Subgroups — C₅×#33397 #11956117

Order 1280.

706 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	C₂²×C₅×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₅×#2151	C₂	1	8	yes	maximal, complemented
4	C₅×#2150	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	—	2	8	no	—
9	C₂×C₅×M₃₂	C₂²	1	4	yes	—
10	C₂×C₅×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
12	C₅×#405	C₄	1	4	yes	—
13	C₅×#405	C₂²	1	4	yes	—
14	C₂×C₅×#97	C₄	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₅×C₄⋊C₁₆	C₂²	1	16	yes	complemented
17	#33397	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	4	yes	—
25	C₂²×C₅×C₈	Q₈	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₅×#117	C₈	1	4	yes	—
29	C₅×#117	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
31	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₅×M₃₂	—	2	8	no	—
34	C₅×M₃₂	—	2	8	no	—
35	C₂×C₅×C₁₆	—	2	8	no	—
36	C₂×C₅×C₁₆	C₂³	1	8	yes	—
37	C₅×#97	C₂×C₄	1	8	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₄×C₅×C₈	C₂³	1	4	yes	—
40	C₂²×M₃₂	C₂×C₅	1	2	yes	—
41	C₂×#405	C₂×C₅	1	1	yes	characteristic
42	#2151	C₂×C₅	1	8	yes	complemented
43	#2150	C₂×C₅	1	4	yes	complemented
44	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅	M₁₆	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₅×M₁₆	—	2	8	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
55	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
60	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
62	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
63	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
64	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
65	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
66	C₄²×C₅	C₂×C₈	1	4	yes	—
67	C₄²×C₅	C₂²×C₄	1	4	yes	—
68	C₅×C₁₆	—	2	8	no	—
69	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
70	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×#117	C₄×C₅	1	1	yes	characteristic
72	C₂×M₃₂	—	2	8	no	—
73	C₂×M₃₂	C₂²×C₅	1	4	yes	—
74	C₂×M₃₂	C₂²×C₅	1	4	yes	—
75	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
76	#405	C₄×C₅	1	4	yes	—
77	#405	C₂²×C₅	1	4	yes	—
78	C₂×#97	C₄×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
80	C₄⋊C₁₆	C₂²×C₅	1	16	yes	complemented
81	C₂³×C₅	—	4	2	no	—
82	C₂³×C₅	C₄⋊C₈	1	4	yes	—
83	C₂³×C₅	C₂²×C₈	1	2	yes	—
84	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
87	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
91	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₅×C₈	—	2	4	no	—
97	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
98	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
99	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
100	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
101	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×M₁₆	—	2	4	no	—
103	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
104	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
107	C₂²×C₈	C₅×D₄	1	4	yes	—
108	C₂²×C₈	C₅×Q₈	1	4	yes	—
109	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₅	1	2	yes	—
111	#117	C₅×C₈	1	4	yes	—
112	#117	C₂×C₄×C₅	1	4	yes	—
113	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
114	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
116	M₃₂	—	2	8	no	—
117	M₃₂	—	2	8	no	—
118	C₂×C₁₆	—	2	8	no	—
119	C₂×C₁₆	C₂³×C₅	1	8	yes	—
120	#97	C₂×C₄×C₅	1	8	yes	—
121	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
122	C₄×C₈	C₂³×C₅	1	4	yes	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	4	no	—
125	C₂²×C₅	—	4	4	no	—
126	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
127	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
128	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₅	—	2	8	no	—
130	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
131	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
132	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
134	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
135	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
136	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
137	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
139	C₂²×C₄	C₅×M₁₆	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
142	M₁₆	—	2	8	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
145	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
146	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
147	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
149	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
150	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
151	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
152	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
153	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
155	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
156	C₄²	C₂×C₅×C₈	1	4	yes	—
157	C₄²	C₂²×C₄×C₅	1	4	yes	—
158	C₁₆	—	2	8	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	4	no	—
161	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₅	#3407	1	2	yes	—
163	C₂³	—	4	2	no	—
164	C₂³	C₅×C₄⋊C₈	1	4	yes	—
165	C₂³	C₂²×C₅×C₈	1	2	yes	—
166	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
169	C₂×C₄	C₅×C₄⋊C₈	1	8	yes	—
170	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
171	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
172	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
173	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
175	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
176	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
177	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
180	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
181	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
182	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
183	C₅	#33397	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₅×C₄⋊C₈	1	2	yes	—
188	C₂²	C₂×C₅×C₄⋊C₈	1	4	yes	—
189	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
192	C₄	C₂×C₅×C₄⋊C₈	1	4	yes	—
193	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₅×C₄⋊C₈	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	C₂⁴×C₅	C₂×C₈
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
16. C₅×C₄⋊C₁₆	C₂²	semidirect
17. #33397	C₅	direct (N × H)
42. #2151	C₂×C₅	semidirect
43. #2150	C₂×C₅	semidirect
80. C₄⋊C₁₆	C₂²×C₅	semidirect
183. C₅	#33397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂×C₅×#405	2
C₅×#2151	2
C₅×#2150	2
#33397	5

Subgroups — C5×#33397 #11956117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33397 #11956117