Subgroups — C₅×#11322 #11934042

Order 1280.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	3	yes	maximal
2	C₂×C₅×#336	C₂	1	3	yes	maximal
3	C₄×C₅×#95	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₅×C₈	C₂²	1	3	yes	—
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₂²	1	3	yes	—
7	C₂×C₅×C₄⋊C₈	—	2	12	no	—
8	C₂×C₅×C₄⋊C₈	—	2	6	no	—
9	C₂×C₅×#95	C₄	1	3	yes	—
10	C₂×C₅×#95	C₄	1	1	yes	characteristic
11	C₅×#336	—	2	6	no	—
12	#11322	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
15	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
16	C₂²×C₅×C₈	—	2	3	no	—
17	C₂²×C₅×C₈	—	2	3	no	—
18	C₂²×C₅×C₈	—	2	6	no	—
19	C₂²×C₅×C₈	—	2	6	no	—
20	C₂²×C₅×C₈	—	2	3	no	—
21	C₂²×C₅×C₈	—	2	3	no	—
22	C₂²×C₅×C₈	D₄	1	6	yes	—
23	C₂²×C₅×C₈	Q₈	1	2	yes	—
24	C₂×C₄²×C₅	—	2	6	no	—
25	C₂×C₄²×C₅	—	2	6	no	—
26	C₂×C₄²×C₅	D₄	1	6	yes	—
27	C₂×C₄²×C₅	Q₈	1	6	yes	—
28	C₅×C₄⋊C₈	—	2	24	no	—
29	C₅×#95	—	2	6	no	—
30	C₅×#95	C₈	1	4	yes	complemented
31	C₂²×C₄⋊C₈	C₂×C₅	1	3	yes	—
32	C₂×#336	C₂×C₅	1	3	yes	—
33	C₄×#95	C₂×C₅	1	1	yes	characteristic
34	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	—	2	6	no	—
36	C₂²×C₄×C₅	—	2	6	no	—
37	C₂²×C₄×C₅	—	2	6	no	—
38	C₂²×C₄×C₅	—	2	6	no	—
39	C₂²×C₄×C₅	—	2	3	no	—
40	C₂²×C₄×C₅	—	2	3	no	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	6	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	6	yes	—
43	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
44	C₂²×C₄×C₅	M₁₆	1	6	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
49	C₂²×C₄×C₅	#41	1	1	yes	characteristic
50	C₂²×C₄×C₅	#41	1	3	yes	—
51	C₂×C₅×C₈	—	2	12	no	—
52	C₂×C₅×C₈	—	2	6	no	—
53	C₂×C₅×C₈	—	2	6	no	—
54	C₂×C₅×C₈	—	4	3	no	—
55	C₂×C₅×C₈	—	4	6	no	—
56	C₂×C₅×C₈	—	4	3	no	—
57	C₂×C₅×C₈	—	4	3	no	—
58	C₂×C₅×C₈	—	4	1	no	—
59	C₄²×C₅	—	2	6	no	—
60	C₄²×C₅	—	2	12	no	—
61	C₄²×C₅	—	2	6	no	—
62	C₂³×C₈	C₂²×C₅	1	3	yes	—
63	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₄²	C₂²×C₅	1	3	yes	—
65	C₂×C₄⋊C₈	—	2	12	no	—
66	C₂×C₄⋊C₈	—	2	6	no	—
67	C₂×#95	C₄×C₅	1	3	yes	—
68	C₂×#95	C₄×C₅	1	1	yes	characteristic
69	#336	—	2	6	no	—
70	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
71	C₂³×C₅	C₄×D₄	1	3	yes	—
72	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
73	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
74	C₂³×C₅	C₂×M₁₆	1	3	yes	—
75	C₂³×C₅	#131	1	3	yes	—
76	C₂³×C₅	#131	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	6	no	—
78	C₂×C₄×C₅	—	2	6	no	—
79	C₂×C₄×C₅	—	2	12	no	—
80	C₂×C₄×C₅	—	2	12	no	—
81	C₂×C₄×C₅	—	2	6	no	—
82	C₂×C₄×C₅	—	2	6	no	—
83	C₂×C₄×C₅	—	2	6	no	—
84	C₂×C₄×C₅	—	4	6	no	—
85	C₂×C₄×C₅	—	4	6	no	—
86	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	complemented
87	C₂×C₄×C₅	#120	1	3	yes	—
88	C₂×C₄×C₅	#121	1	3	yes	—
89	C₂×C₄×C₅	C₂²⋊Q₈	1	6	yes	—
90	C₂×C₄×C₅	C₂²⋊Q₈	1	3	yes	—
91	C₂×C₄×C₅	#123	1	3	yes	—
92	C₂×C₄×C₅	#125	1	3	yes	—
93	C₂×C₄×C₅	#127	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄⋊Q₈	1	3	yes	—
95	C₂×C₄×C₅	C₄⋊Q₈	1	3	yes	—
96	C₅×C₈	—	4	6	no	—
97	C₅×C₈	—	4	2	no	—
98	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
99	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
100	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
101	C₂²×C₈	—	2	3	no	—
102	C₂²×C₈	—	2	3	no	—
103	C₂²×C₈	—	2	6	no	—
104	C₂²×C₈	—	2	6	no	—
105	C₂²×C₈	—	2	3	no	—
106	C₂²×C₈	—	2	3	no	—
107	C₂²×C₈	C₅×D₄	1	6	yes	—
108	C₂²×C₈	C₅×Q₈	1	2	yes	—
109	C₂×C₄²	—	2	6	no	—
110	C₂×C₄²	—	2	6	no	—
111	C₂×C₄²	C₅×D₄	1	6	yes	—
112	C₂×C₄²	C₅×Q₈	1	6	yes	—
113	C₄⋊C₈	—	2	24	no	—
114	#95	—	2	6	no	—
115	#95	C₅×C₈	1	4	yes	complemented
116	C₂²×C₅	#384	1	1	yes	characteristic
117	C₂²×C₅	#385	1	3	yes	—
118	C₂²×C₅	#389	1	3	yes	—
119	C₂²×C₅	C₂×C₄⋊C₈	1	3	yes	—
120	C₂²×C₅	#423	1	3	yes	—
121	C₂²×C₅	#424	1	3	yes	—
122	C₂²×C₅	#424	1	3	yes	—
123	C₂²×C₅	C₈×D₄	1	3	yes	—
124	C₂²×C₅	#435	1	6	yes	—
125	C₂²×C₅	#436	1	3	yes	—
126	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
127	C₂²×C₅	#446	1	3	yes	—
128	C₄×C₅	—	2	12	no	—
129	C₄×C₅	—	4	6	no	—
130	C₄×C₅	#392	1	1	yes	characteristic
131	C₄×C₅	#395	1	3	yes	—
132	C₄×C₅	#398	1	3	yes	—
133	C₄×C₅	#399	1	1	yes	characteristic
134	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₄	—	2	6	no	—
136	C₂²×C₄	—	2	6	no	—
137	C₂²×C₄	—	2	6	no	—
138	C₂²×C₄	—	2	6	no	—
139	C₂²×C₄	—	2	3	no	—
140	C₂²×C₄	—	2	3	no	—
141	C₂²×C₄	C₅×C₄⋊C₄	1	6	yes	—
142	C₂²×C₄	C₅×C₄⋊C₄	1	6	yes	—
143	C₂²×C₄	C₂×C₅×C₈	1	6	yes	—
144	C₂²×C₄	C₅×M₁₆	1	6	yes	—
145	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
146	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
147	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₅×Q₈	1	3	yes	—
149	C₂²×C₄	C₅×#41	1	1	yes	characteristic
150	C₂²×C₄	C₅×#41	1	3	yes	—
151	C₂×C₈	—	2	12	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	2	6	no	—
154	C₂×C₈	—	4	3	no	—
155	C₂×C₈	—	4	6	no	—
156	C₂×C₈	—	4	3	no	—
157	C₂×C₈	—	4	3	no	—
158	C₂×C₈	—	4	1	no	—
159	C₄²	—	2	6	no	—
160	C₄²	—	2	12	no	—
161	C₄²	—	2	6	no	—
162	C₂×C₅	#1444	1	1	yes	characteristic
163	C₂×C₅	#1842	1	1	yes	characteristic
164	C₂×C₅	#1843	1	1	yes	characteristic
165	C₂×C₅	#1850	1	3	yes	—
166	C₂×C₅	#1851	1	3	yes	—
167	C₂×C₅	#1939	1	3	yes	—
168	C₂×C₅	#1940	1	3	yes	—
169	C₂³	C₂×C₅×C₄⋊C₄	1	3	yes	—
170	C₂³	C₄×C₅×D₄	1	3	yes	—
171	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
172	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
173	C₂³	C₂×C₅×M₁₆	1	3	yes	—
174	C₂³	C₅×#131	1	3	yes	—
175	C₂³	C₅×#131	1	1	yes	characteristic
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	6	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	4	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	C₅×C₄⋊C₈	1	12	yes	complemented
186	C₂×C₄	C₅×#120	1	3	yes	—
187	C₂×C₄	C₅×#121	1	3	yes	—
188	C₂×C₄	C₅×C₂²⋊Q₈	1	6	yes	—
189	C₂×C₄	C₅×C₂²⋊Q₈	1	3	yes	—
190	C₂×C₄	C₅×#123	1	3	yes	—
191	C₂×C₄	C₅×#125	1	3	yes	—
192	C₂×C₄	C₅×#127	1	1	yes	characteristic
193	C₂×C₄	C₅×C₄⋊Q₈	1	3	yes	—
194	C₂×C₄	C₅×C₄⋊Q₈	1	3	yes	—
195	C₈	—	4	6	no	—
196	C₈	—	4	2	no	—
197	C₅	#11322	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	C₅×#384	1	1	yes	characteristic
199	C₂²	C₅×#385	1	3	yes	—
200	C₂²	C₅×#389	1	3	yes	—
201	C₂²	C₂×C₅×C₄⋊C₈	1	3	yes	—
202	C₂²	C₅×#423	1	3	yes	—
203	C₂²	C₅×#424	1	3	yes	—
204	C₂²	C₅×#424	1	3	yes	—
205	C₂²	C₅×C₈×D₄	1	3	yes	—
206	C₂²	C₅×#435	1	6	yes	—
207	C₂²	C₅×#436	1	3	yes	—
208	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
209	C₂²	C₅×#446	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	C₅×#392	1	1	yes	characteristic
213	C₄	C₅×#395	1	3	yes	—
214	C₄	C₅×#398	1	3	yes	—
215	C₄	C₅×#399	1	1	yes	characteristic
216	C₂	C₅×#1444	1	1	yes	characteristic
217	C₂	C₅×#1842	1	1	yes	characteristic
218	C₂	C₅×#1843	1	1	yes	characteristic
219	C₂	C₅×#1850	1	3	yes	—
220	C₂	C₅×#1851	1	3	yes	—
221	C₂	C₅×#1939	1	3	yes	—
222	C₂	C₅×#1940	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	C₂³
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle, prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. #11322	C₅	direct (N × H)
30. C₅×#95	C₈	semidirect
86. C₂×C₄×C₅	C₄⋊C₈	semidirect
115. #95	C₅×C₈	semidirect
185. C₂×C₄	C₅×C₄⋊C₈	semidirect
197. C₅	#11322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₂×C₅×#336	2
C₄×C₅×#95	2
#11322	5

Subgroups — C5×#11322 #11934042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11322 #11934042