Subgroups — C₅×#21736 #11944456

Order 1280.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅×#2961	C₂	1	2	yes	maximal
3	C₅×#1994	C₂	1	2	yes	maximal
4	C₅×#1946	C₂	1	2	yes	maximal
5	C₅×#1945	C₂	1	2	yes	maximal
6	C₅×#1849	C₂	1	1	yes	maximal, characteristic
7	C₅×#1773	C₂	1	2	yes	maximal
8	C₅×#1772	C₂	1	2	yes	maximal
9	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
10	C₅×#533	C₂²	1	2	yes	—
11	C₅×#501	C₄	1	8	yes	complemented
12	C₅×#446	—	2	8	no	—
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₂×C₅×#108	C₂²	1	2	yes	—
15	C₅×#427	C₂²	1	2	yes	—
16	C₅×#427	C₂²	1	2	yes	—
17	C₅×#424	C₂²	1	4	yes	—
18	C₅×#423	C₂²	1	2	yes	—
19	C₅×#405	C₂²	1	4	yes	—
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
22	C₂×C₅×#97	C₂²	1	1	yes	characteristic
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₄³×C₅	C₂²	1	1	yes	characteristic
27	C₅×#339	C₂²	1	4	yes	—
28	C₅×#339	C₂²	1	4	yes	—
29	#21736	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
36	C₅×#125	C₂×C₄	1	4	yes	—
37	C₅×C₂²⋊Q₈	—	2	4	no	—
38	C₄×C₅×Q₈	—	2	4	no	—
39	C₅×#117	C₂³	1	4	yes	—
40	C₂×C₄²×C₅	—	2	2	no	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#108	—	2	4	no	—
45	C₅×#108	—	2	4	no	—
46	C₅×#107	C₂×C₄	1	4	yes	—
47	C₅×#107	C₂×C₄	1	4	yes	—
48	C₅×#106	C₂×C₄	1	4	yes	—
49	C₅×#106	C₂×C₄	1	4	yes	—
50	C₅×C₄⋊C₈	—	2	8	no	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×#97	—	2	4	no	—
53	C₅×#97	—	2	2	no	—
54	C₅×#97	C₂×C₄	1	4	yes	—
55	C₅×#97	D₄	1	4	yes	—
56	C₅×#97	Q₈	1	4	yes	complemented
57	C₄×C₅×C₈	—	2	4	no	—
58	C₄×C₅×C₈	—	2	2	no	—
59	C₄×C₅×C₈	Q₈	1	4	yes	complemented
60	#3162	C₂×C₅	1	1	yes	characteristic
61	#2961	C₂×C₅	1	2	yes	—
62	#1994	C₂×C₅	1	2	yes	—
63	#1946	C₂×C₅	1	2	yes	—
64	#1945	C₂×C₅	1	2	yes	—
65	#1849	C₂×C₅	1	1	yes	characteristic
66	#1773	C₂×C₅	1	2	yes	—
67	#1772	C₂×C₅	1	2	yes	—
68	C₄×#97	C₂×C₅	1	1	yes	characteristic
69	C₂×C₅×Q₈	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₅×M₁₆	—	2	4	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
84	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
85	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
86	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
87	C₂×C₅×C₈	#41	1	2	yes	—
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	—	4	4	no	—
97	C₄²×C₅	C₂²×C₄	1	2	yes	—
98	C₄²×C₅	C₂×D₄	1	2	yes	—
99	C₄²×C₅	C₂×Q₈	1	2	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	#533	C₂²×C₅	1	2	yes	—
102	#501	C₄×C₅	1	8	yes	complemented
103	#446	—	2	8	no	—
104	C₂×#108	C₂²×C₅	1	2	yes	—
105	C₂×#108	C₂²×C₅	1	2	yes	—
106	#427	C₂²×C₅	1	2	yes	—
107	#427	C₂²×C₅	1	2	yes	—
108	#424	C₂²×C₅	1	4	yes	—
109	#423	C₂²×C₅	1	2	yes	—
110	#405	C₂²×C₅	1	4	yes	—
111	C₂×#97	C₂²×C₅	1	1	yes	characteristic
112	C₂×#97	C₂²×C₅	1	1	yes	characteristic
113	C₂×#97	C₂²×C₅	1	1	yes	characteristic
114	C₂×#97	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₄³	C₂²×C₅	1	1	yes	characteristic
118	#339	C₂²×C₅	1	4	yes	—
119	#339	C₂²×C₅	1	4	yes	—
120	C₂³×C₅	C₂×#41	1	1	yes	characteristic
121	C₅×Q₈	—	8	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
129	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	4	4	no	—
137	C₅×C₈	C₄×Q₈	1	4	yes	—
138	C₅×C₈	C₄×Q₈	1	4	yes	—
139	C₅×C₈	C₄⋊Q₈	1	4	yes	—
140	C₂×M₁₆	C₂³×C₅	1	4	yes	—
141	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
145	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
146	#125	C₂×C₄×C₅	1	4	yes	—
147	C₂²⋊Q₈	—	2	4	no	—
148	C₄×Q₈	—	2	4	no	—
149	#117	C₂³×C₅	1	4	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
154	#108	—	2	4	no	—
155	#108	—	2	4	no	—
156	#107	C₂×C₄×C₅	1	4	yes	—
157	#107	C₂×C₄×C₅	1	4	yes	—
158	#106	C₂×C₄×C₅	1	4	yes	—
159	#106	C₂×C₄×C₅	1	4	yes	—
160	C₄⋊C₈	—	2	8	no	—
161	C₄⋊C₈	—	2	4	no	—
162	#97	—	2	4	no	—
163	#97	—	2	2	no	—
164	#97	C₂×C₄×C₅	1	4	yes	—
165	#97	C₅×D₄	1	4	yes	—
166	#97	C₅×Q₈	1	4	yes	complemented
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₅×Q₈	1	4	yes	complemented
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₅	#533	1	1	yes	characteristic
173	C₂²×C₅	#533	1	1	yes	characteristic
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	—	4	8	no	—
178	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₄×C₅	#532	1	1	yes	characteristic
182	C₂×Q₈	—	2	4	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	C₅×#41	1	2	yes	—
186	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
187	M₁₆	—	2	4	no	—
188	M₁₆	—	2	4	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
194	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
195	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
196	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
197	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
198	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
199	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
200	C₂×C₈	C₅×#41	1	2	yes	—
201	C₂×C₈	C₅×#41	1	2	yes	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	4	no	—
210	C₄²	C₂²×C₄×C₅	1	2	yes	—
211	C₄²	C₂×C₅×D₄	1	2	yes	—
212	C₄²	C₂×C₅×Q₈	1	2	yes	—
213	C₄²	C₂×C₅×Q₈	1	2	yes	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
217	C₂×C₅	#2956	1	2	yes	—
218	C₂³	C₂×C₅×#41	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
227	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	C₄×C₅×Q₈	1	4	yes	—
236	C₈	C₄×C₅×Q₈	1	4	yes	—
237	C₈	C₅×C₄⋊Q₈	1	4	yes	—
238	C₅	#21736	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
241	C₂²	C₅×#533	1	1	yes	characteristic
242	C₂²	C₅×#533	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	8	no	—
247	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
248	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
249	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
250	C₄	C₅×#532	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
254	C₂	C₅×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#501	C₄	semidirect
29. #21736	C₅	direct (N × H)
56. C₅×#97	Q₈	semidirect
59. C₄×C₅×C₈	Q₈	semidirect
102. #501	C₄×C₅	semidirect
166. #97	C₅×Q₈	semidirect
169. C₄×C₈	C₅×Q₈	semidirect
238. C₅	#21736	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×#3162	2
C₅×#2961	2
C₅×#1994	2
C₅×#1946	2
C₅×#1945	2
C₅×#1849	2
C₅×#1773	2
C₅×#1772	2
C₄×C₅×#97	2
#21736	5

Subgroups — C5×#21736 #11944456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21736 #11944456