Subgroups — C₅×#33398 #11956118

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	2	yes	maximal
2	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
3	C₅×#2151	C₂	1	6	yes	maximal, complemented
4	C₅×#2150	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×C₄⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₄×C₅×#41	C₄	1	1	yes	characteristic
9	C₅×#504	—	2	8	no	—
10	C₂×C₅×M₃₂	C₂²	1	6	yes	—
11	C₂²×C₅×C₁₆	C₂²	1	6	yes	—
12	C₅×#405	C₄	1	3	yes	—
13	C₅×#405	C₂²	1	3	yes	—
14	C₄×C₅×M₁₆	C₄	1	3	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
16	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₁₆	C₂²	1	3	yes	—
18	C₅×C₄⋊C₁₆	C₂²	1	6	yes	—
19	C₅×C₄⋊C₁₆	C₂²	1	6	yes	—
20	#33398	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
22	C₅×#131	—	2	4	no	—
23	C₅×#131	D₄	1	4	yes	—
24	C₅×#131	Q₈	1	4	yes	—
25	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
26	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
28	C₂²×C₅×C₈	C₂³	1	3	yes	—
29	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
30	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
31	C₄×C₅×D₄	C₈	1	3	yes	—
32	C₄×C₅×D₄	C₈	1	3	yes	—
33	C₅×#117	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
35	C₅×M₃₂	—	2	12	no	—
36	C₂×C₅×C₁₆	—	2	12	no	—
37	C₂×C₅×C₁₆	C₂³	1	2	yes	—
38	C₂×C₅×C₁₆	C₂³	1	6	yes	—
39	C₅×#97	C₂×C₄	1	6	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂³	1	3	yes	—
44	C₂×#504	C₂×C₅	1	2	yes	—
45	C₄×#131	C₂×C₅	1	1	yes	characteristic
46	#2151	C₂×C₅	1	6	yes	complemented
47	#2150	C₂×C₅	1	3	yes	complemented
48	C₂×C₄⋊C₁₆	C₂×C₅	1	3	yes	complemented
49	C₅×#41	C₄⋊C₄	1	4	yes	—
50	C₅×#41	M₁₆	1	4	yes	—
51	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
56	C₅×M₁₆	—	2	6	no	—
57	C₅×M₁₆	C₄⋊C₄	1	12	yes	—
58	C₂×C₅×C₈	—	2	6	no	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
63	C₂×C₅×C₈	C₂×Q₈	1	6	yes	—
64	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂×C₈	1	3	yes	—
66	C₅×C₄⋊C₄	C₂×C₈	1	3	yes	—
67	C₅×C₂²⋊C₄	C₂×C₈	1	6	yes	—
68	C₄²×C₅	C₂×C₈	1	3	yes	—
69	C₄²×C₅	C₂×C₈	1	3	yes	—
70	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅	C₂²×C₄	1	3	yes	—
72	C₅×C₁₆	—	2	2	no	—
73	C₅×C₁₆	—	2	6	no	—
74	C₂×#131	C₄×C₅	1	1	yes	characteristic
75	C₂×#131	C₂²×C₅	1	1	yes	characteristic
76	C₄×#41	C₄×C₅	1	1	yes	characteristic
77	#504	—	2	8	no	—
78	C₂×M₃₂	C₂²×C₅	1	6	yes	—
79	C₂²×C₁₆	C₂²×C₅	1	6	yes	—
80	#405	C₄×C₅	1	3	yes	—
81	#405	C₂²×C₅	1	3	yes	—
82	C₄×M₁₆	C₄×C₅	1	3	yes	—
83	C₂×C₄×C₈	C₂²×C₅	1	3	yes	—
84	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
85	C₄⋊C₁₆	C₂²×C₅	1	3	yes	—
86	C₄⋊C₁₆	C₂²×C₅	1	6	yes	—
87	C₄⋊C₁₆	C₂²×C₅	1	6	yes	—
88	C₂³×C₅	C₂²×C₈	1	3	yes	—
89	C₅×Q₈	C₄⋊C₈	1	4	yes	—
90	C₅×D₄	C₄⋊C₈	1	12	yes	—
91	C₂×C₄×C₅	—	2	6	no	—
92	C₂×C₄×C₅	—	4	3	no	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
94	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
96	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
97	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
98	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
100	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	—	2	3	no	—
103	C₅×C₈	C₂×C₄⋊C₄	1	6	yes	—
104	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
105	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
106	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
107	#131	—	2	4	no	—
108	#131	C₅×D₄	1	4	yes	—
109	#131	C₅×Q₈	1	4	yes	—
110	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
111	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
112	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
113	C₂²×C₈	C₂³×C₅	1	3	yes	—
114	C₄×Q₈	C₅×C₈	1	1	yes	characteristic
115	C₄×Q₈	C₅×C₈	1	1	yes	characteristic
116	C₄×D₄	C₅×C₈	1	3	yes	—
117	C₄×D₄	C₅×C₈	1	3	yes	—
118	#117	C₂×C₄×C₅	1	3	yes	—
119	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
120	M₃₂	—	2	12	no	—
121	C₂×C₁₆	—	2	12	no	—
122	C₂×C₁₆	C₂³×C₅	1	2	yes	—
123	C₂×C₁₆	C₂³×C₅	1	6	yes	—
124	#97	C₂×C₄×C₅	1	6	yes	—
125	C₄×C₈	C₂×C₄×C₅	1	3	yes	—
126	C₄×C₈	C₂×C₄×C₅	1	3	yes	—
127	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
128	C₄×C₈	C₂³×C₅	1	3	yes	—
129	C₂²×C₅	—	2	3	no	—
130	C₂²×C₅	—	2	3	no	—
131	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
132	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅	—	2	3	no	—
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	—	2	3	no	—
136	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
137	C₄×C₅	#504	1	2	yes	complemented
138	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
140	#41	C₅×C₄⋊C₄	1	4	yes	—
141	#41	C₅×M₁₆	1	4	yes	—
142	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
143	C₂×D₄	C₂×C₅×C₈	1	3	yes	—
144	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
145	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
146	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
147	M₁₆	—	2	6	no	—
148	M₁₆	C₅×C₄⋊C₄	1	12	yes	—
149	C₂×C₈	—	2	6	no	—
150	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
152	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
153	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
154	C₂×C₈	C₂×C₅×Q₈	1	6	yes	—
155	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₅×C₈	1	3	yes	—
157	C₄⋊C₄	C₂×C₅×C₈	1	3	yes	—
158	C₂²⋊C₄	C₂×C₅×C₈	1	6	yes	—
159	C₄²	C₂×C₅×C₈	1	3	yes	—
160	C₄²	C₂×C₅×C₈	1	3	yes	—
161	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
162	C₄²	C₂²×C₄×C₅	1	3	yes	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	2	6	no	—
165	C₂×C₅	—	2	6	no	—
166	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
167	C₂×C₅	C₂×#504	1	1	yes	characteristic
168	C₂×C₅	#3407	1	1	yes	characteristic
169	C₂³	C₂²×C₅×C₈	1	3	yes	—
170	Q₈	C₅×C₄⋊C₈	1	4	yes	—
171	D₄	C₅×C₄⋊C₈	1	12	yes	—
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	4	3	no	—
174	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
175	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
177	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
178	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
179	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
181	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	3	no	—
184	C₈	C₂×C₅×C₄⋊C₄	1	6	yes	—
185	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
186	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
187	C₅	#33398	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₅×C₄⋊C₈	1	6	yes	—
191	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₅×C₄⋊C₈	1	6	yes	—
196	C₄	C₅×#504	1	2	yes	complemented
197	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
201	C₂	C₂×C₅×#504	1	1	yes	characteristic
202	C₂	C₅×#3407	1	1	yes	characteristic

	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	C₂×C₅×#41	C₂×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
3. C₅×#2151	C₂	semidirect
4. C₅×#2150	C₂	semidirect
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
20. #33398	C₅	direct (N × H)
46. #2151	C₂×C₅	semidirect
47. #2150	C₂×C₅	semidirect
48. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
137. C₄×C₅	#504	semidirect
187. C₅	#33398	direct (N × H)
196. C₄	C₅×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
C₄×C₅×#131	2
C₅×#2151	2
C₅×#2150	2
C₂×C₅×C₄⋊C₁₆	2
#33398	5

Subgroups — C5×#33398 #11956118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33398 #11956118