Subgroups — C₃×#91815 #13088023

Order 1440.

358 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#6719	C₂	1	2	yes	maximal
3	C₃×C₄⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×Dic₁₅	C₂	1	2	yes	maximal, complemented
5	#91815	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃×#4072	—	3	1	no	maximal
7	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₅	C₄	1	4	yes	complemented
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₁₅	C₂²	1	4	yes	—
12	C₃×#926	—	5	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	—	2	1	no	—
14	C₂²×C₃×C₄×C₅	D₃	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
16	#6719	C₂×C₃	1	2	yes	—
17	C₄⋊Dic₁₅	C₂×C₃	1	2	yes	complemented
18	C₄×Dic₁₅	C₂×C₃	1	2	yes	complemented
19	C₃×C₂²⋊Dic₅	—	3	2	no	—
20	C₃×C₄⋊Dic₅	—	3	2	no	—
21	C₃×C₄×Dic₅	—	3	2	no	—
22	C₂²×C₃²×C₅	—	2	2	no	—
23	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
25	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₃×Dic₁₅	—	2	4	no	—
28	#4072	—	3	1	no	Hall
29	C₂²×C₃²×C₄	D₅	1	1	yes	characteristic
30	C₃×#217	—	5	2	no	—
31	C₃×C₄⋊Dic₃	—	5	2	no	—
32	C₃×C₄×Dic₃	—	5	2	no	—
33	C₂³×C₃×C₅	—	2	1	no	—
34	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
35	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	—	2	4	no	—
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	complemented
40	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	complemented
41	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×Dic₁₅	C₂²×C₃	1	4	yes	—
46	C₂×C₃×Dic₅	—	3	4	no	—
47	C₃×#117	—	15	1	no	—
48	#926	—	5	1	no	—
49	C₂×C₃²×C₅	—	2	2	no	—
50	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃²×C₅	#41	1	2	yes	—
52	C₂²×C₄×C₅	C₃×D₃	1	1	yes	characteristic
53	C₂²⋊Dic₅	—	3	2	no	—
54	C₄⋊Dic₅	—	3	2	no	—
55	C₄×Dic₅	—	3	2	no	—
56	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
57	C₂×C₃²×C₄	Dic₅	1	4	yes	complemented
58	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
60	C₂×C₃×Dic₃	—	5	4	no	—
61	C₂²×C₃×C₅	—	2	4	no	—
62	C₂²×C₃×C₅	—	2	2	no	—
63	C₂²×C₃×C₅	—	2	2	no	—
64	C₂²×C₃×C₅	—	2	2	no	—
65	C₂²×C₃×C₅	—	2	1	no	—
66	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
67	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
68	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×C₄×C₅	—	2	2	no	—
72	C₃×C₄×C₅	C₂×Dic₃	1	2	yes	—
73	C₃×C₄×C₅	C₂×Dic₃	1	2	yes	—
74	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
76	Dic₁₅	—	2	4	no	—
77	C₃×Dic₅	—	6	4	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	C₃×D₅	1	1	yes	characteristic
80	C₂²×C₃×C₄	D₁₅	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	15	2	no	—
82	C₃×C₂²⋊C₄	—	15	2	no	—
83	C₃×C₄²	—	15	2	no	—
84	#217	—	5	2	no	—
85	C₄⋊Dic₃	—	5	2	no	—
86	C₄×Dic₃	—	5	2	no	—
87	C₃²×C₅	#117	1	1	yes	characteristic, complemented, Hall
88	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
89	C₂×C₄×C₅	C₃×Dic₃	1	4	yes	complemented
90	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
92	C₂×Dic₅	—	3	4	no	—
93	C₂²×C₃²	—	2	2	no	—
94	C₂²×C₃²	C₂×Dic₅	1	2	yes	—
95	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
96	C₃²×C₄	C₂×Dic₅	1	2	yes	—
97	C₃²×C₄	C₂×Dic₅	1	2	yes	—
98	C₃×Dic₃	—	10	4	no	—
99	#117	—	15	1	no	Sylow
100	C₂×C₃×C₅	—	2	4	no	—
101	C₂×C₃×C₅	—	2	2	no	—
102	C₂×C₃×C₅	—	2	2	no	—
103	C₂×C₃×C₅	—	2	2	no	—
104	C₂×C₃×C₅	—	2	1	no	—
105	C₂×C₃×C₅	#235	1	2	yes	—
106	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
107	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×C₅	C₃×#41	1	2	yes	—
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
111	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	15	4	no	—
116	C₂×C₃×C₄	C₃×Dic₅	1	4	yes	complemented
117	C₂×C₃×C₄	Dic₁₅	1	4	yes	complemented
118	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
122	C₂×Dic₃	—	5	4	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	C₂×C₃×Dic₃	1	2	yes	—
125	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
126	C₄×C₅	C₂×C₃×Dic₃	1	2	yes	—
127	C₄×C₅	C₂×C₃×Dic₃	1	2	yes	—
128	Dic₅	—	6	4	no	—
129	C₂×C₃²	—	2	2	no	—
130	C₂×C₃²	#709	1	2	yes	—
131	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
132	C₂²×C₄	C₃×D₁₅	1	1	yes	characteristic
133	C₄⋊C₄	—	15	2	no	—
134	C₂²⋊C₄	—	15	2	no	—
135	C₄²	—	15	2	no	—
136	C₃×C₅	—	2	1	no	—
137	C₃×C₅	#926	1	1	yes	characteristic, complemented
138	C₃×C₅	C₃×#117	1	1	yes	characteristic, complemented
139	C₂²×C₃	—	2	4	no	—
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	—	2	1	no	—
144	C₂²×C₃	C₂×C₃×Dic₅	1	2	yes	—
145	C₂²×C₃	C₂×Dic₁₅	1	2	yes	—
146	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
147	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	30	4	no	—
151	C₃×C₄	C₂×C₃×Dic₅	1	2	yes	—
152	C₃×C₄	C₂×C₃×Dic₅	1	2	yes	—
153	C₃×C₄	C₂×Dic₁₅	1	2	yes	—
154	C₃×C₄	C₂×Dic₁₅	1	2	yes	—
155	Dic₃	—	10	4	no	—
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	C₃×#235	1	2	yes	—
158	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
159	C₃²	#4072	1	1	yes	characteristic, complemented, Sylow
160	C₂³	C₂×C₃×D₁₅	1	1	yes	characteristic
161	C₂×C₄	—	15	4	no	—
162	C₂×C₄	C₃×Dic₁₅	1	4	yes	complemented
163	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
164	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
165	C₂×C₃	—	2	4	no	—
166	C₂×C₃	—	2	2	no	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	—	2	2	no	—
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	C₃×#709	1	2	yes	—
171	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
172	C₂×C₃	#6817	1	2	yes	—
173	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
174	C₅	C₃×#926	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	C₂×C₃×Dic₁₅	1	2	yes	—
177	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
178	C₄	—	30	4	no	—
179	C₄	C₂×C₃×Dic₁₅	1	2	yes	—
180	C₄	C₂×C₃×Dic₁₅	1	2	yes	—
181	C₃	—	2	1	no	—
182	C₃	C₃×#4072	1	1	yes	characteristic, complemented
183	C₃	#91815	1	1	yes	characteristic, direct product factor
184	C₂	—	2	2	no	—
185	C₂	C₃×#6817	1	2	yes	—
186	C₂	C₂²×C₃×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting	C₂²×C₃²×C₄×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂³×C₃²×C₅	C₂²
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₂²×C₃×C₄	D₁₅
hypocenter	C₃×C₅	C₃×#117
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊Dic₁₅	C₂	semidirect
4. C₃×C₄×Dic₁₅	C₂	semidirect
5. #91815	C₃	direct (N × H)
8. C₂×C₃²×C₄×C₅	C₄	semidirect
17. C₄⋊Dic₁₅	C₂×C₃	semidirect
18. C₄×Dic₁₅	C₂×C₃	semidirect
39. C₂×C₃×C₄×C₅	Dic₃	semidirect
40. C₂×C₃×C₄×C₅	C₃×C₄	semidirect
57. C₂×C₃²×C₄	Dic₅	semidirect
87. C₃²×C₅	#117	semidirect
89. C₂×C₄×C₅	C₃×Dic₃	semidirect
116. C₂×C₃×C₄	C₃×Dic₅	semidirect
117. C₂×C₃×C₄	Dic₁₅	semidirect
137. C₃×C₅	#926	semidirect
138. C₃×C₅	C₃×#117	semidirect
159. C₃²	#4072	semidirect
162. C₂×C₄	C₃×Dic₁₅	semidirect
174. C₅	C₃×#926	semidirect
182. C₃	C₃×#4072	semidirect
183. C₃	#91815	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×C₃²×C₄×C₅	2
C₃×#6719	2
C₃×C₄⋊Dic₁₅	2
C₃×C₄×Dic₁₅	2
#91815	3
C₃×#4072	3
C₃×#926	5

Subgroups — C3×#91815 #13088023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#91815 #13088023