Subgroups — C₅×#17991 #11940711

Order 1280.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic
2	C₅×#2985	C₂	1	2	yes	maximal
3	C₅×#1587	C₂	1	4	yes	maximal, complemented
4	C₅×#1584	C₂	1	4	yes	maximal, complemented
5	C₅×#1568	C₂	1	1	yes	maximal, characteristic
6	C₅×#1497	C₂	1	2	yes	maximal
7	C₅×#1452	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	2	yes	—
9	C₅×#500	C₄	1	1	yes	characteristic
10	C₅×#494	C₄	1	1	yes	characteristic
11	C₅×#487	C₄	1	2	yes	—
12	C₅×#467	C₄	1	2	yes	—
13	C₅×#467	C₄	1	2	yes	—
14	C₅×#446	—	2	4	no	—
15	C₅×#446	C₂²	1	4	yes	—
16	C₅×C₈×Q₈	C₂²	1	4	yes	—
17	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
18	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
19	C₅×#433	—	2	4	no	—
20	C₅×#432	C₂²	1	2	yes	—
21	C₅×#431	C₂²	1	1	yes	characteristic
22	C₅×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#335	C₂²	1	2	yes	—
25	C₅×#334	C₂²	1	2	yes	—
26	C₅×Q₈⋊C₈	C₂²	1	8	yes	—
27	C₅×#322	C₂²	1	2	yes	—
28	C₅×#322	C₂²	1	2	yes	—
29	#17991	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₅×#125	—	2	2	no	—
37	C₅×C₂²⋊Q₈	—	2	2	no	—
38	C₅×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
39	C₄×C₅×Q₈	—	2	2	no	—
40	C₄×C₅×Q₈	C₂³	1	4	yes	—
41	C₅×#117	—	2	2	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#107	C₂×C₄	1	2	yes	—
44	C₅×#107	C₂×C₄	1	2	yes	—
45	C₅×C₄⋊C₈	—	2	4	no	—
46	C₅×C₄⋊C₈	—	2	4	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	D₄	1	2	yes	—
49	C₅×C₄⋊C₈	C₂³	1	4	yes	—
50	C₅×#103	C₂×C₄	1	4	yes	—
51	C₅×#103	C₂×C₄	1	4	yes	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×C₂²⋊C₈	D₄	1	2	yes	—
55	C₅×#97	—	2	4	no	—
56	C₅×#97	—	2	2	no	—
57	C₄×C₅×C₈	—	2	4	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	2	yes	—
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	2	yes	—
64	#3170	C₂×C₅	1	1	yes	characteristic
65	#2985	C₂×C₅	1	2	yes	—
66	#1587	C₂×C₅	1	4	yes	complemented
67	#1584	C₂×C₅	1	4	yes	complemented
68	#1568	C₂×C₅	1	1	yes	characteristic
69	#1497	C₂×C₅	1	2	yes	—
70	#1452	C₂×C₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	1	yes	characteristic
76	C₅×Q₁₆	—	4	2	no	—
77	C₅×Q₁₆	—	4	2	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₅×C₂²⋊C₄	—	2	4	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	#41	1	1	yes	characteristic
98	C₄²×C₅	C₂⁴	1	1	yes	characteristic
99	#533	C₂²×C₅	1	2	yes	—
100	#500	C₄×C₅	1	1	yes	characteristic
101	#494	C₄×C₅	1	1	yes	characteristic
102	#487	C₄×C₅	1	2	yes	—
103	#467	C₄×C₅	1	2	yes	—
104	#467	C₄×C₅	1	2	yes	—
105	#446	—	2	4	no	—
106	#446	C₂²×C₅	1	4	yes	—
107	C₈×Q₈	C₂²×C₅	1	4	yes	—
108	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
109	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
110	#433	—	2	4	no	—
111	#432	C₂²×C₅	1	2	yes	—
112	#431	C₂²×C₅	1	1	yes	characteristic
113	#431	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#335	C₂²×C₅	1	2	yes	—
116	#334	C₂²×C₅	1	2	yes	—
117	Q₈⋊C₈	C₂²×C₅	1	8	yes	—
118	#322	C₂²×C₅	1	2	yes	—
119	#322	C₂²×C₅	1	2	yes	—
120	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
121	C₅×Q₈	—	2	4	no	—
122	C₅×Q₈	—	2	4	no	—
123	C₅×Q₈	—	8	2	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	4	no	—
141	C₅×C₈	—	4	2	no	—
142	C₅×C₈	—	4	4	no	—
143	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
144	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
149	#125	—	2	2	no	—
150	C₂²⋊Q₈	—	2	2	no	—
151	C₂²⋊Q₈	C₂×C₄×C₅	1	4	yes	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	C₂³×C₅	1	4	yes	—
154	#117	—	2	2	no	—
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	#107	C₂×C₄×C₅	1	2	yes	—
157	#107	C₂×C₄×C₅	1	2	yes	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	C₅×D₄	1	2	yes	—
162	C₄⋊C₈	C₂³×C₅	1	4	yes	—
163	#103	C₂×C₄×C₅	1	4	yes	—
164	#103	C₂×C₄×C₅	1	4	yes	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	C₂²⋊C₈	C₅×D₄	1	2	yes	—
168	#97	—	2	4	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₅	1	2	yes	—
174	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₅	1	2	yes	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	#568	1	1	yes	characteristic
188	C₄×C₅	#568	1	1	yes	characteristic
189	C₄×C₅	#578	1	2	yes	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₅×#41	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	2	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
216	C₄²	C₅×#41	1	1	yes	characteristic
217	C₄²	C₂⁴×C₅	1	1	yes	characteristic
218	C₂×C₅	—	4	1	no	—
219	C₂×C₅	#2932	1	1	yes	characteristic
220	C₂×C₅	#2950	1	1	yes	characteristic
221	C₂×C₅	#2957	1	1	yes	characteristic
222	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
223	Q₈	—	2	4	no	—
224	Q₈	—	2	4	no	—
225	Q₈	—	8	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
233	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	4	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	4	no	—
245	C₅	#17991	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	C₅×#568	1	1	yes	characteristic
257	C₄	C₅×#568	1	1	yes	characteristic
258	C₄	C₅×#578	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	C₅×#2932	1	1	yes	characteristic
261	C₂	C₅×#2950	1	1	yes	characteristic
262	C₂	C₅×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1587	C₂	semidirect
4. C₅×#1584	C₂	semidirect
29. #17991	C₅	direct (N × H)
66. #1587	C₂×C₅	semidirect
67. #1584	C₂×C₅	semidirect
245. C₅	#17991	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×#3170	2
C₅×#2985	2
C₅×#1587	2
C₅×#1584	2
C₅×#1568	2
C₅×#1497	2
C₅×#1452	2
#17991	5

Subgroups — C5×#17991 #11940711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17991 #11940711