Subgroups — C₅×#33768 #11956488

Order 1280.

552 conjugacy classes of subgroups.

The subgroup lattice has 250 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	2	yes	maximal
2	C₅×#2237	C₂	1	4	yes	maximal, complemented
3	C₅×#2219	C₂	1	2	yes	maximal, complemented
4	C₅×#2216	C₂	1	2	yes	maximal, complemented
5	C₅×#2188	C₂	1	4	yes	maximal, complemented
6	C₅×#2113	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#536	—	2	2	no	—
10	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₅×#470	—	2	2	no	—
15	C₅×#470	—	2	2	no	—
16	C₅×#467	C₂²	1	4	yes	—
17	C₅×#467	C₂²	1	4	yes	—
18	C₅×#427	C₂²	1	2	yes	—
19	C₅×#419	—	2	2	no	—
20	C₅×#367	C₂²	1	2	yes	—
21	C₅×#366	C₂²	1	2	yes	—
22	C₅×#358	C₂²	1	4	yes	—
23	C₅×#358	C₂²	1	4	yes	—
24	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	#33768	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	D₄	1	1	yes	characteristic
27	C₂²×C₅×Q₈	—	2	2	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	—	4	2	no	—
30	C₂×C₅×Q₁₆	—	4	2	no	—
31	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
32	C₅×#131	—	4	1	no	—
33	C₅×#131	—	4	1	no	—
34	C₂×C₅×M₁₆	D₄	1	2	yes	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊Q₈	—	4	2	no	—
39	C₅×#124	—	4	2	no	—
40	C₅×#123	—	4	2	no	—
41	C₅×#123	—	4	2	no	—
42	C₅×C₂²⋊Q₈	—	2	4	no	—
43	C₅×C₂²⋊Q₈	—	2	4	no	—
44	C₅×#117	—	2	2	no	—
45	C₅×Q₃₂	—	4	2	no	—
46	C₅×Q₃₂	—	4	2	no	—
47	C₅×M₃₂	—	4	1	no	—
48	C₂×C₅×C₁₆	—	4	1	no	—
49	C₂×C₅×C₁₆	C₂³	1	2	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₅×#107	C₂³	1	4	yes	—
52	C₅×#106	—	2	2	no	—
53	C₅×#103	—	2	4	no	—
54	C₅×#103	—	2	4	no	—
55	#3059	C₂×C₅	1	2	yes	—
56	#2237	C₂×C₅	1	4	yes	complemented
57	#2219	C₂×C₅	1	2	yes	complemented
58	#2216	C₂×C₅	1	2	yes	complemented
59	#2188	C₂×C₅	1	4	yes	complemented
60	#2113	C₂×C₅	1	1	yes	characteristic
61	C₅×#41	—	4	1	no	—
62	C₅×#41	—	4	1	no	—
63	C₂×C₅×Q₈	—	2	4	no	—
64	C₂×C₅×Q₈	—	4	4	no	—
65	C₂×C₅×Q₈	—	4	4	no	—
66	C₂×C₅×Q₈	—	4	2	no	—
67	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
68	C₂×C₅×D₄	D₈	1	1	yes	characteristic
69	C₂×C₅×D₄	D₈	1	2	yes	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	4	no	—
74	C₅×Q₁₆	—	2	4	no	—
75	C₅×Q₁₆	—	4	4	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	4	4	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₂²⋊C₄	—	4	2	no	—
89	C₅×C₂²⋊C₄	—	4	4	no	—
90	C₅×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₅	—	4	2	no	—
92	C₅×C₁₆	—	2	2	no	—
93	C₅×C₁₆	—	2	2	no	—
94	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
95	C₂×#131	C₂²×C₅	1	1	yes	characteristic
96	#536	—	2	2	no	—
97	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
98	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
99	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
100	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
101	#470	—	2	2	no	—
102	#470	—	2	2	no	—
103	#467	C₂²×C₅	1	4	yes	—
104	#467	C₂²×C₅	1	4	yes	—
105	#427	C₂²×C₅	1	2	yes	—
106	#419	—	2	2	no	—
107	#367	C₂²×C₅	1	2	yes	—
108	#366	C₂²×C₅	1	2	yes	—
109	#358	C₂²×C₅	1	4	yes	—
110	#358	C₂²×C₅	1	4	yes	—
111	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
112	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
113	C₂³×C₅	C₂×D₈	1	2	yes	—
114	C₅×Q₈	—	4	1	no	—
115	C₅×Q₈	—	4	4	no	—
116	C₅×Q₈	—	4	4	no	—
117	C₅×Q₈	—	8	4	no	—
118	C₅×D₄	—	4	1	no	—
119	C₅×D₄	—	4	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	8	4	no	—
127	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₅×C₈	—	2	1	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
133	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×#41	C₅×D₄	1	1	yes	characteristic
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
140	#131	—	4	1	no	—
141	#131	—	4	1	no	—
142	C₂×M₁₆	C₅×D₄	1	2	yes	—
143	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	2	yes	—
145	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
146	C₄⋊Q₈	—	4	2	no	—
147	#124	—	4	2	no	—
148	#123	—	4	2	no	—
149	#123	—	4	2	no	—
150	C₂²⋊Q₈	—	2	4	no	—
151	C₂²⋊Q₈	—	2	4	no	—
152	#117	—	2	2	no	—
153	Q₃₂	—	4	2	no	—
154	Q₃₂	—	4	2	no	—
155	M₃₂	—	4	1	no	—
156	C₂×C₁₆	—	4	1	no	—
157	C₂×C₁₆	C₂³×C₅	1	2	yes	—
158	C₂×C₁₆	C₂³×C₅	1	2	yes	—
159	#107	C₂³×C₅	1	4	yes	—
160	#106	—	2	2	no	—
161	#103	—	2	4	no	—
162	#103	—	2	4	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	8	4	no	—
171	C₄×C₅	—	8	4	no	—
172	C₄×C₅	#534	1	1	yes	characteristic
173	C₄×C₅	#575	1	1	yes	characteristic
174	#41	—	4	1	no	—
175	#41	—	4	1	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂×Q₈	—	4	4	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
181	C₂×D₄	C₅×D₈	1	1	yes	characteristic
182	C₂×D₄	C₅×D₈	1	2	yes	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	2	4	no	—
188	Q₁₆	—	4	4	no	—
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
197	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	—	4	2	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	—	2	2	no	—
207	C₂×C₅	—	4	2	no	—
208	C₂×C₅	—	4	1	no	—
209	C₂×C₅	#3186	1	1	yes	characteristic
210	C₂×C₅	#3417	1	2	yes	—
211	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
212	C₂³	C₂×C₅×D₈	1	2	yes	—
213	Q₈	—	4	1	no	—
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	8	4	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
227	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	1	no	—
231	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
232	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
233	C₅	#33768	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	4	no	—
243	C₄	C₅×#534	1	1	yes	characteristic
244	C₄	C₅×#575	1	1	yes	characteristic
245	C₂	—	4	2	no	—
246	C₂	—	4	1	no	—
247	C₂	C₅×#3186	1	1	yes	characteristic
248	C₂	C₅×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2237	C₂	semidirect
3. C₅×#2219	C₂	semidirect
4. C₅×#2216	C₂	semidirect
5. C₅×#2188	C₂	semidirect
25. #33768	C₅	direct (N × H)
56. #2237	C₂×C₅	semidirect
57. #2219	C₂×C₅	semidirect
58. #2216	C₂×C₅	semidirect
59. #2188	C₂×C₅	semidirect
233. C₅	#33768	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×#2237	2
C₅×#2219	2
C₅×#2216	2
C₅×#2188	2
C₅×#2113	2
#33768	5

Subgroups — C5×#33768 #11956488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33768 #11956488