Subgroups — Dic₅×#368 #12014714

Order 1280.

314 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×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#64726	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	Dic₅×#108	C₂	1	2	yes	maximal
5	#10598219	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	C₂²	1	2	yes	—
8	M₁₆×Dic₅	—	2	2	no	—
9	C₂×#3951	C₂²	1	2	yes	—
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	C₅×#368	C₄	1	4	yes	complemented
13	#64778	—	2	2	no	—
14	#64726	C₄	1	4	yes	complemented
15	C₁₆×Dic₅	C₄	1	4	yes	—
16	C₄×#368	—	5	1	no	maximal, Sylow
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
20	C₂×#681	—	2	2	no	—
21	C₂²×#168	D₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₅×#108	—	2	2	no	—
26	C₅×#108	C₂×C₄	1	4	yes	—
27	#3951	—	2	2	no	—
28	#3951	C₂×C₄	1	4	yes	—
29	#3947	—	4	2	no	—
30	C₈×Dic₅	—	4	2	no	—
31	C₈×Dic₅	C₂×C₄	1	2	yes	—
32	C₈×Dic₅	Q₈	1	2	yes	—
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	C₂×C₄	1	2	yes	—
36	C₂×#368	—	5	1	no	—
37	C₂×#368	D₅	1	1	yes	characteristic
38	C₂×C₄×C₁₆	—	5	1	no	—
39	C₄×#108	—	5	2	no	—
40	#1381	—	5	2	no	—
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅	Q₁₆	1	2	yes	—
43	C₅×M₁₆	—	2	4	no	—
44	C₅×M₁₆	—	4	2	no	—
45	C₂×C₅×C₈	—	4	2	no	—
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	1	yes	characteristic
51	C₂×C₅×C₈	#41	1	1	yes	characteristic
52	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₅	D₈	1	1	yes	characteristic
54	C₄×Dic₅	D₈	1	1	yes	characteristic
55	#681	—	2	4	no	—
56	#681	—	4	2	no	—
57	C₂×#168	—	2	1	no	—
58	C₂×#168	—	2	1	no	—
59	C₂×#168	—	4	2	no	—
60	C₂×#168	C₄⋊C₄	1	2	yes	—
61	C₅×C₁₆	C₄²	1	4	yes	—
62	#672	—	2	2	no	—
63	C₂²×C₁₆	—	5	1	no	—
64	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×#108	—	5	2	no	—
66	C₂×#108	C₂×D₅	1	2	yes	—
67	C₄×M₁₆	—	10	2	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	#368	—	5	4	no	—
70	#368	Dic₅	1	4	yes	direct product factor
71	C₄×C₁₆	—	5	4	no	—
72	#341	—	10	2	no	—
73	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅	#135	1	1	yes	characteristic
79	C₂×C₄×C₅	#135	1	1	yes	characteristic
80	C₂×Dic₅	—	2	4	no	—
81	C₂×Dic₅	#107	1	2	yes	—
82	C₂×Dic₅	#107	1	2	yes	—
83	C₅×C₈	—	4	4	no	—
84	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
85	C₅×C₈	#117	1	1	yes	characteristic
86	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
87	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
88	#168	—	2	2	no	—
89	#168	—	4	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	10	2	no	—
92	C₂²×C₈	—	5	1	no	—
93	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
94	C₂×C₄²	—	5	1	no	—
95	C₂×C₁₆	—	5	2	no	—
96	C₂×C₁₆	—	10	1	no	—
97	C₂×C₁₆	—	10	1	no	—
98	C₂×C₁₆	C₄×D₅	1	2	yes	—
99	C₂×C₁₆	C₄×D₅	1	2	yes	—
100	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	5	4	no	—
103	#108	—	10	2	no	—
104	#108	C₂×Dic₅	1	4	yes	—
105	#97	—	20	2	no	—
106	C₄×C₈	—	5	2	no	—
107	C₄×C₈	—	5	2	no	—
108	C₄×C₈	—	20	2	no	—
109	C₂²×C₅	—	2	2	no	—
110	C₂²×C₅	C₂×#107	1	1	yes	characteristic
111	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
112	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
113	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₅	#427	1	1	yes	characteristic
115	C₄×C₅	C₄×D₈	1	1	yes	characteristic
116	C₄×C₅	C₄×D₈	1	1	yes	characteristic
117	Dic₅	—	2	2	no	—
118	Dic₅	#368	1	4	yes	direct product factor
119	C₂²×C₄	—	5	2	no	—
120	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	10	4	no	—
124	M₁₆	—	20	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	5	2	no	—
127	C₂×C₈	—	10	1	no	—
128	C₂×C₈	—	10	1	no	—
129	C₂×C₈	—	20	2	no	—
130	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
132	C₂×C₈	#711	1	1	yes	characteristic
133	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
134	C₂×C₈	#713	1	1	yes	characteristic
135	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
136	C₄²	—	5	2	no	—
137	C₄²	—	5	1	no	—
138	C₄²	—	5	1	no	—
139	C₁₆	—	10	2	no	—
140	C₁₆	C₄×Dic₅	1	4	yes	—
141	C₂×C₅	—	2	1	no	—
142	C₂×C₅	—	2	1	no	—
143	C₂×C₅	C₄×#107	1	1	yes	characteristic
144	C₂×C₅	C₂×#368	1	2	yes	—
145	C₂³	D₅×Q₁₆	1	1	yes	characteristic
146	C₂×C₄	—	5	2	no	—
147	C₂×C₄	—	5	2	no	—
148	C₂×C₄	—	10	4	no	—
149	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#4039	1	1	yes	characteristic
151	C₂×C₄	D₅×D₈	1	1	yes	characteristic
152	C₂×C₄	#4058	1	1	yes	characteristic
153	C₂×C₄	#4065	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
155	C₈	—	4	4	no	—
156	C₈	—	10	2	no	—
157	C₈	—	20	4	no	—
158	C₈	#4033	1	1	yes	characteristic
159	C₈	#4038	1	1	yes	characteristic
160	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
161	C₈	Q₈×Dic₅	1	1	yes	characteristic
162	C₅	C₄×#368	1	1	yes	characteristic, complemented, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	#65162	1	1	yes	characteristic
165	C₂²	D₅×#107	1	1	yes	characteristic
166	C₂²	Q₁₆×Dic₅	1	1	yes	characteristic
167	C₄	—	5	4	no	—
168	C₄	—	10	2	no	—
169	C₄	#65161	1	1	yes	characteristic
170	C₄	#65169	1	1	yes	characteristic
171	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
172	C₄	D₈×Dic₅	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	D₅×#368	1	2	yes	—
176	C₂	Dic₅×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#368	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#368	D₅
hypocenter	C₅	C₄×#368
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
12. C₅×#368	C₄	semidirect
14. #64726	C₄	semidirect
70. #368	Dic₅	direct (N × H)
118. Dic₅	#368	direct (N × H)
162. C₅	C₄×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#368	2
C₂×#64726	2
C₂×C₁₆×Dic₅	2
Dic₅×#108	2
#10598219	2
C₄×#368	5

Subgroups — Dic5×#368 #12014714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#368 #12014714