Subgroups — C₅×#12948 #11935668

Order 1280.

384 conjugacy classes of subgroups.

The subgroup lattice has 288 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×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#360	C₂	1	1	yes	maximal, characteristic
4	C₅×#2143	C₂	1	1	yes	maximal, characteristic
5	C₅×#1951	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1378	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	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#444	—	2	2	no	—
13	C₅×#443	—	2	2	no	—
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#428	—	2	1	no	—
17	C₂×C₅×#106	C₂²	1	1	yes	characteristic
18	C₅×#421	—	2	1	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#365	—	2	2	no	—
21	C₅×#360	—	2	2	no	—
22	C₅×#359	C₄	1	4	yes	complemented
23	C₅×#343	—	2	1	no	—
24	#12948	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	—	4	2	no	—
38	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₅×#117	D₄	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	—	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	C₂×C₄	1	2	yes	—
51	C₅×#108	D₄	1	2	yes	—
52	C₅×#107	—	4	1	no	—
53	C₅×#106	—	2	1	no	—
54	C₅×#106	—	2	1	no	—
55	C₅×#106	—	4	1	no	—
56	C₅×#104	—	2	4	no	—
57	C₅×#97	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	#2957	C₂×C₅	1	1	yes	characteristic
60	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#360	C₂×C₅	1	1	yes	characteristic
62	#2143	C₂×C₅	1	1	yes	characteristic
63	#1951	C₂×C₅	1	1	yes	characteristic, complemented
64	#1385	C₂×C₅	1	1	yes	characteristic, complemented
65	#1378	C₂×C₅	1	1	yes	characteristic, complemented
66	C₅×#41	—	2	2	no	—
67	C₅×#41	—	2	2	no	—
68	C₅×#41	—	4	1	no	—
69	C₅×#41	—	4	1	no	—
70	C₂×C₅×Q₈	—	2	1	no	—
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×Q₁₆	—	2	1	no	—
77	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
78	C₅×SD₁₆	—	2	2	no	—
79	C₅×SD₁₆	—	2	2	no	—
80	C₅×D₈	—	2	1	no	—
81	C₅×D₈	C₄⋊C₄	1	2	yes	—
82	C₅×M₁₆	—	2	1	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	2	2	no	—
85	C₅×M₁₆	—	4	4	no	—
86	C₅×M₁₆	—	4	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	SD₁₆	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	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₅×C₈	#41	1	1	yes	characteristic
98	C₂×C₅×C₈	#41	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	1	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₂²⋊C₄	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₅×C₁₆	—	2	2	no	—
105	C₅×C₁₆	—	4	1	no	—
106	C₂×#135	C₂²×C₅	1	1	yes	characteristic
107	#568	—	2	1	no	—
108	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
109	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
110	#444	—	2	2	no	—
111	#443	—	2	2	no	—
112	C₂×#108	C₂²×C₅	1	1	yes	characteristic
113	C₂×#108	C₂²×C₅	1	1	yes	characteristic
114	#428	—	2	1	no	—
115	C₂×#106	C₂²×C₅	1	1	yes	characteristic
116	#421	—	2	1	no	—
117	#405	C₂²×C₅	1	1	yes	characteristic
118	#365	—	2	2	no	—
119	#360	—	2	2	no	—
120	#359	C₄×C₅	1	4	yes	complemented
121	#343	—	2	1	no	—
122	C₂³×C₅	—	4	1	no	—
123	C₂³×C₅	#135	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×Q₈	—	4	1	no	—
126	C₅×D₄	—	2	2	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	8	1	no	—
135	C₂×C₄×C₅	—	16	1	no	—
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#120	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₅×C₈	—	2	2	no	—
143	C₅×C₈	—	4	2	no	—
144	C₅×C₈	—	4	2	no	—
145	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₅×C₈	C₄×D₄	1	1	yes	characteristic
147	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
148	C₅×C₈	#123	1	1	yes	characteristic
149	C₂×#41	—	2	1	no	—
150	#135	—	2	1	no	—
151	#135	—	2	1	no	—
152	#135	C₂×C₄×C₅	1	2	yes	—
153	#135	C₅×D₄	1	2	yes	—
154	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic
155	C₂×SD₁₆	—	2	1	no	—
156	C₂×D₈	C₅×Q₈	1	1	yes	characteristic
157	#131	—	2	4	no	—
158	#131	—	4	2	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	4	2	no	—
162	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
164	#117	C₅×D₄	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	2	1	no	—
168	M₃₂	—	2	1	no	—
169	M₃₂	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	C₂×C₄×C₅	1	2	yes	—
175	#108	C₅×D₄	1	2	yes	—
176	#107	—	4	1	no	—
177	#106	—	2	1	no	—
178	#106	—	2	1	no	—
179	#106	—	4	1	no	—
180	#104	—	2	4	no	—
181	#97	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₂²×C₅	—	4	1	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	16	1	no	—
186	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
187	C₂²×C₅	#449	1	1	yes	characteristic
188	C₂²×C₅	#478	1	1	yes	characteristic
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	—	8	1	no	—
191	C₄×C₅	—	16	1	no	—
192	C₄×C₅	#385	1	1	yes	characteristic
193	C₄×C₅	#442	1	1	yes	characteristic
194	C₄×C₅	#450	1	1	yes	characteristic
195	C₄×C₅	#476	1	1	yes	characteristic
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	C₂×Q₈	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	Q₁₆	—	2	1	no	—
207	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
208	SD₁₆	—	2	2	no	—
209	SD₁₆	—	2	2	no	—
210	D₈	—	2	1	no	—
211	D₈	C₅×C₄⋊C₄	1	2	yes	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	C₅×SD₁₆	1	2	yes	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
227	C₂×C₈	C₅×#41	1	1	yes	characteristic
228	C₂×C₈	C₅×#41	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	1	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	2	1	no	—
238	C₂×C₅	—	2	1	no	—
239	C₂×C₅	—	8	2	no	—
240	C₂×C₅	#1864	1	1	yes	characteristic
241	C₂³	—	4	1	no	—
242	C₂³	C₅×#135	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	4	1	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	16	1	no	—
255	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	C₅×#120	1	1	yes	characteristic
257	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
259	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
265	C₈	C₄×C₅×D₄	1	1	yes	characteristic
266	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
267	C₈	C₅×#123	1	1	yes	characteristic
268	C₅	#12948	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	4	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	16	1	no	—
272	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
273	C₂²	C₅×#449	1	1	yes	characteristic
274	C₂²	C₅×#478	1	1	yes	characteristic
275	C₄	—	4	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	16	1	no	—
278	C₄	C₅×#385	1	1	yes	characteristic
279	C₄	C₅×#442	1	1	yes	characteristic
280	C₄	C₅×#450	1	1	yes	characteristic
281	C₄	C₅×#476	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	8	2	no	—
286	C₂	C₅×#1864	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1864
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#365	C₂	semidirect
5. C₅×#1951	C₂	semidirect
6. C₅×#1385	C₂	semidirect
7. C₅×#1378	C₂	semidirect
22. C₅×#359	C₄	semidirect
24. #12948	C₅	direct (N × H)
60. C₂×#365	C₂×C₅	semidirect
63. #1951	C₂×C₅	semidirect
64. #1385	C₂×C₅	semidirect
65. #1378	C₂×C₅	semidirect
120. #359	C₄×C₅	semidirect
268. C₅	#12948	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×#365	2
C₂×C₅×#360	2
C₅×#2143	2
C₅×#1951	2
C₅×#1385	2
C₅×#1378	2
#12948	5

Subgroups — C5×#12948 #11935668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12948 #11935668