Subgroups — C₅×#21715 #11944435

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 210 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	2	yes	maximal
2	C₅×#2399	C₂	1	1	yes	maximal, characteristic
3	C₅×#1985	C₂	1	2	yes	maximal
4	C₅×#1940	C₂	1	4	yes	maximal
5	C₅×#1843	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×C₄⋊C₄	C₂	1	4	yes	maximal
7	C₅×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
9	C₅×C₈×Q₈	—	2	8	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₅×#97	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#398	C₄	1	4	yes	—
19	C₅×#395	C₄	1	4	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
21	C₅×#376	C₂²	1	1	yes	characteristic
22	C₅×#336	C₂²	1	8	yes	—
23	#21715	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂³	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₅×Q₈	—	2	4	no	—
30	C₄×C₅×Q₈	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	4	yes	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₅×C₄⋊C₈	—	2	8	no	—
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	8	no	—
40	C₅×#97	—	2	2	no	—
41	C₅×#97	—	2	4	no	—
42	C₄×C₅×C₈	—	2	8	no	—
43	C₄×C₅×C₈	D₄	1	4	yes	—
44	C₄×C₅×C₈	Q₈	1	8	yes	complemented
45	C₅×#95	C₂×C₄	1	4	yes	—
46	C₅×#95	C₂×C₄	1	4	yes	—
47	C₅×#95	C₂×C₄	1	4	yes	—
48	C₂×C₈×Q₈	C₂×C₅	1	2	yes	—
49	#2399	C₂×C₅	1	1	yes	characteristic
50	#1985	C₂×C₅	1	2	yes	—
51	#1940	C₂×C₅	1	4	yes	—
52	#1843	C₂×C₅	1	1	yes	characteristic
53	C₈×C₄⋊C₄	C₂×C₅	1	4	yes	—
54	#1726	C₂×C₅	1	1	yes	characteristic
55	C₂×C₅×Q₈	—	2	4	no	—
56	C₂×C₅×Q₈	—	2	8	no	—
57	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	2	8	no	—
67	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
71	C₅×C₄⋊C₄	—	2	8	no	—
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₅×C₄⋊C₄	—	2	16	no	—
74	C₄²×C₅	—	2	4	no	—
75	C₄²×C₅	—	2	4	no	—
76	C₄²×C₅	C₂×D₄	1	2	yes	—
77	C₄²×C₅	C₂×Q₈	1	4	yes	—
78	C₄²×C₅	#41	1	2	yes	—
79	C₄²×C₅	#41	1	4	yes	—
80	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
81	C₈×Q₈	—	2	8	no	—
82	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
84	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
85	C₂×#97	C₂²×C₅	1	2	yes	—
86	C₂×#97	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
90	#398	C₄×C₅	1	4	yes	—
91	#395	C₄×C₅	1	4	yes	—
92	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
93	#376	C₂²×C₅	1	1	yes	characteristic
94	#336	C₂²×C₅	1	8	yes	—
95	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₅×Q₈	—	2	16	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	8	no	—
103	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
105	C₂×C₄×C₅	#131	1	8	yes	—
106	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
108	C₂×C₄×C₅	C₂×#41	1	2	yes	—
109	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	—	2	4	no	—
112	C₅×C₈	—	4	4	no	—
113	C₅×C₈	C₄⋊Q₈	1	4	yes	—
114	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₈	C₂³×C₅	1	4	yes	—
116	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₅	1	2	yes	—
118	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
119	C₄×Q₈	—	2	4	no	—
120	C₄×Q₈	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
124	C₂×C₄²	C₂³×C₅	1	4	yes	—
125	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₄²	C₂³×C₅	1	2	yes	—
127	C₄⋊C₈	—	2	8	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	8	no	—
130	#97	—	2	2	no	—
131	#97	—	2	4	no	—
132	C₄×C₈	—	2	8	no	—
133	C₄×C₈	C₅×D₄	1	4	yes	—
134	C₄×C₈	C₅×Q₈	1	8	yes	complemented
135	#95	C₂×C₄×C₅	1	4	yes	—
136	#95	C₂×C₄×C₅	1	4	yes	—
137	#95	C₂×C₄×C₅	1	4	yes	—
138	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
140	C₂²×C₅	C₂×#131	1	4	yes	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	2	8	no	—
143	C₄×C₅	—	2	8	no	—
144	C₄×C₅	—	4	4	no	—
145	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₄×C₅	#532	1	1	yes	characteristic
147	C₄×C₅	#533	1	2	yes	—
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	2	8	no	—
150	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
154	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
161	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
162	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
163	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	16	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₂×C₅×D₄	1	2	yes	—
170	C₄²	C₂×C₅×Q₈	1	4	yes	—
171	C₄²	C₅×#41	1	2	yes	—
172	C₄²	C₅×#41	1	4	yes	—
173	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₅	#2929	1	2	yes	—
175	C₂×C₅	#2960	1	4	yes	—
176	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
177	Q₈	—	2	16	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
185	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
186	C₂×C₄	C₅×#131	1	8	yes	—
187	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
189	C₂×C₄	C₂×C₅×#41	1	2	yes	—
190	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	2	4	no	—
193	C₈	—	4	4	no	—
194	C₈	C₅×C₄⋊Q₈	1	4	yes	—
195	C₅	#21715	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
197	C₂²	C₂×C₄×C₅×Q₈	1	2	yes	—
198	C₂²	C₂×C₅×#131	1	4	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	8	no	—
201	C₄	—	2	8	no	—
202	C₄	—	4	4	no	—
203	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
204	C₄	C₅×#532	1	1	yes	characteristic
205	C₄	C₅×#533	1	2	yes	—
206	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
207	C₂	C₅×#2929	1	2	yes	—
208	C₂	C₅×#2960	1	4	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
23. #21715	C₅	direct (N × H)
44. C₄×C₅×C₈	Q₈	semidirect
134. C₄×C₈	C₅×Q₈	semidirect
195. C₅	#21715	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
C₅×#2399	2
C₅×#1985	2
C₅×#1940	2
C₅×#1843	2
C₅×C₈×C₄⋊C₄	2
C₅×#1726	2
#21715	5

Subgroups — C5×#21715 #11944435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21715 #11944435