Subgroups — C₅×#13263 #11935983

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1374	C₂	1	4	yes	maximal
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×C₈×Q₈	—	2	2	no	—
9	C₅×C₈×Q₈	—	2	2	no	—
10	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	—	2	2	no	—
15	C₄×C₅×C₁₆	—	2	2	no	—
16	#13263	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₄×C₅×Q₈	—	2	2	no	—
22	C₄×C₅×Q₈	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	—	2	2	no	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₂×C₅×C₁₆	—	2	4	no	—
31	C₂×C₅×C₁₆	Q₈	1	4	yes	complemented
32	C₅×C₄⋊C₈	—	2	4	no	—
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
39	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
40	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
41	#1374	C₂×C₅	1	4	yes	—
42	C₂×C₅×Q₈	—	2	2	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×Q₈	C₁₆	1	4	yes	complemented
45	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₅×C₈	#41	1	2	yes	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	2	2	no	—
60	C₄²×C₅	—	2	2	no	—
61	C₄²×C₅	—	2	2	no	—
62	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅	M₁₆	1	2	yes	—
64	C₅×C₁₆	—	2	4	no	—
65	C₅×C₁₆	—	4	2	no	—
66	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
68	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
69	C₈×Q₈	—	2	2	no	—
70	C₈×Q₈	—	2	2	no	—
71	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
72	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₄⋊C₁₆	—	2	2	no	—
76	C₄×C₁₆	—	2	2	no	—
77	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
78	C₅×Q₈	—	2	8	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
85	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
86	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
87	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
88	C₂×C₄×C₅	M₃₂	1	2	yes	—
89	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
91	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅	#131	1	2	yes	—
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	—	4	2	no	—
95	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
96	C₅×C₈	#124	1	1	yes	characteristic
97	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
98	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
102	C₄×Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	C₅×Q₈	1	4	yes	complemented
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₅×D₄	1	2	yes	—
118	C₄×C₈	C₅×D₄	1	2	yes	—
119	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₅	C₈×Q₈	1	2	yes	—
121	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
122	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
123	C₂²×C₅	#504	1	2	yes	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	4	2	no	—
127	C₄×C₅	C₂²⋊C₁₆	1	4	yes	—
128	C₄×C₅	#391	1	1	yes	characteristic
129	C₄×C₅	#408	1	1	yes	characteristic
130	C₄×C₅	#446	1	2	yes	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	C₅×C₁₆	1	4	yes	complemented
134	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
135	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
137	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
143	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
146	C₂×C₈	C₅×#41	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
152	C₄²	C₅×M₁₆	1	2	yes	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	4	2	no	—
155	C₂×C₅	#1984	1	1	yes	characteristic
156	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
157	C₂×C₅	#2113	1	1	yes	characteristic
158	C₂×C₅	Q₈×C₁₆	1	2	yes	—
159	C₂×C₅	#2183	1	2	yes	—
160	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
161	Q₈	—	2	8	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
168	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
169	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	—
170	C₂×C₄	C₂×C₅×C₁₆	1	2	yes	—
171	C₂×C₄	C₅×M₃₂	1	2	yes	—
172	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
174	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
175	C₂×C₄	C₅×#131	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	2	no	—
178	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
179	C₈	C₅×#124	1	1	yes	characteristic
180	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
181	C₅	#13263	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
183	C₂²	C₅×C₈×Q₈	1	2	yes	—
184	C₂²	C₂²×C₅×C₁₆	1	1	yes	characteristic
185	C₂²	C₂×C₅×M₃₂	1	1	yes	characteristic
186	C₂²	C₅×#504	1	2	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	C₅×C₂²⋊C₁₆	1	4	yes	—
191	C₄	C₅×#391	1	1	yes	characteristic
192	C₄	C₅×#408	1	1	yes	characteristic
193	C₄	C₅×#446	1	2	yes	—
194	C₂	C₅×#1984	1	1	yes	characteristic
195	C₂	C₂×C₅×C₂²⋊C₁₆	1	1	yes	characteristic
196	C₂	C₅×#2113	1	1	yes	characteristic
197	C₂	C₅×Q₈×C₁₆	1	2	yes	—
198	C₂	C₅×#2183	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂²×C₅×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
16. #13263	C₅	direct (N × H)
31. C₂×C₅×C₁₆	Q₈	semidirect
44. C₂×C₅×Q₈	C₁₆	semidirect
112. C₂×C₁₆	C₅×Q₈	semidirect
133. C₂×Q₈	C₅×C₁₆	semidirect
181. C₅	#13263	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×C₈×Q₈	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₄×C₅×C₁₆	2
C₅×#1374	2
#13263	5

Subgroups — C5×#13263 #11935983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13263 #11935983