Subgroups — C₅×#7288 #11930008

Order 1280.

202 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	C₅×#2162	C₂	1	1	yes	maximal, characteristic
2	C₅×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#423	—	2	1	no	—
5	C₂×C₅×C₄⋊C₈	—	2	1	no	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×C₄⋊C₁₆	—	2	1	no	—
8	C₅×C₂²⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₅×C₁₆	—	2	1	no	—
10	C₅×#335	—	2	1	no	—
11	C₅×#335	—	2	1	no	—
12	C₅×#335	C₄	1	2	yes	—
13	C₅×#335	C₄	1	2	yes	—
14	#7288	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂×C₅×M₁₆	—	4	1	no	—
16	C₂²×C₅×C₈	—	4	1	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	—	2	2	no	—
21	C₅×C₄⋊C₈	—	2	2	no	—
22	C₅×C₄⋊C₈	—	2	2	no	—
23	C₅×C₄⋊C₈	—	4	1	no	—
24	C₅×C₄⋊C₈	—	4	1	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	#2162	C₂×C₅	1	1	yes	characteristic
30	#1567	C₂×C₅	1	1	yes	characteristic
31	C₂×#335	C₂×C₅	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
35	C₅×M₁₆	—	8	1	no	—
36	C₅×M₁₆	—	8	1	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	—	4	1	no	—
39	C₂×C₅×C₈	—	4	2	no	—
40	C₂×C₅×C₈	—	4	2	no	—
41	C₂×C₅×C₈	—	8	1	no	—
42	C₂×C₅×C₈	—	8	1	no	—
43	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	SD₁₆	1	1	yes	characteristic
50	C₄²×C₅	SD₁₆	1	1	yes	characteristic
51	C₅×C₁₆	—	4	2	no	—
52	#423	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
55	C₄⋊C₁₆	—	2	1	no	—
56	C₂²⋊C₁₆	C₄×C₅	1	2	yes	—
57	C₄×C₁₆	—	2	1	no	—
58	#335	—	2	1	no	—
59	#335	—	2	1	no	—
60	#335	C₄×C₅	1	2	yes	—
61	#335	C₄×C₅	1	2	yes	—
62	C₂³×C₅	#107	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	1	no	—
64	C₂×C₄×C₅	—	2	1	no	—
65	C₂×C₄×C₅	—	4	1	no	—
66	C₂×C₄×C₅	#95	1	1	yes	characteristic
67	C₂×C₄×C₅	#102	1	1	yes	characteristic
68	C₂×C₄×C₅	#102	1	1	yes	characteristic
69	C₂×C₄×C₅	#103	1	1	yes	characteristic
70	C₂×C₄×C₅	#103	1	1	yes	characteristic
71	C₂×C₄×C₅	#106	1	1	yes	characteristic
72	C₂×C₄×C₅	D₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅	SD₃₂	1	1	yes	characteristic
74	C₂×C₄×C₅	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₅	Q₃₂	1	1	yes	characteristic
76	C₅×C₈	—	2	1	no	—
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	—	8	2	no	—
80	C₅×C₈	—	8	2	no	—
81	C₅×C₈	#108	1	2	yes	—
82	C₂×M₁₆	—	4	1	no	—
83	C₂²×C₈	—	4	1	no	—
84	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
86	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
87	C₂×C₁₆	—	2	2	no	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
94	C₄×C₈	C₅×D₄	1	1	yes	characteristic
95	C₄×C₈	C₅×D₄	1	1	yes	characteristic
96	C₂²×C₅	—	2	1	no	—
97	C₂²×C₅	—	2	1	no	—
98	C₂²×C₅	#340	1	1	yes	characteristic
99	C₂²×C₅	#366	1	1	yes	characteristic
100	C₂²×C₅	#367	1	1	yes	characteristic
101	C₄×C₅	—	2	1	no	—
102	C₄×C₅	—	2	1	no	—
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	#341	1	1	yes	characteristic
105	C₄×C₅	#344	1	1	yes	characteristic
106	C₄×C₅	#357	1	1	yes	characteristic
107	C₄×C₅	#357	1	1	yes	characteristic
108	C₄×C₅	#358	1	1	yes	characteristic
109	C₄×C₅	#358	1	1	yes	characteristic
110	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
112	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
113	M₁₆	—	8	1	no	—
114	M₁₆	—	8	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	1	no	—
120	C₂×C₈	—	8	1	no	—
121	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
122	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
125	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
126	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
127	C₄²	C₅×SD₁₆	1	1	yes	characteristic
128	C₄²	C₅×SD₁₆	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₂×C₅	—	2	1	no	—
131	C₂×C₅	—	2	1	no	—
132	C₂×C₅	#1301	1	1	yes	characteristic
133	C₂×C₅	#1380	1	1	yes	characteristic
134	C₂×C₅	#1388	1	1	yes	characteristic
135	C₂³	C₅×#107	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₅×#95	1	1	yes	characteristic
140	C₂×C₄	C₅×#102	1	1	yes	characteristic
141	C₂×C₄	C₅×#102	1	1	yes	characteristic
142	C₂×C₄	C₅×#103	1	1	yes	characteristic
143	C₂×C₄	C₅×#103	1	1	yes	characteristic
144	C₂×C₄	C₅×#106	1	1	yes	characteristic
145	C₂×C₄	C₅×D₁₆	1	1	yes	characteristic
146	C₂×C₄	C₅×SD₃₂	1	1	yes	characteristic
147	C₂×C₄	C₅×SD₃₂	1	1	yes	characteristic
148	C₂×C₄	C₅×Q₃₂	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	8	2	no	—
153	C₈	—	8	2	no	—
154	C₈	C₅×#108	1	2	yes	—
155	C₅	#7288	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₅×#340	1	1	yes	characteristic
159	C₂²	C₅×#366	1	1	yes	characteristic
160	C₂²	C₅×#367	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₅×#341	1	1	yes	characteristic
165	C₄	C₅×#344	1	1	yes	characteristic
166	C₄	C₅×#357	1	1	yes	characteristic
167	C₄	C₅×#357	1	1	yes	characteristic
168	C₄	C₅×#358	1	1	yes	characteristic
169	C₄	C₅×#358	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₅×#1301	1	1	yes	characteristic
173	C₂	C₅×#1380	1	1	yes	characteristic
174	C₂	C₅×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#340
prime radical	C₂³×C₅	#107
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
14. #7288	C₅	direct (N × H)
155. C₅	#7288	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×#2162	2
C₅×#1567	2
C₂×C₅×#335	2
#7288	5

Subgroups — C5×#7288 #11930008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7288 #11930008