Subgroups — C₅×#33659 #11956379

Order 1280.

546 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3060	C₂	1	1	yes	maximal, characteristic
2	C₅×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2239	C₂	1	4	yes	maximal, complemented
4	C₅×#2238	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#365	C₂	1	2	yes	maximal, complemented
6	C₅×#2148	C₂	1	2	yes	maximal, complemented
7	C₅×#2115	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₅×#568	—	2	1	no	—
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₅×#537	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₅×#470	C₂²	1	2	yes	—
14	C₅×#469	C₂²	1	2	yes	—
15	C₅×#465	—	2	2	no	—
16	C₅×#430	—	2	1	no	—
17	C₅×#430	C₂²	1	4	yes	—
18	C₂×C₅×#108	—	2	2	no	—
19	C₂×C₅×#108	C₂²	1	2	yes	—
20	C₂×C₅×#106	C₂²	1	1	yes	characteristic
21	C₅×#417	—	2	1	no	—
22	C₅×#365	C₂²	1	8	yes	—
23	C₅×#362	C₂²	1	4	yes	—
24	C₅×#361	C₂²	1	4	yes	—
25	C₅×#349	C₂²	1	4	yes	—
26	#33659	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	1	no	—
28	C₂×C₅×#41	D₄	1	1	yes	characteristic
29	C₅×#135	—	2	2	no	—
30	C₅×#135	—	2	2	no	—
31	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
34	C₅×#131	—	2	2	no	—
35	C₅×#131	—	2	2	no	—
36	C₅×#131	—	4	4	no	—
37	C₂×C₅×M₁₆	—	2	4	no	—
38	C₂×C₅×M₁₆	—	2	2	no	—
39	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	D₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₂²⋊Q₈	—	2	2	no	—
44	C₅×C₂²⋊Q₈	—	4	1	no	—
45	C₅×#121	—	2	2	no	—
46	C₅×#121	—	4	1	no	—
47	C₅×#121	—	4	2	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₅×M₃₂	—	2	2	no	—
50	C₅×M₃₂	C₂³	1	4	yes	—
51	C₂×C₅×C₁₆	—	2	2	no	—
52	C₅×#108	—	2	2	no	—
53	C₅×#108	—	2	4	no	—
54	C₅×#108	C₂³	1	4	yes	—
55	C₅×#106	—	2	1	no	—
56	C₅×#106	C₂³	1	2	yes	—
57	C₅×#103	—	2	2	no	—
58	C₅×#102	—	2	2	no	—
59	#3060	C₂×C₅	1	1	yes	characteristic
60	#2916	C₂×C₅	1	1	yes	characteristic, complemented
61	#2239	C₂×C₅	1	4	yes	complemented
62	#2238	C₂×C₅	1	4	yes	complemented
63	C₂×#365	C₂×C₅	1	2	yes	complemented
64	#2148	C₂×C₅	1	2	yes	complemented
65	#2115	C₂×C₅	1	1	yes	characteristic, complemented
66	C₅×#41	—	2	2	no	—
67	C₅×#41	—	2	2	no	—
68	C₅×#41	—	4	2	no	—
69	C₅×#41	—	4	2	no	—
70	C₂×C₅×Q₈	—	2	1	no	—
71	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
72	C₂×C₅×D₄	—	2	1	no	—
73	C₂×C₅×D₄	—	4	1	no	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂×C₅×D₄	SD₁₆	1	2	yes	—
76	C₂×C₅×D₄	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	2	2	no	—
82	C₅×SD₁₆	—	4	1	no	—
83	C₅×SD₁₆	—	4	1	no	—
84	C₅×D₈	—	2	2	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	2	4	no	—
88	C₅×M₁₆	—	4	4	no	—
89	C₅×M₁₆	—	4	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	4	4	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	#41	1	2	yes	—
97	C₂×C₅×C₈	#41	1	2	yes	—
98	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₁₆	—	2	4	no	—
105	C₂×#135	C₂²×C₅	1	1	yes	characteristic
106	#568	—	2	1	no	—
107	C₂×#131	C₂²×C₅	1	1	yes	characteristic
108	#537	—	2	1	no	—
109	C₂×M₃₂	C₂²×C₅	1	2	yes	—
110	#470	C₂²×C₅	1	2	yes	—
111	#469	C₂²×C₅	1	2	yes	—
112	#465	—	2	2	no	—
113	#430	—	2	1	no	—
114	#430	C₂²×C₅	1	4	yes	—
115	C₂×#108	—	2	2	no	—
116	C₂×#108	C₂²×C₅	1	2	yes	—
117	C₂×#106	C₂²×C₅	1	1	yes	characteristic
118	#417	—	2	1	no	—
119	#365	C₂²×C₅	1	8	yes	—
120	#362	C₂²×C₅	1	4	yes	—
121	#361	C₂²×C₅	1	4	yes	—
122	#349	C₂²×C₅	1	4	yes	—
123	C₂³×C₅	—	4	1	no	—
124	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
125	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₅×Q₈	—	2	2	no	—
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	4	1	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	4	1	no	—
132	C₅×D₄	—	4	1	no	—
133	C₅×D₄	—	8	2	no	—
134	C₅×D₄	—	16	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	8	2	no	—
140	C₂×C₄×C₅	—	16	1	no	—
141	C₂×C₄×C₅	#123	1	2	yes	—
142	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	4	4	no	—
148	C₅×C₈	#123	1	1	yes	characteristic
149	C₅×C₈	#123	1	1	yes	characteristic
150	C₅×C₈	C₂×#41	1	2	yes	—
151	C₂×#41	—	2	1	no	—
152	C₂×#41	C₅×D₄	1	1	yes	characteristic
153	#135	—	2	2	no	—
154	#135	—	2	2	no	—
155	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
156	C₂×SD₁₆	—	2	1	no	—
157	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
158	#131	—	2	2	no	—
159	#131	—	2	2	no	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
164	C₂×M₁₆	C₂³×C₅	1	2	yes	—
165	C₂²×C₈	C₅×D₄	1	2	yes	—
166	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
167	C₂²⋊Q₈	—	2	2	no	—
168	C₂²⋊Q₈	—	4	1	no	—
169	#121	—	2	2	no	—
170	#121	—	4	1	no	—
171	#121	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	M₃₂	—	2	2	no	—
174	M₃₂	C₂³×C₅	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	#108	—	2	2	no	—
177	#108	—	2	4	no	—
178	#108	C₂³×C₅	1	4	yes	—
179	#106	—	2	1	no	—
180	#106	C₂³×C₅	1	2	yes	—
181	#103	—	2	2	no	—
182	#102	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	8	1	no	—
187	C₂²×C₅	—	8	1	no	—
188	C₂²×C₅	—	8	1	no	—
189	C₂²×C₅	#481	1	2	yes	—
190	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	8	2	no	—
193	C₄×C₅	—	8	1	no	—
194	C₄×C₅	#481	1	2	yes	—
195	C₄×C₅	C₂×#123	1	1	yes	characteristic
196	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	2	no	—
200	#41	—	4	2	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	C₅×SD₁₆	1	2	yes	—
207	C₂×D₄	C₅×SD₁₆	1	1	yes	characteristic
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	Q₁₆	—	2	2	no	—
213	SD₁₆	—	4	1	no	—
214	SD₁₆	—	4	1	no	—
215	D₈	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	4	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
226	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₈	C₅×#41	1	2	yes	—
228	C₂×C₈	C₅×#41	1	2	yes	—
229	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₁₆	—	2	4	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	—	4	2	no	—
239	C₂×C₅	—	16	1	no	—
240	C₂×C₅	C₂×#481	1	1	yes	characteristic
241	C₂³	—	4	1	no	—
242	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
243	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	8	2	no	—
252	D₄	—	16	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	16	1	no	—
259	C₂×C₄	C₅×#123	1	2	yes	—
260	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
261	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
262	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
263	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	4	no	—
266	C₈	C₅×#123	1	1	yes	characteristic
267	C₈	C₅×#123	1	1	yes	characteristic
268	C₈	C₂×C₅×#41	1	2	yes	—
269	C₅	#33659	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	8	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	1	no	—
276	C₂²	C₅×#481	1	2	yes	—
277	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	1	no	—
281	C₄	C₅×#481	1	2	yes	—
282	C₄	C₂×C₅×#123	1	1	yes	characteristic
283	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	16	1	no	—
288	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	C₅×#3060	C₂
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
2. C₅×#2916	C₂	semidirect
3. C₅×#2239	C₂	semidirect
4. C₅×#2238	C₂	semidirect
5. C₂×C₅×#365	C₂	semidirect
6. C₅×#2148	C₂	semidirect
7. C₅×#2115	C₂	semidirect
26. #33659	C₅	direct (N × H)
60. #2916	C₂×C₅	semidirect
61. #2239	C₂×C₅	semidirect
62. #2238	C₂×C₅	semidirect
63. C₂×#365	C₂×C₅	semidirect
64. #2148	C₂×C₅	semidirect
65. #2115	C₂×C₅	semidirect
269. C₅	#33659	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×#3060	2
C₅×#2916	2
C₅×#2239	2
C₅×#2238	2
C₂×C₅×#365	2
C₅×#2148	2
C₅×#2115	2
#33659	5

Subgroups — C5×#33659 #11956379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33659 #11956379