Subgroups — C₂×C₅×#2658 #11957350

Order 1280.

3590 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
2	C₂²×C₅×#124	C₂	1	3	yes	maximal
3	C₅×#2658	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#399	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅×Q₈	C₂²	1	3	yes	—
7	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊Q₈	—	2	24	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	24	yes	complemented
10	C₂×C₅×#124	—	2	24	no	—
11	C₂×C₅×#124	C₂²	1	24	yes	—
12	C₂²×C₅×C₄⋊C₄	C₂²	1	12	yes	—
13	C₂²×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₄²×C₅	C₂²	1	3	yes	—
15	C₅×#399	C₂²	1	64	yes	complemented
16	C₅×#376	C₂²	1	8	yes	complemented
17	C₂×C₅×#95	C₂²	1	4	yes	—
18	C₂×#2658	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂⁵×C₅	C₂³	1	2	yes	—
20	C₂²×C₅×Q₈	—	2	12	no	—
21	C₂²×C₅×Q₈	—	2	24	no	—
22	C₂²×C₅×Q₈	C₂³	1	12	yes	—
23	C₂²×C₅×D₄	—	2	12	no	—
24	C₂²×C₅×D₄	C₂³	1	4	yes	—
25	C₂³×C₄×C₅	C₂³	1	6	yes	—
26	C₂³×C₄×C₅	C₂³	1	6	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	—	2	96	no	—
29	C₅×#124	—	2	96	no	—
30	C₂×C₅×C₄⋊C₄	—	2	48	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	48	yes	—
32	C₂×C₅×C₂²⋊C₄	—	2	48	no	—
33	C₂×C₅×C₂²⋊C₄	C₂³	1	48	yes	—
34	C₂×C₄²×C₅	D₄	1	24	yes	—
35	C₂×C₄²×C₅	C₂³	1	12	yes	—
36	C₅×#95	C₂³	1	16	yes	—
37	C₂²×C₄⋊Q₈	C₂×C₅	1	3	yes	complemented
38	C₂²×#124	C₂×C₅	1	3	yes	—
39	#2658	C₂×C₅	1	16	yes	direct product factor
40	C₂×#399	C₂×C₅	1	8	yes	complemented
41	C₂×#376	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂⁴×C₅	—	2	12	no	—
43	C₂⁴×C₅	—	2	16	no	—
44	C₂⁴×C₅	C₂⁴	1	4	yes	—
45	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
46	C₂×C₅×Q₈	—	2	24	no	—
47	C₂×C₅×Q₈	—	2	48	no	—
48	C₂×C₅×Q₈	—	4	48	no	—
49	C₂×C₅×D₄	—	2	24	no	—
50	C₂×C₅×D₄	—	4	16	no	—
51	C₂²×C₄×C₅	—	2	24	no	—
52	C₂²×C₄×C₅	—	2	12	no	—
53	C₂²×C₄×C₅	—	2	24	no	—
54	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
55	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
58	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
59	C₅×C₄⋊C₄	—	2	96	no	—
60	C₅×C₂²⋊C₄	—	2	96	no	—
61	C₄²×C₅	C₂×D₄	1	48	yes	—
62	C₂³×Q₈	C₂²×C₅	1	3	yes	—
63	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₄⋊Q₈	—	2	24	no	—
65	C₂×C₄⋊Q₈	C₂²×C₅	1	24	yes	complemented
66	C₂×#124	—	2	24	no	—
67	C₂×#124	C₂²×C₅	1	24	yes	—
68	C₂²×C₄⋊C₄	C₂²×C₅	1	12	yes	—
69	C₂²×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
70	C₂²×C₄²	C₂²×C₅	1	3	yes	—
71	#399	C₂²×C₅	1	64	yes	complemented
72	#376	C₂²×C₅	1	8	yes	complemented
73	C₂×#95	C₂²×C₅	1	4	yes	—
74	C₂³×C₅	—	2	12	no	—
75	C₂³×C₅	—	2	16	no	—
76	C₂³×C₅	—	4	8	no	—
77	C₂³×C₅	—	4	48	no	—
78	C₂³×C₅	C₂²×D₄	1	6	yes	—
79	C₂³×C₅	2₋¹⁺⁴	1	8	yes	—
80	C₂³×C₅	C₂⁵	1	1	yes	characteristic
81	C₅×Q₈	—	4	48	no	—
82	C₅×D₄	—	4	16	no	—
83	C₂×C₄×C₅	—	2	48	no	—
84	C₂×C₄×C₅	—	2	24	no	—
85	C₂×C₄×C₅	—	2	12	no	—
86	C₂×C₄×C₅	—	2	24	no	—
87	C₂×C₄×C₅	—	4	24	no	—
88	C₂×C₄×C₅	#127	1	16	yes	—
89	C₂×C₄×C₅	C₂²×D₄	1	24	yes	—
90	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
91	C₂⁵	C₂³×C₅	1	2	yes	—
92	C₂²×Q₈	—	2	12	no	—
93	C₂²×Q₈	—	2	24	no	—
94	C₂²×Q₈	C₂³×C₅	1	12	yes	—
95	C₂²×D₄	—	2	12	no	—
96	C₂²×D₄	C₂³×C₅	1	4	yes	—
97	C₂³×C₄	C₂³×C₅	1	6	yes	—
98	C₂³×C₄	C₂³×C₅	1	6	yes	—
99	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	96	no	—
101	#124	—	2	96	no	—
102	C₂×C₄⋊C₄	—	2	48	no	—
103	C₂×C₄⋊C₄	C₂³×C₅	1	48	yes	—
104	C₂×C₂²⋊C₄	—	2	48	no	—
105	C₂×C₂²⋊C₄	C₂³×C₅	1	48	yes	—
106	C₂×C₄²	C₅×D₄	1	24	yes	—
107	C₂×C₄²	C₂³×C₅	1	12	yes	—
108	#95	C₂³×C₅	1	16	yes	—
109	C₂²×C₅	—	4	4	no	—
110	C₂²×C₅	—	4	24	no	—
111	C₂²×C₅	—	8	16	no	—
112	C₂²×C₅	C₂×#127	1	4	yes	—
113	C₂²×C₅	#536	1	24	yes	—
114	C₂²×C₅	C₂³×D₄	1	3	yes	—
115	C₂²×C₅	C₂×2₋¹⁺⁴	1	4	yes	—
116	C₄×C₅	—	2	24	no	—
117	C₄×C₅	—	4	12	no	—
118	C₄×C₅	C₂×#127	1	8	yes	—
119	C₂⁴	—	2	12	no	—
120	C₂⁴	—	2	16	no	—
121	C₂⁴	C₂⁴×C₅	1	4	yes	—
122	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
123	C₂×Q₈	—	2	24	no	—
124	C₂×Q₈	—	2	48	no	—
125	C₂×Q₈	—	4	48	no	—
126	C₂×D₄	—	2	24	no	—
127	C₂×D₄	—	4	16	no	—
128	C₂²×C₄	—	2	24	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	C₂×C₅×D₄	1	24	yes	—
132	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
133	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
134	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
135	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
136	C₄⋊C₄	—	2	96	no	—
137	C₂²⋊C₄	—	2	96	no	—
138	C₄²	C₂×C₅×D₄	1	48	yes	—
139	C₂×C₅	—	8	4	no	—
140	C₂×C₅	#2658	1	8	yes	direct product factor
141	C₂×C₅	C₂²×#127	1	1	yes	characteristic
142	C₂×C₅	C₂×#536	1	6	yes	—
143	C₂³	—	2	12	no	—
144	C₂³	—	2	16	no	—
145	C₂³	—	4	8	no	—
146	C₂³	—	4	48	no	—
147	C₂³	C₂²×C₅×D₄	1	6	yes	—
148	C₂³	C₅×2₋¹⁺⁴	1	8	yes	—
149	C₂³	C₂⁵×C₅	1	1	yes	characteristic
150	Q₈	—	4	48	no	—
151	D₄	—	4	16	no	—
152	C₂×C₄	—	2	48	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	—	2	24	no	—
156	C₂×C₄	—	4	24	no	—
157	C₂×C₄	C₅×#127	1	16	yes	—
158	C₂×C₄	C₂²×C₅×D₄	1	24	yes	—
159	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
160	C₅	C₂×#2658	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²	—	4	4	no	—
162	C₂²	—	4	24	no	—
163	C₂²	—	8	16	no	—
164	C₂²	C₂×C₅×#127	1	4	yes	—
165	C₂²	C₅×#536	1	24	yes	—
166	C₂²	C₂³×C₅×D₄	1	3	yes	—
167	C₂²	C₂×C₅×2₋¹⁺⁴	1	4	yes	—
168	C₄	—	2	24	no	—
169	C₄	—	4	12	no	—
170	C₄	C₂×C₅×#127	1	8	yes	—
171	C₂	—	8	4	no	—
172	C₂	C₅×#2658	1	8	yes	direct product factor
173	C₂	C₂²×C₅×#127	1	1	yes	characteristic
174	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. C₂²×C₅×C₄⋊Q₈	C₂	semidirect
3. C₅×#2658	C₂	direct (N × H)
4. C₂×C₅×#399	C₂	semidirect
5. C₂×C₅×#376	C₂	semidirect
9. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
15. C₅×#399	C₂²	semidirect
16. C₅×#376	C₂²	semidirect
18. C₂×#2658	C₅	direct (N × H)
37. C₂²×C₄⋊Q₈	C₂×C₅	semidirect
39. #2658	C₂×C₅	direct (N × H)
40. C₂×#399	C₂×C₅	semidirect
41. C₂×#376	C₂×C₅	semidirect
65. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
71. #399	C₂²×C₅	semidirect
72. #376	C₂²×C₅	semidirect
140. C₂×C₅	#2658	direct (N × H)
160. C₅	C₂×#2658	direct (N × H)
172. C₂	C₅×#2658	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⋊Q₈	2
C₂²×C₅×#124	2
C₅×#2658	2
C₂×C₅×#399	2
C₂×C₅×#376	2
C₂×#2658	5

Subgroups — C2×C5×#2658 #11957350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2658 #11957350