Subgroups — C₅×#13274 #11935994

Order 1280.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
4	C₅×#1382	C₂	1	4	yes	maximal
5	C₅×#533	—	2	1	no	—
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₅×#500	C₄	1	1	yes	characteristic
8	C₅×#500	C₄	1	1	yes	characteristic
9	C₅×#499	—	2	1	no	—
10	C₅×#499	C₄	1	2	yes	—
11	C₅×#498	—	2	1	no	—
12	C₂×C₅×#108	C₂²	1	2	yes	—
13	C₅×#427	—	2	1	no	—
14	C₅×#427	C₂²	1	2	yes	—
15	C₄×C₅×M₁₆	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×#341	—	2	1	no	—
20	#13274	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₅×#125	—	2	2	no	—
27	C₅×C₂²⋊Q₈	—	4	1	no	—
28	C₅×C₂²⋊Q₈	—	4	1	no	—
29	C₄×C₅×Q₈	—	4	1	no	—
30	C₄×C₅×Q₈	—	4	1	no	—
31	C₅×#117	—	2	2	no	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	4	no	—
35	C₅×#108	—	2	2	no	—
36	C₅×#108	Q₈	1	4	yes	complemented
37	C₅×#107	—	2	1	no	—
38	C₅×#107	—	2	1	no	—
39	C₅×#107	C₂×C₄	1	2	yes	—
40	C₅×#107	C₂×C₄	1	2	yes	—
41	C₅×#106	—	2	2	no	—
42	C₅×#106	—	2	2	no	—
43	C₅×#97	—	4	1	no	—
44	C₄×C₅×C₈	—	4	1	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	#3160	C₂×C₅	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
51	C₄×#108	C₂×C₅	1	1	yes	characteristic
52	#1382	C₂×C₅	1	4	yes	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₅×M₁₆	—	2	4	no	—
59	C₅×M₁₆	—	4	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²⋊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	2	yes	—
67	C₂×C₅×C₈	#41	1	2	yes	—
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₅×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₅	—	4	2	no	—
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅	D₈	1	1	yes	characteristic
79	C₄²×C₅	D₈	1	1	yes	characteristic
80	C₅×C₁₆	—	4	2	no	—
81	#533	—	2	1	no	—
82	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
83	#500	C₄×C₅	1	1	yes	characteristic
84	#500	C₄×C₅	1	1	yes	characteristic
85	#499	—	2	1	no	—
86	#499	C₄×C₅	1	2	yes	—
87	#498	—	2	1	no	—
88	C₂×#108	C₂²×C₅	1	2	yes	—
89	#427	—	2	1	no	—
90	#427	C₂²×C₅	1	2	yes	—
91	C₄×M₁₆	—	2	1	no	—
92	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₄⋊C₁₆	—	2	1	no	—
94	C₄⋊C₁₆	—	2	1	no	—
95	#341	—	2	1	no	—
96	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
97	C₅×Q₈	—	8	2	no	—
98	C₅×Q₈	—	8	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	4	1	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	#102	1	1	yes	characteristic
104	C₂×C₄×C₅	#102	1	1	yes	characteristic
105	C₂×C₄×C₅	#102	1	1	yes	characteristic
106	C₂×C₄×C₅	#102	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#137	1	2	yes	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	4	4	no	—
114	C₅×C₈	C₄×Q₈	1	2	yes	—
115	C₅×C₈	#124	1	1	yes	characteristic
116	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	1	no	—
121	C₄⋊Q₈	—	2	1	no	—
122	#125	—	2	2	no	—
123	C₂²⋊Q₈	—	4	1	no	—
124	C₂²⋊Q₈	—	4	1	no	—
125	C₄×Q₈	—	4	1	no	—
126	C₄×Q₈	—	4	1	no	—
127	#117	—	2	2	no	—
128	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	4	no	—
131	#108	—	2	2	no	—
132	#108	C₅×Q₈	1	4	yes	complemented
133	#107	—	2	1	no	—
134	#107	—	2	1	no	—
135	#107	C₂×C₄×C₅	1	2	yes	—
136	#107	C₂×C₄×C₅	1	2	yes	—
137	#106	—	2	2	no	—
138	#106	—	2	2	no	—
139	#97	—	4	1	no	—
140	C₄×C₈	—	4	1	no	—
141	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
142	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₄×C₈	C₅×D₄	1	1	yes	characteristic
144	C₄×C₈	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	C₂×#102	1	1	yes	characteristic
148	C₂²×C₅	#475	1	2	yes	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	8	2	no	—
153	C₄×C₅	—	8	2	no	—
154	C₄×C₅	#359	1	2	yes	—
155	C₄×C₅	#391	1	1	yes	characteristic
156	C₄×C₅	#419	1	1	yes	characteristic
157	C₄×C₅	#474	1	2	yes	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
161	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	M₁₆	—	2	4	no	—
164	M₁₆	—	4	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
172	C₂×C₈	C₅×#41	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₄²	C₅×D₈	1	1	yes	characteristic
184	C₄²	C₅×D₈	1	1	yes	characteristic
185	C₁₆	—	4	2	no	—
186	C₂×C₅	—	2	2	no	—
187	C₂×C₅	#1988	1	1	yes	characteristic
188	C₂×C₅	C₂×#359	1	1	yes	characteristic
189	C₂×C₅	#2143	1	1	yes	characteristic
190	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
191	Q₈	—	8	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₅×#102	1	1	yes	characteristic
198	C₂×C₄	C₅×#102	1	1	yes	characteristic
199	C₂×C₄	C₅×#102	1	1	yes	characteristic
200	C₂×C₄	C₅×#102	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
203	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
204	C₂×C₄	C₅×#137	1	2	yes	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	4	4	no	—
208	C₈	C₄×C₅×Q₈	1	2	yes	—
209	C₈	C₅×#124	1	1	yes	characteristic
210	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
211	C₅	#13274	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×C₅×#102	1	1	yes	characteristic
215	C₂²	C₅×#475	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₅×#359	1	2	yes	—
222	C₄	C₅×#391	1	1	yes	characteristic
223	C₄	C₅×#419	1	1	yes	characteristic
224	C₄	C₅×#474	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	C₅×#1988	1	1	yes	characteristic
227	C₂	C₂×C₅×#359	1	1	yes	characteristic
228	C₂	C₅×#2143	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×#102
prime radical	C₂³×C₅	C₂×SD₁₆
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
20. #13274	C₅	direct (N × H)
36. C₅×#108	Q₈	semidirect
132. #108	C₅×Q₈	semidirect
211. C₅	#13274	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×#3160	2
C₂×C₅×C₄⋊C₁₆	2
C₄×C₅×#108	2
C₅×#1382	2
#13274	5

Subgroups — C5×#13274 #11935994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13274 #11935994