Subgroups — C₅×#13537 #11936257

Order 1280.

294 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1942	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic
3	C₅×#1380	C₂	1	2	yes	maximal
4	C₅×#1380	C₂	1	2	yes	maximal
5	C₅×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	C₂²	1	2	yes	—
9	C₂×C₅×#107	C₂²	1	1	yes	characteristic
10	C₂×C₅×#107	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#389	—	2	1	no	—
13	C₅×#384	—	2	1	no	—
14	C₅×#340	—	2	1	no	—
15	#13537	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	—	2	1	no	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	1	no	—
21	C₂×C₅×C₄⋊C₄	—	4	1	no	—
22	C₂×C₅×C₄⋊C₄	—	4	1	no	—
23	C₂×C₅×C₄⋊C₄	—	4	1	no	—
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	—	2	2	no	—
28	C₅×#107	—	2	1	no	—
29	C₅×#107	—	2	2	no	—
30	C₅×#107	—	2	4	no	—
31	C₅×#107	—	4	2	no	—
32	C₅×#107	Q₈	1	2	yes	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₅×#95	—	4	1	no	—
35	#1942	C₂×C₅	1	1	yes	characteristic
36	#1845	C₂×C₅	1	1	yes	characteristic
37	#1380	C₂×C₅	1	2	yes	—
38	#1380	C₂×C₅	1	2	yes	—
39	#1374	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₄×C₅	—	4	1	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	—	4	1	no	—
43	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	SD₁₆	1	2	yes	—
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	2	yes	—
51	C₂×C₅×C₈	#41	1	2	yes	—
52	C₂×C₅×C₈	#41	1	1	yes	characteristic
53	C₅×C₄⋊C₄	—	2	2	no	—
54	C₅×C₄⋊C₄	—	4	1	no	—
55	C₅×C₄⋊C₄	—	4	1	no	—
56	C₅×C₄⋊C₄	—	4	2	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₅×C₄⋊C₄	—	8	2	no	—
60	C₅×C₄⋊C₄	—	8	2	no	—
61	C₅×C₄⋊C₄	—	8	2	no	—
62	C₄²×C₅	—	2	1	no	—
63	C₄²×C₅	—	2	1	no	—
64	C₅×C₁₆	—	4	2	no	—
65	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	C₂²×C₅	1	2	yes	—
68	C₂×#107	C₂²×C₅	1	1	yes	characteristic
69	C₂×#107	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	#389	—	2	1	no	—
72	#384	—	2	1	no	—
73	#340	—	2	1	no	—
74	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	8	1	no	—
81	C₂×C₄×C₅	—	8	1	no	—
82	C₂×C₄×C₅	—	8	1	no	—
83	C₂×C₄×C₅	—	8	2	no	—
84	C₂×C₄×C₅	—	8	1	no	—
85	C₂×C₄×C₅	—	8	2	no	—
86	C₂×C₄×C₅	—	8	2	no	—
87	C₂×C₄×C₅	—	8	1	no	—
88	C₂×C₄×C₅	D₁₆	1	2	yes	—
89	C₂×C₄×C₅	SD₃₂	1	2	yes	—
90	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	#123	1	1	yes	characteristic
92	C₂×C₄×C₅	#124	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	#137	1	1	yes	characteristic
95	C₂×C₄×C₅	#137	1	1	yes	characteristic
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	#125	1	2	yes	—
98	C₅×C₈	#126	1	2	yes	—
99	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	1	no	—
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄⋊C₄	—	4	1	no	—
107	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#107	—	2	1	no	—
112	#107	—	2	2	no	—
113	#107	—	2	4	no	—
114	#107	—	4	2	no	—
115	#107	C₅×Q₈	1	2	yes	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	4	1	no	—
118	C₂²×C₅	#474	1	1	yes	characteristic
119	C₂²×C₅	#480	1	1	yes	characteristic
120	C₂²×C₅	#486	1	1	yes	characteristic
121	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
122	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
123	C₂²×C₅	#510	1	1	yes	characteristic
124	C₂²×C₅	#510	1	1	yes	characteristic
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	8	2	no	—
128	C₄×C₅	—	16	1	no	—
129	C₄×C₅	—	16	2	no	—
130	C₄×C₅	#400	1	1	yes	characteristic
131	C₄×C₅	#475	1	1	yes	characteristic
132	C₄×C₅	#483	1	1	yes	characteristic
133	C₄×C₅	#491	1	1	yes	characteristic
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
138	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	C₅×SD₁₆	1	2	yes	—
143	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
144	C₂×C₈	C₅×#41	1	2	yes	—
145	C₂×C₈	C₅×#41	1	2	yes	—
146	C₂×C₈	C₅×#41	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	8	2	no	—
154	C₄⋊C₄	—	8	2	no	—
155	C₄⋊C₄	—	8	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	4	2	no	—
159	C₂×C₅	#2099	1	1	yes	characteristic
160	C₂×C₅	#2224	1	1	yes	characteristic
161	C₂×C₅	#2225	1	1	yes	characteristic
162	C₂×C₅	#2232	1	1	yes	characteristic
163	C₂×C₅	#2235	1	1	yes	characteristic
164	C₂×C₅	#2240	1	1	yes	characteristic
165	C₂×C₅	#2245	1	1	yes	characteristic
166	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	C₅×D₁₆	1	2	yes	—
181	C₂×C₄	C₅×SD₃₂	1	2	yes	—
182	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₅×#123	1	1	yes	characteristic
184	C₂×C₄	C₅×#124	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
186	C₂×C₄	C₅×#137	1	1	yes	characteristic
187	C₂×C₄	C₅×#137	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	C₅×#125	1	2	yes	—
190	C₈	C₅×#126	1	2	yes	—
191	C₅	#13537	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₅×#474	1	1	yes	characteristic
193	C₂²	C₅×#480	1	1	yes	characteristic
194	C₂²	C₅×#486	1	1	yes	characteristic
195	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
196	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
197	C₂²	C₅×#510	1	1	yes	characteristic
198	C₂²	C₅×#510	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	16	1	no	—
203	C₄	—	16	2	no	—
204	C₄	C₅×#400	1	1	yes	characteristic
205	C₄	C₅×#475	1	1	yes	characteristic
206	C₄	C₅×#483	1	1	yes	characteristic
207	C₄	C₅×#491	1	1	yes	characteristic
208	C₂	C₅×#2099	1	1	yes	characteristic
209	C₂	C₅×#2224	1	1	yes	characteristic
210	C₂	C₅×#2225	1	1	yes	characteristic
211	C₂	C₅×#2232	1	1	yes	characteristic
212	C₂	C₅×#2235	1	1	yes	characteristic
213	C₂	C₅×#2240	1	1	yes	characteristic
214	C₂	C₅×#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
15. #13537	C₅	direct (N × H)
191. C₅	#13537	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×#1942	2
C₅×#1845	2
C₅×#1380	2
C₅×#1374	2
#13537	5

Subgroups — C5×#13537 #11936257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13537 #11936257