Subgroups — C₅×#9609 #11932329

Order 1280.

294 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#1565	C₂	1	2	yes	maximal
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
4	C₅×#1452	C₂	1	1	yes	maximal, characteristic
5	C₅×#1283	C₂	1	2	yes	maximal
6	C₅×#431	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₄	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#341	C₄	1	1	yes	characteristic
13	C₅×#341	C₄	1	1	yes	characteristic
14	C₅×#335	—	2	2	no	—
15	C₅×#334	—	2	2	no	—
16	C₅×#334	—	2	2	no	—
17	C₅×#322	—	2	2	no	—
18	#9609	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₅×#108	—	2	4	no	—
26	C₅×C₄⋊C₈	—	2	2	no	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	1	no	—
29	C₅×C₄⋊C₈	D₄	1	2	yes	—
30	C₅×C₂²⋊C₈	—	2	1	no	—
31	C₅×C₂²⋊C₈	Q₈	1	2	yes	—
32	C₅×#97	—	2	2	no	—
33	C₄×C₅×C₈	—	2	1	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	C₄×C₅×C₈	Q₈	1	2	yes	—
38	C₄×#108	C₂×C₅	1	1	yes	characteristic
39	#1565	C₂×C₅	1	2	yes	—
40	C₂×#334	C₂×C₅	1	1	yes	characteristic
41	#1452	C₂×C₅	1	1	yes	characteristic
42	#1283	C₂×C₅	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	1	yes	characteristic
46	C₅×M₁₆	—	2	2	no	—
47	C₅×M₁₆	—	2	2	no	—
48	C₅×M₁₆	—	2	2	no	—
49	C₅×M₁₆	M₁₆	1	4	yes	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	4	2	no	—
57	C₂×C₅×C₈	—	8	2	no	—
58	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅	#41	1	1	yes	characteristic
64	#431	C₂²×C₅	1	1	yes	characteristic
65	#431	C₂²×C₅	1	1	yes	characteristic
66	C₂×#108	C₄×C₅	1	2	yes	—
67	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
68	C₄×M₁₆	C₂²×C₅	1	2	yes	—
69	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
70	#341	C₄×C₅	1	1	yes	characteristic
71	#341	C₄×C₅	1	1	yes	characteristic
72	#335	—	2	2	no	—
73	#334	—	2	2	no	—
74	#334	—	2	2	no	—
75	#322	—	2	2	no	—
76	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	4	1	no	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅	#121	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	#125	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅	#131	1	2	yes	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	2	4	no	—
90	C₅×C₈	—	4	2	no	—
91	C₅×C₈	—	4	2	no	—
92	C₅×C₈	—	4	2	no	—
93	C₅×C₈	—	8	2	no	—
94	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
95	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
99	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
100	#108	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	C₅×D₄	1	2	yes	—
105	C₂²⋊C₈	—	2	1	no	—
106	C₂²⋊C₈	C₅×Q₈	1	2	yes	—
107	#97	—	2	2	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₅×D₄	1	2	yes	—
112	C₄×C₈	C₅×Q₈	1	2	yes	—
113	C₂²×C₅	—	2	1	no	—
114	C₂²×C₅	—	2	1	no	—
115	C₂²×C₅	#389	1	1	yes	characteristic
116	C₂²×C₅	#446	1	2	yes	—
117	C₄×C₅	—	2	1	no	—
118	C₄×C₅	—	2	1	no	—
119	C₄×C₅	—	2	1	no	—
120	C₄×C₅	#423	1	1	yes	characteristic
121	C₄×C₅	#424	1	1	yes	characteristic
122	C₄×C₅	#435	1	2	yes	—
123	C₄×C₅	#471	1	2	yes	—
124	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
126	C₂²×C₄	C₅×#41	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	C₅×M₁₆	1	4	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
140	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
141	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
142	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
143	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
144	C₄²	C₅×#41	1	1	yes	characteristic
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	#1940	1	1	yes	characteristic
147	C₂×C₅	#1951	1	1	yes	characteristic
148	C₂×C₅	#1953	1	1	yes	characteristic
149	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
154	C₂×C₄	C₅×#121	1	1	yes	characteristic
155	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄	C₅×#125	1	1	yes	characteristic
157	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
159	C₂×C₄	C₅×#131	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	4	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	8	2	no	—
167	C₅	#9609	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₅×#389	1	1	yes	characteristic
171	C₂²	C₅×#446	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₅×#423	1	1	yes	characteristic
176	C₄	C₅×#424	1	1	yes	characteristic
177	C₄	C₅×#435	1	2	yes	—
178	C₄	C₅×#471	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₅×#1940	1	1	yes	characteristic
181	C₂	C₅×#1951	1	1	yes	characteristic
182	C₂	C₅×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
18. #9609	C₅	direct (N × H)
167. C₅	#9609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#108	2
C₅×#1565	2
C₂×C₅×#334	2
C₅×#1452	2
C₅×#1283	2
#9609	5

Subgroups — C5×#9609 #11932329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9609 #11932329