Subgroups — C₂×C₄²×Dic₁₅ #14319322

Order 1920.

2544 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₅	C₂	1	6	yes	maximal
3	C₄²×Dic₁₅	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×Dic₅	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
6	C₂³×Dic₁₅	C₂²	1	4	yes	—
7	C₂×C₃×C₄²×C₅	C₄	1	8	yes	complemented
8	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₅	C₄	1	48	yes	direct product factor
10	C₂×C₄×Dic₁₅	C₂²	1	24	yes	—
11	C₂×C₄²×Dic₃	—	5	1	no	maximal, Hall
12	C₂²×C₄²×C₅	D₃	1	1	yes	characteristic
13	C₂²×C₄×Dic₅	—	3	6	no	—
14	C₄²×Dic₅	—	3	8	no	—
15	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	24	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	12	yes	—
18	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
19	C₂²×Dic₁₅	C₂×C₄	1	48	yes	—
20	C₂²×Dic₁₅	C₂³	1	8	yes	—
21	C₃×C₄²×C₅	C₂×C₄	1	16	yes	complemented
22	C₄×Dic₁₅	C₂×C₄	1	96	yes	direct product factor
23	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic
24	C₂²×C₄×Dic₃	—	5	6	no	—
25	C₄²×Dic₃	—	5	8	no	—
26	C₂³×C₄×C₅	C₂×D₃	1	3	yes	—
27	C₂³×Dic₅	—	3	4	no	—
28	C₂×C₄²×C₅	Dic₃	1	8	yes	complemented
29	C₂×C₄²×C₅	C₂×D₃	1	4	yes	—
30	C₂×C₄×Dic₅	—	3	48	no	—
31	C₂×C₄×Dic₅	—	3	24	no	—
32	C₂×C₄³	—	15	1	no	Sylow
33	C₂³×C₃×C₅	C₂²×C₄	1	12	yes	—
34	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
35	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	C₄²	1	48	yes	complemented
37	C₂×C₃×C₄×C₅	C₂²×C₄	1	24	yes	—
38	C₂×C₃×C₄×C₅	C₂²×C₄	1	12	yes	—
39	C₂×Dic₁₅	C₄²	1	64	yes	direct product factor
40	C₂×Dic₁₅	C₂²×C₄	1	48	yes	—
41	C₂³×C₃×C₄	C₂×D₅	1	3	yes	—
42	C₂³×Dic₃	—	5	4	no	—
43	C₂×C₃×C₄²	Dic₅	1	8	yes	complemented
44	C₂×C₃×C₄²	C₂×D₅	1	4	yes	—
45	C₂×C₄×Dic₃	—	5	48	no	—
46	C₂×C₄×Dic₃	—	5	24	no	—
47	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₄×D₃	1	24	yes	—
49	C₂²×C₄×C₅	C₂×Dic₃	1	12	yes	—
50	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
51	C₂²×Dic₅	—	3	48	no	—
52	C₂²×Dic₅	—	3	8	no	—
53	C₄²×C₅	C₂×Dic₃	1	16	yes	complemented
54	C₄×Dic₅	—	3	96	no	—
55	C₂²×C₄²	—	15	6	no	—
56	C₂²×C₄²	D₁₅	1	1	yes	characteristic
57	C₄³	—	15	8	no	—
58	C₂²×C₃×C₅	C₂×C₄²	1	16	yes	—
59	C₂²×C₃×C₅	C₂×C₄²	1	12	yes	—
60	C₂²×C₃×C₅	C₂³×C₄	1	6	yes	—
61	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
62	C₃×C₄×C₅	C₂×C₄²	1	24	yes	complemented
63	Dic₁₅	C₂×C₄²	1	32	yes	direct product factor
64	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₄×D₅	1	24	yes	—
66	C₂²×C₃×C₄	C₂×Dic₅	1	12	yes	—
67	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
68	C₂²×Dic₃	—	5	48	no	—
69	C₂²×Dic₃	—	5	8	no	—
70	C₃×C₄²	C₂×Dic₅	1	16	yes	complemented
71	C₄×Dic₃	—	5	96	no	—
72	C₂³×C₅	C₂×C₄×D₃	1	12	yes	—
73	C₂³×C₅	C₂²×Dic₃	1	2	yes	—
74	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
75	C₂×C₄×C₅	C₄×Dic₃	1	48	yes	complemented
76	C₂×C₄×C₅	C₂×C₄×D₃	1	24	yes	—
77	C₂×C₄×C₅	C₂²×Dic₃	1	12	yes	—
78	C₂×Dic₅	—	3	64	no	—
79	C₂×Dic₅	—	3	48	no	—
80	C₂³×C₄	—	15	4	no	—
81	C₂³×C₄	C₂×D₁₅	1	3	yes	—
82	C₂×C₄²	—	15	48	no	—
83	C₂×C₄²	—	15	24	no	—
84	C₂×C₄²	Dic₁₅	1	8	yes	direct product factor
85	C₂×C₄²	C₂×D₁₅	1	4	yes	—
86	C₂×C₃×C₅	C₄³	1	8	yes	complemented
87	C₂×C₃×C₅	C₂²×C₄²	1	4	yes	—
88	C₂×C₃×C₅	C₂²×C₄²	1	3	yes	—
89	C₂³×C₃	C₂×C₄×D₅	1	12	yes	—
90	C₂³×C₃	C₂²×Dic₅	1	2	yes	—
91	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
92	C₂×C₃×C₄	C₄×Dic₅	1	48	yes	complemented
93	C₂×C₃×C₄	C₂×C₄×D₅	1	24	yes	—
94	C₂×C₃×C₄	C₂²×Dic₅	1	12	yes	—
95	C₂×Dic₃	—	5	48	no	—
96	C₂×Dic₃	—	5	64	no	—
97	C₂²×C₅	C₄²×D₃	1	16	yes	—
98	C₂²×C₅	C₂×C₄×Dic₃	1	12	yes	—
99	C₂²×C₅	C₂²×C₄×D₃	1	6	yes	—
100	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
101	C₄×C₅	C₂×C₄×Dic₃	1	24	yes	complemented
102	Dic₅	—	3	32	no	—
103	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
104	C₂²×C₄	—	15	48	no	—
105	C₂²×C₄	—	15	8	no	—
106	C₂²×C₄	C₄×D₁₅	1	24	yes	—
107	C₂²×C₄	C₂×Dic₁₅	1	12	yes	—
108	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
109	C₄²	—	15	96	no	—
110	C₄²	C₂×Dic₁₅	1	16	yes	direct product factor
111	C₃×C₅	C₂×C₄³	1	1	yes	characteristic, complemented, Hall
112	C₂²×C₃	C₄²×D₅	1	16	yes	—
113	C₂²×C₃	C₂×C₄×Dic₅	1	12	yes	—
114	C₂²×C₃	C₂²×C₄×D₅	1	6	yes	—
115	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
116	C₃×C₄	C₂×C₄×Dic₅	1	24	yes	complemented
117	Dic₃	—	5	32	no	—
118	C₂×C₅	C₄²×Dic₃	1	8	yes	complemented
119	C₂×C₅	C₂×C₄²×D₃	1	4	yes	—
120	C₂×C₅	C₂²×C₄×Dic₃	1	3	yes	—
121	C₂³	C₂×C₄×D₁₅	1	12	yes	—
122	C₂³	C₂²×Dic₁₅	1	2	yes	—
123	C₂³	C₂³×D₁₅	1	1	yes	characteristic
124	C₂×C₄	—	15	64	no	—
125	C₂×C₄	—	15	48	no	—
126	C₂×C₄	C₄×Dic₁₅	1	48	yes	direct product factor
127	C₂×C₄	C₂×C₄×D₁₅	1	24	yes	—
128	C₂×C₄	C₂²×Dic₁₅	1	12	yes	—
129	C₂×C₃	C₄²×Dic₅	1	8	yes	complemented
130	C₂×C₃	C₂×C₄²×D₅	1	4	yes	—
131	C₂×C₃	C₂²×C₄×Dic₅	1	3	yes	—
132	C₅	C₂×C₄²×Dic₃	1	1	yes	characteristic, complemented, Sylow
133	C₂²	C₄²×D₁₅	1	16	yes	—
134	C₂²	C₂×C₄×Dic₁₅	1	12	yes	—
135	C₂²	C₂²×C₄×D₁₅	1	6	yes	—
136	C₂²	C₂³×Dic₁₅	1	1	yes	characteristic
137	C₄	—	15	32	no	—
138	C₄	C₂×C₄×Dic₁₅	1	24	yes	direct product factor
139	C₃	C₂×C₄²×Dic₅	1	1	yes	characteristic, complemented, Sylow
140	C₂	C₄²×Dic₁₅	1	8	yes	direct product factor
141	C₂	C₂×C₄²×D₁₅	1	4	yes	—
142	C₂	C₂²×C₄×Dic₁₅	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₁₅
commutator, hypocenter	C₃×C₅	C₂×C₄³
Frattini	C₂³	C₂³×D₁₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle, prime radical	C₂⁴×C₃×C₅	C₂³
cosocle	C₂³×C₃×C₅	C₂⁴
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄²×Dic₁₅	C₂	direct (N × H)
7. C₂×C₃×C₄²×C₅	C₄	semidirect
9. C₂×C₄×Dic₁₅	C₄	direct (N × H)
21. C₃×C₄²×C₅	C₂×C₄	semidirect
22. C₄×Dic₁₅	C₂×C₄	direct (N × H)
28. C₂×C₄²×C₅	Dic₃	semidirect
36. C₂×C₃×C₄×C₅	C₄²	semidirect
39. C₂×Dic₁₅	C₄²	direct (N × H)
43. C₂×C₃×C₄²	Dic₅	semidirect
53. C₄²×C₅	C₂×Dic₃	semidirect
62. C₃×C₄×C₅	C₂×C₄²	semidirect
63. Dic₁₅	C₂×C₄²	direct (N × H)
70. C₃×C₄²	C₂×Dic₅	semidirect
75. C₂×C₄×C₅	C₄×Dic₃	semidirect
84. C₂×C₄²	Dic₁₅	direct (N × H)
86. C₂×C₃×C₅	C₄³	semidirect
92. C₂×C₃×C₄	C₄×Dic₅	semidirect
101. C₄×C₅	C₂×C₄×Dic₃	semidirect
110. C₄²	C₂×Dic₁₅	direct (N × H)
111. C₃×C₅	C₂×C₄³	semidirect
116. C₃×C₄	C₂×C₄×Dic₅	semidirect
118. C₂×C₅	C₄²×Dic₃	semidirect
126. C₂×C₄	C₄×Dic₁₅	direct (N × H)
129. C₂×C₃	C₄²×Dic₅	semidirect
132. C₅	C₂×C₄²×Dic₃	semidirect
138. C₄	C₂×C₄×Dic₁₅	direct (N × H)
139. C₃	C₂×C₄²×Dic₅	semidirect
140. C₂	C₄²×Dic₁₅	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×C₃×C₄²×C₅	2
C₂²×C₄×Dic₁₅	2
C₄²×Dic₁₅	2
C₂×C₄²×Dic₅	3
C₂×C₄²×Dic₃	5

Subgroups — C2×C42×Dic15 #14319322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Dic₁₅ #14319322