Subgroups — C₅×#33494 #11956214

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#2168	C₂	1	8	yes	maximal, complemented
3	C₅×#2163	C₂	1	2	yes	maximal, complemented
4	C₅×#2113	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#131	C₂²	1	2	yes	—
6	C₅×#532	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	4	yes	—
8	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
9	C₅×#436	C₄	1	4	yes	—
10	C₅×C₈×D₄	C₂²	1	4	yes	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#408	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
15	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	—
17	C₅×#346	C₂²	1	4	yes	—
18	#33494	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×#41	C₂×C₄	1	2	yes	—
20	C₅×#131	—	4	2	no	—
21	C₅×#131	—	4	2	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	4	yes	—
27	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
28	C₅×#127	C₈	1	1	yes	characteristic
29	C₅×#124	C₈	1	2	yes	—
30	C₅×#121	C₈	1	4	yes	—
31	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₅×M₃₂	—	4	4	no	—
34	C₂×C₅×C₁₆	—	4	4	no	—
35	C₂×C₅×C₁₆	C₂³	1	8	yes	—
36	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
37	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₅×C₈	C₂³	1	2	yes	—
40	#2931	C₂×C₅	1	1	yes	characteristic
41	#2168	C₂×C₅	1	8	yes	complemented
42	#2163	C₂×C₅	1	2	yes	complemented
43	#2113	C₂×C₅	1	4	yes	complemented
44	C₅×#41	—	4	2	no	—
45	C₅×#41	—	4	2	no	—
46	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
47	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
48	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
49	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
53	C₅×M₁₆	—	4	2	no	—
54	C₅×M₁₆	—	4	4	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	—	4	1	no	—
59	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
62	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
64	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
65	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₅	C₂²×C₄	1	2	yes	—
68	C₅×C₁₆	—	2	8	no	—
69	C₂×#131	C₂²×C₅	1	2	yes	—
70	#532	C₄×C₅	1	1	yes	characteristic
71	C₂×M₃₂	C₂²×C₅	1	4	yes	—
72	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
73	#436	C₄×C₅	1	4	yes	—
74	C₈×D₄	C₂²×C₅	1	4	yes	—
75	#423	C₄×C₅	1	1	yes	characteristic
76	#408	C₄×C₅	1	2	yes	—
77	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
79	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
80	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	—
81	#346	C₂²×C₅	1	4	yes	—
82	C₂³×C₅	C₂²×C₈	1	4	yes	—
83	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₅×Q₈	—	4	2	no	—
85	C₅×D₄	—	4	4	no	—
86	C₅×D₄	—	4	2	no	—
87	C₅×D₄	—	4	4	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
98	C₅×C₈	—	2	4	no	—
99	C₅×C₈	—	2	1	no	—
100	C₅×C₈	—	2	2	no	—
101	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
102	C₂×#41	C₂×C₄×C₅	1	2	yes	—
103	#131	—	4	2	no	—
104	#131	—	4	2	no	—
105	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
106	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
107	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₅	1	4	yes	—
110	C₄⋊Q₈	C₅×C₈	1	1	yes	characteristic
111	#127	C₅×C₈	1	1	yes	characteristic
112	#124	C₅×C₈	1	2	yes	—
113	#121	C₅×C₈	1	4	yes	—
114	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
115	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
116	M₃₂	—	4	4	no	—
117	C₂×C₁₆	—	4	4	no	—
118	C₂×C₁₆	C₂³×C₅	1	8	yes	—
119	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
120	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
121	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
122	C₄×C₈	C₂³×C₅	1	2	yes	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	8	no	—
125	C₂²×C₅	—	2	1	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	—	2	4	no	—
132	C₄×C₅	#518	1	1	yes	characteristic
133	C₄×C₅	#568	1	1	yes	characteristic
134	#41	—	4	2	no	—
135	#41	—	4	2	no	—
136	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
137	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
138	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
139	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
140	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
142	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
143	M₁₆	—	4	4	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
151	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
152	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
153	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
154	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
155	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
156	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
157	C₄²	C₂²×C₄×C₅	1	2	yes	—
158	C₁₆	—	2	8	no	—
159	C₂×C₅	—	4	1	no	—
160	C₂×C₅	—	4	4	no	—
161	C₂×C₅	#2972	1	1	yes	characteristic
162	C₂×C₅	#3407	1	2	yes	—
163	C₂³	C₂²×C₅×C₈	1	4	yes	—
164	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
165	Q₈	—	4	2	no	—
166	D₄	—	4	2	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
176	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
177	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
178	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	2	4	no	—
181	C₈	—	2	2	no	—
182	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
183	C₅	#33494	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	8	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	4	no	—
188	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	4	no	—
193	C₄	C₅×#518	1	1	yes	characteristic
194	C₄	C₅×#568	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	C₅×#2972	1	1	yes	characteristic
198	C₂	C₅×#3407	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×#2168	C₂	semidirect
3. C₅×#2163	C₂	semidirect
4. C₅×#2113	C₂	semidirect
18. #33494	C₅	direct (N × H)
41. #2168	C₂×C₅	semidirect
42. #2163	C₂×C₅	semidirect
43. #2113	C₂×C₅	semidirect
183. C₅	#33494	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×#2168	2
C₅×#2163	2
C₅×#2113	2
#33494	5

Subgroups — C5×#33494 #11956214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33494 #11956214