Subgroups — C₅×#33464 #11956184

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3414	C₂	1	1	yes	maximal, characteristic
2	C₅×#2957	C₂	1	2	yes	maximal, complemented
3	C₅×#2179	C₂	1	4	yes	maximal, complemented
4	C₅×#2178	C₂	1	4	yes	maximal, complemented
5	C₅×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#359	C₂	1	2	yes	maximal, complemented
7	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	2	yes	—
9	C₅×#568	—	2	2	no	—
10	C₅×#510	C₄	1	4	yes	complemented
11	C₅×#509	C₄	1	4	yes	complemented
12	C₅×#508	C₂²	1	4	yes	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#444	—	2	4	no	—
16	C₅×#443	C₂²	1	8	yes	complemented
17	C₂×C₅×#108	C₂²	1	2	yes	—
18	C₅×#421	—	2	2	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#368	C₂²	1	4	yes	complemented
21	C₅×#359	C₂²	1	4	yes	complemented
22	C₅×#359	C₂²	1	4	yes	complemented
23	C₅×#346	C₂²	1	2	yes	complemented
24	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
25	#33464	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	2	no	—
27	C₅×#135	—	2	2	no	—
28	C₅×#135	—	2	2	no	—
29	C₅×#135	C₂×C₄	1	4	yes	—
30	C₅×#135	C₂³	1	4	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₅×SD₁₆	—	2	2	no	—
33	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
34	C₅×#131	—	2	8	no	—
35	C₅×#131	—	4	4	no	—
36	C₂×C₅×M₁₆	—	2	2	no	—
37	C₂×C₅×M₁₆	—	4	4	no	—
38	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₅×#117	D₄	1	1	yes	characteristic
41	C₅×Q₃₂	C₂×C₄	1	4	yes	—
42	C₅×SD₃₂	C₂×C₄	1	8	yes	—
43	C₅×D₁₆	C₂×C₄	1	4	yes	—
44	C₅×M₃₂	C₂×C₄	1	4	yes	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂³	1	2	yes	—
49	C₅×#108	—	2	2	no	—
50	C₅×#108	C₂³	1	4	yes	—
51	C₅×#104	—	2	8	no	—
52	C₅×#97	D₄	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	#3414	C₂×C₅	1	1	yes	characteristic
55	#2957	C₂×C₅	1	2	yes	complemented
56	#2179	C₂×C₅	1	4	yes	complemented
57	#2178	C₂×C₅	1	4	yes	complemented
58	#2161	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#359	C₂×C₅	1	2	yes	complemented
60	#2108	C₂×C₅	1	1	yes	characteristic, complemented
61	C₅×#41	—	2	4	no	—
62	C₅×#41	—	2	4	no	—
63	C₅×#41	—	4	2	no	—
64	C₅×#41	—	4	2	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×Q₁₆	—	2	2	no	—
71	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
72	C₅×SD₁₆	—	2	4	no	—
73	C₅×SD₁₆	—	2	4	no	—
74	C₅×D₈	—	2	2	no	—
75	C₅×D₈	C₂²×C₄	1	4	yes	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₅×M₁₆	—	4	8	no	—
79	C₅×M₁₆	D₈	1	4	yes	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	8	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₂²⋊C₄	—	2	1	no	—
89	C₄²×C₅	C₂×D₄	1	2	yes	—
90	C₅×C₁₆	—	2	1	no	—
91	C₅×C₁₆	—	2	1	no	—
92	C₅×C₁₆	C₂²×C₄	1	4	yes	—
93	C₂×#135	C₂²×C₅	1	2	yes	—
94	#568	—	2	2	no	—
95	#510	C₄×C₅	1	4	yes	complemented
96	#509	C₄×C₅	1	4	yes	complemented
97	#508	C₂²×C₅	1	4	yes	—
98	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
100	#444	—	2	4	no	—
101	#443	C₂²×C₅	1	8	yes	complemented
102	C₂×#108	C₂²×C₅	1	2	yes	—
103	#421	—	2	2	no	—
104	#405	C₂²×C₅	1	1	yes	characteristic
105	#368	C₂²×C₅	1	4	yes	complemented
106	#359	C₂²×C₅	1	4	yes	complemented
107	#359	C₂²×C₅	1	4	yes	complemented
108	#346	C₂²×C₅	1	2	yes	complemented
109	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	#135	1	1	yes	characteristic
112	C₅×Q₈	—	2	4	no	—
113	C₅×Q₈	—	4	2	no	—
114	C₅×D₄	—	2	4	no	—
115	C₅×D₄	—	4	4	no	—
116	C₅×D₄	—	4	4	no	—
117	C₅×D₄	—	4	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅	#135	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₅×C₈	—	4	4	no	—
127	C₅×C₈	C₄×D₄	1	2	yes	—
128	C₅×C₈	C₂×D₈	1	4	yes	—
129	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
130	C₅×C₈	C₂×#41	1	1	yes	characteristic
131	C₂×#41	—	2	2	no	—
132	#135	—	2	2	no	—
133	#135	—	2	2	no	—
134	#135	C₂×C₄×C₅	1	4	yes	—
135	#135	C₂³×C₅	1	4	yes	—
136	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
137	C₂×SD₁₆	—	2	2	no	—
138	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
139	#131	—	2	8	no	—
140	#131	—	4	4	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	4	4	no	—
143	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
145	#117	C₅×D₄	1	1	yes	characteristic
146	Q₃₂	C₂×C₄×C₅	1	4	yes	—
147	SD₃₂	C₂×C₄×C₅	1	8	yes	—
148	D₁₆	C₂×C₄×C₅	1	4	yes	—
149	M₃₂	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
152	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
153	C₂×C₁₆	C₂³×C₅	1	2	yes	—
154	#108	—	2	2	no	—
155	#108	C₂³×C₅	1	4	yes	—
156	#104	—	2	8	no	—
157	#97	C₅×D₄	1	2	yes	—
158	C₄×C₈	C₂³×C₅	1	2	yes	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	—	4	4	no	—
161	C₂²×C₅	—	16	2	no	—
162	C₂²×C₅	C₄×D₈	1	2	yes	—
163	C₂²×C₅	C₂×#135	1	1	yes	characteristic
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	C₄×D₈	1	2	yes	—
167	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
169	#41	—	2	4	no	—
170	#41	—	2	4	no	—
171	#41	—	4	2	no	—
172	#41	—	4	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	Q₁₆	—	2	2	no	—
179	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
180	SD₁₆	—	2	4	no	—
181	SD₁₆	—	2	4	no	—
182	D₈	—	2	2	no	—
183	D₈	C₂²×C₄×C₅	1	4	yes	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	4	2	no	—
186	M₁₆	—	4	8	no	—
187	M₁₆	C₅×D₈	1	4	yes	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	8	no	—
190	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
191	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
193	C₂×C₈	C₅×#41	1	2	yes	—
194	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
195	C₄⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₄²	C₂×C₅×D₄	1	2	yes	—
198	C₁₆	—	2	1	no	—
199	C₁₆	—	2	1	no	—
200	C₁₆	C₂²×C₄×C₅	1	4	yes	—
201	C₂×C₅	—	2	1	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	—	8	4	no	—
204	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
205	C₂³	—	4	2	no	—
206	C₂³	C₅×#135	1	1	yes	characteristic
207	Q₈	—	2	4	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
218	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
219	C₂×C₄	C₅×#135	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
221	C₈	—	4	4	no	—
222	C₈	C₄×C₅×D₄	1	2	yes	—
223	C₈	C₂×C₅×D₈	1	4	yes	—
224	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
225	C₈	C₂×C₅×#41	1	1	yes	characteristic
226	C₅	#33464	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	C₄×C₅×D₈	1	2	yes	—
231	C₂²	C₂×C₅×#135	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₄×C₅×D₈	1	2	yes	—
235	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
236	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×C₄×D₈
prime radical	C₅×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2957	C₂	semidirect
3. C₅×#2179	C₂	semidirect
4. C₅×#2178	C₂	semidirect
5. C₅×#2161	C₂	semidirect
6. C₂×C₅×#359	C₂	semidirect
7. C₅×#2108	C₂	semidirect
10. C₅×#510	C₄	semidirect
11. C₅×#509	C₄	semidirect
16. C₅×#443	C₂²	semidirect
20. C₅×#368	C₂²	semidirect
21. C₅×#359	C₂²	semidirect
22. C₅×#359	C₂²	semidirect
23. C₅×#346	C₂²	semidirect
24. C₄×C₅×C₁₆	C₂²	semidirect
25. #33464	C₅	direct (N × H)
55. #2957	C₂×C₅	semidirect
56. #2179	C₂×C₅	semidirect
57. #2178	C₂×C₅	semidirect
58. #2161	C₂×C₅	semidirect
59. C₂×#359	C₂×C₅	semidirect
60. #2108	C₂×C₅	semidirect
95. #510	C₄×C₅	semidirect
96. #509	C₄×C₅	semidirect
101. #443	C₂²×C₅	semidirect
105. #368	C₂²×C₅	semidirect
106. #359	C₂²×C₅	semidirect
107. #359	C₂²×C₅	semidirect
108. #346	C₂²×C₅	semidirect
109. C₄×C₁₆	C₂²×C₅	semidirect
226. C₅	#33464	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×#3414	2
C₅×#2957	2
C₅×#2179	2
C₅×#2178	2
C₅×#2161	2
C₂×C₅×#359	2
C₅×#2108	2
#33464	5

Subgroups — C5×#33464 #11956184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33464 #11956184