Subgroups — C₅×#9608 #11932328

Order 1280.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#1565	C₂	1	2	yes	maximal
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#1450	C₂	1	1	yes	maximal, characteristic
5	C₅×#1283	C₂	1	2	yes	maximal
6	C₅×#431	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₄	1	1	yes	characteristic
9	C₂×C₅×#108	C₄	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#341	C₄	1	2	yes	—
14	C₅×#335	—	2	2	no	—
15	C₅×#335	—	2	1	no	—
16	C₅×#335	—	2	1	no	—
17	C₅×#334	—	2	2	no	—
18	C₅×#322	—	2	1	no	—
19	C₅×C₈²	—	2	1	no	—
20	#9608	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×#108	—	2	2	no	—
28	C₅×#108	C₈	1	4	yes	complemented
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	1	no	—
32	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
34	C₅×C₂²⋊C₈	—	2	1	no	—
35	C₅×C₂²⋊C₈	Q₈	1	2	yes	—
36	C₅×#97	—	2	2	no	—
37	C₄×C₅×C₈	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₄×C₅×C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
43	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
44	C₄×#108	C₂×C₅	1	1	yes	characteristic
45	#1565	C₂×C₅	1	2	yes	—
46	C₂×#335	C₂×C₅	1	1	yes	characteristic
47	#1450	C₂×C₅	1	1	yes	characteristic
48	#1283	C₂×C₅	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	#41	1	1	yes	characteristic
52	C₅×M₁₆	—	2	2	no	—
53	C₅×M₁₆	—	2	2	no	—
54	C₅×M₁₆	—	2	2	no	—
55	C₅×M₁₆	C₂×C₈	1	4	yes	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	8	2	no	—
65	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
70	C₂×C₅×C₈	M₁₆	1	2	yes	—
71	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅	#41	1	1	yes	characteristic
75	#431	C₂²×C₅	1	1	yes	characteristic
76	#431	C₂²×C₅	1	1	yes	characteristic
77	C₂×#108	C₄×C₅	1	1	yes	characteristic
78	C₂×#108	C₄×C₅	1	1	yes	characteristic
79	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
80	C₄×M₁₆	C₂²×C₅	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	#341	C₄×C₅	1	2	yes	—
83	#335	—	2	2	no	—
84	#335	—	2	1	no	—
85	#335	—	2	1	no	—
86	#334	—	2	2	no	—
87	#322	—	2	1	no	—
88	C₈²	—	2	1	no	—
89	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	4	1	no	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅	#121	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅	#125	1	1	yes	characteristic
98	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅	#131	1	1	yes	characteristic
102	C₂×C₄×C₅	#131	1	1	yes	characteristic
103	C₅×C₈	—	2	1	no	—
104	C₅×C₈	—	2	1	no	—
105	C₅×C₈	—	2	4	no	—
106	C₅×C₈	—	4	2	no	—
107	C₅×C₈	—	4	2	no	—
108	C₅×C₈	—	4	1	no	—
109	C₅×C₈	—	4	1	no	—
110	C₅×C₈	—	8	2	no	—
111	C₅×C₈	C₄⋊C₈	1	2	yes	—
112	C₅×C₈	C₄⋊C₈	1	2	yes	—
113	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
114	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
119	#108	—	2	2	no	—
120	#108	C₅×C₈	1	4	yes	complemented
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
125	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
126	C₂²⋊C₈	—	2	1	no	—
127	C₂²⋊C₈	C₅×Q₈	1	2	yes	—
128	#97	—	2	2	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₅×D₄	1	1	yes	characteristic
133	C₄×C₈	C₅×D₄	1	1	yes	characteristic
134	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
135	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	#389	1	1	yes	characteristic
139	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
140	C₂²×C₅	#446	1	1	yes	characteristic
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₄×C₅	#424	1	1	yes	characteristic
146	C₄×C₅	C₈×D₄	1	1	yes	characteristic
147	C₄×C₅	#436	1	1	yes	characteristic
148	C₄×C₅	#472	1	1	yes	characteristic
149	C₄×C₅	#473	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
152	C₂²×C₄	C₅×#41	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×C₅×C₈	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
171	C₂×C₈	C₅×M₁₆	1	2	yes	—
172	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
173	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
174	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₄²	C₅×#41	1	1	yes	characteristic
176	C₂×C₅	—	2	1	no	—
177	C₂×C₅	—	2	1	no	—
178	C₂×C₅	#1939	1	1	yes	characteristic
179	C₂×C₅	#1952	1	1	yes	characteristic
180	C₂×C₅	#1953	1	1	yes	characteristic
181	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
187	C₂×C₄	C₅×#121	1	1	yes	characteristic
188	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×C₄	C₅×#125	1	1	yes	characteristic
190	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
193	C₂×C₄	C₅×#131	1	1	yes	characteristic
194	C₂×C₄	C₅×#131	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	4	no	—
198	C₈	—	4	2	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	1	no	—
201	C₈	—	4	1	no	—
202	C₈	—	8	2	no	—
203	C₈	C₅×C₄⋊C₈	1	2	yes	—
204	C₈	C₅×C₄⋊C₈	1	2	yes	—
205	C₅	#9608	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₅×#389	1	1	yes	characteristic
209	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
210	C₂²	C₅×#446	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
215	C₄	C₅×#424	1	1	yes	characteristic
216	C₄	C₅×C₈×D₄	1	1	yes	characteristic
217	C₄	C₅×#436	1	1	yes	characteristic
218	C₄	C₅×#472	1	1	yes	characteristic
219	C₄	C₅×#473	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₅×#1939	1	1	yes	characteristic
223	C₂	C₅×#1952	1	1	yes	characteristic
224	C₂	C₅×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #9608	C₅	direct (N × H)
28. C₅×#108	C₈	semidirect
120. #108	C₅×C₈	semidirect
205. C₅	#9608	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#108	2
C₅×#1565	2
C₂×C₅×#335	2
C₅×#1450	2
C₅×#1283	2
#9608	5

Subgroups — C5×#9608 #11932328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9608 #11932328