Subgroups — C₅×#33329 #11956049

Order 1280.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3407	C₂	1	2	yes	maximal
2	C₅×#2875	C₂	1	1	yes	maximal, characteristic
3	C₅×#2110	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×#347	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#131	C₄	1	1	yes	characteristic
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₄×C₅×#41	C₄	1	1	yes	characteristic
8	C₅×#504	C₄	1	16	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	12	yes	—
10	C₅×#405	C₄	1	3	yes	—
11	C₅×#405	C₂²	1	3	yes	—
12	C₄×C₅×M₁₆	C₄	1	3	yes	—
13	C₂×C₅×#97	C₂²	1	3	yes	—
14	C₅×#347	C₂²	1	4	yes	—
15	C₅×#347	C₂²	1	12	yes	—
16	#33329	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
18	C₅×#131	—	2	4	no	—
19	C₅×#131	C₂×C₄	1	8	yes	—
20	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
21	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₅×C₈	C₂³	1	3	yes	—
24	C₄×C₅×Q₈	—	2	1	no	—
25	C₄×C₅×D₄	—	2	3	no	—
26	C₅×#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
28	C₅×M₃₂	C₂×C₄	1	24	yes	complemented
29	C₅×M₃₂	C₂³	1	8	yes	—
30	C₂×C₅×C₁₆	C₂×C₄	1	24	yes	complemented
31	C₅×#97	C₂×C₄	1	3	yes	—
32	C₅×#97	C₂×C₄	1	3	yes	—
33	C₅×#97	C₂³	1	1	yes	characteristic
34	C₅×#97	C₂³	1	3	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	6	yes	—
36	#3407	C₂×C₅	1	2	yes	—
37	#2875	C₂×C₅	1	1	yes	characteristic
38	#2110	C₂×C₅	1	6	yes	complemented
39	C₂×#347	C₂×C₅	1	6	yes	complemented
40	C₅×#41	C₄²	1	4	yes	—
41	C₅×#41	M₁₆	1	4	yes	—
42	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
43	C₂×C₅×D₄	M₁₆	1	3	yes	—
44	C₂²×C₄×C₅	—	2	3	no	—
45	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
46	C₅×M₁₆	—	2	6	no	—
47	C₅×M₁₆	C₄²	1	12	yes	—
48	C₂×C₅×C₈	—	2	6	no	—
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	12	yes	—
53	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
54	C₅×C₄⋊C₄	—	2	3	no	—
55	C₅×C₂²⋊C₄	—	2	3	no	—
56	C₄²×C₅	—	2	3	no	—
57	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅	C₂²×C₄	1	3	yes	—
59	C₅×C₁₆	C₂²×C₄	1	16	yes	—
60	C₂×#131	C₄×C₅	1	1	yes	characteristic
61	C₂×#131	C₂²×C₅	1	1	yes	characteristic
62	C₄×#41	C₄×C₅	1	1	yes	characteristic
63	#504	C₄×C₅	1	16	yes	complemented
64	C₂×M₃₂	C₂²×C₅	1	12	yes	—
65	#405	C₄×C₅	1	3	yes	—
66	#405	C₂²×C₅	1	3	yes	—
67	C₄×M₁₆	C₄×C₅	1	3	yes	—
68	C₂×#97	C₂²×C₅	1	3	yes	—
69	#347	C₂²×C₅	1	4	yes	—
70	#347	C₂²×C₅	1	12	yes	—
71	C₂³×C₅	C₂×M₁₆	1	3	yes	—
72	C₅×Q₈	#97	1	4	yes	—
73	C₅×D₄	#97	1	12	yes	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	3	no	—
76	C₂×C₄×C₅	—	8	3	no	—
77	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
79	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
80	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
81	C₅×C₈	—	2	1	no	—
82	C₅×C₈	—	2	3	no	—
83	C₅×C₈	C₂×C₄²	1	6	yes	—
84	C₅×C₈	C₂³×C₄	1	2	yes	—
85	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
86	#131	—	2	4	no	—
87	#131	C₂×C₄×C₅	1	8	yes	—
88	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
89	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
90	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
91	C₂²×C₈	C₂³×C₅	1	3	yes	—
92	C₄×Q₈	—	2	1	no	—
93	C₄×D₄	—	2	3	no	—
94	#117	C₂×C₄×C₅	1	3	yes	—
95	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
96	M₃₂	C₂×C₄×C₅	1	24	yes	complemented
97	M₃₂	C₂³×C₅	1	8	yes	—
98	C₂×C₁₆	C₂×C₄×C₅	1	24	yes	complemented
99	#97	C₂×C₄×C₅	1	3	yes	—
100	#97	C₂×C₄×C₅	1	3	yes	—
101	#97	C₂³×C₅	1	1	yes	characteristic
102	#97	C₂³×C₅	1	3	yes	—
103	C₄×C₈	C₂×C₄×C₅	1	6	yes	—
104	C₂²×C₅	—	4	3	no	—
105	C₂²×C₅	C₂×#97	1	6	yes	—
106	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
107	C₄×C₅	—	4	3	no	—
108	C₄×C₅	—	4	1	no	—
109	C₄×C₅	C₂×#97	1	6	yes	—
110	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
111	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
112	#41	C₄²×C₅	1	4	yes	—
113	#41	C₅×M₁₆	1	4	yes	—
114	C₂×Q₈	C₅×M₁₆	1	1	yes	characteristic
115	C₂×D₄	C₅×M₁₆	1	3	yes	—
116	C₂²×C₄	—	2	3	no	—
117	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
118	M₁₆	—	2	6	no	—
119	M₁₆	C₄²×C₅	1	12	yes	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
122	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
124	C₂×C₈	C₂²×C₄×C₅	1	12	yes	—
125	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
126	C₄⋊C₄	—	2	3	no	—
127	C₂²⋊C₄	—	2	3	no	—
128	C₄²	—	2	3	no	—
129	C₄²	C₂²×C₄×C₅	1	3	yes	—
130	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₁₆	C₂²×C₄×C₅	1	16	yes	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	6	no	—
134	C₂×C₅	C₂²×#97	1	1	yes	characteristic
135	C₂³	C₂×C₅×M₁₆	1	3	yes	—
136	Q₈	C₅×#97	1	4	yes	—
137	D₄	C₅×#97	1	12	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	8	3	no	—
141	C₂×C₄	C₂×C₄²×C₅	1	6	yes	—
142	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
143	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
144	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	3	no	—
147	C₈	C₂×C₄²×C₅	1	6	yes	—
148	C₈	C₂³×C₄×C₅	1	2	yes	—
149	C₅	#33329	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	—	4	3	no	—
151	C₂²	C₂×C₅×#97	1	6	yes	—
152	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×C₅×#97	1	6	yes	—
156	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
157	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×C₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂²×#97
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2110	C₂	semidirect
4. C₂×C₅×#347	C₂	semidirect
8. C₅×#504	C₄	semidirect
16. #33329	C₅	direct (N × H)
28. C₅×M₃₂	C₂×C₄	semidirect
30. C₂×C₅×C₁₆	C₂×C₄	semidirect
38. #2110	C₂×C₅	semidirect
39. C₂×#347	C₂×C₅	semidirect
63. #504	C₄×C₅	semidirect
96. M₃₂	C₂×C₄×C₅	semidirect
98. C₂×C₁₆	C₂×C₄×C₅	semidirect
149. C₅	#33329	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×#3407	2
C₅×#2875	2
C₅×#2110	2
C₂×C₅×#347	2
#33329	5

Subgroups — C5×#33329 #11956049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33329 #11956049