Subgroups — C₄×C₅×#358 #11935356

Order 1280.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#358	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
5	C₅×#1380	C₂	1	2	yes	maximal
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₄×C₅×Q₁₆	—	2	2	no	—
10	C₄×C₅×Q₁₆	C₄	1	4	yes	complemented
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#103	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	—	2	1	no	—
16	C₅×#358	C₄	1	8	yes	direct product factor
17	C₄×C₅×C₁₆	—	2	2	no	—
18	C₅×#340	—	2	1	no	—
19	C₄×#358	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	1	no	—
21	C₂×C₅×Q₁₆	—	2	1	no	—
22	C₂×C₅×Q₁₆	—	2	2	no	—
23	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₅×Q₈	—	2	4	no	—
28	C₄×C₅×Q₈	—	4	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₅	—	4	1	no	—
33	C₂×C₄²×C₅	—	4	1	no	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
38	C₅×#107	—	2	1	no	—
39	C₅×#107	—	2	2	no	—
40	C₅×#107	C₂×C₄	1	4	yes	—
41	C₅×#107	C₂×C₄	1	2	yes	—
42	C₅×#103	—	2	4	no	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	D₄	1	2	yes	—
45	C₅×#95	—	4	1	no	—
46	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
47	C₂×#358	C₂×C₅	1	2	yes	—
48	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
49	C₄×#107	C₂×C₅	1	1	yes	characteristic
50	#1380	C₂×C₅	1	2	yes	—
51	C₂×C₅×Q₈	—	2	2	no	—
52	C₂×C₅×Q₈	—	2	4	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₅×Q₁₆	—	2	2	no	—
62	C₅×Q₁₆	—	2	4	no	—
63	C₅×Q₁₆	C₄²	1	4	yes	complemented
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×C₈	#41	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₄⋊C₄	—	4	4	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₄²×C₅	—	4	4	no	—
76	C₄²×C₅	—	8	2	no	—
77	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅	SD₁₆	1	2	yes	—
79	C₅×C₁₆	—	2	4	no	—
80	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₄×Q₈	—	2	1	no	—
82	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
83	C₄×Q₁₆	—	2	2	no	—
84	C₄×Q₁₆	C₄×C₅	1	4	yes	complemented
85	C₂×#107	C₂²×C₅	1	2	yes	—
86	C₂×#107	C₂²×C₅	1	1	yes	characteristic
87	C₂×#103	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	2	1	no	—
90	#358	C₄×C₅	1	8	yes	direct product factor
91	C₄×C₁₆	—	2	2	no	—
92	#340	—	2	1	no	—
93	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
94	C₅×Q₈	—	2	4	no	—
95	C₅×Q₈	—	4	2	no	—
96	C₅×Q₈	—	4	4	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	4	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	—	8	2	no	—
103	C₂×C₄×C₅	—	8	2	no	—
104	C₂×C₄×C₅	—	8	2	no	—
105	C₂×C₄×C₅	#102	1	2	yes	—
106	C₂×C₄×C₅	#102	1	2	yes	—
107	C₂×C₄×C₅	SD₃₂	1	2	yes	—
108	C₂×C₄×C₅	Q₃₂	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	#135	1	2	yes	—
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
115	C₅×C₈	#117	1	1	yes	characteristic
116	C₅×C₈	C₄×D₄	1	2	yes	—
117	C₂²×Q₈	—	2	1	no	—
118	C₂×Q₁₆	—	2	1	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
121	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
122	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
124	C₄×Q₈	—	2	4	no	—
125	C₄×Q₈	—	4	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄²	—	4	1	no	—
130	C₂×C₄²	—	4	1	no	—
131	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
135	#107	—	2	1	no	—
136	#107	—	2	2	no	—
137	#107	C₂×C₄×C₅	1	4	yes	—
138	#107	C₂×C₄×C₅	1	2	yes	—
139	#103	—	2	4	no	—
140	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
141	C₄×C₈	C₅×D₄	1	2	yes	—
142	#95	—	4	1	no	—
143	C₂²×C₅	C₂×#102	1	1	yes	characteristic
144	C₂²×C₅	C₄×D₈	1	2	yes	—
145	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
146	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
147	C₂²×C₅	#508	1	2	yes	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	4	4	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	—	8	4	no	—
152	C₄×C₅	#358	1	4	yes	direct product factor
153	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₅	#416	1	1	yes	characteristic
155	C₄×C₅	C₄×SD₁₆	1	2	yes	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
164	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	Q₁₆	—	2	2	no	—
167	Q₁₆	—	2	4	no	—
168	Q₁₆	C₄²×C₅	1	4	yes	complemented
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
171	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
173	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₈	C₅×#41	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄²	—	4	4	no	—
181	C₄²	—	8	2	no	—
182	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
183	C₄²	C₅×SD₁₆	1	2	yes	—
184	C₁₆	—	2	4	no	—
185	C₂×C₅	C₄×#102	1	1	yes	characteristic
186	C₂×C₅	C₂×#358	1	1	yes	characteristic
187	C₂×C₅	#2138	1	1	yes	characteristic
188	C₂×C₅	C₄×SD₃₂	1	2	yes	—
189	C₂×C₅	C₄×Q₃₂	1	2	yes	—
190	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
191	Q₈	—	2	4	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	4	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₅×#102	1	2	yes	—
203	C₂×C₄	C₅×#102	1	2	yes	—
204	C₂×C₄	C₅×SD₃₂	1	2	yes	—
205	C₂×C₄	C₅×Q₃₂	1	2	yes	—
206	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
208	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
209	C₂×C₄	C₅×#135	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
212	C₈	C₅×#117	1	1	yes	characteristic
213	C₈	C₄×C₅×D₄	1	2	yes	—
214	C₅	C₄×#358	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₂×C₅×#102	1	1	yes	characteristic
216	C₂²	C₄×C₅×D₈	1	2	yes	—
217	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
218	C₂²	C₂×C₅×Q₃₂	1	1	yes	characteristic
219	C₂²	C₅×#508	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	—	8	4	no	—
224	C₄	C₅×#358	1	4	yes	direct product factor
225	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₄	C₅×#416	1	1	yes	characteristic
227	C₄	C₄×C₅×SD₁₆	1	2	yes	—
228	C₂	C₄×C₅×#102	1	1	yes	characteristic
229	C₂	C₂×C₅×#358	1	1	yes	characteristic
230	C₂	C₅×#2138	1	1	yes	characteristic
231	C₂	C₄×C₅×SD₃₂	1	2	yes	—
232	C₂	C₄×C₅×Q₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
10. C₄×C₅×Q₁₆	C₄	semidirect
16. C₅×#358	C₄	direct (N × H)
19. C₄×#358	C₅	direct (N × H)
63. C₅×Q₁₆	C₄²	semidirect
84. C₄×Q₁₆	C₄×C₅	semidirect
90. #358	C₄×C₅	direct (N × H)
152. C₄×C₅	#358	direct (N × H)
168. Q₁₆	C₄²×C₅	semidirect
214. C₅	C₄×#358	direct (N × H)
224. C₄	C₅×#358	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
C₂×C₅×#358	2
C₂×C₄×C₅×C₁₆	2
C₄×C₅×#107	2
C₅×#1380	2
C₄×#358	5

Subgroups — C4×C5×#358 #11935356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#358 #11935356