Subgroups — C₅×#13268 #11935988

Order 1280.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅×#1377	C₂	1	4	yes	maximal
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×M₃₂	C₂²	1	2	yes	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₅×#446	—	2	1	no	—
11	C₅×#446	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	—	2	1	no	—
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×#346	—	2	1	no	—
19	C₄×C₅×C₁₆	—	2	1	no	—
20	#13268	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	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
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	C₅×M₃₂	—	2	2	no	—
34	C₅×M₃₂	—	2	4	no	—
35	C₅×M₃₂	Q₈	1	4	yes	complemented
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×#97	—	2	2	no	—
43	C₅×#97	—	2	2	no	—
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂×#446	C₂×C₅	1	1	yes	characteristic
49	#2150	C₂×C₅	1	1	yes	characteristic
50	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
51	#1377	C₂×C₅	1	4	yes	—
52	C₂×C₅×Q₈	—	2	1	no	—
53	C₂×C₅×Q₈	—	2	1	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₅×C₈	#41	1	2	yes	—
69	C₅×C₄⋊C₄	—	2	1	no	—
70	C₅×C₄⋊C₄	—	2	1	no	—
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	M₁₆	1	1	yes	characteristic
78	C₄²×C₅	M₁₆	1	1	yes	characteristic
79	C₅×C₁₆	—	2	4	no	—
80	C₅×C₁₆	—	4	2	no	—
81	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
82	C₂×M₃₂	C₂²×C₅	1	2	yes	—
83	C₂×M₃₂	C₂²×C₅	1	2	yes	—
84	#446	—	2	1	no	—
85	#446	—	2	1	no	—
86	#446	—	2	1	no	—
87	#446	—	2	1	no	—
88	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
89	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
90	C₂×#97	C₂²×C₅	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	#346	—	2	1	no	—
95	C₄×C₁₆	—	2	1	no	—
96	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₅×Q₈	—	4	2	no	—
98	C₅×Q₈	—	4	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	#131	1	2	yes	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	4	2	no	—
116	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
117	C₅×C₈	#124	1	1	yes	characteristic
118	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
119	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	2	1	no	—
127	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	4	no	—
133	M₃₂	C₅×Q₈	1	4	yes	complemented
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	#97	—	2	2	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	C₅×D₄	1	1	yes	characteristic
143	C₄×C₈	C₅×D₄	1	1	yes	characteristic
144	C₄×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₅	C₈×Q₈	1	2	yes	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	—	4	2	no	—
155	C₄×C₅	#349	1	2	yes	—
156	C₄×C₅	#391	1	1	yes	characteristic
157	C₄×C₅	#408	1	1	yes	characteristic
158	C₄×C₅	#446	1	2	yes	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
164	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
165	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
175	C₂×C₈	C₅×#41	1	2	yes	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₄²	C₅×M₁₆	1	1	yes	characteristic
185	C₄²	C₅×M₁₆	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₁₆	—	4	2	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	#1984	1	1	yes	characteristic
190	C₂×C₅	C₂×#349	1	1	yes	characteristic
191	C₂×C₅	#2115	1	1	yes	characteristic
192	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
204	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
205	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
207	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
208	C₂×C₄	C₅×#131	1	2	yes	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
213	C₈	C₅×#124	1	1	yes	characteristic
214	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
215	C₅	#13268	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
219	C₂²	C₅×C₈×Q₈	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₅×#349	1	2	yes	—
226	C₄	C₅×#391	1	1	yes	characteristic
227	C₄	C₅×#408	1	1	yes	characteristic
228	C₄	C₅×#446	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₅×#1984	1	1	yes	characteristic
231	C₂	C₂×C₅×#349	1	1	yes	characteristic
232	C₂	C₅×#2115	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. #13268	C₅	direct (N × H)
35. C₅×M₃₂	Q₈	semidirect
133. M₃₂	C₅×Q₈	semidirect
215. C₅	#13268	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×#2150	2
C₄×C₅×M₃₂	2
C₅×#1377	2
#13268	5

Subgroups — C5×#13268 #11935988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13268 #11935988