Subgroups — C₅×#8617 #11931337

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₅×#1801	C₂	1	2	yes	maximal
3	C₅×#1567	C₂	1	2	yes	maximal
4	C₅×#1292	C₂	1	2	yes	maximal
5	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
6	C₅×C₈×Q₈	—	2	2	no	—
7	C₅×C₈×Q₈	—	2	2	no	—
8	C₅×#423	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#101	—	2	2	no	—
12	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#343	—	2	2	no	—
15	C₅×#335	—	2	2	no	—
16	C₅×#335	—	2	2	no	—
17	#8617	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅×Q₈	—	2	2	no	—
24	C₄×C₅×Q₈	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×C₄⋊C₈	—	2	4	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	4	no	—
32	C₅×C₄⋊C₈	—	2	4	no	—
33	C₅×#101	—	4	4	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	#1801	C₂×C₅	1	2	yes	—
40	#1567	C₂×C₅	1	2	yes	—
41	#1292	C₂×C₅	1	2	yes	—
42	C₂×C₅×Q₈	—	2	2	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×Q₈	D₈	1	2	yes	—
45	C₂×C₅×Q₈	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₅×M₁₆	—	4	4	no	—
52	C₅×M₁₆	—	4	4	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	4	1	no	—
56	C₂×C₅×C₈	—	4	2	no	—
57	C₂×C₅×C₈	—	4	4	no	—
58	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	2	2	no	—
62	C₅×C₄⋊C₄	—	2	4	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	C₂×D₄	1	2	yes	—
66	C₄²×C₅	#41	1	2	yes	—
67	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
68	C₈×Q₈	—	2	2	no	—
69	C₈×Q₈	—	2	2	no	—
70	#423	C₂²×C₅	1	4	yes	—
71	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
72	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×#101	—	2	2	no	—
74	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
75	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	#343	—	2	2	no	—
77	#335	—	2	2	no	—
78	#335	—	2	2	no	—
79	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₅×Q₈	—	2	4	no	—
81	C₅×Q₈	—	2	4	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	4	1	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	#107	1	4	yes	—
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	#117	1	1	yes	characteristic
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	#135	1	2	yes	—
97	C₅×C₈	—	2	1	no	—
98	C₅×C₈	—	2	2	no	—
99	C₅×C₈	—	4	2	no	—
100	C₅×C₈	—	8	4	no	—
101	C₅×C₈	#101	1	2	yes	complemented
102	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
103	C₂×M₁₆	—	2	4	no	—
104	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₄×Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
111	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	#101	—	4	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₅×D₄	1	2	yes	—
121	C₄×C₈	C₅×D₄	1	2	yes	—
122	C₂²×C₅	—	2	1	no	—
123	C₂²×C₅	—	2	1	no	—
124	C₂²×C₅	#380	1	1	yes	characteristic
125	C₂²×C₅	C₂×#107	1	1	yes	characteristic
126	C₂²×C₅	#427	1	1	yes	characteristic
127	C₄×C₅	—	2	1	no	—
128	C₄×C₅	—	2	4	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	C₂×#101	1	1	yes	characteristic
132	C₄×C₅	#413	1	1	yes	characteristic
133	C₄×C₅	#466	1	2	yes	—
134	C₄×C₅	#467	1	2	yes	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	C₅×D₈	1	2	yes	—
138	C₂×Q₈	C₅×Q₁₆	1	2	yes	—
139	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
140	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
144	M₁₆	—	4	4	no	—
145	M₁₆	—	4	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
153	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	C₂×C₅×D₄	1	2	yes	—
159	C₄²	C₅×#41	1	2	yes	—
160	C₂×C₅	—	2	1	no	—
161	C₂×C₅	—	2	1	no	—
162	C₂×C₅	#1801	1	1	yes	characteristic
163	C₂×C₅	#1808	1	1	yes	characteristic
164	C₂×C₅	#1813	1	1	yes	characteristic
165	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
166	Q₈	—	2	4	no	—
167	Q₈	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₅×#107	1	4	yes	—
176	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₅×#117	1	1	yes	characteristic
178	C₂×C₄	C₅×#121	1	2	yes	—
179	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
180	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
181	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
182	C₂×C₄	C₅×#135	1	2	yes	—
183	C₈	—	2	1	no	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	—	8	4	no	—
187	C₈	C₅×#101	1	2	yes	complemented
188	C₅	#8617	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₅×#380	1	1	yes	characteristic
192	C₂²	C₂×C₅×#107	1	1	yes	characteristic
193	C₂²	C₅×#427	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×C₅×#101	1	1	yes	characteristic
199	C₄	C₅×#413	1	1	yes	characteristic
200	C₄	C₅×#466	1	2	yes	—
201	C₄	C₅×#467	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	C₅×#1801	1	1	yes	characteristic
205	C₂	C₅×#1808	1	1	yes	characteristic
206	C₂	C₅×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
17. #8617	C₅	direct (N × H)
101. C₅×C₈	#101	semidirect
187. C₈	C₅×#101	semidirect
188. C₅	#8617	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×#1801	2
C₅×#1567	2
C₅×#1292	2
#8617	5

Subgroups — C5×#8617 #11931337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8617 #11931337