Subgroups — C₅×#33658 #11956378

Order 1280.

534 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3059	C₂	1	1	yes	maximal, characteristic
2	C₅×#2916	C₂	1	1	yes	maximal, characteristic
3	C₅×#2239	C₂	1	8	yes	maximal, complemented
4	C₅×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#362	C₂	1	2	yes	maximal, complemented
6	C₅×#2115	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₅×#568	—	2	1	no	—
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₅×#536	—	2	1	no	—
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#470	C₂²	1	4	yes	—
13	C₅×#467	—	2	2	no	—
14	C₅×#430	—	2	1	no	—
15	C₅×#430	C₂²	1	4	yes	—
16	C₂×C₅×#108	—	2	2	no	—
17	C₂×C₅×#108	C₂²	1	2	yes	—
18	C₅×#427	C₂²	1	1	yes	characteristic
19	C₅×#419	—	2	1	no	—
20	C₅×#365	C₂²	1	8	yes	—
21	C₅×#362	C₂²	1	8	yes	—
22	C₅×#349	C₂²	1	4	yes	—
23	#33658	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	D₄	1	1	yes	characteristic
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂×C₅×Q₁₆	—	2	1	no	—
27	C₂×C₅×Q₁₆	—	2	2	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
30	C₅×#131	—	2	2	no	—
31	C₅×#131	—	2	2	no	—
32	C₅×#131	—	4	4	no	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊Q₈	—	4	1	no	—
40	C₅×#124	—	4	1	no	—
41	C₅×#123	—	4	2	no	—
42	C₅×C₂²⋊Q₈	—	2	4	no	—
43	C₅×#117	—	2	1	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	C₂³	1	4	yes	—
46	C₂×C₅×C₁₆	—	2	2	no	—
47	C₅×#108	—	2	2	no	—
48	C₅×#108	—	2	4	no	—
49	C₅×#108	C₂³	1	4	yes	—
50	C₅×#107	—	2	1	no	—
51	C₅×#106	C₂³	1	2	yes	—
52	C₅×#103	—	2	4	no	—
53	#3059	C₂×C₅	1	1	yes	characteristic
54	#2916	C₂×C₅	1	1	yes	characteristic
55	#2239	C₂×C₅	1	8	yes	complemented
56	#2155	C₂×C₅	1	2	yes	complemented
57	C₂×#362	C₂×C₅	1	2	yes	complemented
58	#2115	C₂×C₅	1	1	yes	characteristic
59	C₅×#41	—	2	2	no	—
60	C₅×#41	—	2	2	no	—
61	C₂×C₅×Q₈	—	2	2	no	—
62	C₂×C₅×Q₈	—	4	1	no	—
63	C₂×C₅×Q₈	—	4	2	no	—
64	C₂×C₅×Q₈	—	4	2	no	—
65	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
66	C₂×C₅×D₄	SD₁₆	1	2	yes	—
67	C₂×C₅×D₄	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₅×Q₁₆	—	2	4	no	—
72	C₅×Q₁₆	—	4	1	no	—
73	C₅×Q₁₆	—	4	1	no	—
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	4	4	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	4	4	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	#41	1	2	yes	—
87	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₂²⋊C₄	—	4	1	no	—
91	C₅×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₅	—	4	1	no	—
93	C₅×C₁₆	—	2	4	no	—
94	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
95	#568	—	2	1	no	—
96	C₂×#131	C₂²×C₅	1	1	yes	characteristic
97	#536	—	2	1	no	—
98	C₂×M₃₂	C₂²×C₅	1	2	yes	—
99	#470	C₂²×C₅	1	4	yes	—
100	#467	—	2	2	no	—
101	#430	—	2	1	no	—
102	#430	C₂²×C₅	1	4	yes	—
103	C₂×#108	—	2	2	no	—
104	C₂×#108	C₂²×C₅	1	2	yes	—
105	#427	C₂²×C₅	1	1	yes	characteristic
106	#419	—	2	1	no	—
107	#365	C₂²×C₅	1	8	yes	—
108	#362	C₂²×C₅	1	8	yes	—
109	#349	C₂²×C₅	1	4	yes	—
110	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
111	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₅×Q₈	—	2	2	no	—
113	C₅×Q₈	—	4	2	no	—
114	C₅×Q₈	—	4	2	no	—
115	C₅×Q₈	—	8	2	no	—
116	C₅×D₄	—	2	4	no	—
117	C₅×D₄	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	8	2	no	—
123	C₂×C₄×C₅	#123	1	2	yes	—
124	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	4	4	no	—
130	C₅×C₈	#123	1	1	yes	characteristic
131	C₅×C₈	#123	1	1	yes	characteristic
132	C₅×C₈	C₂×#41	1	2	yes	—
133	C₂×#41	C₅×D₄	1	1	yes	characteristic
134	C₂²×Q₈	—	2	1	no	—
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
139	#131	—	2	2	no	—
140	#131	—	2	2	no	—
141	#131	—	4	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
145	C₂×M₁₆	C₂³×C₅	1	2	yes	—
146	C₂²×C₈	C₅×D₄	1	2	yes	—
147	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
148	C₄⋊Q₈	—	4	1	no	—
149	#124	—	4	1	no	—
150	#123	—	4	2	no	—
151	C₂²⋊Q₈	—	2	4	no	—
152	#117	—	2	1	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	C₂³×C₅	1	4	yes	—
155	C₂×C₁₆	—	2	2	no	—
156	#108	—	2	2	no	—
157	#108	—	2	4	no	—
158	#108	C₂³×C₅	1	4	yes	—
159	#107	—	2	1	no	—
160	#106	C₂³×C₅	1	2	yes	—
161	#103	—	2	4	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	4	1	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	#481	1	2	yes	—
166	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	8	2	no	—
169	C₄×C₅	—	8	2	no	—
170	C₄×C₅	#481	1	2	yes	—
171	C₄×C₅	C₂×#123	1	1	yes	characteristic
172	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	4	1	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
180	C₂×D₄	C₅×SD₁₆	1	2	yes	—
181	C₂×D₄	C₅×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	1	no	—
187	Q₁₆	—	4	1	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
198	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂×C₈	C₅×#41	1	2	yes	—
200	C₂×C₈	C₅×#41	1	2	yes	—
201	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₂²⋊C₄	—	4	1	no	—
205	C₂²⋊C₄	—	4	4	no	—
206	C₄²	—	4	1	no	—
207	C₁₆	—	2	4	no	—
208	C₂×C₅	—	2	1	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	4	2	no	—
211	C₂×C₅	C₂×#481	1	1	yes	characteristic
212	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
213	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	8	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₅×#123	1	2	yes	—
226	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
227	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
228	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
229	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	C₅×#123	1	1	yes	characteristic
233	C₈	C₅×#123	1	1	yes	characteristic
234	C₈	C₂×C₅×#41	1	2	yes	—
235	C₅	#33658	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	C₅×#481	1	2	yes	—
240	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	C₅×#481	1	2	yes	—
245	C₄	C₂×C₅×#123	1	1	yes	characteristic
246	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×C₅×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2239	C₂	semidirect
4. C₅×#2155	C₂	semidirect
5. C₂×C₅×#362	C₂	semidirect
23. #33658	C₅	direct (N × H)
55. #2239	C₂×C₅	semidirect
56. #2155	C₂×C₅	semidirect
57. C₂×#362	C₂×C₅	semidirect
235. C₅	#33658	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×#3059	2
C₅×#2916	2
C₅×#2239	2
C₅×#2155	2
C₂×C₅×#362	2
C₅×#2115	2
#33658	5

Subgroups — C5×#33658 #11956378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33658 #11956378