Subgroups — C₂×C₄×C₅×#106 #11943167

Order 1280.

1358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#106	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₄×C₅×#106	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#340	C₂	1	2	yes	maximal
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₄²×C₅	—	2	2	no	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×#106	C₄	1	16	yes	direct product factor
12	C₂×C₅×#106	C₂²	1	8	yes	—
13	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₄×C₅×C₄⋊C₄	—	2	4	no	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	8	yes	—
17	C₂×C₅×#95	—	2	2	no	—
18	C₅×#340	C₂²	1	8	yes	—
19	C₂×C₄×#106	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	—	2	4	no	—
21	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
33	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
34	C₂×C₄²×C₅	—	2	8	no	—
35	C₂×C₄²×C₅	—	4	8	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	Q₈	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	2	yes	—
40	C₅×#106	C₂×C₄	1	32	yes	direct product factor
41	C₄×C₅×C₈	C₂×C₄	1	16	yes	complemented
42	C₅×#95	—	2	8	no	—
43	C₂²×#106	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
46	C₄×#106	C₂×C₅	1	8	yes	direct product factor
47	C₂×#340	C₂×C₅	1	2	yes	—
48	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	16	no	—
51	C₂²×C₄×C₅	—	2	8	no	—
52	C₂²×C₄×C₅	—	4	8	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅	#41	1	4	yes	—
64	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
65	C₂×C₅×C₈	—	2	8	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	C₄²	1	16	yes	complemented
68	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
71	C₅×C₄⋊C₄	—	2	16	no	—
72	C₅×C₄⋊C₄	—	2	8	no	—
73	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
74	C₄²×C₅	—	4	16	no	—
75	C₄²×C₅	C₄⋊C₄	1	8	yes	—
76	C₄²×C₅	C₂²×C₄	1	4	yes	—
77	C₄²×C₅	C₂×Q₈	1	4	yes	—
78	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
81	C₂²×C₄²	—	2	2	no	—
82	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
83	C₂×#106	C₄×C₅	1	16	yes	direct product factor
84	C₂×#106	C₂²×C₅	1	8	yes	—
85	C₂×C₄×C₈	C₄×C₅	1	8	yes	complemented
86	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
87	C₄×C₄⋊C₄	—	2	4	no	—
88	C₄×C₄⋊C₄	C₂²×C₅	1	8	yes	—
89	C₂×#95	—	2	2	no	—
90	#340	C₂²×C₅	1	8	yes	—
91	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₅	C₄×D₄	1	4	yes	—
93	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
94	C₂³×C₅	#135	1	4	yes	—
95	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	16	no	—
99	C₂×C₄×C₅	—	4	8	no	—
100	C₂×C₄×C₅	—	4	8	no	—
101	C₂×C₄×C₅	—	4	32	no	—
102	C₂×C₄×C₅	#106	1	16	yes	direct product factor
103	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₅	#117	1	4	yes	—
108	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₅	C₂×SD₁₆	1	8	yes	—
111	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×#41	1	2	yes	—
115	C₅×C₈	—	2	4	no	—
116	C₅×C₈	C₂×C₄²	1	8	yes	complemented
117	C₂³×C₄	—	2	4	no	—
118	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
119	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
120	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
123	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
124	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
125	C₂²×C₈	C₂³×C₅	1	2	yes	—
126	C₂²×C₈	C₂³×C₅	1	2	yes	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	8	no	—
129	C₂×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
130	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
131	C₂×C₄²	—	2	8	no	—
132	C₂×C₄²	—	4	8	no	—
133	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
134	C₂×C₄²	C₅×D₄	1	2	yes	—
135	C₂×C₄²	C₅×Q₈	1	4	yes	—
136	C₂×C₄²	C₂³×C₅	1	2	yes	—
137	#106	C₂×C₄×C₅	1	32	yes	direct product factor
138	C₄×C₈	C₂×C₄×C₅	1	16	yes	complemented
139	#95	—	2	8	no	—
140	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
141	C₂²×C₅	C₂×#106	1	4	yes	—
142	C₂²×C₅	#427	1	4	yes	—
143	C₂²×C₅	C₄×SD₁₆	1	16	yes	—
144	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₅	C₂×C₄×D₄	1	2	yes	—
146	C₂²×C₅	C₂²×SD₁₆	1	2	yes	—
147	C₂²×C₅	C₂×#135	1	2	yes	—
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	4	16	no	—
150	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
151	C₄×C₅	C₂×#106	1	8	yes	direct product factor
152	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₅	C₂×#117	1	1	yes	characteristic
154	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
155	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	16	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	4	8	no	—
161	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
162	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
163	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
164	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
166	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
167	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
169	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
170	C₂²×C₄	C₅×#41	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
172	C₂×C₈	—	2	8	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	C₄²×C₅	1	16	yes	complemented
175	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
176	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
177	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	C₂²×C₄×C₅	1	16	yes	—
181	C₄²	—	4	16	no	—
182	C₄²	C₅×C₄⋊C₄	1	8	yes	—
183	C₄²	C₂²×C₄×C₅	1	4	yes	—
184	C₄²	C₂×C₅×Q₈	1	4	yes	—
185	C₂×C₅	C₄×#106	1	8	yes	direct product factor
186	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₅	C₂²×#106	1	1	yes	characteristic
188	C₂×C₅	C₂×#427	1	1	yes	characteristic
189	C₂×C₅	C₂×C₄×SD₁₆	1	4	yes	—
190	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
191	C₂³	C₄×C₅×D₄	1	4	yes	—
192	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
193	C₂³	C₅×#135	1	4	yes	—
194	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	4	32	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	C₅×#106	1	16	yes	direct product factor
202	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
203	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
204	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
206	C₂×C₄	C₅×#117	1	4	yes	—
207	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
208	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
209	C₂×C₄	C₂×C₅×SD₁₆	1	8	yes	—
210	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
211	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
212	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×#41	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	C₂×C₄²×C₅	1	8	yes	complemented
216	C₅	C₂×C₄×#106	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	C₄×C₅×C₄⋊C₄	1	4	yes	—
218	C₂²	C₂×C₅×#106	1	4	yes	—
219	C₂²	C₅×#427	1	4	yes	—
220	C₂²	C₄×C₅×SD₁₆	1	16	yes	—
221	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
222	C₂²	C₂×C₄×C₅×D₄	1	2	yes	—
223	C₂²	C₂²×C₅×SD₁₆	1	2	yes	—
224	C₂²	C₂×C₅×#135	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	4	16	no	—
227	C₄	C₄×C₅×C₄⋊C₄	1	4	yes	—
228	C₄	C₂×C₅×#106	1	8	yes	direct product factor
229	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
230	C₄	C₂×C₅×#117	1	1	yes	characteristic
231	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
232	C₂	C₄×C₅×#106	1	8	yes	direct product factor
233	C₂	C₂×C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂	C₂²×C₅×#106	1	1	yes	characteristic
235	C₂	C₂×C₅×#427	1	1	yes	characteristic
236	C₂	C₂×C₄×C₅×SD₁₆	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₅×#106	C₂	direct (N × H)
11. C₂×C₅×#106	C₄	direct (N × H)
13. C₂×C₄×C₅×C₈	C₄	semidirect
19. C₂×C₄×#106	C₅	direct (N × H)
40. C₅×#106	C₂×C₄	direct (N × H)
41. C₄×C₅×C₈	C₂×C₄	semidirect
46. C₄×#106	C₂×C₅	direct (N × H)
67. C₂×C₅×C₈	C₄²	semidirect
83. C₂×#106	C₄×C₅	direct (N × H)
85. C₂×C₄×C₈	C₄×C₅	semidirect
102. C₂×C₄×C₅	#106	direct (N × H)
116. C₅×C₈	C₂×C₄²	semidirect
137. #106	C₂×C₄×C₅	direct (N × H)
138. C₄×C₈	C₂×C₄×C₅	semidirect
151. C₄×C₅	C₂×#106	direct (N × H)
174. C₂×C₈	C₄²×C₅	semidirect
185. C₂×C₅	C₄×#106	direct (N × H)
201. C₂×C₄	C₅×#106	direct (N × H)
215. C₈	C₂×C₄²×C₅	semidirect
216. C₅	C₂×C₄×#106	direct (N × H)
228. C₄	C₂×C₅×#106	direct (N × H)
232. C₂	C₄×C₅×#106	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×#106	2
C₂²×C₄×C₅×C₈	2
C₂×C₄×C₅×C₄⋊C₄	2
C₄×C₅×#106	2
C₂×C₅×#340	2
C₂×C₄×#106	5

Subgroups — C2×C4×C5×#106 #11943167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×#106 #11943167