—	2	2	no	—
59	C₂²×C₁₆	—	2	4	no	—
60	C₂²×C₁₆	C₄×C₅	1	4	yes	complemented
61	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
62	#427	—	2	4	no	—
63	#427	C₄×C₅	1	8	yes	complemented
64	C₂×#107	C₂²×C₅	1	4	yes	—
65	C₂×#106	—	2	2	no	—
66	C₂³×C₅	—	2	4	no	—
67	C₂³×C₅	#102	1	4	yes	—
68	C₂³×C₅	#107	1	2	yes	—
69	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅	—	4	8	no	—
71	C₂×C₄×C₅	—	8	8	no	—
72	C₂×C₄×C₅	#95	1	4	yes	—
73	C₂×C₄×C₅	#102	1	4	yes	—
74	C₂×C₄×C₅	#103	1	8	yes	—
75	C₂×C₄×C₅	#106	1	4	yes	—
76	C₂×C₄×C₅	#107	1	2	yes	—
77	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
81	C₅×C₈	#95	1	4	yes	—
82	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
83	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
87	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
88	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₅×D₄	1	4	yes	—
90	C₂²×C₈	C₅×Q₈	1	2	yes	—
91	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
92	#117	—	2	8	no	—
93	#117	—	4	4	no	—
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₂²⋊C₄	—	4	2	no	—
96	C₂×C₄²	—	4	2	no	—
97	C₂×C₁₆	—	2	8	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
100	#107	—	2	4	no	—
101	#107	C₂×C₄×C₅	1	8	yes	—
102	#106	—	2	8	no	—
103	C₂²×C₅	—	2	8	no	—
104	C₂²×C₅	—	2	4	no	—
105	C₂²×C₅	#340	1	4	yes	—
106	C₂²×C₅	C₂×#102	1	2	yes	—
107	C₂²×C₅	C₂×#107	1	1	yes	characteristic
108	C₂²×C₅	#508	1	4	yes	—
109	C₄×C₅	—	8	8	no	—
110	C₄×C₅	#340	1	4	yes	—
111	C₄×C₅	C₂×#95	1	1	yes	characteristic
112	C₄×C₅	C₂×#103	1	2	yes	—
113	C₄×C₅	C₂×#106	1	1	yes	characteristic
114	C₂⁴	C₅×D₈	1	1	yes	characteristic
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
117	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
118	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
119	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
120	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
121	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
122	C₂×C₈	C₄²×C₅	1	4	yes	—
123	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
124	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
125	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
126	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
127	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
129	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
130	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
131	C₄⋊C₄	—	2	8	no	—
132	C₄⋊C₄	—	4	4	no	—
133	C₂²⋊C₄	—	4	8	no	—
134	C₄²	—	4	8	no	—
135	C₁₆	—	2	4	no	—
136	C₂×C₅	—	2	4	no	—
137	C₂×C₅	C₂×#340	1	1	yes	characteristic
138	C₂×C₅	#2138	1	4	yes	—
139	C₂×C₅	#2158	1	2	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	C₅×#102	1	4	yes	—
142	C₂³	C₅×#107	1	2	yes	—
143	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
144	C₂×C₄	—	4	8	no	—
145	C₂×C₄	—	8	8	no	—
146	C₂×C₄	C₅×#95	1	4	yes	—
147	C₂×C₄	C₅×#102	1	4	yes	—
148	C₂×C₄	C₅×#103	1	8	yes	—
149	C₂×C₄	C₅×#106	1	4	yes	—
150	C₂×C₄	C₅×#107	1	2	yes	—
151	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
154	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
155	C₈	C₅×#95	1	4	yes	—
156	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
157	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
158	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
159	C₅	#12556	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²	—	2	8	no	—
161	C₂²	—	2	4	no	—
162	C₂²	C₅×#340	1	4	yes	—
163	C₂²	C₂×C₅×#102	1	2	yes	—
164	C₂²	C₂×C₅×#107	1	1	yes	characteristic
165	C₂²	C₅×#508	1	4	yes	—
166	C₄	—	8	8	no	—
167	C₄	C₅×#340	1	4	yes	—
168	C₄	C₂×C₅×#95	1	1	yes	characteristic
169	C₄	C₂×C₅×#103	1	2	yes	—
170	C₄	C₂×C₅×#106	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×C₅×#340	1	1	yes	characteristic
173	C₂	C₅×#2138	1	4	yes	—
174	C₂	C₅×#2158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂×D₈
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1380	C₂	semidirect
7. C₂²×C₅×C₁₆	C₄	semidirect
10. C₅×#427	C₄	semidirect
13. #12556	C₅	direct (N × H)
34. #1380	C₂×C₅	semidirect
60. C₂²×C₁₆	C₄×C₅	semidirect
63. #427	C₄×C₅	semidirect
159. C₅	#12556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂×C₅×#427	2
C₅×#1380	2
#12556	5

Subgroups — C5×#12556 #11935276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12556 #11935276