Subgroups — C₃×C₅×#2176 #14275210

Order 1920.

296 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₁₆	C₂	1	2	yes	maximal
3	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#358	C₂	1	2	yes	maximal
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic
6	C₅×#2176	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₃×C₅×Q₃₂	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₅×#103	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#2176	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic
17	C₄×C₅×Q₁₆	C₂×C₃	1	2	yes	—
18	C₅×#367	C₂×C₃	1	1	yes	characteristic
19	C₅×#358	C₂×C₃	1	2	yes	—
20	C₅×#346	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₃×C₅×Q₁₆	—	2	2	no	—
23	C₃×C₅×Q₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	—	2	2	no	—
27	C₃×C₅×C₄⋊C₄	—	4	2	no	—
28	C₃×C₄²×C₅	—	4	2	no	—
29	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	1	no	—
31	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic
33	C₃×C₄×Q₁₆	C₂×C₅	1	2	yes	—
34	C₃×#367	C₂×C₅	1	1	yes	characteristic
35	C₃×#358	C₂×C₅	1	2	yes	—
36	C₃×#346	C₂×C₅	1	1	yes	characteristic
37	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
38	C₄×C₅×Q₈	—	2	2	no	—
39	C₅×Q₃₂	C₃×C₄	1	4	yes	complemented
40	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₅×#107	C₂²×C₃	1	2	yes	—
43	C₅×#103	—	2	2	no	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	#2176	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₃×C₅×Q₈	—	2	4	no	—
47	C₃×C₅×Q₈	—	4	2	no	—
48	C₂×C₃×C₄×C₅	—	4	2	no	—
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₃×C₅×C₈	—	2	1	no	—
53	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₅×C₈	#41	1	1	yes	characteristic
55	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
56	C₃×C₄×Q₈	—	2	2	no	—
57	C₃×Q₃₂	C₄×C₅	1	4	yes	complemented
58	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
59	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
60	C₃×#107	C₂²×C₅	1	2	yes	—
61	C₃×#103	—	2	2	no	—
62	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₅×Q₁₆	—	2	2	no	—
65	C₅×Q₁₆	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₄²×C₅	—	4	2	no	—
71	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
72	C₅×C₁₆	—	2	1	no	—
73	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	—
74	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic
75	C₄×Q₁₆	C₂×C₃×C₅	1	2	yes	—
76	#367	C₂×C₃×C₅	1	1	yes	characteristic
77	#358	C₂×C₃×C₅	1	2	yes	—
78	#346	C₂×C₃×C₅	1	1	yes	characteristic
79	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
80	C₃×C₄×C₅	—	2	2	no	—
81	C₃×C₄×C₅	—	4	4	no	—
82	C₃×C₄×C₅	—	8	2	no	—
83	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
84	C₃×C₄×C₅	#135	1	1	yes	characteristic
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₃×Q₁₆	—	2	2	no	—
87	C₃×Q₁₆	C₂×C₄×C₅	1	4	yes	—
88	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	—	4	2	no	—
92	C₃×C₄²	—	4	2	no	—
93	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
94	C₃×C₁₆	—	2	1	no	—
95	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
96	C₅×Q₈	—	2	4	no	—
97	C₅×Q₈	—	4	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
101	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
102	C₅×C₈	—	2	1	no	—
103	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
104	C₅×C₈	C₃×#41	1	1	yes	characteristic
105	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
106	C₄×Q₈	—	2	2	no	—
107	Q₃₂	C₃×C₄×C₅	1	4	yes	complemented
108	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
110	#107	C₂²×C₃×C₅	1	2	yes	—
111	#103	—	2	2	no	—
112	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
113	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
114	C₂×C₃×C₅	#510	1	2	yes	—
115	C₃×Q₈	—	2	4	no	—
116	C₃×Q₈	—	4	2	no	—
117	C₂×C₃×C₄	—	4	2	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
120	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
121	C₃×C₈	—	2	1	no	—
122	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
123	C₃×C₈	C₅×#41	1	1	yes	characteristic
124	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	4	4	no	—
127	C₄×C₅	—	8	2	no	—
128	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
129	C₄×C₅	C₃×#135	1	1	yes	characteristic
130	C₂×Q₈	—	2	2	no	—
131	Q₁₆	—	2	2	no	—
132	Q₁₆	C₂×C₃×C₄×C₅	1	4	yes	—
133	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	—	4	2	no	—
138	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	—
141	C₃×C₅	#2176	1	1	yes	characteristic, direct product factor, Hall
142	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	4	4	no	—
145	C₃×C₄	—	8	2	no	—
146	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
147	C₃×C₄	C₅×#135	1	1	yes	characteristic
148	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
149	C₂×C₅	C₃×#510	1	2	yes	—
150	Q₈	—	2	4	no	—
151	Q₈	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
155	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
158	C₈	C₃×C₅×#41	1	1	yes	characteristic
159	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
160	C₂×C₃	C₅×#510	1	2	yes	—
161	C₅	C₃×#2176	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	2	no	—
166	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
167	C₄	C₃×C₅×#135	1	1	yes	characteristic
168	C₃	C₅×#2176	1	1	yes	characteristic, direct product factor, Sylow
169	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
170	C₂	C₃×C₅×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2176	C₃	direct (N × H)
9. C₃×C₅×Q₃₂	C₄	semidirect
15. C₃×#2176	C₅	direct (N × H)
39. C₅×Q₃₂	C₃×C₄	semidirect
45. #2176	C₃×C₅	direct (N × H)
57. C₃×Q₃₂	C₄×C₅	semidirect
107. Q₃₂	C₃×C₄×C₅	semidirect
141. C₃×C₅	#2176	direct (N × H)
161. C₅	C₃×#2176	direct (N × H)
168. C₃	C₅×#2176	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×Q₃₂	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#367	2
C₃×C₅×#358	2
C₃×C₅×#346	2
C₅×#2176	3
C₃×#2176	5

Subgroups — C3×C5×#2176 #14275210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2176 #14275210