Subgroups — C₂×C₃×#66054 #14495179

Order 1920.

2688 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4085	C₂	1	6	yes	maximal, complemented
3	C₂×C₃×#4083	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×#4081	C₂	1	3	yes	maximal, complemented
5	C₃×#66054	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃×#4022	C₂	1	2	yes	maximal, complemented
7	C₂×#66054	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	complemented
11	C₂²×C₃×C₂²⋊D₅	C₂²	1	6	yes	—
12	C₂³×C₃×Dic₅	C₂²	1	3	yes	complemented
13	C₃×C₅×#127	C₂²	1	8	yes	complemented
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃×C₂²⋊Dic₅	C₂²	1	3	yes	—
16	C₂×C₃×C₂²⋊Dic₅	C₂²	1	3	yes	complemented
17	C₃×#4085	C₂²	1	48	yes	complemented
18	C₃×#4083	C₂²	1	24	yes	complemented
19	C₃×#4081	C₂²	1	24	yes	complemented
20	C₂×C₃×#685	C₂²	1	6	yes	—
21	C₂×C₃×#683	C₂²	1	6	yes	complemented
22	C₃×#4022	C₂²	1	16	yes	complemented
23	C₂×C₃×#560	—	5	1	no	maximal, Hall
24	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#4085	C₂×C₃	1	6	yes	complemented
26	C₂×#4083	C₂×C₃	1	3	yes	complemented
27	C₂×#4081	C₂×C₃	1	3	yes	complemented
28	#66054	C₂×C₃	1	16	yes	direct product factor
29	C₂×#4022	C₂×C₃	1	2	yes	complemented
30	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
32	C₂³×C₃×D₅	—	4	6	no	—
33	C₂³×C₃×D₅	C₂³	1	2	yes	—
34	C₂³×C₃×D₅	C₂³	1	4	yes	—
35	C₂×C₃×C₅×D₄	—	4	6	no	—
36	C₂×C₃×C₅×D₄	—	4	6	no	—
37	C₂×C₃×C₅×D₄	C₂³	1	12	yes	—
38	C₂×C₃×C₅×D₄	C₂³	1	12	yes	complemented
39	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
40	C₂×C₃×C₂²⋊D₅	—	4	12	no	—
41	C₂×C₃×C₂²⋊D₅	C₂³	1	24	yes	—
42	C₂²×C₃×Dic₅	—	4	6	no	—
43	C₂²×C₃×Dic₅	C₂³	1	6	yes	—
44	C₂²×C₃×Dic₅	C₂³	1	12	yes	complemented
45	C₃×C₄²×C₅	C₂³	1	4	yes	complemented
46	C₃×C₂²⋊Dic₅	C₂³	1	12	yes	—
47	C₃×C₂²⋊Dic₅	C₂³	1	12	yes	complemented
48	C₃×#685	C₂³	1	24	yes	—
49	C₃×#683	C₂³	1	24	yes	complemented
50	C₃×#560	—	5	16	no	—
51	C₂×C₃×#127	D₅	1	1	yes	characteristic, complemented
52	C₂×C₃×#126	—	5	2	no	—
53	C₂×C₃×#123	—	5	3	no	—
54	C₂×C₃×#121	—	5	6	no	—
55	C₂×C₃×#120	—	5	3	no	—
56	C₂⁴×D₅	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
58	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	complemented
59	C₂²×C₂²⋊D₅	C₂²×C₃	1	6	yes	—
60	C₂³×Dic₅	C₂²×C₃	1	3	yes	complemented
61	C₅×#127	C₂²×C₃	1	8	yes	complemented
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
63	C₂×C₂²⋊Dic₅	C₂²×C₃	1	3	yes	—
64	C₂×C₂²⋊Dic₅	C₂²×C₃	1	3	yes	complemented
65	#4085	C₂²×C₃	1	48	yes	complemented
66	#4083	C₂²×C₃	1	24	yes	complemented
67	#4081	C₂²×C₃	1	24	yes	complemented
68	C₂×#685	C₂²×C₃	1	6	yes	—
69	C₂×#683	C₂²×C₃	1	6	yes	complemented
70	#4022	C₂²×C₃	1	16	yes	complemented
71	C₂×#560	—	5	1	no	Sylow
72	C₂³×C₃×C₅	—	2	6	no	—
73	C₂³×C₃×C₅	—	2	12	no	—
74	C₂³×C₃×C₅	—	2	6	no	—
75	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
76	C₂³×C₃×C₅	C₂⁴	1	6	yes	—
77	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂²×C₃×D₅	—	2	12	no	—
79	C₂²×C₃×D₅	—	4	12	no	—
80	C₂²×C₃×D₅	—	4	16	no	—
81	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
82	C₃×C₅×D₄	—	4	12	no	—
83	C₃×C₅×D₄	—	4	12	no	—
84	C₂×C₃×C₄×C₅	—	2	6	no	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
86	C₃×C₂²⋊D₅	—	4	24	no	—
87	C₂×C₃×Dic₅	—	2	12	no	—
88	C₂×C₃×Dic₅	—	4	12	no	—
89	C₂×C₃×Dic₅	C₂⁴	1	12	yes	—
90	C₂⁵×C₃	—	5	1	no	—
91	C₂²×C₃×D₄	—	5	6	no	—
92	C₂²×C₃×D₄	C₂×D₅	1	3	yes	—
93	C₂²×C₃×D₄	C₂×D₅	1	3	yes	complemented
94	C₂³×C₃×C₄	—	5	3	no	—
95	C₃×#127	C₂×D₅	1	8	yes	complemented
96	C₃×#126	—	5	16	no	—
97	C₃×#123	—	5	24	no	—
98	C₃×#121	—	5	48	no	—
99	C₃×#120	—	5	24	no	—
100	C₂×C₃×C₄⋊C₄	—	5	6	no	—
101	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
102	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
103	C₂×C₃×C₂²⋊C₄	—	5	6	no	—
104	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
105	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
107	C₂³×D₅	—	4	6	no	—
108	C₂³×D₅	C₂³×C₃	1	2	yes	—
109	C₂³×D₅	C₂³×C₃	1	4	yes	—
110	C₂×C₅×D₄	—	4	6	no	—
111	C₂×C₅×D₄	—	4	6	no	—
112	C₂×C₅×D₄	C₂³×C₃	1	12	yes	—
113	C₂×C₅×D₄	C₂³×C₃	1	12	yes	complemented
114	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
115	C₂×C₂²⋊D₅	—	4	12	no	—
116	C₂×C₂²⋊D₅	C₂³×C₃	1	24	yes	—
117	C₂²×Dic₅	—	4	6	no	—
118	C₂²×Dic₅	C₂³×C₃	1	6	yes	—
119	C₂²×Dic₅	C₂³×C₃	1	12	yes	complemented
120	C₄²×C₅	C₂³×C₃	1	4	yes	complemented
121	C₂²⋊Dic₅	C₂³×C₃	1	12	yes	—
122	C₂²⋊Dic₅	C₂³×C₃	1	12	yes	complemented
123	#685	C₂³×C₃	1	24	yes	—
124	#683	C₂³×C₃	1	24	yes	complemented
125	#560	—	5	16	no	—
126	C₂×#127	C₃×D₅	1	1	yes	characteristic, complemented
127	C₂×#126	—	5	2	no	—
128	C₂×#123	—	5	3	no	—
129	C₂×#121	—	5	6	no	—
130	C₂×#120	—	5	3	no	—
131	C₂²×C₃×C₅	—	2	6	no	—
132	C₂²×C₃×C₅	—	2	12	no	—
133	C₂²×C₃×C₅	—	2	6	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	4	12	no	—
136	C₂²×C₃×C₅	H(5,𝔽₂)	1	6	yes	—
137	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
138	C₂×C₃×D₅	—	2	12	no	—
139	C₂×C₃×D₅	—	4	16	no	—
140	C₂×C₃×D₅	—	4	8	no	—
141	C₃×C₄×C₅	—	2	6	no	—
142	C₃×Dic₅	—	2	12	no	—
143	C₂⁴×C₃	—	5	2	no	—
144	C₂⁴×C₃	—	5	4	no	—
145	C₂⁴×C₃	—	20	6	no	—
146	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
147	C₂⁴×C₃	C₂²×D₅	1	3	yes	—
148	C₂×C₃×D₄	—	4	6	no	—
149	C₂×C₃×D₄	—	4	6	no	—
150	C₂×C₃×D₄	—	5	24	no	—
151	C₂×C₃×D₄	—	20	12	no	—
152	C₂×C₃×D₄	C₂²×D₅	1	12	yes	—
153	C₂×C₃×D₄	C₂²×D₅	1	12	yes	complemented
154	C₂²×C₃×C₄	—	5	6	no	—
155	C₂²×C₃×C₄	—	5	12	no	—
156	C₂²×C₃×C₄	—	20	6	no	—
157	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
158	C₃×C₄⋊C₄	—	5	24	no	—
159	C₃×C₂²⋊C₄	—	5	12	no	—
160	C₃×C₂²⋊C₄	—	5	12	no	—
161	C₃×C₂²⋊C₄	—	5	24	no	—
162	C₃×C₄²	C₂²×D₅	1	4	yes	complemented
163	C₂³×C₅	—	2	6	no	—
164	C₂³×C₅	—	2	12	no	—
165	C₂³×C₅	—	2	6	no	—
166	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
167	C₂³×C₅	C₂⁴×C₃	1	6	yes	—
168	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
169	C₂²×D₅	—	2	12	no	—
170	C₂²×D₅	—	4	12	no	—
171	C₂²×D₅	—	4	16	no	—
172	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
173	C₅×D₄	—	4	12	no	—
174	C₅×D₄	—	4	12	no	—
175	C₂×C₄×C₅	—	2	6	no	—
176	C₂×C₄×C₅	C₂⁴×C₃	1	6	yes	—
177	C₂²⋊D₅	—	4	24	no	—
178	C₂×Dic₅	—	2	12	no	—
179	C₂×Dic₅	—	4	12	no	—
180	C₂×Dic₅	C₂⁴×C₃	1	12	yes	—
181	C₂⁵	—	5	1	no	—
182	C₂²×D₄	—	5	6	no	—
183	C₂²×D₄	C₂×C₃×D₅	1	3	yes	—
184	C₂²×D₄	C₂×C₃×D₅	1	3	yes	complemented
185	C₂³×C₄	—	5	3	no	—
186	#127	C₂×C₃×D₅	1	8	yes	complemented
187	#126	—	5	16	no	—
188	#123	—	5	24	no	—
189	#121	—	5	48	no	—
190	#120	—	5	24	no	—
191	C₂×C₄⋊C₄	—	5	6	no	—
192	C₂×C₂²⋊C₄	—	5	3	no	—
193	C₂×C₂²⋊C₄	—	5	3	no	—
194	C₂×C₂²⋊C₄	—	5	6	no	—
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic, complemented
196	C₂×C₃×C₅	—	4	2	no	—
197	C₂×C₃×C₅	—	4	6	no	—
198	C₂×C₃×C₅	#560	1	4	yes	complemented
199	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
200	C₃×D₅	—	4	4	no	—
201	C₂³×C₃	—	2	6	no	—
202	C₂³×C₃	—	2	12	no	—
203	C₂³×C₃	—	2	6	no	—
204	C₂³×C₃	—	5	4	no	—
205	C₂³×C₃	—	10	12	no	—
206	C₂³×C₃	—	20	12	no	—
207	C₂³×C₃	—	20	16	no	—
208	C₂³×C₃	C₂³×D₅	1	2	yes	—
209	C₂³×C₃	C₂³×D₅	1	6	yes	—
210	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
211	C₃×D₄	—	4	12	no	—
212	C₃×D₄	—	4	12	no	—
213	C₃×D₄	—	20	24	no	—
214	C₂×C₃×C₄	—	2	6	no	—
215	C₂×C₃×C₄	—	5	12	no	—
216	C₂×C₃×C₄	—	10	12	no	—
217	C₂×C₃×C₄	—	20	12	no	—
218	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
219	C₂²×C₅	—	2	6	no	—
220	C₂²×C₅	—	2	12	no	—
221	C₂²×C₅	—	2	6	no	—
222	C₂²×C₅	—	4	4	no	—
223	C₂²×C₅	—	4	12	no	—
224	C₂²×C₅	C₃×H(5,𝔽₂)	1	6	yes	—
225	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
226	C₂×D₅	—	2	12	no	—
227	C₂×D₅	—	4	16	no	—
228	C₂×D₅	—	4	8	no	—
229	C₄×C₅	—	2	6	no	—
230	Dic₅	—	2	12	no	—
231	C₂⁴	—	5	2	no	—
232	C₂⁴	—	5	4	no	—
233	C₂⁴	—	20	6	no	—
234	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
235	C₂⁴	C₂²×C₃×D₅	1	3	yes	—
236	C₂×D₄	—	4	6	no	—
237	C₂×D₄	—	4	6	no	—
238	C₂×D₄	—	5	24	no	—
239	C₂×D₄	—	20	12	no	—
240	C₂×D₄	C₂²×C₃×D₅	1	12	yes	—
241	C₂×D₄	C₂²×C₃×D₅	1	12	yes	complemented
242	C₂²×C₄	—	5	6	no	—
243	C₂²×C₄	—	5	12	no	—
244	C₂²×C₄	—	20	6	no	—
245	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
246	C₄⋊C₄	—	5	24	no	—
247	C₂²⋊C₄	—	5	12	no	—
248	C₂²⋊C₄	—	5	12	no	—
249	C₂²⋊C₄	—	5	24	no	—
250	C₄²	C₂²×C₃×D₅	1	4	yes	complemented
251	C₃×C₅	C₂×#560	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	2	6	no	—
253	C₂²×C₃	—	2	12	no	—
254	C₂²×C₃	—	2	6	no	—
255	C₂²×C₃	—	4	12	no	—
256	C₂²×C₃	—	4	4	no	—
257	C₂²×C₃	—	10	12	no	—
258	C₂²×C₃	—	20	16	no	—
259	C₂²×C₃	—	20	8	no	—
260	C₂²×C₃	#4144	1	6	yes	—
261	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
262	C₃×C₄	—	2	6	no	—
263	C₃×C₄	—	10	12	no	—
264	C₂×C₅	—	4	2	no	—
265	C₂×C₅	—	4	6	no	—
266	C₂×C₅	C₃×#560	1	4	yes	complemented
267	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	3	yes	—
268	D₅	—	4	4	no	—
269	C₂³	—	2	6	no	—
270	C₂³	—	2	12	no	—
271	C₂³	—	2	6	no	—
272	C₂³	—	5	4	no	—
273	C₂³	—	10	12	no	—
274	C₂³	—	20	12	no	—
275	C₂³	—	20	16	no	—
276	C₂³	C₂³×C₃×D₅	1	2	yes	—
277	C₂³	C₂³×C₃×D₅	1	6	yes	—
278	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
279	D₄	—	4	12	no	—
280	D₄	—	4	12	no	—
281	D₄	—	20	24	no	—
282	C₂×C₄	—	2	6	no	—
283	C₂×C₄	—	5	12	no	—
284	C₂×C₄	—	10	12	no	—
285	C₂×C₄	—	20	12	no	—
286	C₂×C₄	C₂³×C₃×D₅	1	6	yes	—
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	—	4	6	no	—
289	C₂×C₃	—	20	4	no	—
290	C₂×C₃	#66054	1	4	yes	direct product factor
291	C₂×C₃	C₂×#4144	1	3	yes	—
292	C₅	C₂×C₃×#560	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	6	no	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	6	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	12	no	—
298	C₂²	—	10	12	no	—
299	C₂²	—	20	16	no	—
300	C₂²	—	20	8	no	—
301	C₂²	C₃×#4144	1	6	yes	—
302	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
303	C₄	—	2	6	no	—
304	C₄	—	10	12	no	—
305	C₃	C₂×#66054	1	1	yes	characteristic, direct product factor, Sylow
306	C₂	—	4	2	no	—
307	C₂	—	4	6	no	—
308	C₂	—	20	4	no	—
309	C₂	C₃×#66054	1	4	yes	direct product factor
310	C₂	C₂×C₃×#4144	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₂×C₃×#560
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂×C₃×#4085	C₂	semidirect
3. C₂×C₃×#4083	C₂	semidirect
4. C₂×C₃×#4081	C₂	semidirect
5. C₃×#66054	C₂	direct (N × H)
6. C₂×C₃×#4022	C₂	semidirect
7. C₂×#66054	C₃	direct (N × H)
10. C₂²×C₃×C₅×D₄	C₂²	semidirect
12. C₂³×C₃×Dic₅	C₂²	semidirect
13. C₃×C₅×#127	C₂²	semidirect
14. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
17. C₃×#4085	C₂²	semidirect
18. C₃×#4083	C₂²	semidirect
19. C₃×#4081	C₂²	semidirect
21. C₂×C₃×#683	C₂²	semidirect
22. C₃×#4022	C₂²	semidirect
24. C₂×C₅×#127	C₂×C₃	semidirect
25. C₂×#4085	C₂×C₃	semidirect
26. C₂×#4083	C₂×C₃	semidirect
27. C₂×#4081	C₂×C₃	semidirect
28. #66054	C₂×C₃	direct (N × H)
29. C₂×#4022	C₂×C₃	semidirect
38. C₂×C₃×C₅×D₄	C₂³	semidirect
44. C₂²×C₃×Dic₅	C₂³	semidirect
45. C₃×C₄²×C₅	C₂³	semidirect
47. C₃×C₂²⋊Dic₅	C₂³	semidirect
49. C₃×#683	C₂³	semidirect
51. C₂×C₃×#127	D₅	semidirect
58. C₂²×C₅×D₄	C₂²×C₃	semidirect
60. C₂³×Dic₅	C₂²×C₃	semidirect
61. C₅×#127	C₂²×C₃	semidirect
62. C₂×C₄²×C₅	C₂²×C₃	semidirect
64. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
65. #4085	C₂²×C₃	semidirect
66. #4083	C₂²×C₃	semidirect
67. #4081	C₂²×C₃	semidirect
69. C₂×#683	C₂²×C₃	semidirect
70. #4022	C₂²×C₃	semidirect
93. C₂²×C₃×D₄	C₂×D₅	semidirect
95. C₃×#127	C₂×D₅	semidirect
104. C₂×C₃×C₄²	C₂×D₅	semidirect
113. C₂×C₅×D₄	C₂³×C₃	semidirect
119. C₂²×Dic₅	C₂³×C₃	semidirect
120. C₄²×C₅	C₂³×C₃	semidirect
122. C₂²⋊Dic₅	C₂³×C₃	semidirect
124. #683	C₂³×C₃	semidirect
126. C₂×#127	C₃×D₅	semidirect
153. C₂×C₃×D₄	C₂²×D₅	semidirect
162. C₃×C₄²	C₂²×D₅	semidirect
184. C₂²×D₄	C₂×C₃×D₅	semidirect
186. #127	C₂×C₃×D₅	semidirect
195. C₂×C₄²	C₂×C₃×D₅	semidirect
198. C₂×C₃×C₅	#560	semidirect
241. C₂×D₄	C₂²×C₃×D₅	semidirect
250. C₄²	C₂²×C₃×D₅	semidirect
251. C₃×C₅	C₂×#560	semidirect
266. C₂×C₅	C₃×#560	semidirect
290. C₂×C₃	#66054	direct (N × H)
292. C₅	C₂×C₃×#560	semidirect
305. C₃	C₂×#66054	direct (N × H)
309. C₂	C₃×#66054	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×#127	2
C₂×C₃×#4085	2
C₂×C₃×#4083	2
C₂×C₃×#4081	2
C₃×#66054	2
C₂×C₃×#4022	2
C₂×#66054	3
C₂×C₃×#560	5

Subgroups — C2×C3×#66054 #14495179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66054 #14495179