Subgroups — C₅×#11495 #11934215

Order 1280.

314 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic
3	C₅×#2162	C₂	1	1	yes	maximal, characteristic
4	C₅×#1322	C₂	1	4	yes	maximal
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₅×#446	—	2	1	no	—
7	C₅×#446	—	2	1	no	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×#97	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	—	2	1	no	—
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
17	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₅×#346	—	2	1	no	—
19	C₄×C₅×C₁₆	—	2	1	no	—
20	#11495	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	—	2	1	no	—
28	C₄×C₅×Q₈	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×#97	—	2	2	no	—
40	C₅×#97	—	2	2	no	—
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	D₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂×#446	C₂×C₅	1	1	yes	characteristic
46	#2163	C₂×C₅	1	1	yes	characteristic
47	#2162	C₂×C₅	1	1	yes	characteristic
48	#1322	C₂×C₅	1	4	yes	—
49	C₂×C₅×Q₈	—	2	1	no	—
50	C₂×C₅×Q₈	—	2	1	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	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	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	1	no	—
66	C₅×C₄⋊C₄	—	2	1	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅	M₁₆	1	1	yes	characteristic
74	C₄²×C₅	M₁₆	1	1	yes	characteristic
75	C₅×C₁₆	—	4	2	no	—
76	C₅×C₁₆	—	4	2	no	—
77	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
78	#446	—	2	1	no	—
79	#446	—	2	1	no	—
80	#446	—	2	1	no	—
81	#446	—	2	1	no	—
82	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
83	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
84	C₂×#97	C₄×C₅	1	2	yes	—
85	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	—	2	1	no	—
88	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
89	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
90	#346	—	2	1	no	—
91	C₄×C₁₆	—	2	1	no	—
92	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₅×Q₈	—	4	2	no	—
94	C₅×Q₈	—	4	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	4	2	no	—
113	C₅×C₈	#101	1	2	yes	—
114	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
118	C₄×Q₈	—	2	1	no	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	—	2	1	no	—
122	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
124	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
125	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	#97	—	2	2	no	—
133	#97	—	2	2	no	—
134	C₄×C₈	C₅×D₄	1	1	yes	characteristic
135	C₄×C₈	C₅×D₄	1	1	yes	characteristic
136	C₄×C₈	C₅×D₄	1	1	yes	characteristic
137	C₄×C₈	C₅×D₄	1	1	yes	characteristic
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₅	#504	1	2	yes	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	4	2	no	—
147	C₄×C₅	—	4	2	no	—
148	C₄×C₅	#349	1	2	yes	—
149	C₄×C₅	#410	1	1	yes	characteristic
150	C₄×C₅	C₂×#101	1	1	yes	characteristic
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
155	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
166	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
174	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₄²	C₅×M₁₆	1	1	yes	characteristic
176	C₄²	C₅×M₁₆	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₁₆	—	4	2	no	—
179	C₂×C₅	—	2	2	no	—
180	C₂×C₅	#1461	1	1	yes	characteristic
181	C₂×C₅	#2113	1	1	yes	characteristic
182	C₂×C₅	C₂×#349	1	1	yes	characteristic
183	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	2	no	—
204	C₈	C₅×#101	1	2	yes	—
205	C₅	#11495	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
209	C₂²	C₅×#504	1	2	yes	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₅×#349	1	2	yes	—
217	C₄	C₅×#410	1	1	yes	characteristic
218	C₄	C₂×C₅×#101	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₅×#1461	1	1	yes	characteristic
221	C₂	C₅×#2113	1	1	yes	characteristic
222	C₂	C₂×C₅×#349	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
20. #11495	C₅	direct (N × H)
205. C₅	#11495	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#446	2
C₅×#2163	2
C₅×#2162	2
C₅×#1322	2
#11495	5

Subgroups — C5×#11495 #11934215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11495 #11934215