Subgroups — C₅×#13164 #11935884

Order 1280.

284 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	C₅×#2153	C₂	1	2	yes	maximal
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅×#1772	C₂	1	1	yes	maximal, characteristic
5	C₅×#1378	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₅×#108	C₄	1	1	yes	characteristic
10	C₅×#427	C₄	1	1	yes	characteristic
11	C₅×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#364	—	2	4	no	—
14	C₅×#364	—	2	1	no	—
15	C₅×#364	—	2	1	no	—
16	C₅×#346	—	2	1	no	—
17	C₄×C₅×C₁₆	—	2	1	no	—
18	C₅×#339	C₄	1	2	yes	—
19	#13164	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₅×#117	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₅×M₃₂	—	2	1	no	—
26	C₅×M₃₂	—	2	1	no	—
27	C₅×M₃₂	—	2	2	no	—
28	C₅×M₃₂	—	2	2	no	—
29	C₅×M₃₂	D₄	1	2	yes	—
30	C₅×M₃₂	Q₈	1	2	yes	—
31	C₂×C₅×C₁₆	—	2	1	no	—
32	C₂×C₅×C₁₆	—	2	1	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₅×#108	C₈	1	2	yes	—
35	C₅×#108	C₈	1	2	yes	—
36	C₅×#107	—	2	1	no	—
37	C₅×#106	—	2	1	no	—
38	C₅×#97	—	2	2	no	—
39	C₄×C₅×C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
43	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
44	#2153	C₂×C₅	1	2	yes	—
45	C₂×#364	C₂×C₅	1	1	yes	characteristic
46	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
47	#1772	C₂×C₅	1	1	yes	characteristic
48	#1378	C₂×C₅	1	2	yes	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₅×M₁₆	—	2	2	no	—
52	C₅×M₁₆	C₂×C₈	1	4	yes	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
59	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	1	yes	characteristic
65	C₂×C₅×C₈	#41	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₂²⋊C₄	—	2	2	no	—
68	C₄²×C₅	—	2	1	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₅×C₁₆	—	2	2	no	—
73	C₅×C₁₆	—	2	2	no	—
74	C₅×C₁₆	—	4	2	no	—
75	C₂×M₃₂	C₂²×C₅	1	2	yes	—
76	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
77	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
78	C₂×#108	C₄×C₅	1	1	yes	characteristic
79	#427	C₄×C₅	1	1	yes	characteristic
80	#405	C₂²×C₅	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	#364	—	2	4	no	—
83	#364	—	2	1	no	—
84	#364	—	2	1	no	—
85	#346	—	2	1	no	—
86	C₄×C₁₆	—	2	1	no	—
87	#339	C₄×C₅	1	2	yes	—
88	C₂³×C₅	#131	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	4	2	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	#131	1	1	yes	characteristic
99	C₅×C₈	—	2	4	no	—
100	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
101	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
102	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
103	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
104	C₅×C₈	#121	1	1	yes	characteristic
105	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₅×C₈	#125	1	1	yes	characteristic
107	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
109	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
111	#117	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
113	M₃₂	—	2	1	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	M₃₂	C₅×D₄	1	2	yes	—
118	M₃₂	C₅×Q₈	1	2	yes	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	#108	C₅×C₈	1	2	yes	—
123	#108	C₅×C₈	1	2	yes	—
124	#107	—	2	1	no	—
125	#106	—	2	1	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	C₅×D₄	1	1	yes	characteristic
129	C₄×C₈	C₅×D₄	1	1	yes	characteristic
130	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
131	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
132	C₂²×C₅	—	4	1	no	—
133	C₂²×C₅	#424	1	1	yes	characteristic
134	C₂²×C₅	#436	1	1	yes	characteristic
135	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	—	4	1	no	—
138	C₄×C₅	—	8	2	no	—
139	C₄×C₅	#389	1	1	yes	characteristic
140	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₄×C₅	C₈×D₄	1	1	yes	characteristic
142	C₄×C₅	#446	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	C₂×C₅×C₈	1	4	yes	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
151	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
152	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
153	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
154	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
157	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
158	C₂×C₈	C₅×#41	1	1	yes	characteristic
159	C₂×C₈	C₅×#41	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
165	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
166	C₁₆	—	2	2	no	—
167	C₁₆	—	2	2	no	—
168	C₁₆	—	4	2	no	—
169	C₂×C₅	—	2	1	no	—
170	C₂×C₅	—	2	1	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	#1939	1	1	yes	characteristic
173	C₂³	C₅×#131	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
180	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
182	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
183	C₂×C₄	C₅×#131	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
186	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
187	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
188	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
189	C₈	C₅×#121	1	1	yes	characteristic
190	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
191	C₈	C₅×#125	1	1	yes	characteristic
192	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
193	C₅	#13164	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	4	1	no	—
195	C₂²	C₅×#424	1	1	yes	characteristic
196	C₂²	C₅×#436	1	1	yes	characteristic
197	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₅×#389	1	1	yes	characteristic
202	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
203	C₄	C₅×C₈×D₄	1	1	yes	characteristic
204	C₄	C₅×#446	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₅×#1939	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1939
prime radical	C₂³×C₅	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
19. #13164	C₅	direct (N × H)
193. C₅	#13164	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2153	2
C₂×C₅×#364	2
C₄×C₅×M₃₂	2
C₅×#1772	2
C₅×#1378	2
#13164	5

Subgroups — C5×#13164 #11935884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13164 #11935884