Subgroups — C₂³×C₅×#215 #14509658

Order 1920.

3440 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#215	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×SD₁₆	—	3	1	no	maximal, Hall
6	C₂⁴×C₅×D₃	—	2	1	no	—
7	C₂²×C₃×C₅×Q₈	—	2	7	no	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	14	yes	complemented
9	C₂³×C₃×C₄×C₅	—	2	1	no	—
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₅×D₁₂	—	2	7	no	—
12	C₂²×C₅×D₁₂	C₂²	1	14	yes	—
13	C₂×C₅×#215	C₂²	1	112	yes	direct product factor
14	C₂²×C₅×C₃⋊C₈	C₂²	1	14	yes	complemented
15	C₂³×#215	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
17	C₂³×C₅×D₄	—	3	1	no	—
18	C₂²×C₅×SD₁₆	—	3	28	no	—
19	C₂³×C₅×C₈	—	3	1	no	—
20	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
21	C₂³×C₅×D₃	—	2	14	no	—
22	C₂³×C₅×D₃	—	4	8	no	—
23	C₂×C₃×C₅×Q₈	—	2	42	no	—
24	C₂×C₃×C₅×Q₈	C₂³	1	28	yes	complemented
25	C₂²×C₃×C₄×C₅	—	2	14	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	7	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	7	yes	—
28	C₂×C₅×D₁₂	—	2	42	no	—
29	C₂×C₅×D₁₂	C₂³	1	28	yes	—
30	C₅×#215	C₂³	1	64	yes	direct product factor
31	C₂×C₅×C₃⋊C₈	C₂³	1	28	yes	complemented
32	C₂³×C₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
33	C₂³×D₁₂	C₂×C₅	1	1	yes	characteristic
34	C₂²×#215	C₂×C₅	1	28	yes	direct product factor
35	C₂³×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂⁵×C₅	—	6	1	no	—
37	C₂²×C₅×Q₈	—	2	7	no	—
38	C₂²×C₅×Q₈	C₂×D₃	1	14	yes	complemented
39	C₂²×C₅×D₄	—	3	14	no	—
40	C₂²×C₅×D₄	—	6	7	no	—
41	C₂³×C₄×C₅	—	2	1	no	—
42	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₅×SD₁₆	—	3	112	no	—
44	C₂²×C₅×C₈	—	3	14	no	—
45	C₂³×SD₁₆	—	3	1	no	Sylow
46	C₂³×C₃×C₅	SD₁₆	1	8	yes	complemented
47	C₂³×C₃×C₅	C₂×D₄	1	7	yes	—
48	C₂²×C₅×D₃	—	2	28	no	—
49	C₂²×C₅×D₃	—	4	56	no	—
50	C₃×C₅×Q₈	—	2	28	no	—
51	C₃×C₅×Q₈	C₂⁴	1	8	yes	complemented
52	C₂×C₃×C₄×C₅	—	2	28	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	21	yes	—
54	C₂×C₃×C₄×C₅	C₂⁴	1	7	yes	—
55	C₅×D₁₂	—	2	28	no	—
56	C₅×D₁₂	C₂⁴	1	8	yes	—
57	C₅×C₃⋊C₈	C₂⁴	1	8	yes	complemented
58	C₂⁴×D₃	—	2	1	no	—
59	C₂²×C₃×Q₈	—	2	7	no	—
60	C₂²×C₃×Q₈	C₂²×C₅	1	14	yes	complemented
61	C₂³×C₃×C₄	—	2	1	no	—
62	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
63	C₂²×D₁₂	—	2	7	no	—
64	C₂²×D₁₂	C₂²×C₅	1	14	yes	—
65	C₂×#215	C₂²×C₅	1	112	yes	direct product factor
66	C₂²×C₃⋊C₈	C₂²×C₅	1	14	yes	complemented
67	C₂⁴×C₅	—	6	14	no	—
68	C₂⁴×C₅	—	12	8	no	—
69	C₂⁴×C₅	#67	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	2	42	no	—
71	C₂×C₅×Q₈	C₂²×D₃	1	28	yes	complemented
72	C₂×C₅×D₄	—	3	28	no	—
73	C₂×C₅×D₄	—	6	42	no	—
74	C₂²×C₄×C₅	—	2	14	no	—
75	C₂²×C₄×C₅	#67	1	7	yes	—
76	C₂²×C₄×C₅	C₂²×D₃	1	7	yes	—
77	C₅×SD₁₆	—	3	64	no	—
78	C₂×C₅×C₈	—	3	28	no	—
79	C₂³×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
80	C₂³×D₄	—	3	1	no	—
81	C₂²×SD₁₆	—	3	28	no	—
82	C₂³×C₈	—	3	1	no	—
83	C₂²×C₃×C₅	C₂×SD₁₆	1	28	yes	complemented
84	C₂²×C₃×C₅	C₂²×D₄	1	7	yes	—
85	C₂×C₅×D₃	—	2	8	no	—
86	C₂×C₅×D₃	—	4	56	no	—
87	C₃×C₄×C₅	—	2	8	no	—
88	C₃×C₄×C₅	C₂²×D₄	1	7	yes	—
89	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
90	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
91	C₂³×D₃	—	2	14	no	—
92	C₂³×D₃	—	4	8	no	—
93	C₂×C₃×Q₈	—	2	42	no	—
94	C₂×C₃×Q₈	C₂³×C₅	1	28	yes	complemented
95	C₂²×C₃×C₄	—	2	14	no	—
96	C₂²×C₃×C₄	C₅×D₄	1	7	yes	—
97	C₂²×C₃×C₄	C₂³×C₅	1	7	yes	—
98	C₂×D₁₂	—	2	42	no	—
99	C₂×D₁₂	C₂³×C₅	1	28	yes	—
100	#215	C₂³×C₅	1	64	yes	direct product factor
101	C₂×C₃⋊C₈	C₂³×C₅	1	28	yes	complemented
102	C₂³×C₅	—	6	28	no	—
103	C₂³×C₅	—	12	56	no	—
104	C₂³×C₅	#215	1	8	yes	direct product factor
105	C₂³×C₅	C₂×#67	1	7	yes	—
106	C₅×Q₈	—	2	28	no	—
107	C₅×Q₈	C₂³×D₃	1	8	yes	complemented
108	C₅×D₄	—	3	8	no	—
109	C₅×D₄	—	6	28	no	—
110	C₂×C₄×C₅	—	2	28	no	—
111	C₂×C₄×C₅	C₂×#67	1	21	yes	—
112	C₂×C₄×C₅	C₂³×D₃	1	7	yes	—
113	C₅×C₈	—	3	8	no	—
114	C₂⁵	—	6	1	no	—
115	C₂²×Q₈	—	2	7	no	—
116	C₂²×Q₈	C₂×C₅×D₃	1	14	yes	complemented
117	C₂²×D₄	—	3	14	no	—
118	C₂²×D₄	—	6	7	no	—
119	C₂³×C₄	—	2	1	no	—
120	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
121	C₂×SD₁₆	—	3	112	no	—
122	C₂²×C₈	—	3	14	no	—
123	C₂×C₃×C₅	C₂²×SD₁₆	1	14	yes	complemented
124	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
125	C₅×D₃	—	4	8	no	—
126	C₂³×C₃	C₅×SD₁₆	1	8	yes	complemented
127	C₂³×C₃	C₂×C₅×D₄	1	7	yes	—
128	C₂²×D₃	—	2	28	no	—
129	C₂²×D₃	—	4	56	no	—
130	C₃×Q₈	—	2	28	no	—
131	C₃×Q₈	C₂⁴×C₅	1	8	yes	complemented
132	C₂×C₃×C₄	—	2	28	no	—
133	C₂×C₃×C₄	C₂×C₅×D₄	1	21	yes	—
134	C₂×C₃×C₄	C₂⁴×C₅	1	7	yes	—
135	D₁₂	—	2	28	no	—
136	D₁₂	C₂⁴×C₅	1	8	yes	—
137	C₃⋊C₈	C₂⁴×C₅	1	8	yes	complemented
138	C₂²×C₅	—	6	8	no	—
139	C₂²×C₅	—	12	56	no	—
140	C₂²×C₅	C₂×#215	1	28	yes	direct product factor
141	C₂²×C₅	C₂²×#67	1	7	yes	—
142	C₄×C₅	—	2	8	no	—
143	C₄×C₅	C₂²×#67	1	7	yes	—
144	C₄×C₅	C₂⁴×D₃	1	1	yes	characteristic
145	C₂⁴	—	6	14	no	—
146	C₂⁴	—	12	8	no	—
147	C₂⁴	C₅×#67	1	1	yes	characteristic
148	C₂×Q₈	—	2	42	no	—
149	C₂×Q₈	C₂²×C₅×D₃	1	28	yes	complemented
150	C₂×D₄	—	3	28	no	—
151	C₂×D₄	—	6	42	no	—
152	C₂²×C₄	—	2	14	no	—
153	C₂²×C₄	C₅×#67	1	7	yes	—
154	C₂²×C₄	C₂²×C₅×D₃	1	7	yes	—
155	SD₁₆	—	3	64	no	—
156	C₂×C₈	—	3	28	no	—
157	C₃×C₅	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Hall
158	C₂²×C₃	C₂×C₅×SD₁₆	1	28	yes	complemented
159	C₂²×C₃	C₂²×C₅×D₄	1	7	yes	—
160	C₂×D₃	—	2	8	no	—
161	C₂×D₃	—	4	56	no	—
162	C₃×C₄	—	2	8	no	—
163	C₃×C₄	C₂²×C₅×D₄	1	7	yes	—
164	C₃×C₄	C₂⁵×C₅	1	1	yes	characteristic
165	C₂×C₅	—	12	8	no	—
166	C₂×C₅	C₂²×#215	1	14	yes	direct product factor
167	C₂×C₅	C₂³×#67	1	1	yes	characteristic
168	C₂³	—	6	28	no	—
169	C₂³	—	12	56	no	—
170	C₂³	C₅×#215	1	8	yes	direct product factor
171	C₂³	C₂×C₅×#67	1	7	yes	—
172	Q₈	—	2	28	no	—
173	Q₈	C₂³×C₅×D₃	1	8	yes	complemented
174	D₄	—	3	8	no	—
175	D₄	—	6	28	no	—
176	C₂×C₄	—	2	28	no	—
177	C₂×C₄	C₂×C₅×#67	1	21	yes	—
178	C₂×C₄	C₂³×C₅×D₃	1	7	yes	—
179	C₈	—	3	8	no	—
180	C₂×C₃	C₂²×C₅×SD₁₆	1	14	yes	complemented
181	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
182	D₃	—	4	8	no	—
183	C₅	C₂³×#215	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	6	8	no	—
185	C₂²	—	12	56	no	—
186	C₂²	C₂×C₅×#215	1	28	yes	direct product factor
187	C₂²	C₂²×C₅×#67	1	7	yes	—
188	C₄	—	2	8	no	—
189	C₄	C₂²×C₅×#67	1	7	yes	—
190	C₄	C₂⁴×C₅×D₃	1	1	yes	characteristic
191	C₃	C₂³×C₅×SD₁₆	1	1	yes	characteristic, complemented, Sylow
192	C₂	—	12	8	no	—
193	C₂	C₂²×C₅×#215	1	14	yes	direct product factor
194	C₂	C₂³×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₅	#67
commutator, cosocle	C₃×C₄	C₂⁵×C₅
Frattini	C₄	C₂⁴×C₅×D₃
Fitting	C₂³×C₃×C₅×Q₈	C₂
socle	C₂⁴×C₃×C₅	D₄
prime radical	C₂³×C₅×D₁₂	C₂
hypercenter	C₂³×C₅×Q₈	D₃
hypocenter	C₃	C₂³×C₅×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×Q₈	C₂	semidirect
3. C₂²×C₅×#215	C₂	direct (N × H)
4. C₂³×C₅×C₃⋊C₈	C₂	semidirect
8. C₂²×C₃×C₅×Q₈	C₂²	semidirect
13. C₂×C₅×#215	C₂²	direct (N × H)
14. C₂²×C₅×C₃⋊C₈	C₂²	semidirect
15. C₂³×#215	C₅	direct (N × H)
16. C₂³×C₅×Q₈	D₃	semidirect
24. C₂×C₃×C₅×Q₈	C₂³	semidirect
30. C₅×#215	C₂³	direct (N × H)
31. C₂×C₅×C₃⋊C₈	C₂³	semidirect
32. C₂³×C₃×Q₈	C₂×C₅	semidirect
34. C₂²×#215	C₂×C₅	direct (N × H)
35. C₂³×C₃⋊C₈	C₂×C₅	semidirect
38. C₂²×C₅×Q₈	C₂×D₃	semidirect
46. C₂³×C₃×C₅	SD₁₆	semidirect
51. C₃×C₅×Q₈	C₂⁴	semidirect
57. C₅×C₃⋊C₈	C₂⁴	semidirect
60. C₂²×C₃×Q₈	C₂²×C₅	semidirect
65. C₂×#215	C₂²×C₅	direct (N × H)
66. C₂²×C₃⋊C₈	C₂²×C₅	semidirect
71. C₂×C₅×Q₈	C₂²×D₃	semidirect
79. C₂³×Q₈	C₅×D₃	semidirect
83. C₂²×C₃×C₅	C₂×SD₁₆	semidirect
94. C₂×C₃×Q₈	C₂³×C₅	semidirect
100. #215	C₂³×C₅	direct (N × H)
101. C₂×C₃⋊C₈	C₂³×C₅	semidirect
104. C₂³×C₅	#215	direct (N × H)
107. C₅×Q₈	C₂³×D₃	semidirect
116. C₂²×Q₈	C₂×C₅×D₃	semidirect
123. C₂×C₃×C₅	C₂²×SD₁₆	semidirect
126. C₂³×C₃	C₅×SD₁₆	semidirect
131. C₃×Q₈	C₂⁴×C₅	semidirect
137. C₃⋊C₈	C₂⁴×C₅	semidirect
140. C₂²×C₅	C₂×#215	direct (N × H)
149. C₂×Q₈	C₂²×C₅×D₃	semidirect
157. C₃×C₅	C₂³×SD₁₆	semidirect
158. C₂²×C₃	C₂×C₅×SD₁₆	semidirect
166. C₂×C₅	C₂²×#215	direct (N × H)
170. C₂³	C₅×#215	direct (N × H)
173. Q₈	C₂³×C₅×D₃	semidirect
180. C₂×C₃	C₂²×C₅×SD₁₆	semidirect
183. C₅	C₂³×#215	direct (N × H)
186. C₂²	C₂×C₅×#215	direct (N × H)
191. C₃	C₂³×C₅×SD₁₆	semidirect
193. C₂	C₂²×C₅×#215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂³×C₅×D₁₂	2
C₂²×C₅×#215	2
C₂³×C₅×C₃⋊C₈	2
C₂³×C₅×SD₁₆	3
C₂³×#215	5

Subgroups — C23×C5×#215 #14509658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×#215 #14509658