Subgroups — C₅×#9610 #11932330

Order 1280.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#1565	C₂	1	2	yes	maximal
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×#1452	C₂	1	1	yes	maximal, characteristic
5	C₅×#1283	C₂	1	2	yes	maximal
6	C₅×#431	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₄	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#341	C₄	1	1	yes	characteristic
13	C₅×#341	C₄	1	1	yes	characteristic
14	C₅×#335	—	2	2	no	—
15	C₅×#335	—	2	2	no	—
16	C₅×#334	—	2	2	no	—
17	C₅×#322	—	2	2	no	—
18	#9610	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₅×#108	—	2	4	no	—
26	C₅×C₄⋊C₈	—	2	2	no	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	1	no	—
29	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
30	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
31	C₅×C₂²⋊C₈	—	2	1	no	—
32	C₅×C₂²⋊C₈	Q₈	1	2	yes	—
33	C₅×#97	—	2	2	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	—	2	2	no	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	C₄×C₅×C₈	Q₈	1	2	yes	—
39	C₄×#108	C₂×C₅	1	1	yes	characteristic
40	#1565	C₂×C₅	1	2	yes	—
41	C₂×#335	C₂×C₅	1	1	yes	characteristic
42	#1452	C₂×C₅	1	1	yes	characteristic
43	#1283	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	#41	1	1	yes	characteristic
47	C₅×M₁₆	—	2	2	no	—
48	C₅×M₁₆	—	2	2	no	—
49	C₅×M₁₆	—	2	2	no	—
50	C₅×M₁₆	M₁₆	1	4	yes	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	—	2	1	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	8	2	no	—
60	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅	#41	1	1	yes	characteristic
66	#431	C₂²×C₅	1	1	yes	characteristic
67	#431	C₂²×C₅	1	1	yes	characteristic
68	C₂×#108	C₄×C₅	1	2	yes	—
69	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
70	C₄×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	#341	C₄×C₅	1	1	yes	characteristic
73	#341	C₄×C₅	1	1	yes	characteristic
74	#335	—	2	2	no	—
75	#335	—	2	2	no	—
76	#334	—	2	2	no	—
77	#322	—	2	2	no	—
78	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	4	1	no	—
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅	#121	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	#125	1	1	yes	characteristic
87	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₅	#131	1	2	yes	—
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	—	2	2	no	—
92	C₅×C₈	—	2	4	no	—
93	C₅×C₈	—	4	2	no	—
94	C₅×C₈	—	4	2	no	—
95	C₅×C₈	—	4	1	no	—
96	C₅×C₈	—	4	1	no	—
97	C₅×C₈	—	8	2	no	—
98	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
99	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
104	#108	—	2	4	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
109	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
110	C₂²⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₅×Q₈	1	2	yes	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	C₅×D₄	1	2	yes	—
117	C₄×C₈	C₅×Q₈	1	2	yes	—
118	C₂²×C₅	—	2	1	no	—
119	C₂²×C₅	—	2	1	no	—
120	C₂²×C₅	#389	1	1	yes	characteristic
121	C₂²×C₅	#446	1	2	yes	—
122	C₄×C₅	—	2	1	no	—
123	C₄×C₅	—	2	1	no	—
124	C₄×C₅	—	2	1	no	—
125	C₄×C₅	#423	1	1	yes	characteristic
126	C₄×C₅	#424	1	1	yes	characteristic
127	C₄×C₅	#435	1	2	yes	—
128	C₄×C₅	#472	1	1	yes	characteristic
129	C₄×C₅	#473	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
132	C₂²×C₄	C₅×#41	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	C₅×M₁₆	1	4	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
147	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
148	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
149	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
150	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
151	C₄²	C₅×#41	1	1	yes	characteristic
152	C₂×C₅	—	2	2	no	—
153	C₂×C₅	#1940	1	1	yes	characteristic
154	C₂×C₅	#1952	1	1	yes	characteristic
155	C₂×C₅	#1953	1	1	yes	characteristic
156	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
162	C₂×C₄	C₅×#121	1	1	yes	characteristic
163	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₄	C₅×#125	1	1	yes	characteristic
165	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
167	C₂×C₄	C₅×#131	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	2	4	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	8	2	no	—
176	C₅	#9610	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	C₅×#389	1	1	yes	characteristic
180	C₂²	C₅×#446	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₅×#423	1	1	yes	characteristic
185	C₄	C₅×#424	1	1	yes	characteristic
186	C₄	C₅×#435	1	2	yes	—
187	C₄	C₅×#472	1	1	yes	characteristic
188	C₄	C₅×#473	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	C₅×#1940	1	1	yes	characteristic
191	C₂	C₅×#1952	1	1	yes	characteristic
192	C₂	C₅×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
18. #9610	C₅	direct (N × H)
176. C₅	#9610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#108	2
C₅×#1565	2
C₂×C₅×#335	2
C₅×#1452	2
C₅×#1283	2
#9610	5

Subgroups — C5×#9610 #11932330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9610 #11932330