Subgroups — C₅×#12658 #11935378

Order 1280.

350 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2150	C₂	1	2	yes	maximal
2	C₂×C₅×#347	C₂	1	2	yes	maximal
3	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₅×#1375	C₂	1	2	yes	maximal
5	C₂×C₅×M₃₂	C₂²	1	4	yes	—
6	C₂×C₅×#97	C₄	1	1	yes	characteristic
7	C₂×C₅×#97	C₄	1	1	yes	characteristic
8	C₂×C₅×#97	C₂²	1	1	yes	characteristic
9	C₂×C₅×#97	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄³×C₅	C₄	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	C₄	1	8	yes	complemented
14	C₅×#347	—	2	4	no	—
15	#12658	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅	—	2	2	no	—
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×M₃₂	—	2	4	no	—
25	C₅×M₃₂	—	2	4	no	—
26	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	—
27	C₅×#97	—	2	2	no	—
28	C₅×#97	—	2	2	no	—
29	C₅×#97	—	2	2	no	—
30	C₅×#97	D₄	1	2	yes	—
31	C₅×#97	Q₈	1	2	yes	—
32	C₄×C₅×C₈	—	2	1	no	—
33	C₄×C₅×C₈	—	2	1	no	—
34	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
35	#2150	C₂×C₅	1	2	yes	—
36	C₂×#347	C₂×C₅	1	2	yes	—
37	C₄×#97	C₂×C₅	1	1	yes	characteristic
38	#1375	C₂×C₅	1	2	yes	—
39	C₂²×C₄×C₅	—	2	1	no	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	M₁₆	1	2	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	—	2	1	no	—
44	C₂×C₅×C₈	—	2	1	no	—
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	2	1	no	—
49	C₂×C₅×C₈	C₄²	1	2	yes	—
50	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₅×C₈	M₁₆	1	2	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	2	yes	—
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	—	2	1	no	—
58	C₄²×C₅	—	4	4	no	—
59	C₄²×C₅	C₄²	1	2	yes	—
60	C₄²×C₅	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅	M₁₆	1	2	yes	—
62	C₄²×C₅	M₁₆	1	1	yes	characteristic
63	C₄²×C₅	M₁₆	1	1	yes	characteristic
64	C₅×C₁₆	—	2	8	no	—
65	C₂×M₃₂	C₂²×C₅	1	4	yes	—
66	C₂×#97	C₄×C₅	1	1	yes	characteristic
67	C₂×#97	C₄×C₅	1	1	yes	characteristic
68	C₂×#97	C₂²×C₅	1	1	yes	characteristic
69	C₂×#97	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₄³	C₄×C₅	1	1	yes	characteristic
73	C₄⋊C₁₆	C₄×C₅	1	8	yes	complemented
74	#347	—	2	4	no	—
75	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	#97	1	2	yes	—
82	C₂×C₄×C₅	#97	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅	#117	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
89	C₅×C₈	—	2	2	no	—
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	—	2	2	no	—
92	C₅×C₈	C₄×D₄	1	2	yes	—
93	C₅×C₈	C₄×Q₈	1	2	yes	—
94	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
98	C₂×C₄²	—	2	2	no	—
99	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
102	M₃₂	—	2	4	no	—
103	M₃₂	—	2	4	no	—
104	C₂×C₁₆	C₂×C₄×C₅	1	8	yes	—
105	#97	—	2	2	no	—
106	#97	—	2	2	no	—
107	#97	—	2	2	no	—
108	#97	C₅×D₄	1	2	yes	—
109	#97	C₅×Q₈	1	2	yes	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
113	C₂²×C₅	—	2	2	no	—
114	C₂²×C₅	C₂×#97	1	1	yes	characteristic
115	C₂²×C₅	#423	1	1	yes	characteristic
116	C₂²×C₅	#432	1	1	yes	characteristic
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	—	4	2	no	—
119	C₄×C₅	—	4	2	no	—
120	C₄×C₅	#347	1	4	yes	complemented
121	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₅	#423	1	1	yes	characteristic
124	C₄×C₅	#432	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₅×M₁₆	1	2	yes	—
128	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	C₄²×C₅	1	2	yes	—
136	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
137	C₂×C₈	C₅×M₁₆	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
139	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
141	C₂×C₈	C₅×#41	1	2	yes	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₄²×C₅	1	2	yes	—
146	C₄²	C₅×C₄⋊C₄	1	2	yes	—
147	C₄²	C₅×M₁₆	1	2	yes	—
148	C₄²	C₅×M₁₆	1	1	yes	characteristic
149	C₄²	C₅×M₁₆	1	1	yes	characteristic
150	C₁₆	—	2	8	no	—
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	1	no	—
153	C₂×C₅	#1767	1	1	yes	characteristic
154	C₂×C₅	C₂×#347	1	2	yes	—
155	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	C₅×#97	1	2	yes	—
162	C₂×C₄	C₅×#97	1	2	yes	—
163	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
164	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₅×#117	1	1	yes	characteristic
166	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
167	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
168	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	C₄×C₅×D₄	1	2	yes	—
173	C₈	C₄×C₅×Q₈	1	2	yes	—
174	C₅	#12658	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	C₂×C₅×#97	1	1	yes	characteristic
177	C₂²	C₅×#423	1	1	yes	characteristic
178	C₂²	C₅×#432	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	C₅×#347	1	4	yes	complemented
183	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
185	C₄	C₅×#423	1	1	yes	characteristic
186	C₄	C₅×#432	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₅×#1767	1	1	yes	characteristic
190	C₂	C₂×C₅×#347	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×C₄⋊C₁₆	C₄	semidirect
15. #12658	C₅	direct (N × H)
73. C₄⋊C₁₆	C₄×C₅	semidirect
120. C₄×C₅	#347	semidirect
174. C₅	#12658	direct (N × H)
182. C₄	C₅×#347	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2150	2
C₂×C₅×#347	2
C₄×C₅×#97	2
C₅×#1375	2
#12658	5

Subgroups — C5×#12658 #11935378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12658 #11935378