Subgroups — C₅×#12596 #11935316

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#347	C₂	1	2	yes	maximal
2	C₅×#1726	C₂	1	1	yes	maximal, characteristic
3	C₅×#1377	C₂	1	2	yes	maximal
4	C₅×#1376	C₂	1	2	yes	maximal
5	C₂×C₅×M₃₂	C₂²	1	4	yes	—
6	C₂×C₅×#97	C₄	1	1	yes	characteristic
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×C₅×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#376	C₄	1	1	yes	characteristic
13	C₅×#347	C₄	1	8	yes	complemented
14	#12596	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₅×M₃₂	—	2	4	no	—
23	C₅×M₃₂	C₂×C₄	1	8	yes	—
24	C₂×C₅×C₁₆	—	2	4	no	—
25	C₅×#97	—	2	2	no	—
26	C₅×#97	—	2	2	no	—
27	C₅×#97	C₂×C₄	1	4	yes	—
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	—	2	2	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₅×#95	—	2	2	no	—
32	C₂×#347	C₂×C₅	1	2	yes	—
33	#1726	C₂×C₅	1	1	yes	characteristic
34	#1377	C₂×C₅	1	2	yes	—
35	#1376	C₂×C₅	1	2	yes	—
36	C₂²×C₄×C₅	—	2	1	no	—
37	C₂²×C₄×C₅	—	2	1	no	—
38	C₂²×C₄×C₅	M₁₆	1	2	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	—	2	1	no	—
41	C₂×C₅×C₈	—	2	1	no	—
42	C₂×C₅×C₈	—	2	1	no	—
43	C₂×C₅×C₈	—	2	1	no	—
44	C₂×C₅×C₈	—	2	1	no	—
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	C₄²	1	2	yes	—
47	C₂×C₅×C₈	M₁₆	1	2	yes	—
48	C₂×C₅×C₈	M₁₆	1	2	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₅×C₈	#41	1	2	yes	—
51	C₂×C₅×C₈	#41	1	2	yes	—
52	C₄²×C₅	—	2	1	no	—
53	C₄²×C₅	—	2	1	no	—
54	C₄²×C₅	—	2	1	no	—
55	C₄²×C₅	—	2	1	no	—
56	C₄²×C₅	C₄²	1	2	yes	—
57	C₄²×C₅	M₁₆	1	2	yes	—
58	C₅×C₁₆	—	2	8	no	—
59	C₂×M₃₂	C₂²×C₅	1	4	yes	—
60	C₂×#97	C₄×C₅	1	1	yes	characteristic
61	C₂×#97	C₂²×C₅	1	1	yes	characteristic
62	C₂×#97	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
64	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	#376	C₄×C₅	1	1	yes	characteristic
67	#347	C₄×C₅	1	8	yes	complemented
68	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	4	2	no	—
73	C₂×C₄×C₅	—	4	2	no	—
74	C₂×C₄×C₅	—	4	2	no	—
75	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅	#117	1	1	yes	characteristic
77	C₂×C₄×C₅	#117	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
81	C₅×C₈	—	2	2	no	—
82	C₅×C₈	—	2	2	no	—
83	C₅×C₈	—	2	2	no	—
84	C₅×C₈	#117	1	4	yes	—
85	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
89	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
91	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
92	M₃₂	—	2	4	no	—
93	M₃₂	C₂×C₄×C₅	1	8	yes	—
94	C₂×C₁₆	—	2	4	no	—
95	#97	—	2	2	no	—
96	#97	—	2	2	no	—
97	#97	C₂×C₄×C₅	1	4	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	2	no	—
101	#95	—	2	2	no	—
102	C₂²×C₅	—	2	2	no	—
103	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
104	C₂²×C₅	#432	1	1	yes	characteristic
105	C₂²×C₅	#432	1	1	yes	characteristic
106	C₄×C₅	—	4	2	no	—
107	C₄×C₅	—	4	2	no	—
108	C₄×C₅	—	4	2	no	—
109	C₄×C₅	#376	1	1	yes	characteristic
110	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₅	#432	1	1	yes	characteristic
112	C₄×C₅	#432	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	C₅×M₁₆	1	2	yes	—
116	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	C₄²×C₅	1	2	yes	—
124	C₂×C₈	C₅×M₁₆	1	2	yes	—
125	C₂×C₈	C₅×M₁₆	1	2	yes	—
126	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
127	C₂×C₈	C₅×#41	1	2	yes	—
128	C₂×C₈	C₅×#41	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	C₄²×C₅	1	2	yes	—
134	C₄²	C₅×M₁₆	1	2	yes	—
135	C₁₆	—	2	8	no	—
136	C₂×C₅	—	2	1	no	—
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	#1745	1	1	yes	characteristic
139	C₂×C₅	#2166	1	2	yes	—
140	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
148	C₂×C₄	C₅×#117	1	1	yes	characteristic
149	C₂×C₄	C₅×#117	1	1	yes	characteristic
150	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
151	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
152	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	C₅×#117	1	4	yes	—
157	C₅	#12596	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	2	no	—
159	C₂²	C₄×C₅×M₁₆	1	1	yes	characteristic
160	C₂²	C₅×#432	1	1	yes	characteristic
161	C₂²	C₅×#432	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	C₅×#376	1	1	yes	characteristic
166	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
167	C₄	C₅×#432	1	1	yes	characteristic
168	C₄	C₅×#432	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₅×#1745	1	1	yes	characteristic
172	C₂	C₅×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	#432
prime radical	C₂³×C₅	C₂×M₁₆
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
13. C₅×#347	C₄	semidirect
14. #12596	C₅	direct (N × H)
67. #347	C₄×C₅	semidirect
157. C₅	#12596	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×C₅×#347	2
C₅×#1726	2
C₅×#1377	2
C₅×#1376	2
#12596	5

Subgroups — C5×#12596 #11935316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12596 #11935316