Subgroups — C₃×C₅×#3293 #14276327

Order 1920.

888 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#578	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#473	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#456	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#443	C₂	1	2	yes	maximal, complemented
6	C₅×#3293	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×2₋¹⁺⁴	C₂²	1	4	yes	—
8	C₃×C₅×#137	—	2	8	no	—
9	C₃×C₅×#137	C₂²	1	8	yes	—
10	C₃×C₅×#135	—	2	4	no	—
11	C₃×C₅×#135	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
13	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
14	C₃×C₅×#131	C₂²	1	4	yes	—
15	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₃×C₅×#108	C₂²	1	2	yes	—
17	C₃×C₅×#104	C₂²	1	4	yes	—
18	C₃×C₅×#101	C₂²	1	4	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×#3293	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#578	C₂×C₃	1	4	yes	complemented
22	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#473	C₂×C₃	1	4	yes	complemented
24	C₅×#456	C₂×C₃	1	4	yes	complemented
25	C₅×#443	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	8	no	—
27	C₃×C₅×#41	—	4	4	no	—
28	C₃×C₅×#41	C₂³	1	4	yes	—
29	C₂×C₃×C₅×Q₈	—	2	8	no	—
30	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
31	C₃×C₅×Q₁₆	—	2	8	no	—
32	C₃×C₅×Q₁₆	—	2	4	no	—
33	C₃×C₅×Q₁₆	—	2	4	no	—
34	C₃×C₅×Q₁₆	D₄	1	2	yes	—
35	C₃×C₅×SD₁₆	—	2	8	no	—
36	C₃×C₅×SD₁₆	D₄	1	4	yes	—
37	C₃×C₅×D₈	D₄	1	2	yes	—
38	C₃×C₅×M₁₆	—	2	4	no	—
39	C₃×C₅×M₁₆	C₂³	1	4	yes	—
40	C₂×C₃×C₅×C₈	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	4	2	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×#578	C₂×C₅	1	4	yes	complemented
45	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
46	C₃×#473	C₂×C₅	1	4	yes	complemented
47	C₃×#456	C₂×C₅	1	4	yes	complemented
48	C₃×#443	C₂×C₅	1	2	yes	complemented
49	C₅×2₋¹⁺⁴	C₂²×C₃	1	4	yes	—
50	C₅×#137	—	2	8	no	—
51	C₅×#137	C₂²×C₃	1	8	yes	—
52	C₅×#135	—	2	4	no	—
53	C₅×#135	C₂²×C₃	1	2	yes	—
54	C₂×C₅×Q₁₆	—	2	4	no	—
55	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
56	C₅×#131	C₂²×C₃	1	4	yes	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
58	C₅×#108	C₂²×C₃	1	2	yes	—
59	C₅×#104	C₂²×C₃	1	4	yes	—
60	C₅×#101	C₂²×C₃	1	4	yes	—
61	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
62	#3293	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₃×C₅×Q₈	—	2	8	no	—
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
66	C₃×C₅×D₄	—	2	4	no	—
67	C₃×C₅×D₄	—	4	4	no	—
68	C₃×C₅×D₄	C₂×D₄	1	4	yes	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	4	8	no	—
73	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₃×C₅×C₈	—	2	4	no	—
75	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₃×2₋¹⁺⁴	C₂²×C₅	1	4	yes	—
78	C₃×#137	—	2	8	no	—
79	C₃×#137	C₂²×C₅	1	8	yes	—
80	C₃×#135	—	2	4	no	—
81	C₃×#135	C₂²×C₅	1	2	yes	—
82	C₂×C₃×Q₁₆	—	2	4	no	—
83	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
84	C₃×#131	C₂²×C₅	1	4	yes	—
85	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
86	C₃×#108	C₂²×C₅	1	2	yes	—
87	C₃×#104	C₂²×C₅	1	4	yes	—
88	C₃×#101	C₂²×C₅	1	4	yes	—
89	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₅×#41	—	2	8	no	—
91	C₅×#41	—	4	4	no	—
92	C₅×#41	C₂³×C₃	1	4	yes	—
93	C₂×C₅×Q₈	—	2	8	no	—
94	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
95	C₅×Q₁₆	—	2	8	no	—
96	C₅×Q₁₆	—	2	4	no	—
97	C₅×Q₁₆	—	2	4	no	—
98	C₅×Q₁₆	C₃×D₄	1	2	yes	—
99	C₅×SD₁₆	—	2	8	no	—
100	C₅×SD₁₆	C₃×D₄	1	4	yes	—
101	C₅×D₈	C₃×D₄	1	2	yes	—
102	C₅×M₁₆	—	2	4	no	—
103	C₅×M₁₆	C₂³×C₃	1	4	yes	—
104	C₂×C₅×C₈	—	2	4	no	—
105	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
108	#578	C₂×C₃×C₅	1	4	yes	complemented
109	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	#473	C₂×C₃×C₅	1	4	yes	complemented
111	#456	C₂×C₃×C₅	1	4	yes	complemented
112	#443	C₂×C₃×C₅	1	2	yes	complemented
113	C₂²×C₃×C₅	—	2	4	no	—
114	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	—	2	4	no	—
117	C₃×C₄×C₅	—	4	4	no	—
118	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
119	C₃×#41	—	2	8	no	—
120	C₃×#41	—	4	4	no	—
121	C₃×#41	C₂³×C₅	1	4	yes	—
122	C₂×C₃×Q₈	—	2	8	no	—
123	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
124	C₃×Q₁₆	—	2	8	no	—
125	C₃×Q₁₆	—	2	4	no	—
126	C₃×Q₁₆	—	2	4	no	—
127	C₃×Q₁₆	C₅×D₄	1	2	yes	—
128	C₃×SD₁₆	—	2	8	no	—
129	C₃×SD₁₆	C₅×D₄	1	4	yes	—
130	C₃×D₈	C₅×D₄	1	2	yes	—
131	C₃×M₁₆	—	2	4	no	—
132	C₃×M₁₆	C₂³×C₅	1	4	yes	—
133	C₂×C₃×C₈	—	2	4	no	—
134	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
135	C₃×C₄⋊C₄	—	4	2	no	—
136	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
137	C₅×Q₈	—	2	8	no	—
138	C₅×Q₈	—	4	4	no	—
139	C₅×Q₈	C₂×C₃×D₄	1	4	yes	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	C₂×C₃×D₄	1	4	yes	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	4	8	no	—
147	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
148	C₅×C₈	—	2	4	no	—
149	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
150	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
151	2₋¹⁺⁴	C₂²×C₃×C₅	1	4	yes	—
152	#137	—	2	8	no	—
153	#137	C₂²×C₃×C₅	1	8	yes	—
154	#135	—	2	4	no	—
155	#135	C₂²×C₃×C₅	1	2	yes	—
156	C₂×Q₁₆	—	2	4	no	—
157	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
158	#131	C₂²×C₃×C₅	1	4	yes	—
159	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
160	#108	C₂²×C₃×C₅	1	2	yes	—
161	#104	C₂²×C₃×C₅	1	4	yes	—
162	#101	C₂²×C₃×C₅	1	4	yes	—
163	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₂×C₃×C₅	—	2	1	no	—
165	C₂×C₃×C₅	—	4	4	no	—
166	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
167	C₃×Q₈	—	2	8	no	—
168	C₃×Q₈	—	4	4	no	—
169	C₃×Q₈	C₂×C₅×D₄	1	4	yes	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	4	4	no	—
172	C₃×D₄	C₂×C₅×D₄	1	4	yes	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	4	8	no	—
177	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
178	C₃×C₈	—	2	4	no	—
179	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
180	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
187	#41	—	2	8	no	—
188	#41	—	4	4	no	—
189	#41	C₂³×C₃×C₅	1	4	yes	—
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
192	Q₁₆	—	2	8	no	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	4	no	—
195	Q₁₆	C₃×C₅×D₄	1	2	yes	—
196	SD₁₆	—	2	8	no	—
197	SD₁₆	C₃×C₅×D₄	1	4	yes	—
198	D₈	C₃×C₅×D₄	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	C₂³×C₃×C₅	1	4	yes	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
205	C₃×C₅	#3293	1	1	yes	characteristic, direct product factor, Hall
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	4	4	no	—
211	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	4	4	no	—
214	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
215	Q₈	—	2	8	no	—
216	Q₈	—	4	4	no	—
217	Q₈	C₂×C₃×C₅×D₄	1	4	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	4	4	no	—
220	D₄	C₂×C₃×C₅×D₄	1	4	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
226	C₈	—	2	4	no	—
227	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
228	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	4	4	no	—
231	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
232	C₅	C₃×#3293	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	—	2	4	no	—
234	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	4	no	—
238	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
239	C₃	C₅×#3293	1	1	yes	characteristic, direct product factor, Sylow
240	C₂	—	2	1	no	—
241	C₂	—	4	4	no	—
242	C₂	C₃×C₅×D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#578	C₂	semidirect
2. C₃×C₅×#494	C₂	semidirect
3. C₃×C₅×#473	C₂	semidirect
4. C₃×C₅×#456	C₂	semidirect
5. C₃×C₅×#443	C₂	semidirect
6. C₅×#3293	C₃	direct (N × H)
20. C₃×#3293	C₅	direct (N × H)
21. C₅×#578	C₂×C₃	semidirect
22. C₅×#494	C₂×C₃	semidirect
23. C₅×#473	C₂×C₃	semidirect
24. C₅×#456	C₂×C₃	semidirect
25. C₅×#443	C₂×C₃	semidirect
44. C₃×#578	C₂×C₅	semidirect
45. C₃×#494	C₂×C₅	semidirect
46. C₃×#473	C₂×C₅	semidirect
47. C₃×#456	C₂×C₅	semidirect
48. C₃×#443	C₂×C₅	semidirect
62. #3293	C₃×C₅	direct (N × H)
108. #578	C₂×C₃×C₅	semidirect
109. #494	C₂×C₃×C₅	semidirect
110. #473	C₂×C₃×C₅	semidirect
111. #456	C₂×C₃×C₅	semidirect
112. #443	C₂×C₃×C₅	semidirect
205. C₃×C₅	#3293	direct (N × H)
232. C₅	C₃×#3293	direct (N × H)
239. C₃	C₅×#3293	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#578	2
C₃×C₅×#494	2
C₃×C₅×#473	2
C₃×C₅×#456	2
C₃×C₅×#443	2
C₅×#3293	3
C₃×#3293	5

Subgroups — C3×C5×#3293 #14276327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3293 #14276327