Subgroups — C₅×#3486 #10619271

Order 640.

1492 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₄×Q₈	C₂	1	8	yes	maximal, complemented
4	C₅×#548	C₂	1	8	yes	maximal, complemented
5	C₅×#536	C₂	1	6	yes	maximal, complemented
6	C₅×#525	C₂	1	6	yes	maximal, complemented
7	C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×2₋¹⁺⁴	—	2	8	no	—
9	C₂×C₅×#41	C₂²	1	6	yes	—
10	C₂×C₅×#41	C₂²	1	4	yes	complemented
11	C₂²×C₅×Q₈	—	2	8	no	—
12	C₂²×C₅×Q₈	C₂²	1	3	yes	—
13	C₂²×C₅×Q₈	C₂²	1	8	yes	—
14	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Q₈	C₂²	1	4	yes	complemented
16	C₂³×C₄×C₅	C₂²	1	3	yes	—
17	C₅×C₄⋊Q₈	C₂²	1	12	yes	complemented
18	C₅×#124	C₂²	1	12	yes	complemented
19	C₅×#123	C₂²	1	12	yes	—
20	C₅×C₂²⋊Q₈	C₂²	1	12	yes	—
21	C₅×C₂²⋊Q₈	C₂²	1	24	yes	complemented
22	C₅×#121	C₂²	1	12	yes	—
23	C₅×#120	C₂²	1	4	yes	—
24	C₄×C₅×Q₈	C₂²	1	8	yes	complemented
25	C₄×C₅×D₄	C₂²	1	24	yes	—
26	C₅×#117	C₂²	1	6	yes	—
27	#3486	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁴×C₅	C₂³	1	1	yes	characteristic
29	C₅×#41	—	2	16	no	—
30	C₅×#41	—	2	24	no	—
31	C₂×C₅×Q₈	—	2	32	no	—
32	C₂×C₅×Q₈	—	2	16	no	—
33	C₂×C₅×Q₈	—	2	12	no	—
34	C₂×C₅×Q₈	D₄	1	8	yes	complemented
35	C₂×C₅×Q₈	C₂³	1	6	yes	—
36	C₂×C₅×Q₈	C₂³	1	12	yes	—
37	C₂×C₅×Q₈	C₂³	1	4	yes	—
38	C₂×C₅×D₄	C₂³	1	8	yes	—
39	C₂×C₅×D₄	C₂³	1	4	yes	—
40	C₂×C₅×D₄	C₂³	1	6	yes	—
41	C₂²×C₄×C₅	—	2	12	no	—
42	C₂²×C₄×C₅	C₂³	1	12	yes	—
43	C₂²×C₄×C₅	C₂³	1	12	yes	—
44	C₂²×C₄×C₅	C₂³	1	3	yes	—
45	C₂²×C₄×C₅	C₂³	1	3	yes	—
46	C₅×C₄⋊C₄	C₂³	1	12	yes	—
47	C₅×C₄⋊C₄	C₂³	1	24	yes	—
48	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
49	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
50	C₅×C₂²⋊C₄	C₂³	1	24	yes	—
51	C₄²×C₅	C₂³	1	12	yes	—
52	C₂×2₋¹⁺⁴	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂³×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
54	D₄×Q₈	C₂×C₅	1	8	yes	complemented
55	#548	C₂×C₅	1	8	yes	complemented
56	#536	C₂×C₅	1	6	yes	complemented
57	#525	C₂×C₅	1	6	yes	complemented
58	#519	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂³×C₅	—	2	2	no	—
60	C₂³×C₅	—	2	4	no	—
61	C₂³×C₅	C₂⁴	1	4	yes	—
62	C₂³×C₅	C₂⁴	1	1	yes	characteristic
63	C₂³×C₅	C₂⁴	1	2	yes	—
64	C₅×Q₈	—	2	24	no	—
65	C₅×Q₈	—	2	12	no	—
66	C₅×Q₈	C₂×D₄	1	16	yes	—
67	C₅×D₄	—	2	8	no	—
68	C₅×D₄	—	2	16	no	—
69	C₅×D₄	—	2	12	no	—
70	C₂×C₄×C₅	—	2	6	no	—
71	C₂×C₄×C₅	—	2	24	no	—
72	C₂×C₄×C₅	—	2	24	no	—
73	C₂×C₄×C₅	C₂×D₄	1	12	yes	—
74	C₂×C₄×C₅	C₂⁴	1	8	yes	—
75	C₂×C₄×C₅	C₂⁴	1	6	yes	—
76	C₂×C₄×C₅	C₂⁴	1	4	yes	—
77	C₂×C₄×C₅	C₂⁴	1	6	yes	—
78	2₋¹⁺⁴	—	2	8	no	—
79	C₂×#41	C₂²×C₅	1	6	yes	—
80	C₂×#41	C₂²×C₅	1	4	yes	complemented
81	C₂²×Q₈	—	2	8	no	—
82	C₂²×Q₈	C₂²×C₅	1	3	yes	—
83	C₂²×Q₈	C₂²×C₅	1	8	yes	—
84	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₂²×Q₈	C₂²×C₅	1	4	yes	complemented
86	C₂³×C₄	C₂²×C₅	1	3	yes	—
87	C₄⋊Q₈	C₂²×C₅	1	12	yes	complemented
88	#124	C₂²×C₅	1	12	yes	complemented
89	#123	C₂²×C₅	1	12	yes	—
90	C₂²⋊Q₈	C₂²×C₅	1	12	yes	—
91	C₂²⋊Q₈	C₂²×C₅	1	24	yes	complemented
92	#121	C₂²×C₅	1	12	yes	—
93	#120	C₂²×C₅	1	4	yes	—
94	C₄×Q₈	C₂²×C₅	1	8	yes	complemented
95	C₄×D₄	C₂²×C₅	1	24	yes	—
96	#117	C₂²×C₅	1	6	yes	—
97	C₂²×C₅	—	2	8	no	—
98	C₂²×C₅	—	2	2	no	—
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	—	2	4	no	—
102	C₂²×C₅	—	2	8	no	—
103	C₂²×C₅	C₂²×D₄	1	2	yes	—
104	C₂²×C₅	2₋¹⁺⁴	1	4	yes	complemented
105	C₂²×C₅	C₂⁵	1	1	yes	characteristic
106	C₄×C₅	—	2	4	no	—
107	C₄×C₅	—	2	6	no	—
108	C₄×C₅	—	2	8	no	—
109	C₄×C₅	C₂²×D₄	1	12	yes	—
110	C₂⁴	C₂³×C₅	1	1	yes	characteristic
111	#41	—	2	16	no	—
112	#41	—	2	24	no	—
113	C₂×Q₈	—	2	32	no	—
114	C₂×Q₈	—	2	16	no	—
115	C₂×Q₈	—	2	12	no	—
116	C₂×Q₈	C₅×D₄	1	8	yes	complemented
117	C₂×Q₈	C₂³×C₅	1	6	yes	—
118	C₂×Q₈	C₂³×C₅	1	12	yes	—
119	C₂×Q₈	C₂³×C₅	1	4	yes	—
120	C₂×D₄	C₂³×C₅	1	8	yes	—
121	C₂×D₄	C₂³×C₅	1	4	yes	—
122	C₂×D₄	C₂³×C₅	1	6	yes	—
123	C₂²×C₄	—	2	12	no	—
124	C₂²×C₄	C₂³×C₅	1	12	yes	—
125	C₂²×C₄	C₂³×C₅	1	12	yes	—
126	C₂²×C₄	C₂³×C₅	1	3	yes	—
127	C₂²×C₄	C₂³×C₅	1	3	yes	—
128	C₄⋊C₄	C₂³×C₅	1	12	yes	—
129	C₄⋊C₄	C₂³×C₅	1	24	yes	—
130	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
131	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
132	C₂²⋊C₄	C₂³×C₅	1	24	yes	—
133	C₄²	C₂³×C₅	1	12	yes	—
134	C₂×C₅	—	2	2	no	—
135	C₂×C₅	—	2	4	no	—
136	C₂×C₅	—	4	4	no	—
137	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
138	C₂×C₅	C₂×2₋¹⁺⁴	1	2	yes	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	4	no	—
141	C₂³	C₂⁴×C₅	1	4	yes	—
142	C₂³	C₂⁴×C₅	1	1	yes	characteristic
143	C₂³	C₂⁴×C₅	1	2	yes	—
144	Q₈	—	2	24	no	—
145	Q₈	—	2	12	no	—
146	Q₈	C₂×C₅×D₄	1	16	yes	—
147	D₄	—	2	8	no	—
148	D₄	—	2	16	no	—
149	D₄	—	2	12	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	24	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	C₂×C₅×D₄	1	12	yes	—
154	C₂×C₄	C₂⁴×C₅	1	8	yes	—
155	C₂×C₄	C₂⁴×C₅	1	6	yes	—
156	C₂×C₄	C₂⁴×C₅	1	4	yes	—
157	C₂×C₄	C₂⁴×C₅	1	6	yes	—
158	C₅	#3486	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²	—	2	8	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂²×C₅×D₄	1	2	yes	—
166	C₂²	C₅×2₋¹⁺⁴	1	4	yes	complemented
167	C₂²	C₂⁵×C₅	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	—	2	8	no	—
171	C₄	C₂²×C₅×D₄	1	12	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂³×C₅×D₄	1	1	yes	characteristic
176	C₂	C₂×C₅×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×2₋¹⁺⁴	C₂	semidirect
2. C₂³×C₅×Q₈	C₂	semidirect
3. C₅×D₄×Q₈	C₂	semidirect
4. C₅×#548	C₂	semidirect
5. C₅×#536	C₂	semidirect
6. C₅×#525	C₂	semidirect
7. C₅×#519	C₂	semidirect
10. C₂×C₅×#41	C₂²	semidirect
15. C₂²×C₅×Q₈	C₂²	semidirect
17. C₅×C₄⋊Q₈	C₂²	semidirect
18. C₅×#124	C₂²	semidirect
21. C₅×C₂²⋊Q₈	C₂²	semidirect
24. C₄×C₅×Q₈	C₂²	semidirect
27. #3486	C₅	direct (N × H)
34. C₂×C₅×Q₈	D₄	semidirect
52. C₂×2₋¹⁺⁴	C₂×C₅	semidirect
53. C₂³×Q₈	C₂×C₅	semidirect
54. D₄×Q₈	C₂×C₅	semidirect
55. #548	C₂×C₅	semidirect
56. #536	C₂×C₅	semidirect
57. #525	C₂×C₅	semidirect
58. #519	C₂×C₅	semidirect
80. C₂×#41	C₂²×C₅	semidirect
85. C₂²×Q₈	C₂²×C₅	semidirect
87. C₄⋊Q₈	C₂²×C₅	semidirect
88. #124	C₂²×C₅	semidirect
91. C₂²⋊Q₈	C₂²×C₅	semidirect
94. C₄×Q₈	C₂²×C₅	semidirect
104. C₂²×C₅	2₋¹⁺⁴	semidirect
116. C₂×Q₈	C₅×D₄	semidirect
158. C₅	#3486	direct (N × H)
166. C₂²	C₅×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×2₋¹⁺⁴	2
C₂³×C₅×Q₈	2
C₅×D₄×Q₈	2
C₅×#548	2
C₅×#536	2
C₅×#525	2
C₅×#519	2
#3486	5

Subgroups — C5×#3486 #10619271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3486 #10619271