Subgroups — C₅×#11646 #11934366

Order 1280.

314 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#1332	C₂	1	4	yes	maximal, complemented
5	C₅×#533	—	2	1	no	—
6	C₅×#500	C₄	1	1	yes	characteristic
7	C₅×#494	C₄	1	1	yes	characteristic
8	C₅×#487	—	2	1	no	—
9	C₅×#467	—	2	1	no	—
10	C₅×#467	C₄	1	2	yes	—
11	C₄×C₅×Q₁₆	—	2	1	no	—
12	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	C₂²	1	2	yes	—
18	C₅×#335	—	2	1	no	—
19	C₅×#335	C₂²	1	2	yes	—
20	#11646	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×Q₁₆	—	2	1	no	—
22	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	—	4	1	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×#125	—	4	1	no	—
29	C₅×C₂²⋊Q₈	—	2	2	no	—
30	C₅×C₂²⋊Q₈	—	4	1	no	—
31	C₄×C₅×Q₈	—	2	2	no	—
32	C₄×C₅×Q₈	—	4	1	no	—
33	C₅×#117	—	4	1	no	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₅×#107	—	2	1	no	—
37	C₅×#107	C₂×C₄	1	2	yes	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	4	1	no	—
40	C₅×#103	—	2	2	no	—
41	C₅×#103	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
46	#3170	C₂×C₅	1	1	yes	characteristic
47	#2162	C₂×C₅	1	1	yes	characteristic
48	C₂×#335	C₂×C₅	1	1	yes	characteristic
49	#1332	C₂×C₅	1	4	yes	complemented
50	C₂×C₅×Q₈	—	2	2	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
55	C₅×Q₁₆	—	2	4	no	—
56	C₅×Q₁₆	—	4	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	4	1	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	8	2	no	—
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	4	1	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	1	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₅	—	4	2	no	—
73	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₅	D₈	1	1	yes	characteristic
75	C₄²×C₅	SD₁₆	1	1	yes	characteristic
76	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×C₁₆	—	4	2	no	—
78	#533	—	2	1	no	—
79	#500	C₄×C₅	1	1	yes	characteristic
80	#494	C₄×C₅	1	1	yes	characteristic
81	#487	—	2	1	no	—
82	#467	—	2	1	no	—
83	#467	C₄×C₅	1	2	yes	—
84	C₄×Q₁₆	—	2	1	no	—
85	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
86	C₂×C₄⋊C₈	—	2	1	no	—
87	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₄×C₁₆	C₂²×C₅	1	2	yes	—
91	#335	—	2	1	no	—
92	#335	C₂²×C₅	1	2	yes	—
93	C₂³×C₅	#102	1	1	yes	characteristic
94	C₅×Q₈	—	4	2	no	—
95	C₅×Q₈	—	4	2	no	—
96	C₅×Q₈	—	8	2	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	4	1	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	#102	1	1	yes	characteristic
103	C₂×C₄×C₅	#102	1	1	yes	characteristic
104	C₂×C₄×C₅	#102	1	1	yes	characteristic
105	C₂×C₄×C₅	SD₃₂	1	2	yes	—
106	C₂×C₄×C₅	Q₃₂	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	8	2	no	—
114	C₅×C₈	#131	1	2	yes	—
115	C₂×Q₁₆	—	2	1	no	—
116	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₈	—	4	1	no	—
118	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	1	no	—
121	C₄⋊Q₈	—	2	1	no	—
122	#125	—	4	1	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₂²⋊Q₈	—	4	1	no	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	4	1	no	—
127	#117	—	4	1	no	—
128	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	#107	—	2	1	no	—
131	#107	C₂×C₄×C₅	1	2	yes	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	4	1	no	—
134	#103	—	2	2	no	—
135	#103	—	2	2	no	—
136	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
137	C₄×C₈	C₅×D₄	1	1	yes	characteristic
138	C₄×C₈	C₅×D₄	1	1	yes	characteristic
139	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	#358	1	2	yes	—
143	C₂²×C₅	C₂×#102	1	1	yes	characteristic
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	8	2	no	—
148	C₄×C₅	—	8	2	no	—
149	C₄×C₅	#358	1	2	yes	—
150	C₄×C₅	#408	1	1	yes	characteristic
151	C₄×C₅	#421	1	1	yes	characteristic
152	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
153	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
158	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
159	Q₁₆	—	2	4	no	—
160	Q₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	8	2	no	—
165	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
178	C₄²	C₅×D₈	1	1	yes	characteristic
179	C₄²	C₅×SD₁₆	1	1	yes	characteristic
180	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
181	C₁₆	—	4	2	no	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	#1480	1	1	yes	characteristic
184	C₂×C₅	C₂×#358	1	1	yes	characteristic
185	C₂×C₅	#2143	1	1	yes	characteristic
186	C₂³	C₅×#102	1	1	yes	characteristic
187	Q₈	—	4	2	no	—
188	Q₈	—	4	2	no	—
189	Q₈	—	8	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₅×#102	1	1	yes	characteristic
196	C₂×C₄	C₅×#102	1	1	yes	characteristic
197	C₂×C₄	C₅×#102	1	1	yes	characteristic
198	C₂×C₄	C₅×SD₃₂	1	2	yes	—
199	C₂×C₄	C₅×Q₃₂	1	2	yes	—
200	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	8	2	no	—
207	C₈	C₅×#131	1	2	yes	—
208	C₅	#11646	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₅×#358	1	2	yes	—
212	C₂²	C₂×C₅×#102	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	C₅×#358	1	2	yes	—
219	C₄	C₅×#408	1	1	yes	characteristic
220	C₄	C₅×#421	1	1	yes	characteristic
221	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
222	C₄	C₂×C₅×Q₃₂	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₅×#1480	1	1	yes	characteristic
225	C₂	C₂×C₅×#358	1	1	yes	characteristic
226	C₂	C₅×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1332	C₂	semidirect
20. #11646	C₅	direct (N × H)
49. #1332	C₂×C₅	semidirect
208. C₅	#11646	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×#2162	2
C₂×C₅×#335	2
C₅×#1332	2
#11646	5

Subgroups — C5×#11646 #11934366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11646 #11934366