Subgroups — C₂×C₅×#1363 #11934653

Order 1280.

318 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×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
4	C₅×#1363	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#125	—	2	1	no	—
6	C₂²×C₅×C₁₆	—	2	1	no	—
7	C₅×#499	—	2	1	no	—
8	C₅×#499	—	2	1	no	—
9	C₅×#499	C₄	1	2	yes	—
10	C₅×#499	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₄	1	1	yes	characteristic
12	C₂×C₅×#107	C₄	1	1	yes	characteristic
13	C₂×C₅×#106	—	2	1	no	—
14	C₂×C₅×C₄⋊C₈	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	—	2	1	no	—
17	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₅×#334	—	2	1	no	—
19	C₅×#334	C₂²	1	2	yes	—
20	C₂×#1363	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×C₈	—	4	1	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₅×#125	—	2	2	no	—
25	C₅×#125	—	2	2	no	—
26	C₅×#125	—	4	1	no	—
27	C₅×#125	—	4	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	4	1	no	—
31	C₂×C₅×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	4	2	no	—
35	C₅×#107	—	2	1	no	—
36	C₅×#107	—	2	1	no	—
37	C₅×#107	C₂×C₄	1	2	yes	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×#106	—	2	2	no	—
40	C₅×#106	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	4	1	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂×#499	C₂×C₅	1	1	yes	characteristic
48	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
49	C₂×#334	C₂×C₅	1	1	yes	characteristic
50	#1363	C₂×C₅	1	4	yes	direct product factor
51	C₂²×C₄×C₅	—	4	1	no	—
52	C₂²×C₄×C₅	—	4	1	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	4	2	no	—
58	C₂×C₅×C₈	—	8	2	no	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	SD₁₆	1	2	yes	—
62	C₂×C₅×C₈	Q₁₆	1	2	yes	—
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	—	2	2	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	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	C₂×#125	—	2	1	no	—
79	C₂²×C₁₆	—	2	1	no	—
80	#499	—	2	1	no	—
81	#499	—	2	1	no	—
82	#499	C₄×C₅	1	2	yes	—
83	#499	C₂²×C₅	1	2	yes	—
84	C₂×#107	C₄×C₅	1	1	yes	characteristic
85	C₂×#107	C₄×C₅	1	1	yes	characteristic
86	C₂×#106	—	2	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
91	#334	—	2	1	no	—
92	#334	C₂²×C₅	1	2	yes	—
93	C₂³×C₅	#102	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	8	2	no	—
99	C₂×C₄×C₅	—	8	2	no	—
100	C₂×C₄×C₅	#101	1	2	yes	—
101	C₂×C₄×C₅	#102	1	1	yes	characteristic
102	C₂×C₄×C₅	#102	1	1	yes	characteristic
103	C₂×C₄×C₅	#102	1	1	yes	characteristic
104	C₂×C₄×C₅	#103	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	8	2	no	—
111	C₅×C₈	#103	1	2	yes	—
112	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
113	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
117	#125	—	2	2	no	—
118	#125	—	2	2	no	—
119	#125	—	4	1	no	—
120	#125	—	4	1	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	4	1	no	—
124	C₂×C₄⋊C₄	—	4	1	no	—
125	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	4	2	no	—
128	#107	—	2	1	no	—
129	#107	—	2	1	no	—
130	#107	C₂×C₄×C₅	1	2	yes	—
131	#107	C₂×C₄×C₅	1	2	yes	—
132	#106	—	2	2	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	4	1	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	#332	1	2	yes	—
141	C₂²×C₅	#359	1	2	yes	—
142	C₂²×C₅	#360	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	—	2	1	no	—
148	C₄×C₅	—	8	2	no	—
149	C₄×C₅	—	8	2	no	—
150	C₄×C₅	#332	1	2	yes	—
151	C₄×C₅	C₂×#101	1	1	yes	characteristic
152	C₄×C₅	C₂×#103	1	1	yes	characteristic
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
157	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	8	2	no	—
161	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₅×SD₁₆	1	2	yes	—
164	C₂×C₈	C₅×Q₁₆	1	2	yes	—
165	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
166	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	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	2	no	—
175	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₄²	C₅×D₈	1	1	yes	characteristic
177	C₄²	C₅×SD₁₆	1	1	yes	characteristic
178	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
179	C₁₆	—	4	2	no	—
180	C₂×C₅	#1363	1	4	yes	direct product factor
181	C₂×C₅	C₂×#332	1	1	yes	characteristic
182	C₂×C₅	C₂×#359	1	1	yes	characteristic
183	C₂×C₅	C₂×#360	1	1	yes	characteristic
184	C₂³	C₅×#102	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₅×#101	1	2	yes	—
192	C₂×C₄	C₅×#102	1	1	yes	characteristic
193	C₂×C₄	C₅×#102	1	1	yes	characteristic
194	C₂×C₄	C₅×#102	1	1	yes	characteristic
195	C₂×C₄	C₅×#103	1	2	yes	—
196	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
198	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	8	2	no	—
202	C₈	C₅×#103	1	2	yes	—
203	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
204	C₈	C₂×C₅×Q₁₆	1	1	yes	characteristic
205	C₅	C₂×#1363	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	C₅×#332	1	2	yes	—
207	C₂²	C₅×#359	1	2	yes	—
208	C₂²	C₅×#360	1	2	yes	—
209	C₂²	C₂×C₅×#102	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	2	no	—
216	C₄	C₅×#332	1	2	yes	—
217	C₄	C₂×C₅×#101	1	1	yes	characteristic
218	C₄	C₂×C₅×#103	1	1	yes	characteristic
219	C₂	C₅×#1363	1	4	yes	direct product factor
220	C₂	C₂×C₅×#332	1	1	yes	characteristic
221	C₂	C₂×C₅×#359	1	1	yes	characteristic
222	C₂	C₂×C₅×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#1363	C₂	direct (N × H)
20. C₂×#1363	C₅	direct (N × H)
50. #1363	C₂×C₅	direct (N × H)
180. C₂×C₅	#1363	direct (N × H)
205. C₅	C₂×#1363	direct (N × H)
219. C₂	C₅×#1363	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×#499	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#334	2
C₅×#1363	2
C₂×#1363	5

Subgroups — C2×C5×#1363 #11934653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1363 #11934653