Finite Group Databasebeta

← C₂⁴×C₅⋊S₄ #14514713

Subgroups — C₂⁴×C₅⋊S₄ #14514713

Order 1920.

10254 conjugacy classes of subgroups; only the 709 normal subgroups are shown.

normal subgroups only

This group has 10254 conjugacy classes of subgroups — too many to draw. Only its 709 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅⋊S₄	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₂²⋊D₅	—	3	1	no	maximal, Hall
4	C₂⁴×D₁₅	—	4	1	no	maximal
5	C₂³×C₅×A₄	C₂²	1	15	yes	complemented
6	C₂²×C₅⋊S₄	C₂²	1	140	yes	direct product factor
7	C₂⁴×S₄	—	5	1	no	maximal, Hall
8	C₂⁶×C₅	D₃	1	1	yes	characteristic, complemented
9	C₂⁵×D₅	—	3	1	no	—
10	C₂³×C₂²⋊D₅	—	3	30	no	—
11	C₂³×C₂²⋊D₅	—	3	30	no	—
12	C₂⁴×Dic₅	—	3	1	no	—
13	C₂⁴×C₃×C₅	—	4	1	no	—
14	C₂³×D₁₅	—	4	30	no	—
15	C₂²×C₅×A₄	C₂³	1	35	yes	complemented
16	C₂×C₅⋊S₄	C₂³	1	120	yes	direct product factor
17	C₂⁴×A₄	D₅	1	1	yes	characteristic, complemented
18	C₂³×S₄	—	5	30	no	—
19	C₂⁵×C₅	—	3	1	no	—
20	C₂⁵×C₅	—	3	15	no	—
21	C₂⁵×C₅	C₂×D₃	1	15	yes	complemented
22	C₂⁴×D₅	—	3	30	no	—
23	C₂⁴×D₅	—	6	15	no	—
24	C₂⁴×D₅	—	6	1	no	—
25	C₂²×C₂²⋊D₅	—	3	420	no	—
26	C₂²×C₂²⋊D₅	—	3	140	no	—
27	C₂³×Dic₅	—	3	30	no	—
28	C₂⁴×D₄	—	15	1	no	Sylow
29	C₂³×C₃×C₅	—	4	15	no	—
30	C₂²×D₁₅	—	4	140	no	—
31	C₂×C₅×A₄	C₂⁴	1	15	yes	complemented
32	C₅⋊S₄	C₂⁴	1	16	yes	direct product factor
33	C₂⁴×D₃	—	20	1	no	—
34	C₂³×A₄	C₂×D₅	1	15	yes	complemented
35	C₂²×S₄	—	5	140	no	—
36	C₂⁴×C₅	—	3	15	no	—
37	C₂⁴×C₅	—	3	15	no	—
38	C₂⁴×C₅	—	3	105	no	—
39	C₂⁴×C₅	—	6	35	no	—
40	C₂⁴×C₅	S₄	1	1	yes	characteristic, complemented
41	C₂⁴×C₅	C₂²×D₃	1	35	yes	complemented
42	C₂³×D₅	—	3	140	no	—
43	C₂³×D₅	—	6	210	no	—
44	C₂³×D₅	—	6	30	no	—
45	C₂×C₂²⋊D₅	—	3	840	no	—
46	C₂×C₂²⋊D₅	—	3	120	no	—
47	C₂²×Dic₅	—	3	140	no	—
48	C₂⁶	—	15	1	no	—
49	C₂⁶	D₁₅	1	1	yes	characteristic, complemented
50	C₂³×D₄	—	15	30	no	—
51	C₂³×D₄	—	15	30	no	—
52	C₂⁴×C₄	—	15	1	no	—
53	C₂²×C₃×C₅	—	4	35	no	—
54	C₂×D₁₅	—	4	120	no	—
55	C₅×A₄	C₂⁵	1	1	yes	characteristic, complemented
56	C₂⁴×C₃	—	4	1	no	—
57	C₂³×D₃	—	20	30	no	—
58	C₂²×A₄	C₂²×D₅	1	35	yes	complemented
59	C₂×S₄	—	5	120	no	—
60	C₂³×C₅	—	3	35	no	—
61	C₂³×C₅	—	3	105	no	—
62	C₂³×C₅	—	3	105	no	—
63	C₂³×C₅	—	6	105	no	—
64	C₂³×C₅	C₂×S₄	1	15	yes	complemented
65	C₂³×C₅	C₂³×D₃	1	15	yes	complemented
66	C₂²×D₅	—	3	120	no	—
67	C₂²×D₅	—	6	420	no	—
68	C₂²×D₅	—	6	140	no	—
69	C₂²⋊D₅	—	3	240	no	—
70	C₂²⋊D₅	—	3	16	no	—
71	C₂×Dic₅	—	3	120	no	—
72	C₂⁵	—	3	1	no	—
73	C₂⁵	—	3	15	no	—
74	C₂⁵	—	15	30	no	—
75	C₂⁵	—	30	15	no	—
76	C₂⁵	—	30	1	no	—
77	C₂⁵	C₂×D₁₅	1	15	yes	complemented
78	C₂²×D₄	—	15	420	no	—
79	C₂²×D₄	—	15	140	no	—
80	C₂³×C₄	—	15	30	no	—
81	C₂×C₃×C₅	—	4	15	no	—
82	D₁₅	—	4	16	no	—
83	C₂³×C₃	—	4	15	no	—
84	C₂²×D₃	—	20	140	no	—
85	C₂×A₄	C₂³×D₅	1	15	yes	complemented
86	S₄	—	5	16	no	—
87	C₂²×C₅	—	3	15	no	—
88	C₂²×C₅	—	3	15	no	—
89	C₂²×C₅	—	3	105	no	—
90	C₂²×C₅	—	6	35	no	—
91	C₂²×C₅	C₂²×S₄	1	35	yes	complemented
92	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic, complemented
93	C₂×D₅	—	3	16	no	—
94	C₂×D₅	—	6	120	no	—
95	C₂×D₅	—	6	120	no	—
96	Dic₅	—	3	16	no	—
97	C₂⁴	—	3	15	no	—
98	C₂⁴	—	3	15	no	—
99	C₂⁴	—	3	105	no	—
100	C₂⁴	—	6	35	no	—
101	C₂⁴	—	15	140	no	—
102	C₂⁴	—	30	210	no	—
103	C₂⁴	—	30	30	no	—
104	C₂⁴	C₅⋊S₄	1	1	yes	characteristic, direct product factor
105	C₂⁴	C₂²×D₁₅	1	35	yes	complemented
106	C₂×D₄	—	15	840	no	—
107	C₂×D₄	—	15	120	no	—
108	C₂²×C₄	—	15	140	no	—
109	C₃×C₅	—	4	1	no	Hall
110	C₂²×C₃	—	4	35	no	—
111	C₂×D₃	—	20	120	no	—
112	A₄	C₂⁴×D₅	1	1	yes	characteristic, complemented
113	C₂×C₅	—	3	1	no	—
114	C₂×C₅	—	3	15	no	—
115	C₂×C₅	C₂³×S₄	1	15	yes	complemented
116	D₅	—	6	16	no	—
117	C₂³	—	3	35	no	—
118	C₂³	—	3	105	no	—
119	C₂³	—	3	105	no	—
120	C₂³	—	6	105	no	—
121	C₂³	—	15	120	no	—
122	C₂³	—	30	420	no	—
123	C₂³	—	30	140	no	—
124	C₂³	C₂×C₅⋊S₄	1	15	yes	direct product factor
125	C₂³	C₂³×D₁₅	1	15	yes	complemented
126	D₄	—	15	240	no	—
127	D₄	—	15	16	no	—
128	C₂×C₄	—	15	120	no	—
129	C₂×C₃	—	4	15	no	—
130	D₃	—	20	16	no	—
131	C₅	C₂⁴×S₄	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	3	15	no	—
133	C₂²	—	3	15	no	—
134	C₂²	—	3	105	no	—
135	C₂²	—	6	35	no	—
136	C₂²	—	15	16	no	—
137	C₂²	—	30	120	no	—
138	C₂²	—	30	120	no	—
139	C₂²	C₂²×C₅⋊S₄	1	35	yes	direct product factor
140	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic, complemented
141	C₄	—	15	16	no	—
142	C₃	—	4	1	no	Sylow
143	C₂	—	3	1	no	—
144	C₂	—	3	15	no	—
145	C₂	—	30	16	no	—
146	C₂	C₂³×C₅⋊S₄	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	C₅⋊S₄
commutator, cosocle, hypocenter	C₅×A₄	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁶×C₅	D₃
LCS	G ▷ C₅×A₄
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×A₄	C₂	semidirect
2. C₂³×C₅⋊S₄	C₂	direct (N × H)
5. C₂³×C₅×A₄	C₂²	semidirect
6. C₂²×C₅⋊S₄	C₂²	direct (N × H)
8. C₂⁶×C₅	D₃	semidirect
15. C₂²×C₅×A₄	C₂³	semidirect
16. C₂×C₅⋊S₄	C₂³	direct (N × H)
17. C₂⁴×A₄	D₅	semidirect
21. C₂⁵×C₅	C₂×D₃	semidirect
31. C₂×C₅×A₄	C₂⁴	semidirect
32. C₅⋊S₄	C₂⁴	direct (N × H)
34. C₂³×A₄	C₂×D₅	semidirect
40. C₂⁴×C₅	S₄	semidirect
41. C₂⁴×C₅	C₂²×D₃	semidirect
49. C₂⁶	D₁₅	semidirect
55. C₅×A₄	C₂⁵	semidirect
58. C₂²×A₄	C₂²×D₅	semidirect
64. C₂³×C₅	C₂×S₄	semidirect
65. C₂³×C₅	C₂³×D₃	semidirect
77. C₂⁵	C₂×D₁₅	semidirect
85. C₂×A₄	C₂³×D₅	semidirect
91. C₂²×C₅	C₂²×S₄	semidirect
92. C₂²×C₅	C₂⁴×D₃	semidirect
104. C₂⁴	C₅⋊S₄	direct (N × H)
105. C₂⁴	C₂²×D₁₅	semidirect
112. A₄	C₂⁴×D₅	semidirect
115. C₂×C₅	C₂³×S₄	semidirect
124. C₂³	C₂×C₅⋊S₄	direct (N × H)
125. C₂³	C₂³×D₁₅	semidirect
131. C₅	C₂⁴×S₄	semidirect
139. C₂²	C₂²×C₅⋊S₄	direct (N × H)
140. C₂²	C₂⁴×D₁₅	semidirect
146. C₂	C₂³×C₅⋊S₄	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×A₄	2
C₂³×C₅⋊S₄	2
C₂⁴×C₂²⋊D₅	3
C₂⁴×D₁₅	4
C₂⁴×S₄	5