Subgroups — C₅×#12947 #11935667

Order 1280.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2957	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#359	C₂	1	1	yes	maximal, characteristic
5	C₅×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1379	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×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₅×#444	—	2	2	no	—
13	C₅×#443	—	2	2	no	—
14	C₅×#430	—	2	1	no	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₅×#421	—	2	1	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#368	—	2	2	no	—
21	C₅×#359	—	2	2	no	—
22	C₅×#359	C₄	1	4	yes	complemented
23	C₅×#344	—	2	1	no	—
24	#12947	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	C₂×C₄	1	2	yes	—
29	C₅×#135	D₄	1	2	yes	—
30	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic
31	C₂×C₅×SD₁₆	—	2	1	no	—
32	C₂×C₅×D₈	Q₈	1	1	yes	characteristic
33	C₅×#131	—	2	4	no	—
34	C₅×#131	—	4	2	no	—
35	C₂×C₅×M₁₆	—	2	1	no	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	—	2	1	no	—
38	C₂×C₅×M₁₆	—	4	2	no	—
39	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×#117	D₄	1	1	yes	characteristic
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
48	C₅×#108	—	2	1	no	—
49	C₅×#108	—	2	1	no	—
50	C₅×#108	—	2	1	no	—
51	C₅×#108	—	2	1	no	—
52	C₅×#108	—	4	2	no	—
53	C₅×#108	C₂×C₄	1	2	yes	—
54	C₅×#108	D₄	1	2	yes	—
55	C₅×#104	—	2	4	no	—
56	C₅×#97	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	#2957	C₂×C₅	1	1	yes	characteristic
59	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
60	#2143	C₂×C₅	1	1	yes	characteristic
61	C₂×#359	C₂×C₅	1	1	yes	characteristic
62	#1953	C₂×C₅	1	1	yes	characteristic, complemented
63	#1388	C₂×C₅	1	1	yes	characteristic, complemented
64	#1379	C₂×C₅	1	1	yes	characteristic, complemented
65	C₅×#41	—	2	2	no	—
66	C₅×#41	—	2	2	no	—
67	C₅×#41	—	4	1	no	—
68	C₅×#41	—	4	1	no	—
69	C₂×C₅×Q₈	—	2	1	no	—
70	C₂×C₅×D₄	—	2	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₅×Q₁₆	—	2	1	no	—
75	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
76	C₅×SD₁₆	—	2	2	no	—
77	C₅×SD₁₆	—	2	2	no	—
78	C₅×D₈	—	2	1	no	—
79	C₅×D₈	C₄⋊C₄	1	2	yes	—
80	C₅×M₁₆	—	2	1	no	—
81	C₅×M₁₆	—	2	2	no	—
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	4	4	no	—
84	C₅×M₁₆	—	4	1	no	—
85	C₅×M₁₆	—	4	1	no	—
86	C₅×M₁₆	—	4	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	D₈	1	2	yes	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	4	4	no	—
91	C₂×C₅×C₈	—	4	1	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₅×C₈	#41	1	1	yes	characteristic
99	C₂×C₅×C₈	#41	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	1	no	—
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₅×C₁₆	—	2	2	no	—
104	C₅×C₁₆	—	4	1	no	—
105	C₂×#135	C₂²×C₅	1	1	yes	characteristic
106	#568	—	2	1	no	—
107	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
108	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
109	#444	—	2	2	no	—
110	#443	—	2	2	no	—
111	#430	—	2	1	no	—
112	C₂×#108	C₂²×C₅	1	1	yes	characteristic
113	C₂×#108	C₂²×C₅	1	1	yes	characteristic
114	C₂×#108	C₂²×C₅	1	1	yes	characteristic
115	#421	—	2	1	no	—
116	#405	C₂²×C₅	1	1	yes	characteristic
117	#368	—	2	2	no	—
118	#359	—	2	2	no	—
119	#359	C₄×C₅	1	4	yes	complemented
120	#344	—	2	1	no	—
121	C₂³×C₅	—	4	1	no	—
122	C₂³×C₅	#137	1	1	yes	characteristic
123	C₅×Q₈	—	2	2	no	—
124	C₅×Q₈	—	4	1	no	—
125	C₅×D₄	—	2	2	no	—
126	C₅×D₄	—	4	2	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	#120	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅	#135	1	1	yes	characteristic
138	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	—	4	2	no	—
142	C₅×C₈	—	8	1	no	—
143	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₅×C₈	C₄×D₄	1	1	yes	characteristic
145	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
146	C₅×C₈	#123	1	1	yes	characteristic
147	C₂×#41	—	2	1	no	—
148	#135	—	2	1	no	—
149	#135	—	2	1	no	—
150	#135	C₂×C₄×C₅	1	2	yes	—
151	#135	C₅×D₄	1	2	yes	—
152	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic
153	C₂×SD₁₆	—	2	1	no	—
154	C₂×D₈	C₅×Q₈	1	1	yes	characteristic
155	#131	—	2	4	no	—
156	#131	—	4	2	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	4	2	no	—
161	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	#117	C₅×D₄	1	1	yes	characteristic
164	M₃₂	—	2	1	no	—
165	M₃₂	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
170	#108	—	2	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	—	4	2	no	—
175	#108	C₂×C₄×C₅	1	2	yes	—
176	#108	C₅×D₄	1	2	yes	—
177	#104	—	2	4	no	—
178	#97	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₂²×C₅	—	4	1	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	16	1	no	—
183	C₂²×C₅	#440	1	1	yes	characteristic
184	C₂²×C₅	#452	1	1	yes	characteristic
185	C₂²×C₅	#474	1	1	yes	characteristic
186	C₄×C₅	—	4	2	no	—
187	C₄×C₅	—	8	1	no	—
188	C₄×C₅	#385	1	1	yes	characteristic
189	C₄×C₅	C₄×D₈	1	1	yes	characteristic
190	C₄×C₅	#447	1	1	yes	characteristic
191	C₄×C₅	#478	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	Q₁₆	—	2	1	no	—
202	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
203	SD₁₆	—	2	2	no	—
204	SD₁₆	—	2	2	no	—
205	D₈	—	2	1	no	—
206	D₈	C₅×C₄⋊C₄	1	2	yes	—
207	M₁₆	—	2	1	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	C₅×D₈	1	2	yes	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
225	C₂×C₈	C₅×#41	1	1	yes	characteristic
226	C₂×C₈	C₅×#41	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	4	1	no	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	2	1	no	—
235	C₂×C₅	—	8	2	no	—
236	C₂×C₅	#1862	1	1	yes	characteristic
237	C₂³	—	4	1	no	—
238	C₂³	C₅×#137	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
250	C₂×C₄	C₅×#120	1	1	yes	characteristic
251	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
253	C₂×C₄	C₅×#135	1	1	yes	characteristic
254	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	8	1	no	—
259	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
260	C₈	C₄×C₅×D₄	1	1	yes	characteristic
261	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₅×#123	1	1	yes	characteristic
263	C₅	#12947	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₅×#440	1	1	yes	characteristic
268	C₂²	C₅×#452	1	1	yes	characteristic
269	C₂²	C₅×#474	1	1	yes	characteristic
270	C₄	—	4	2	no	—
271	C₄	—	8	1	no	—
272	C₄	C₅×#385	1	1	yes	characteristic
273	C₄	C₄×C₅×D₈	1	1	yes	characteristic
274	C₄	C₅×#447	1	1	yes	characteristic
275	C₄	C₅×#478	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₅×#1862	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#447
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1862
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#368	C₂	semidirect
5. C₅×#1953	C₂	semidirect
6. C₅×#1388	C₂	semidirect
7. C₅×#1379	C₂	semidirect
22. C₅×#359	C₄	semidirect
24. #12947	C₅	direct (N × H)
59. C₂×#368	C₂×C₅	semidirect
62. #1953	C₂×C₅	semidirect
63. #1388	C₂×C₅	semidirect
64. #1379	C₂×C₅	semidirect
119. #359	C₄×C₅	semidirect
263. C₅	#12947	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×#2957	2
C₂×C₅×#368	2
C₅×#2143	2
C₂×C₅×#359	2
C₅×#1953	2
C₅×#1388	2
C₅×#1379	2
#12947	5

Subgroups — C5×#12947 #11935667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12947 #11935667