Subgroups — C₅×#33860 #11956580

Order 1280.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3417	C₂	1	2	yes	maximal, complemented
2	C₅×#3293	C₂	1	2	yes	maximal, complemented
3	C₅×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2222	C₂	1	2	yes	maximal, complemented
5	C₅×#2214	C₂	1	2	yes	maximal, complemented
6	C₅×#2195	C₂	1	4	yes	maximal, complemented
7	C₅×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#578	—	2	2	no	—
10	C₅×#578	C₂²	1	4	yes	—
11	C₅×#510	—	2	4	no	—
12	C₅×#510	C₂²	1	4	yes	—
13	C₅×#508	—	2	2	no	—
14	C₅×#508	C₂²	1	1	yes	characteristic
15	C₂×C₅×Q₃₂	—	2	2	no	—
16	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
18	C₅×#504	C₂²	1	2	yes	—
19	C₅×#494	C₂²	1	2	yes	—
20	C₅×#473	—	2	2	no	—
21	C₅×#473	C₂²	1	4	yes	—
22	C₅×#456	—	2	4	no	—
23	C₅×#443	C₂²	1	2	yes	—
24	C₅×#443	C₂²	1	1	yes	characteristic
25	C₅×#368	C₂²	1	1	yes	characteristic
26	C₅×#364	C₂²	1	1	yes	characteristic
27	C₅×#362	C₂²	1	4	yes	—
28	C₅×#359	C₂²	1	2	yes	—
29	C₅×#350	C₂²	1	2	yes	—
30	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
31	#33860	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₅×2₋¹⁺⁴	—	2	4	no	—
33	C₅×#137	—	2	4	no	—
34	C₅×#137	—	2	4	no	—
35	C₅×#137	—	4	4	no	—
36	C₅×#137	—	4	4	no	—
37	C₅×#135	—	2	4	no	—
38	C₅×#135	—	4	2	no	—
39	C₅×#135	D₄	1	1	yes	characteristic
40	C₅×#135	C₂³	1	2	yes	—
41	C₂×C₅×Q₁₆	—	2	4	no	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×Q₁₆	—	4	2	no	—
44	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
45	C₅×#131	—	2	2	no	—
46	C₅×#131	C₂³	1	2	yes	—
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×Q₃₂	—	2	4	no	—
49	C₅×Q₃₂	—	2	2	no	—
50	C₅×Q₃₂	—	2	2	no	—
51	C₅×Q₃₂	—	4	2	no	—
52	C₅×Q₃₂	D₄	1	1	yes	characteristic
53	C₅×SD₃₂	—	2	4	no	—
54	C₅×SD₃₂	D₄	1	2	yes	—
55	C₅×D₁₆	D₄	1	1	yes	characteristic
56	C₅×M₃₂	—	2	2	no	—
57	C₅×M₃₂	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	—	2	2	no	—
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
61	C₅×#108	D₄	1	1	yes	characteristic
62	C₅×#108	C₂³	1	2	yes	—
63	C₅×#104	—	2	2	no	—
64	C₅×#104	D₄	1	2	yes	—
65	C₅×#101	—	2	4	no	—
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	#3417	C₂×C₅	1	2	yes	complemented
68	#3293	C₂×C₅	1	2	yes	complemented
69	#2247	C₂×C₅	1	1	yes	characteristic, complemented
70	#2222	C₂×C₅	1	2	yes	complemented
71	#2214	C₂×C₅	1	2	yes	complemented
72	#2195	C₂×C₅	1	4	yes	complemented
73	#2178	C₂×C₅	1	1	yes	characteristic, complemented
74	#2170	C₂×C₅	1	1	yes	characteristic, complemented
75	C₅×#41	—	2	2	no	—
76	C₅×#41	—	4	4	no	—
77	C₅×#41	—	4	4	no	—
78	C₅×#41	—	8	4	no	—
79	C₅×#41	C₂×D₄	1	2	yes	—
80	C₂×C₅×Q₈	—	2	4	no	—
81	C₂×C₅×Q₈	—	4	4	no	—
82	C₂×C₅×Q₈	—	4	4	no	—
83	C₅×Q₁₆	—	2	4	no	—
84	C₅×Q₁₆	—	2	4	no	—
85	C₅×Q₁₆	—	4	4	no	—
86	C₅×Q₁₆	—	4	4	no	—
87	C₅×Q₁₆	—	4	2	no	—
88	C₅×Q₁₆	—	4	2	no	—
89	C₅×Q₁₆	D₈	1	1	yes	characteristic
90	C₅×Q₁₆	C₂×D₄	1	2	yes	—
91	C₅×SD₁₆	—	2	2	no	—
92	C₅×SD₁₆	—	4	4	no	—
93	C₅×SD₁₆	—	4	4	no	—
94	C₅×SD₁₆	D₈	1	2	yes	—
95	C₅×D₈	D₈	1	1	yes	characteristic
96	C₅×D₈	C₂×D₄	1	2	yes	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	4	2	no	—
100	C₅×M₁₆	C₂×D₄	1	2	yes	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	4	2	no	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	8	2	no	—
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₅×C₁₆	—	2	2	no	—
108	C₅×C₁₆	—	2	1	no	—
109	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
110	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
111	#578	—	2	2	no	—
112	#578	C₂²×C₅	1	4	yes	—
113	#510	—	2	4	no	—
114	#510	C₂²×C₅	1	4	yes	—
115	#508	—	2	2	no	—
116	#508	C₂²×C₅	1	1	yes	characteristic
117	C₂×Q₃₂	—	2	2	no	—
118	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
119	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
120	#504	C₂²×C₅	1	2	yes	—
121	#494	C₂²×C₅	1	2	yes	—
122	#473	—	2	2	no	—
123	#473	C₂²×C₅	1	4	yes	—
124	#456	—	2	4	no	—
125	#443	C₂²×C₅	1	2	yes	—
126	#443	C₂²×C₅	1	1	yes	characteristic
127	#368	C₂²×C₅	1	1	yes	characteristic
128	#364	C₂²×C₅	1	1	yes	characteristic
129	#362	C₂²×C₅	1	4	yes	—
130	#359	C₂²×C₅	1	2	yes	—
131	#350	C₂²×C₅	1	2	yes	—
132	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
133	C₅×Q₈	—	2	2	no	—
134	C₅×Q₈	—	4	4	no	—
135	C₅×Q₈	—	4	4	no	—
136	C₅×Q₈	—	8	4	no	—
137	C₅×Q₈	C₂×D₈	1	2	yes	—
138	C₅×D₄	—	2	2	no	—
139	C₅×D₄	—	2	2	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	8	4	no	—
142	C₅×D₄	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	8	4	no	—
148	C₂×C₄×C₅	—	8	4	no	—
149	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	C₂×D₈	1	1	yes	characteristic
153	C₅×C₈	C₂×D₈	1	1	yes	characteristic
154	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
155	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
156	2₋¹⁺⁴	—	2	4	no	—
157	#137	—	2	4	no	—
158	#137	—	2	4	no	—
159	#137	—	4	4	no	—
160	#137	—	4	4	no	—
161	#135	—	2	4	no	—
162	#135	—	4	2	no	—
163	#135	C₅×D₄	1	1	yes	characteristic
164	#135	C₂³×C₅	1	2	yes	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	4	2	no	—
168	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
169	#131	—	2	2	no	—
170	#131	C₂³×C₅	1	2	yes	—
171	C₄⋊Q₈	—	2	2	no	—
172	Q₃₂	—	2	4	no	—
173	Q₃₂	—	2	2	no	—
174	Q₃₂	—	2	2	no	—
175	Q₃₂	—	4	2	no	—
176	Q₃₂	C₅×D₄	1	1	yes	characteristic
177	SD₃₂	—	2	4	no	—
178	SD₃₂	C₅×D₄	1	2	yes	—
179	D₁₆	C₅×D₄	1	1	yes	characteristic
180	M₃₂	—	2	2	no	—
181	M₃₂	C₂³×C₅	1	2	yes	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
184	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
185	#108	C₅×D₄	1	1	yes	characteristic
186	#108	C₂³×C₅	1	2	yes	—
187	#104	—	2	2	no	—
188	#104	C₅×D₄	1	2	yes	—
189	#101	—	2	4	no	—
190	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	4	2	no	—
193	C₂²×C₅	#576	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	4	2	no	—
197	C₄×C₅	—	8	4	no	—
198	C₄×C₅	D₄²	1	1	yes	characteristic
199	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
200	#41	—	2	2	no	—
201	#41	—	4	4	no	—
202	#41	—	4	4	no	—
203	#41	—	8	4	no	—
204	#41	C₂×C₅×D₄	1	2	yes	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	4	no	—
208	Q₁₆	—	2	4	no	—
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	4	no	—
211	Q₁₆	—	4	4	no	—
212	Q₁₆	—	4	2	no	—
213	Q₁₆	—	4	2	no	—
214	Q₁₆	C₅×D₈	1	1	yes	characteristic
215	Q₁₆	C₂×C₅×D₄	1	2	yes	—
216	SD₁₆	—	2	2	no	—
217	SD₁₆	—	4	4	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	C₅×D₈	1	2	yes	—
220	D₈	C₅×D₈	1	1	yes	characteristic
221	D₈	C₂×C₅×D₄	1	2	yes	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	C₂×C₅×D₄	1	2	yes	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	8	2	no	—
231	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
235	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	4	2	no	—
238	C₂×C₅	—	8	2	no	—
239	C₂×C₅	D₄×D₈	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	8	4	no	—
244	Q₈	C₂×C₅×D₈	1	2	yes	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	8	4	no	—
249	D₄	C₂×C₅×D₈	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	C₂×C₅×D₈	1	1	yes	characteristic
260	C₈	C₂×C₅×D₈	1	1	yes	characteristic
261	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
262	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
263	C₅	#33860	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₅×#576	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	C₅×D₄²	1	1	yes	characteristic
272	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	—	8	2	no	—
276	C₂	C₅×D₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3417	C₂	semidirect
2. C₅×#3293	C₂	semidirect
3. C₅×#2247	C₂	semidirect
4. C₅×#2222	C₂	semidirect
5. C₅×#2214	C₂	semidirect
6. C₅×#2195	C₂	semidirect
7. C₅×#2178	C₂	semidirect
8. C₅×#2170	C₂	semidirect
31. #33860	C₅	direct (N × H)
67. #3417	C₂×C₅	semidirect
68. #3293	C₂×C₅	semidirect
69. #2247	C₂×C₅	semidirect
70. #2222	C₂×C₅	semidirect
71. #2214	C₂×C₅	semidirect
72. #2195	C₂×C₅	semidirect
73. #2178	C₂×C₅	semidirect
74. #2170	C₂×C₅	semidirect
263. C₅	#33860	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×#3417	2
C₅×#3293	2
C₅×#2247	2
C₅×#2222	2
C₅×#2214	2
C₅×#2195	2
C₅×#2178	2
C₅×#2170	2
#33860	5

Subgroups — C5×#33860 #11956580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33860 #11956580