Subgroups — C₅×#33456 #11956176

Order 1280.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2948	C₂	1	2	yes	maximal, complemented
3	C₅×#2177	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×D₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2140	C₂	1	2	yes	maximal, complemented
7	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	2	yes	—
9	C₅×#518	—	2	2	no	—
10	C₂×C₅×D₁₆	C₄	1	8	yes	complemented
11	C₂×C₅×D₁₆	C₂²	1	2	yes	—
12	C₂×C₅×D₁₆	C₂²	1	2	yes	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#442	—	2	4	no	—
16	C₄×C₅×D₈	C₂²	1	8	yes	complemented
17	C₅×#427	C₂²	1	2	yes	—
18	C₅×#418	—	2	2	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#367	C₂²	1	2	yes	complemented
21	C₅×#366	C₂²	1	2	yes	complemented
22	C₅×#357	C₂²	1	8	yes	complemented
23	C₅×#346	C₂²	1	2	yes	complemented
24	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
25	#33456	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×D₄	—	2	2	no	—
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	—	2	4	no	—
29	C₂×C₅×D₈	C₂×C₄	1	8	yes	complemented
30	C₂×C₅×D₈	C₂³	1	4	yes	—
31	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×D₄	—	2	8	no	—
34	C₄×C₅×D₄	—	4	4	no	—
35	C₅×#117	—	2	2	no	—
36	C₅×#117	D₄	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	4	4	no	—
38	C₅×D₁₆	C₂×C₄	1	8	yes	complemented
39	C₅×D₁₆	C₂×C₄	1	8	yes	complemented
40	C₅×M₃₂	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
42	C₂×C₅×C₁₆	C₂³	1	2	yes	—
43	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
45	C₅×#107	C₂³	1	4	yes	—
46	C₅×#106	—	2	2	no	—
47	C₅×#102	—	2	8	no	—
48	C₅×#97	D₄	1	2	yes	—
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic
51	#2948	C₂×C₅	1	2	yes	complemented
52	#2177	C₂×C₅	1	4	yes	complemented
53	C₄×D₁₆	C₂×C₅	1	4	yes	complemented
54	#2158	C₂×C₅	1	1	yes	characteristic, complemented
55	#2140	C₂×C₅	1	2	yes	complemented
56	#2108	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂⁴×C₅	—	4	2	no	—
58	C₂×C₅×D₄	—	2	8	no	—
59	C₂×C₅×D₄	—	2	4	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂×C₅×D₄	—	4	4	no	—
62	C₂²×C₄×C₅	—	4	8	no	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₅×D₈	—	2	8	no	—
65	C₅×D₈	—	2	4	no	—
66	C₅×D₈	C₂²×C₄	1	8	yes	—
67	C₅×M₁₆	—	2	2	no	—
68	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	2	yes	—
72	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	D₈	1	2	yes	—
75	C₅×C₂²⋊C₄	—	4	2	no	—
76	C₅×C₂²⋊C₄	—	4	8	no	—
77	C₅×C₂²⋊C₄	D₈	1	2	yes	—
78	C₄²×C₅	—	4	2	no	—
79	C₄²×C₅	C₂×D₄	1	2	yes	—
80	C₅×C₁₆	—	2	2	no	—
81	C₅×C₁₆	C₂²×C₄	1	2	yes	—
82	C₅×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂²×D₈	C₂²×C₅	1	2	yes	—
84	#518	—	2	2	no	—
85	C₂×D₁₆	C₄×C₅	1	8	yes	complemented
86	C₂×D₁₆	C₂²×C₅	1	2	yes	—
87	C₂×D₁₆	C₂²×C₅	1	2	yes	—
88	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
90	#442	—	2	4	no	—
91	C₄×D₈	C₂²×C₅	1	8	yes	complemented
92	#427	C₂²×C₅	1	2	yes	—
93	#418	—	2	2	no	—
94	#405	C₂²×C₅	1	1	yes	characteristic
95	#367	C₂²×C₅	1	2	yes	complemented
96	#366	C₂²×C₅	1	2	yes	complemented
97	#357	C₂²×C₅	1	8	yes	complemented
98	#346	C₂²×C₅	1	2	yes	complemented
99	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
100	C₂³×C₅	—	4	4	no	—
101	C₂³×C₅	—	4	8	no	—
102	C₂³×C₅	—	16	4	no	—
103	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
104	C₅×D₄	—	2	8	no	—
105	C₅×D₄	—	4	4	no	—
106	C₅×D₄	—	4	8	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	16	8	no	—
109	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
111	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅	#135	1	2	yes	—
113	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	C₄×D₄	1	2	yes	—
116	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
117	C₅×C₈	C₂×#41	1	1	yes	characteristic
118	C₂²×D₄	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	4	no	—
121	C₂×D₈	C₂×C₄×C₅	1	8	yes	complemented
122	C₂×D₈	C₂³×C₅	1	4	yes	—
123	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
125	C₄×D₄	—	2	8	no	—
126	C₄×D₄	—	4	4	no	—
127	#117	—	2	2	no	—
128	#117	C₅×D₄	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	—	4	4	no	—
130	D₁₆	C₂×C₄×C₅	1	8	yes	complemented
131	D₁₆	C₂×C₄×C₅	1	8	yes	complemented
132	M₃₂	—	2	2	no	—
133	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
134	C₂×C₁₆	C₂³×C₅	1	2	yes	—
135	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
137	#107	C₂³×C₅	1	4	yes	—
138	#106	—	2	2	no	—
139	#102	—	2	8	no	—
140	#97	C₅×D₄	1	2	yes	—
141	C₄×C₈	C₂³×C₅	1	2	yes	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	—	4	8	no	—
144	C₂²×C₅	—	8	8	no	—
145	C₂²×C₅	—	16	8	no	—
146	C₂²×C₅	C₄×D₈	1	2	yes	—
147	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	8	4	no	—
150	C₄×C₅	C₄×D₈	1	2	yes	—
151	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₅	C₂×#135	1	1	yes	characteristic
153	C₂⁴	—	4	2	no	—
154	C₂×D₄	—	2	8	no	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	D₈	—	2	8	no	—
161	D₈	—	2	4	no	—
162	D₈	C₂²×C₄×C₅	1	8	yes	—
163	M₁₆	—	2	2	no	—
164	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
165	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
166	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₈	C₅×#41	1	2	yes	—
168	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	C₅×D₈	1	2	yes	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	8	no	—
173	C₂²⋊C₄	C₅×D₈	1	2	yes	—
174	C₄²	—	4	2	no	—
175	C₄²	C₂×C₅×D₄	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²×C₄×C₅	1	2	yes	—
178	C₁₆	C₂²×C₄×C₅	1	2	yes	—
179	C₂×C₅	—	2	2	no	—
180	C₂×C₅	—	8	8	no	—
181	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
182	C₂×C₅	#3415	1	2	yes	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	8	no	—
185	C₂³	—	16	4	no	—
186	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
187	D₄	—	2	8	no	—
188	D₄	—	4	8	no	—
189	D₄	—	4	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	16	8	no	—
192	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
193	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
194	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
195	C₂×C₄	C₅×#135	1	2	yes	—
196	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	C₄×C₅×D₄	1	2	yes	—
199	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
200	C₈	C₂×C₅×#41	1	1	yes	characteristic
201	C₅	#33456	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	C₄×C₅×D₈	1	2	yes	—
207	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₄×C₅×D₈	1	2	yes	—
211	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
212	C₄	C₂×C₅×#135	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
216	C₂	C₅×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2948	C₂	semidirect
3. C₅×#2177	C₂	semidirect
4. C₄×C₅×D₁₆	C₂	semidirect
5. C₅×#2158	C₂	semidirect
6. C₅×#2140	C₂	semidirect
7. C₅×#2108	C₂	semidirect
10. C₂×C₅×D₁₆	C₄	semidirect
16. C₄×C₅×D₈	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×#366	C₂²	semidirect
22. C₅×#357	C₂²	semidirect
23. C₅×#346	C₂²	semidirect
24. C₄×C₅×C₁₆	C₂²	semidirect
25. #33456	C₅	direct (N × H)
29. C₂×C₅×D₈	C₂×C₄	semidirect
38. C₅×D₁₆	C₂×C₄	semidirect
39. C₅×D₁₆	C₂×C₄	semidirect
41. C₂×C₅×C₁₆	C₂×C₄	semidirect
51. #2948	C₂×C₅	semidirect
52. #2177	C₂×C₅	semidirect
53. C₄×D₁₆	C₂×C₅	semidirect
54. #2158	C₂×C₅	semidirect
55. #2140	C₂×C₅	semidirect
56. #2108	C₂×C₅	semidirect
85. C₂×D₁₆	C₄×C₅	semidirect
91. C₄×D₈	C₂²×C₅	semidirect
95. #367	C₂²×C₅	semidirect
96. #366	C₂²×C₅	semidirect
97. #357	C₂²×C₅	semidirect
98. #346	C₂²×C₅	semidirect
99. C₄×C₁₆	C₂²×C₅	semidirect
121. C₂×D₈	C₂×C₄×C₅	semidirect
130. D₁₆	C₂×C₄×C₅	semidirect
131. D₁₆	C₂×C₄×C₅	semidirect
133. C₂×C₁₆	C₂×C₄×C₅	semidirect
201. C₅	#33456	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×D₁₆	2
C₅×#2948	2
C₅×#2177	2
C₄×C₅×D₁₆	2
C₅×#2158	2
C₅×#2140	2
C₅×#2108	2
#33456	5

Subgroups — C5×#33456 #11956176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33456 #11956176