Subgroups — C₂³×C₅×Dic₃⋊C₄ #14478703

Order 1920.

4976 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅×Dic₃⋊C₄	C₂	1	28	yes	maximal, direct product factor
5	C₂³×C₅×C₄⋊C₄	—	3	1	no	maximal, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	—	2	8	no	—
8	C₂³×C₃×C₄×C₅	C₂²	1	14	yes	—
9	C₂³×C₅×Dic₃	—	2	8	no	—
10	C₂³×C₅×Dic₃	C₄	1	16	yes	complemented
11	C₂³×C₅×Dic₃	C₂²	1	14	yes	—
12	C₂³×C₅×Dic₃	C₂²	1	14	yes	—
13	C₂×C₅×Dic₃⋊C₄	C₂²	1	112	yes	direct product factor
14	C₂³×Dic₃⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂⁴×C₄×C₅	—	3	1	no	—
16	C₂⁴×C₄×C₅	—	3	1	no	—
17	C₂⁴×C₄×C₅	D₃	1	1	yes	characteristic
18	C₂²×C₅×C₄⋊C₄	—	3	28	no	—
19	C₂⁴×C₃×C₅	C₂×C₄	1	8	yes	—
20	C₂⁴×C₃×C₅	D₄	1	8	yes	—
21	C₂⁴×C₃×C₅	Q₈	1	8	yes	—
22	C₂⁴×C₃×C₅	C₂³	1	7	yes	—
23	C₂²×C₃×C₄×C₅	—	2	56	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	28	yes	—
25	C₂²×C₅×Dic₃	—	2	56	no	—
26	C₂²×C₅×Dic₃	C₂×C₄	1	112	yes	complemented
27	C₂²×C₅×Dic₃	C₂³	1	28	yes	—
28	C₂²×C₅×Dic₃	C₂³	1	28	yes	—
29	C₅×Dic₃⋊C₄	C₂³	1	64	yes	direct product factor
30	C₂⁴×C₃×C₄	C₂×C₅	1	1	yes	characteristic
31	C₂⁴×Dic₃	C₂×C₅	1	1	yes	characteristic
32	C₂⁴×Dic₃	C₂×C₅	1	1	yes	characteristic
33	C₂²×Dic₃⋊C₄	C₂×C₅	1	28	yes	direct product factor
34	C₂⁵×C₅	C₂×D₃	1	1	yes	characteristic
35	C₂³×C₄×C₅	—	2	8	no	—
36	C₂³×C₄×C₅	—	3	16	no	—
37	C₂³×C₄×C₅	—	3	14	no	—
38	C₂³×C₄×C₅	—	3	14	no	—
39	C₂³×C₄×C₅	—	6	8	no	—
40	C₂³×C₄×C₅	C₂×D₃	1	14	yes	—
41	C₂×C₅×C₄⋊C₄	—	3	112	no	—
42	C₂³×C₄⋊C₄	—	3	1	no	Sylow
43	C₂³×C₃×C₅	C₄⋊C₄	1	64	yes	complemented
44	C₂³×C₃×C₅	C₂²×C₄	1	28	yes	—
45	C₂³×C₃×C₅	C₂×D₄	1	28	yes	—
46	C₂³×C₃×C₅	C₂×Q₈	1	28	yes	—
47	C₂³×C₃×C₅	C₂⁴	1	7	yes	—
48	C₂×C₃×C₄×C₅	—	2	56	no	—
49	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
50	C₂×C₅×Dic₃	—	2	56	no	—
51	C₂×C₅×Dic₃	C₂²×C₄	1	112	yes	complemented
52	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
53	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
54	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
55	C₂³×C₃×C₄	—	2	8	no	—
56	C₂³×C₃×C₄	C₂²×C₅	1	14	yes	—
57	C₂³×Dic₃	—	2	8	no	—
58	C₂³×Dic₃	C₄×C₅	1	16	yes	complemented
59	C₂³×Dic₃	C₂²×C₅	1	14	yes	—
60	C₂³×Dic₃	C₂²×C₅	1	14	yes	—
61	C₂×Dic₃⋊C₄	C₂²×C₅	1	112	yes	direct product factor
62	C₂⁴×C₅	Dic₆	1	8	yes	—
63	C₂⁴×C₅	C₄×D₃	1	8	yes	—
64	C₂⁴×C₅	#67	1	8	yes	—
65	C₂⁴×C₅	C₂²×D₃	1	7	yes	—
66	C₂²×C₄×C₅	—	2	56	no	—
67	C₂²×C₄×C₅	—	3	112	no	—
68	C₂²×C₄×C₅	—	3	28	no	—
69	C₂²×C₄×C₅	—	3	28	no	—
70	C₂²×C₄×C₅	—	6	56	no	—
71	C₂²×C₄×C₅	C₂²×D₃	1	28	yes	—
72	C₅×C₄⋊C₄	—	3	64	no	—
73	C₂⁴×C₄	—	3	1	no	—
74	C₂⁴×C₄	—	3	1	no	—
75	C₂⁴×C₄	C₅×D₃	1	1	yes	characteristic
76	C₂²×C₄⋊C₄	—	3	28	no	—
77	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	112	yes	complemented
78	C₂²×C₃×C₅	C₂³×C₄	1	14	yes	—
79	C₂²×C₃×C₅	C₂²×D₄	1	14	yes	—
80	C₂²×C₃×C₅	C₂²×Q₈	1	14	yes	—
81	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	8	no	—
83	C₅×Dic₃	—	2	8	no	—
84	C₅×Dic₃	C₂³×C₄	1	16	yes	complemented
85	C₂⁴×C₃	C₂×C₄×C₅	1	8	yes	—
86	C₂⁴×C₃	C₅×D₄	1	8	yes	—
87	C₂⁴×C₃	C₅×Q₈	1	8	yes	—
88	C₂⁴×C₃	C₂³×C₅	1	7	yes	—
89	C₂²×C₃×C₄	—	2	56	no	—
90	C₂²×C₃×C₄	C₂³×C₅	1	28	yes	—
91	C₂²×Dic₃	—	2	56	no	—
92	C₂²×Dic₃	C₂×C₄×C₅	1	112	yes	complemented
93	C₂²×Dic₃	C₂³×C₅	1	28	yes	—
94	C₂²×Dic₃	C₂³×C₅	1	28	yes	—
95	Dic₃⋊C₄	C₂³×C₅	1	64	yes	direct product factor
96	C₂³×C₅	Dic₃⋊C₄	1	64	yes	direct product factor
97	C₂³×C₅	C₂×Dic₆	1	28	yes	—
98	C₂³×C₅	C₂×C₄×D₃	1	28	yes	—
99	C₂³×C₅	C₂×#67	1	28	yes	—
100	C₂³×C₅	C₂³×D₃	1	7	yes	—
101	C₂×C₄×C₅	—	2	56	no	—
102	C₂×C₄×C₅	—	3	112	no	—
103	C₂×C₄×C₅	—	3	8	no	—
104	C₂×C₄×C₅	—	3	8	no	—
105	C₂×C₄×C₅	—	6	56	no	—
106	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
107	C₂⁵	C₂×C₅×D₃	1	1	yes	characteristic
108	C₂³×C₄	—	2	8	no	—
109	C₂³×C₄	—	3	16	no	—
110	C₂³×C₄	—	3	14	no	—
111	C₂³×C₄	—	3	14	no	—
112	C₂³×C₄	—	6	8	no	—
113	C₂³×C₄	C₂×C₅×D₃	1	14	yes	—
114	C₂×C₄⋊C₄	—	3	112	no	—
115	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	28	yes	complemented
116	C₂×C₃×C₅	C₂⁴×C₄	1	1	yes	characteristic
117	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
118	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
119	C₂³×C₃	C₅×C₄⋊C₄	1	64	yes	complemented
120	C₂³×C₃	C₂²×C₄×C₅	1	28	yes	—
121	C₂³×C₃	C₂×C₅×D₄	1	28	yes	—
122	C₂³×C₃	C₂×C₅×Q₈	1	28	yes	—
123	C₂³×C₃	C₂⁴×C₅	1	7	yes	—
124	C₂×C₃×C₄	—	2	56	no	—
125	C₂×C₃×C₄	C₂⁴×C₅	1	8	yes	—
126	C₂×Dic₃	—	2	56	no	—
127	C₂×Dic₃	C₂²×C₄×C₅	1	112	yes	complemented
128	C₂×Dic₃	C₂⁴×C₅	1	8	yes	—
129	C₂×Dic₃	C₂⁴×C₅	1	8	yes	—
130	C₂²×C₅	C₂×Dic₃⋊C₄	1	112	yes	direct product factor
131	C₂²×C₅	C₂²×Dic₆	1	14	yes	—
132	C₂²×C₅	C₂²×C₄×D₃	1	14	yes	—
133	C₂²×C₅	C₂²×#67	1	14	yes	—
134	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
135	C₄×C₅	—	2	8	no	—
136	C₄×C₅	—	3	16	no	—
137	C₄×C₅	—	6	8	no	—
138	C₂⁴	C₅×Dic₆	1	8	yes	—
139	C₂⁴	C₄×C₅×D₃	1	8	yes	—
140	C₂⁴	C₅×#67	1	8	yes	—
141	C₂⁴	C₂²×C₅×D₃	1	7	yes	—
142	C₂²×C₄	—	2	56	no	—
143	C₂²×C₄	—	3	112	no	—
144	C₂²×C₄	—	3	28	no	—
145	C₂²×C₄	—	3	28	no	—
146	C₂²×C₄	—	6	56	no	—
147	C₂²×C₄	C₂²×C₅×D₃	1	28	yes	—
148	C₄⋊C₄	—	3	64	no	—
149	C₃×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
150	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	112	yes	complemented
151	C₂²×C₃	C₂³×C₄×C₅	1	14	yes	—
152	C₂²×C₃	C₂²×C₅×D₄	1	14	yes	—
153	C₂²×C₃	C₂²×C₅×Q₈	1	14	yes	—
154	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
155	C₃×C₄	—	2	8	no	—
156	Dic₃	—	2	8	no	—
157	Dic₃	C₂³×C₄×C₅	1	16	yes	complemented
158	C₂×C₅	C₂²×Dic₃⋊C₄	1	28	yes	direct product factor
159	C₂×C₅	C₂³×Dic₆	1	1	yes	characteristic
160	C₂×C₅	C₂³×C₄×D₃	1	1	yes	characteristic
161	C₂×C₅	C₂³×#67	1	1	yes	characteristic
162	C₂³	C₅×Dic₃⋊C₄	1	64	yes	direct product factor
163	C₂³	C₂×C₅×Dic₆	1	28	yes	—
164	C₂³	C₂×C₄×C₅×D₃	1	28	yes	—
165	C₂³	C₂×C₅×#67	1	28	yes	—
166	C₂³	C₂³×C₅×D₃	1	7	yes	—
167	C₂×C₄	—	2	56	no	—
168	C₂×C₄	—	3	112	no	—
169	C₂×C₄	—	3	8	no	—
170	C₂×C₄	—	3	8	no	—
171	C₂×C₄	—	6	56	no	—
172	C₂×C₄	C₂³×C₅×D₃	1	8	yes	—
173	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	28	yes	complemented
174	C₂×C₃	C₂⁴×C₄×C₅	1	1	yes	characteristic
175	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
176	C₂×C₃	C₂³×C₅×Q₈	1	1	yes	characteristic
177	C₅	C₂³×Dic₃⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²	C₂×C₅×Dic₃⋊C₄	1	112	yes	direct product factor
179	C₂²	C₂²×C₅×Dic₆	1	14	yes	—
180	C₂²	C₂²×C₄×C₅×D₃	1	14	yes	—
181	C₂²	C₂²×C₅×#67	1	14	yes	—
182	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
183	C₄	—	2	8	no	—
184	C₄	—	3	16	no	—
185	C₄	—	6	8	no	—
186	C₃	C₂³×C₅×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
187	C₂	C₂²×C₅×Dic₃⋊C₄	1	28	yes	direct product factor
188	C₂	C₂³×C₅×Dic₆	1	1	yes	characteristic
189	C₂	C₂³×C₄×C₅×D₃	1	1	yes	characteristic
190	C₂	C₂³×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵×C₅	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₄×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂⁴×C₃×C₄×C₅	C₂
socle, prime radical	C₂⁵×C₃×C₅	C₂²
cosocle	C₂²×C₃	C₂⁵×C₅
hypercenter	C₂⁴×C₄×C₅	D₃
hypocenter	C₃	C₂³×C₅×C₄⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵×C₅ ◁ C₂⁴×C₄×C₅

Complements

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

N	H ≅ G/N	type
4. C₂²×C₅×Dic₃⋊C₄	C₂	direct (N × H)
10. C₂³×C₅×Dic₃	C₄	semidirect
13. C₂×C₅×Dic₃⋊C₄	C₂²	direct (N × H)
14. C₂³×Dic₃⋊C₄	C₅	direct (N × H)
26. C₂²×C₅×Dic₃	C₂×C₄	semidirect
29. C₅×Dic₃⋊C₄	C₂³	direct (N × H)
33. C₂²×Dic₃⋊C₄	C₂×C₅	direct (N × H)
43. C₂³×C₃×C₅	C₄⋊C₄	semidirect
51. C₂×C₅×Dic₃	C₂²×C₄	semidirect
58. C₂³×Dic₃	C₄×C₅	semidirect
61. C₂×Dic₃⋊C₄	C₂²×C₅	direct (N × H)
77. C₂²×C₃×C₅	C₂×C₄⋊C₄	semidirect
84. C₅×Dic₃	C₂³×C₄	semidirect
92. C₂²×Dic₃	C₂×C₄×C₅	semidirect
95. Dic₃⋊C₄	C₂³×C₅	direct (N × H)
96. C₂³×C₅	Dic₃⋊C₄	direct (N × H)
115. C₂×C₃×C₅	C₂²×C₄⋊C₄	semidirect
119. C₂³×C₃	C₅×C₄⋊C₄	semidirect
127. C₂×Dic₃	C₂²×C₄×C₅	semidirect
130. C₂²×C₅	C₂×Dic₃⋊C₄	direct (N × H)
149. C₃×C₅	C₂³×C₄⋊C₄	semidirect
150. C₂²×C₃	C₂×C₅×C₄⋊C₄	semidirect
157. Dic₃	C₂³×C₄×C₅	semidirect
158. C₂×C₅	C₂²×Dic₃⋊C₄	direct (N × H)
162. C₂³	C₅×Dic₃⋊C₄	direct (N × H)
173. C₂×C₃	C₂²×C₅×C₄⋊C₄	semidirect
177. C₅	C₂³×Dic₃⋊C₄	direct (N × H)
178. C₂²	C₂×C₅×Dic₃⋊C₄	direct (N × H)
186. C₃	C₂³×C₅×C₄⋊C₄	semidirect
187. C₂	C₂²×C₅×Dic₃⋊C₄	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×C₃×C₄×C₅	2
C₂⁴×C₅×Dic₃	2
C₂²×C₅×Dic₃⋊C₄	2
C₂³×C₅×C₄⋊C₄	3
C₂³×Dic₃⋊C₄	5

Subgroups — C23×C5×Dic3⋊C4 #14478703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×Dic₃⋊C₄ #14478703