Subgroups — C₅×#33493 #11956213

Order 1280.

572 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#2169	C₂	1	4	yes	maximal, complemented
3	C₅×D₄×C₁₆	C₂	1	4	yes	maximal, complemented
4	C₅×#2162	C₂	1	2	yes	maximal, complemented
5	C₅×#2113	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	2	yes	—
7	C₅×#532	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	C₂²	1	4	yes	—
9	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
10	C₅×#436	C₄	1	4	yes	—
11	C₅×C₈×D₄	C₂²	1	4	yes	—
12	C₅×#423	C₄	1	1	yes	characteristic
13	C₅×#408	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
18	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	—
19	C₄×C₅×C₁₆	C₂²	1	4	yes	—
20	#33493	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	C₂×C₄	1	2	yes	—
22	C₅×#131	—	4	2	no	—
23	C₅×#131	—	4	2	no	—
24	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	4	yes	—
29	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₅×#127	C₈	1	1	yes	characteristic
31	C₅×#124	C₈	1	2	yes	—
32	C₅×#121	C₈	1	4	yes	—
33	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₅×M₃₂	—	4	4	no	—
36	C₂×C₅×C₁₆	—	4	4	no	—
37	C₂×C₅×C₁₆	C₂³	1	4	yes	—
38	C₂×C₅×C₁₆	C₂³	1	4	yes	—
39	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
40	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂³	1	2	yes	—
43	#2931	C₂×C₅	1	1	yes	characteristic
44	#2169	C₂×C₅	1	4	yes	complemented
45	D₄×C₁₆	C₂×C₅	1	4	yes	complemented
46	#2162	C₂×C₅	1	2	yes	complemented
47	#2113	C₂×C₅	1	4	yes	complemented
48	C₅×#41	—	4	2	no	—
49	C₅×#41	—	4	2	no	—
50	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
51	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
52	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
53	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
57	C₅×M₁₆	—	4	2	no	—
58	C₅×M₁₆	—	4	4	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	4	1	no	—
64	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
68	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
69	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
70	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅	C₂²×C₄	1	2	yes	—
73	C₅×C₁₆	—	2	4	no	—
74	C₅×C₁₆	—	2	4	no	—
75	C₂×#131	C₂²×C₅	1	2	yes	—
76	#532	C₄×C₅	1	1	yes	characteristic
77	C₂×M₃₂	C₂²×C₅	1	4	yes	—
78	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
79	#436	C₄×C₅	1	4	yes	—
80	C₈×D₄	C₂²×C₅	1	4	yes	—
81	#423	C₄×C₅	1	1	yes	characteristic
82	#408	C₄×C₅	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
85	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
86	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
87	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	—
88	C₄×C₁₆	C₂²×C₅	1	4	yes	—
89	C₂³×C₅	C₂²×C₈	1	4	yes	—
90	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₅×Q₈	—	4	2	no	—
92	C₅×D₄	—	2	8	no	—
93	C₅×D₄	—	4	2	no	—
94	C₅×D₄	—	4	4	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₅×C₈	—	2	4	no	—
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	2	2	no	—
108	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
109	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×#41	C₂×C₄×C₅	1	2	yes	—
111	#131	—	4	2	no	—
112	#131	—	4	2	no	—
113	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
114	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
115	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₅	1	4	yes	—
118	C₄⋊Q₈	C₅×C₈	1	1	yes	characteristic
119	#127	C₅×C₈	1	1	yes	characteristic
120	#124	C₅×C₈	1	2	yes	—
121	#121	C₅×C₈	1	4	yes	—
122	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
124	M₃₂	—	4	4	no	—
125	C₂×C₁₆	—	4	4	no	—
126	C₂×C₁₆	C₂³×C₅	1	4	yes	—
127	C₂×C₁₆	C₂³×C₅	1	4	yes	—
128	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
129	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
130	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
131	C₄×C₈	C₂³×C₅	1	2	yes	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	2	8	no	—
134	C₂²×C₅	—	2	1	no	—
135	C₂²×C₅	—	2	4	no	—
136	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	2	4	no	—
140	C₄×C₅	#504	1	4	yes	—
141	C₄×C₅	#518	1	1	yes	characteristic
142	C₄×C₅	#568	1	1	yes	characteristic
143	#41	—	4	2	no	—
144	#41	—	4	2	no	—
145	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
146	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
147	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
148	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
149	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
152	M₁₆	—	4	4	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
161	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
162	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
163	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
164	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
165	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
166	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
167	C₄²	C₂²×C₄×C₅	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₁₆	—	2	4	no	—
170	C₂×C₅	—	4	1	no	—
171	C₂×C₅	—	4	4	no	—
172	C₂×C₅	#2972	1	1	yes	characteristic
173	C₂×C₅	C₂×#504	1	2	yes	—
174	C₂³	C₂²×C₅×C₈	1	4	yes	—
175	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
176	Q₈	—	4	2	no	—
177	D₄	—	2	8	no	—
178	D₄	—	4	2	no	—
179	D₄	—	4	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
187	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
188	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
189	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
194	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
195	C₅	#33493	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	4	no	—
204	C₄	C₅×#504	1	4	yes	—
205	C₄	C₅×#518	1	1	yes	characteristic
206	C₄	C₅×#568	1	1	yes	characteristic
207	C₂	—	4	1	no	—
208	C₂	—	4	4	no	—
209	C₂	C₅×#2972	1	1	yes	characteristic
210	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₅×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2169	C₂	semidirect
3. C₅×D₄×C₁₆	C₂	semidirect
4. C₅×#2162	C₂	semidirect
5. C₅×#2113	C₂	semidirect
20. #33493	C₅	direct (N × H)
44. #2169	C₂×C₅	semidirect
45. D₄×C₁₆	C₂×C₅	semidirect
46. #2162	C₂×C₅	semidirect
47. #2113	C₂×C₅	semidirect
195. C₅	#33493	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×#2931	2
C₅×#2169	2
C₅×D₄×C₁₆	2
C₅×#2162	2
C₅×#2113	2
#33493	5

Subgroups — C5×#33493 #11956213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33493 #11956213