Subgroups — C₂×C₅×#2614 #11957318

Order 1280.

5194 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#123	C₂	1	3	yes	maximal
3	C₅×#2614	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#395	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#393	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#381	C₂	1	3	yes	maximal, complemented
7	C₂³×C₅×Q₈	—	2	24	no	—
8	C₂³×C₅×Q₈	C₂²	1	4	yes	—
9	C₂³×C₅×Q₈	C₂²	1	8	yes	complemented
10	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅	C₂²	1	3	yes	—
12	C₂×C₅×#123	—	2	24	no	—
13	C₂×C₅×#123	C₂²	1	24	yes	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
17	C₅×#395	C₂²	1	32	yes	complemented
18	C₅×#393	C₂²	1	32	yes	complemented
19	C₅×#381	C₂²	1	24	yes	complemented
20	C₂×C₅×#95	C₂²	1	12	yes	—
21	C₂×#2614	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂⁵×C₅	C₂³	1	1	yes	characteristic
23	C₂⁵×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Q₈	—	2	128	no	—
25	C₂²×C₅×Q₈	—	2	48	no	—
26	C₂²×C₅×Q₈	—	2	96	no	—
27	C₂²×C₅×Q₈	C₂³	1	16	yes	—
28	C₂²×C₅×D₄	—	2	12	no	—
29	C₂²×C₅×D₄	C₂³	1	4	yes	—
30	C₂³×C₄×C₅	—	2	24	no	—
31	C₂³×C₄×C₅	D₄	1	24	yes	—
32	C₂³×C₄×C₅	C₂³	1	6	yes	—
33	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₅	C₂³	1	12	yes	—
35	C₂³×C₄×C₅	C₂³	1	6	yes	—
36	C₅×#123	—	2	96	no	—
37	C₂×C₅×C₄⋊C₄	—	2	24	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
39	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
40	C₂×C₅×C₂²⋊C₄	—	2	24	no	—
41	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
42	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	—
43	C₅×#95	C₂³	1	48	yes	—
44	C₂⁴×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×#123	C₂×C₅	1	3	yes	—
46	#2614	C₂×C₅	1	16	yes	direct product factor
47	C₂×#395	C₂×C₅	1	4	yes	complemented
48	C₂×#393	C₂×C₅	1	4	yes	complemented
49	C₂×#381	C₂×C₅	1	3	yes	complemented
50	C₂⁴×C₅	—	2	8	no	—
51	C₂⁴×C₅	—	2	6	no	—
52	C₂⁴×C₅	—	2	8	no	—
53	C₂⁴×C₅	C₂×D₄	1	12	yes	—
54	C₂⁴×C₅	C₂⁴	1	2	yes	—
55	C₂⁴×C₅	C₂⁴	1	2	yes	—
56	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×C₅×Q₈	—	2	128	no	—
58	C₂×C₅×Q₈	—	2	256	no	—
59	C₂×C₅×Q₈	—	2	96	no	—
60	C₂×C₅×D₄	—	2	24	no	—
61	C₂×C₅×D₄	—	4	16	no	—
62	C₂²×C₄×C₅	—	2	96	no	—
63	C₂²×C₄×C₅	—	2	48	no	—
64	C₂²×C₄×C₅	—	2	24	no	—
65	C₂²×C₄×C₅	—	2	12	no	—
66	C₂²×C₄×C₅	—	2	6	no	—
67	C₂²×C₄×C₅	—	2	24	no	—
68	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	48	yes	—
70	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
73	C₅×C₄⋊C₄	—	2	48	no	—
74	C₅×C₂²⋊C₄	—	2	24	no	—
75	C₅×C₂²⋊C₄	—	2	48	no	—
76	C₂³×Q₈	—	2	24	no	—
77	C₂³×Q₈	C₂²×C₅	1	4	yes	—
78	C₂³×Q₈	C₂²×C₅	1	8	yes	complemented
79	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
80	C₂⁴×C₄	C₂²×C₅	1	3	yes	—
81	C₂×#123	—	2	24	no	—
82	C₂×#123	C₂²×C₅	1	24	yes	—
83	C₂²×C₄⋊C₄	C₂²×C₅	1	6	yes	—
84	C₂²×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
85	C₂²×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
86	#395	C₂²×C₅	1	32	yes	complemented
87	#393	C₂²×C₅	1	32	yes	complemented
88	#381	C₂²×C₅	1	24	yes	complemented
89	C₂×#95	C₂²×C₅	1	12	yes	—
90	C₂³×C₅	—	2	48	no	—
91	C₂³×C₅	—	2	8	no	—
92	C₂³×C₅	—	2	6	no	—
93	C₂³×C₅	—	2	8	no	—
94	C₂³×C₅	—	4	4	no	—
95	C₂³×C₅	—	4	24	no	—
96	C₂³×C₅	#120	1	16	yes	—
97	C₂³×C₅	C₂²×D₄	1	12	yes	—
98	C₂³×C₅	C₂²×D₄	1	6	yes	—
99	C₂³×C₅	2₋¹⁺⁴	1	8	yes	—
100	C₂³×C₅	C₂⁵	1	1	yes	characteristic
101	C₅×Q₈	—	2	128	no	—
102	C₅×D₄	—	4	16	no	—
103	C₂×C₄×C₅	—	2	96	no	—
104	C₂×C₄×C₅	—	2	48	no	—
105	C₂×C₄×C₅	—	2	24	no	—
106	C₂×C₄×C₅	—	2	12	no	—
107	C₂×C₄×C₅	—	2	6	no	—
108	C₂×C₄×C₅	—	2	24	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	24	no	—
111	C₂×C₄×C₅	C₂²×D₄	1	24	yes	—
112	C₂⁵	C₂³×C₅	1	1	yes	characteristic
113	C₂⁵	C₂³×C₅	1	1	yes	characteristic
114	C₂²×Q₈	—	2	128	no	—
115	C₂²×Q₈	—	2	48	no	—
116	C₂²×Q₈	—	2	96	no	—
117	C₂²×Q₈	C₂³×C₅	1	16	yes	—
118	C₂²×D₄	—	2	12	no	—
119	C₂²×D₄	C₂³×C₅	1	4	yes	—
120	C₂³×C₄	—	2	24	no	—
121	C₂³×C₄	C₅×D₄	1	24	yes	—
122	C₂³×C₄	C₂³×C₅	1	6	yes	—
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂³×C₄	C₂³×C₅	1	6	yes	—
125	C₂³×C₄	C₂³×C₅	1	12	yes	—
126	#123	—	2	96	no	—
127	C₂×C₄⋊C₄	—	2	24	no	—
128	C₂×C₄⋊C₄	C₂³×C₅	1	24	yes	—
129	C₂×C₂²⋊C₄	—	2	12	no	—
130	C₂×C₂²⋊C₄	—	2	24	no	—
131	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
132	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	—
133	#95	C₂³×C₅	1	48	yes	—
134	C₂²×C₅	—	2	32	no	—
135	C₂²×C₅	—	2	24	no	—
136	C₂²×C₅	—	4	2	no	—
137	C₂²×C₅	—	4	12	no	—
138	C₂²×C₅	—	8	8	no	—
139	C₂²×C₅	C₂×#120	1	8	yes	—
140	C₂²×C₅	C₂×#120	1	4	yes	—
141	C₂²×C₅	#536	1	24	yes	—
142	C₂²×C₅	C₂³×D₄	1	3	yes	—
143	C₂²×C₅	C₂×2₋¹⁺⁴	1	4	yes	—
144	C₄×C₅	—	2	24	no	—
145	C₄×C₅	—	4	2	no	—
146	C₄×C₅	—	4	12	no	—
147	C₂⁴	—	2	8	no	—
148	C₂⁴	—	2	6	no	—
149	C₂⁴	—	2	8	no	—
150	C₂⁴	C₂×C₅×D₄	1	12	yes	—
151	C₂⁴	C₂⁴×C₅	1	2	yes	—
152	C₂⁴	C₂⁴×C₅	1	2	yes	—
153	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
154	C₂×Q₈	—	2	128	no	—
155	C₂×Q₈	—	2	256	no	—
156	C₂×Q₈	—	2	96	no	—
157	C₂×D₄	—	2	24	no	—
158	C₂×D₄	—	4	16	no	—
159	C₂²×C₄	—	2	96	no	—
160	C₂²×C₄	—	2	48	no	—
161	C₂²×C₄	—	2	24	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	24	no	—
165	C₂²×C₄	C₂×C₅×D₄	1	24	yes	—
166	C₂²×C₄	C₂×C₅×D₄	1	48	yes	—
167	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
168	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
169	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
170	C₄⋊C₄	—	2	48	no	—
171	C₂²⋊C₄	—	2	24	no	—
172	C₂²⋊C₄	—	2	48	no	—
173	C₂×C₅	—	2	8	no	—
174	C₂×C₅	—	8	2	no	—
175	C₂×C₅	#2614	1	8	yes	direct product factor
176	C₂×C₅	C₂²×#120	1	1	yes	characteristic
177	C₂×C₅	C₂×#536	1	6	yes	—
178	C₂³	—	2	48	no	—
179	C₂³	—	2	8	no	—
180	C₂³	—	2	6	no	—
181	C₂³	—	2	8	no	—
182	C₂³	—	4	4	no	—
183	C₂³	—	4	24	no	—
184	C₂³	C₅×#120	1	16	yes	—
185	C₂³	C₂²×C₅×D₄	1	12	yes	—
186	C₂³	C₂²×C₅×D₄	1	6	yes	—
187	C₂³	C₅×2₋¹⁺⁴	1	8	yes	—
188	C₂³	C₂⁵×C₅	1	1	yes	characteristic
189	Q₈	—	2	128	no	—
190	D₄	—	4	16	no	—
191	C₂×C₄	—	2	96	no	—
192	C₂×C₄	—	2	48	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	24	no	—
199	C₂×C₄	C₂²×C₅×D₄	1	24	yes	—
200	C₅	C₂×#2614	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	24	no	—
202	C₂²	—	2	32	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	12	no	—
205	C₂²	—	8	8	no	—
206	C₂²	C₂×C₅×#120	1	8	yes	—
207	C₂²	C₂×C₅×#120	1	4	yes	—
208	C₂²	C₅×#536	1	24	yes	—
209	C₂²	C₂³×C₅×D₄	1	3	yes	—
210	C₂²	C₂×C₅×2₋¹⁺⁴	1	4	yes	—
211	C₄	—	2	24	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	12	no	—
214	C₂	—	2	8	no	—
215	C₂	—	8	2	no	—
216	C₂	C₅×#2614	1	8	yes	direct product factor
217	C₂	C₂²×C₅×#120	1	1	yes	characteristic
218	C₂	C₂×C₅×#536	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×Q₈	C₂	semidirect
3. C₅×#2614	C₂	direct (N × H)
4. C₂×C₅×#395	C₂	semidirect
5. C₂×C₅×#393	C₂	semidirect
6. C₂×C₅×#381	C₂	semidirect
9. C₂³×C₅×Q₈	C₂²	semidirect
17. C₅×#395	C₂²	semidirect
18. C₅×#393	C₂²	semidirect
19. C₅×#381	C₂²	semidirect
21. C₂×#2614	C₅	direct (N × H)
44. C₂⁴×Q₈	C₂×C₅	semidirect
46. #2614	C₂×C₅	direct (N × H)
47. C₂×#395	C₂×C₅	semidirect
48. C₂×#393	C₂×C₅	semidirect
49. C₂×#381	C₂×C₅	semidirect
78. C₂³×Q₈	C₂²×C₅	semidirect
86. #395	C₂²×C₅	semidirect
87. #393	C₂²×C₅	semidirect
88. #381	C₂²×C₅	semidirect
175. C₂×C₅	#2614	direct (N × H)
200. C₅	C₂×#2614	direct (N × H)
216. C₂	C₅×#2614	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×Q₈	2
C₂²×C₅×#123	2
C₅×#2614	2
C₂×C₅×#395	2
C₂×C₅×#393	2
C₂×C₅×#381	2
C₂×#2614	5

Subgroups — C2×C5×#2614 #11957318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2614 #11957318