Subgroups — C₅×#22259 #11944979

Order 1280.

594 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅×#2961	C₂	1	2	yes	maximal
3	C₅×#1994	C₂	1	2	yes	maximal
4	C₅×#1922	C₂	1	8	yes	maximal
5	C₅×#1838	C₂	1	1	yes	maximal, characteristic
6	C₅×#1745	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₂²	1	2	yes	—
8	C₅×#501	C₄	1	8	yes	complemented
9	C₅×#446	—	2	8	no	—
10	C₅×#427	C₂²	1	4	yes	—
11	C₅×#424	C₂²	1	4	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₅×#405	C₂²	1	4	yes	—
14	C₂×C₅×#97	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	C₂²	1	1	yes	characteristic
16	C₄³×C₅	C₂²	1	1	yes	characteristic
17	C₅×#341	C₂²	1	4	yes	—
18	C₅×#339	C₂²	1	8	yes	—
19	C₅×#336	C₂²	1	4	yes	—
20	#22259	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×M₁₆	C₂³	1	4	yes	—
22	C₂²×C₅×C₈	C₂³	1	2	yes	—
23	C₂²×C₅×C₈	C₂³	1	2	yes	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₅×#125	C₂×C₄	1	4	yes	—
26	C₅×C₂²⋊Q₈	—	2	4	no	—
27	C₄×C₅×Q₈	—	2	4	no	—
28	C₅×#117	C₂³	1	4	yes	—
29	C₂×C₄²×C₅	—	2	2	no	—
30	C₂×C₄²×C₅	C₂³	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×#107	C₂×C₄	1	8	yes	—
33	C₅×#106	C₂×C₄	1	8	yes	—
34	C₅×C₄⋊C₈	—	2	8	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×#97	—	2	4	no	—
37	C₅×#97	C₂×C₄	1	4	yes	—
38	C₅×#97	D₄	1	4	yes	—
39	C₄×C₅×C₈	—	2	4	no	—
40	#3162	C₂×C₅	1	1	yes	characteristic
41	#2961	C₂×C₅	1	2	yes	—
42	#1994	C₂×C₅	1	2	yes	—
43	#1922	C₂×C₅	1	8	yes	—
44	#1838	C₂×C₅	1	1	yes	characteristic
45	#1745	C₂×C₅	1	1	yes	characteristic
46	C₂×C₅×Q₈	—	2	4	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	#41	1	2	yes	—
49	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
50	C₅×M₁₆	—	2	8	no	—
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
56	C₅×C₄⋊C₄	—	2	8	no	—
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
59	C₅×C₂²⋊C₄	—	2	4	no	—
60	C₄²×C₅	—	2	4	no	—
61	C₄²×C₅	—	2	4	no	—
62	C₄²×C₅	—	4	4	no	—
63	C₄²×C₅	C₂²×C₄	1	2	yes	—
64	C₄²×C₅	C₂×D₄	1	2	yes	—
65	#533	C₂²×C₅	1	2	yes	—
66	#501	C₄×C₅	1	8	yes	complemented
67	#446	—	2	8	no	—
68	#427	C₂²×C₅	1	4	yes	—
69	#424	C₂²×C₅	1	4	yes	—
70	#423	C₂²×C₅	1	2	yes	—
71	#405	C₂²×C₅	1	4	yes	—
72	C₂×#97	C₂²×C₅	1	1	yes	characteristic
73	C₂×#97	C₂²×C₅	1	1	yes	characteristic
74	C₄³	C₂²×C₅	1	1	yes	characteristic
75	#341	C₂²×C₅	1	4	yes	—
76	#339	C₂²×C₅	1	8	yes	—
77	#336	C₂²×C₅	1	4	yes	—
78	C₂³×C₅	C₂×#41	1	1	yes	characteristic
79	C₅×Q₈	—	8	4	no	—
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	8	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
89	C₅×C₈	—	2	4	no	—
90	C₅×C₈	—	2	4	no	—
91	C₅×C₈	—	4	4	no	—
92	C₂×M₁₆	C₂³×C₅	1	4	yes	—
93	C₂²×C₈	C₂³×C₅	1	2	yes	—
94	C₂²×C₈	C₂³×C₅	1	2	yes	—
95	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
96	#125	C₂×C₄×C₅	1	4	yes	—
97	C₂²⋊Q₈	—	2	4	no	—
98	C₄×Q₈	—	2	4	no	—
99	#117	C₂³×C₅	1	4	yes	—
100	C₂×C₄²	—	2	2	no	—
101	C₂×C₄²	C₂³×C₅	1	2	yes	—
102	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
103	#107	C₂×C₄×C₅	1	8	yes	—
104	#106	C₂×C₄×C₅	1	8	yes	—
105	C₄⋊C₈	—	2	8	no	—
106	C₄⋊C₈	—	2	4	no	—
107	#97	—	2	4	no	—
108	#97	C₂×C₄×C₅	1	4	yes	—
109	#97	C₅×D₄	1	4	yes	—
110	C₄×C₈	—	2	4	no	—
111	C₂²×C₅	—	2	2	no	—
112	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₅	#541	1	2	yes	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	4	4	no	—
116	C₄×C₅	—	4	8	no	—
117	C₄×C₅	#519	1	2	yes	—
118	C₄×C₅	#536	1	2	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₅×#41	1	2	yes	—
122	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
123	M₁₆	—	2	8	no	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
128	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
129	C₄⋊C₄	—	2	8	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	4	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	4	4	no	—
136	C₄²	C₂²×C₄×C₅	1	2	yes	—
137	C₄²	C₂×C₅×D₄	1	2	yes	—
138	C₂×C₅	—	2	2	no	—
139	C₂×C₅	#2327	1	1	yes	characteristic
140	C₂×C₅	#2956	1	2	yes	—
141	C₂³	C₂×C₅×#41	1	1	yes	characteristic
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
149	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
151	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	2	4	no	—
154	C₈	—	4	4	no	—
155	C₅	#22259	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	—	2	2	no	—
157	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
158	C₂²	C₅×#541	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	8	no	—
162	C₄	C₅×#519	1	2	yes	—
163	C₄	C₅×#536	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₅×#2327	1	1	yes	characteristic
166	C₂	C₅×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#501	C₄	semidirect
20. #22259	C₅	direct (N × H)
66. #501	C₄×C₅	semidirect
155. C₅	#22259	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×#3162	2
C₅×#2961	2
C₅×#1994	2
C₅×#1922	2
C₅×#1838	2
C₅×#1745	2
#22259	5

Subgroups — C5×#22259 #11944979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22259 #11944979