Subgroups — C₅×#13179 #11935899

Order 1280.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	2	yes	maximal
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1848	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
5	C₅×#1381	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂×C₅×#108	—	2	1	no	—
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₂×C₅×#108	C₂²	1	1	yes	characteristic
10	C₂×C₅×#108	C₂²	1	1	yes	characteristic
11	C₅×#423	—	2	1	no	—
12	C₄×C₅×M₁₆	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#368	—	2	4	no	—
15	C₅×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₅×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₅×#341	—	2	1	no	—
18	#13179	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×C₁₆	—	2	2	no	—
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
28	C₅×#108	—	2	2	no	—
29	C₅×#108	—	2	1	no	—
30	C₅×#108	—	2	1	no	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	—	4	2	no	—
33	C₅×#108	D₄	1	2	yes	—
34	C₅×#108	Q₈	1	2	yes	—
35	C₅×C₄⋊C₈	—	4	1	no	—
36	C₅×C₄⋊C₈	—	4	1	no	—
37	C₅×#97	—	4	1	no	—
38	C₄×C₅×C₈	—	4	1	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
42	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
43	C₂×#368	C₂×C₅	1	2	yes	—
44	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
45	#1848	C₂×C₅	1	1	yes	characteristic
46	C₄×#108	C₂×C₅	1	1	yes	characteristic
47	#1381	C₂×C₅	1	2	yes	—
48	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₅×M₁₆	—	2	2	no	—
52	C₅×M₁₆	—	2	2	no	—
53	C₅×M₁₆	—	4	1	no	—
54	C₅×M₁₆	—	4	1	no	—
55	C₅×M₁₆	—	4	2	no	—
56	C₅×M₁₆	—	4	2	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₅×C₈	#41	1	1	yes	characteristic
69	C₂×C₅×C₈	#41	1	1	yes	characteristic
70	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	D₈	1	1	yes	characteristic
73	C₄²×C₅	D₈	1	1	yes	characteristic
74	C₅×C₁₆	—	2	4	no	—
75	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
76	C₂×#108	—	2	1	no	—
77	C₂×#108	C₂²×C₅	1	2	yes	—
78	C₂×#108	C₂²×C₅	1	1	yes	characteristic
79	C₂×#108	C₂²×C₅	1	1	yes	characteristic
80	#423	—	2	1	no	—
81	C₄×M₁₆	—	2	1	no	—
82	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	#368	—	2	4	no	—
84	C₄⋊C₁₆	C₄×C₅	1	2	yes	—
85	C₄⋊C₁₆	C₄×C₅	1	2	yes	—
86	#341	—	2	1	no	—
87	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	4	1	no	—
90	C₂×C₄×C₅	#107	1	1	yes	characteristic
91	C₂×C₄×C₅	#107	1	1	yes	characteristic
92	C₂×C₄×C₅	#107	1	1	yes	characteristic
93	C₂×C₄×C₅	#107	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	#121	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₅	#137	1	1	yes	characteristic
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	—	4	2	no	—
103	C₅×C₈	—	4	2	no	—
104	C₅×C₈	—	8	2	no	—
105	C₅×C₈	C₄×D₄	1	1	yes	characteristic
106	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
107	C₅×C₈	#125	1	1	yes	characteristic
108	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×M₁₆	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
114	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	1	no	—
120	#108	—	2	1	no	—
121	#108	—	2	2	no	—
122	#108	—	4	2	no	—
123	#108	C₅×D₄	1	2	yes	—
124	#108	C₅×Q₈	1	2	yes	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄⋊C₈	—	4	1	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
132	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	—	2	1	no	—
135	C₂²×C₅	C₂×#107	1	1	yes	characteristic
136	C₂²×C₅	#462	1	1	yes	characteristic
137	C₂²×C₅	#477	1	1	yes	characteristic
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	#368	1	2	yes	complemented
142	C₄×C₅	#389	1	1	yes	characteristic
143	C₄×C₅	#428	1	1	yes	characteristic
144	C₄×C₅	#459	1	1	yes	characteristic
145	C₄×C₅	#474	1	1	yes	characteristic
146	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
147	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	1	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
163	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
166	C₂×C₈	C₅×#41	1	1	yes	characteristic
167	C₂×C₈	C₅×#41	1	1	yes	characteristic
168	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₄²	C₅×D₈	1	1	yes	characteristic
171	C₄²	C₅×D₈	1	1	yes	characteristic
172	C₁₆	—	2	4	no	—
173	C₂×C₅	—	2	1	no	—
174	C₂×C₅	—	2	1	no	—
175	C₂×C₅	#1947	1	1	yes	characteristic
176	C₂×C₅	C₂×#368	1	1	yes	characteristic
177	C₂×C₅	#2161	1	1	yes	characteristic
178	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	C₅×#107	1	1	yes	characteristic
182	C₂×C₄	C₅×#107	1	1	yes	characteristic
183	C₂×C₄	C₅×#107	1	1	yes	characteristic
184	C₂×C₄	C₅×#107	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₅×#121	1	1	yes	characteristic
187	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
188	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
189	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₄	C₅×#137	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	8	2	no	—
196	C₈	C₄×C₅×D₄	1	1	yes	characteristic
197	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
198	C₈	C₅×#125	1	1	yes	characteristic
199	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
200	C₅	#13179	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₅×#107	1	1	yes	characteristic
204	C₂²	C₅×#462	1	1	yes	characteristic
205	C₂²	C₅×#477	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	C₅×#368	1	2	yes	complemented
210	C₄	C₅×#389	1	1	yes	characteristic
211	C₄	C₅×#428	1	1	yes	characteristic
212	C₄	C₅×#459	1	1	yes	characteristic
213	C₄	C₅×#474	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	C₅×#1947	1	1	yes	characteristic
217	C₂	C₂×C₅×#368	1	1	yes	characteristic
218	C₂	C₅×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
18. #13179	C₅	direct (N × H)
141. C₄×C₅	#368	semidirect
200. C₅	#13179	direct (N × H)
209. C₄	C₅×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#368	2
C₂×C₅×C₄⋊C₁₆	2
C₅×#1848	2
C₄×C₅×#108	2
C₅×#1381	2
#13179	5

Subgroups — C5×#13179 #11935899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13179 #11935899