Subgroups — C₅×#13171 #11935891

Order 1280.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
3	C₅×#2108	C₂	1	1	yes	maximal, characteristic
4	C₅×#1952	C₂	1	2	yes	maximal
5	C₅×#1387	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₅×#428	—	2	2	no	—
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₅×#405	C₂²	1	1	yes	characteristic
12	C₅×#368	C₄	1	4	yes	complemented
13	C₅×#366	—	2	1	no	—
14	C₅×#366	—	2	1	no	—
15	C₅×#346	—	2	1	no	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₅×#343	—	2	2	no	—
18	#13171	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₅×#117	D₄	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₅×M₃₂	—	2	1	no	—
25	C₅×M₃₂	Q₈	1	2	yes	—
26	C₂×C₅×C₁₆	—	2	1	no	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₅×C₁₆	Q₈	1	2	yes	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	C₂×C₄	1	4	yes	—
33	C₅×#107	—	2	2	no	—
34	C₅×#107	—	2	2	no	—
35	C₅×#107	—	4	2	no	—
36	C₅×#106	—	4	2	no	—
37	C₅×#97	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₂×#368	C₂×C₅	1	1	yes	characteristic
40	C₂×#366	C₂×C₅	1	1	yes	characteristic
41	#2108	C₂×C₅	1	1	yes	characteristic
42	#1952	C₂×C₅	1	2	yes	—
43	#1387	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₅	—	4	2	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₅×M₁₆	—	2	1	no	—
47	C₅×M₁₆	—	2	4	no	—
48	C₅×M₁₆	—	4	2	no	—
49	C₅×M₁₆	Q₁₆	1	2	yes	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
61	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
62	C₅×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₅	—	2	1	no	—
64	C₅×C₁₆	—	2	2	no	—
65	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
66	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
67	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₂×#108	C₂²×C₅	1	2	yes	—
70	#428	—	2	2	no	—
71	C₂×#107	C₂²×C₅	1	2	yes	—
72	#405	C₂²×C₅	1	1	yes	characteristic
73	#368	C₄×C₅	1	4	yes	complemented
74	#366	—	2	1	no	—
75	#366	—	2	1	no	—
76	#346	—	2	1	no	—
77	C₄×C₁₆	—	2	1	no	—
78	#343	—	2	2	no	—
79	C₂³×C₅	#135	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	8	2	no	—
83	C₂×C₄×C₅	—	16	2	no	—
84	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅	#121	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅	#135	1	1	yes	characteristic
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	—	4	4	no	—
92	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
95	C₅×C₈	#125	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
99	#117	C₅×D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	M₃₂	—	2	1	no	—
102	M₃₂	C₅×Q₈	1	2	yes	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
105	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
106	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
107	C₂×C₁₆	C₅×Q₈	1	2	yes	—
108	#108	—	2	2	no	—
109	#108	C₂×C₄×C₅	1	4	yes	—
110	#107	—	2	2	no	—
111	#107	—	2	2	no	—
112	#107	—	4	2	no	—
113	#106	—	4	2	no	—
114	#97	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₂²×C₅	—	2	1	no	—
117	C₂²×C₅	—	2	1	no	—
118	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
119	C₂²×C₅	#466	1	1	yes	characteristic
120	C₂²×C₅	#499	1	1	yes	characteristic
121	C₄×C₅	—	4	1	no	—
122	C₄×C₅	—	4	1	no	—
123	C₄×C₅	—	16	2	no	—
124	C₄×C₅	#389	1	1	yes	characteristic
125	C₄×C₅	C₄×D₈	1	1	yes	characteristic
126	C₄×C₅	#467	1	1	yes	characteristic
127	C₄×C₅	#500	1	1	yes	characteristic
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
130	M₁₆	—	2	1	no	—
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	M₁₆	C₅×Q₁₆	1	2	yes	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
138	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
140	C₂×C₈	C₅×#41	1	1	yes	characteristic
141	C₂×C₈	C₅×#41	1	1	yes	characteristic
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
145	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₅×C₄⋊C₄	1	2	yes	—
150	C₁₆	C₅×C₄⋊C₄	1	2	yes	—
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	1	no	—
153	C₂×C₅	#1942	1	1	yes	characteristic
154	C₂×C₅	#2221	1	1	yes	characteristic
155	C₂×C₅	#2222	1	1	yes	characteristic
156	C₂³	C₅×#135	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	16	2	no	—
161	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
162	C₂×C₄	C₅×#121	1	1	yes	characteristic
163	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
164	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
165	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
166	C₂×C₄	C₅×#135	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	4	4	no	—
169	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
171	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
172	C₈	C₅×#125	1	1	yes	characteristic
173	C₅	#13171	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
177	C₂²	C₅×#466	1	1	yes	characteristic
178	C₂²	C₅×#499	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	16	2	no	—
182	C₄	C₅×#389	1	1	yes	characteristic
183	C₄	C₄×C₅×D₈	1	1	yes	characteristic
184	C₄	C₅×#467	1	1	yes	characteristic
185	C₄	C₅×#500	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	C₅×#1942	1	1	yes	characteristic
189	C₂	C₅×#2221	1	1	yes	characteristic
190	C₂	C₅×#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#368	C₄	semidirect
18. #13171	C₅	direct (N × H)
73. #368	C₄×C₅	semidirect
173. C₅	#13171	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×#368	2
C₂×C₅×#366	2
C₅×#2108	2
C₅×#1952	2
C₅×#1387	2
#13171	5

Subgroups — C5×#13171 #11935891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13171 #11935891