Subgroups — C₂²×C₅×#875 #14479659

Order 1920.

3760 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅×(C₂×D₃)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#875	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₅×#124	—	3	1	no	maximal, Hall
7	C₂⁴×C₅×D₃	C₂²	1	2	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₅×D₁₂	—	2	8	no	—
12	C₂²×C₅×D₁₂	C₂²	1	12	yes	—
13	C₂²×C₅×Dic₆	—	2	8	no	—
14	C₂²×C₅×Dic₆	C₂²	1	12	yes	complemented
15	C₂×C₃×C₄²×C₅	C₂²	1	12	yes	complemented
16	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	48	yes	complemented
17	C₅×#875	C₂²	1	64	yes	direct product factor
18	C₂²×#875	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₅×Q₈	—	3	1	no	—
20	C₂³×C₅×D₄	—	3	1	no	—
21	C₂×C₅×#124	—	3	24	no	—
22	C₂²×C₅×C₂²⋊C₄	—	3	4	no	—
23	C₂²×C₄²×C₅	D₃	1	1	yes	characteristic, complemented
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂³×C₅×D₃	—	2	8	no	—
26	C₂³×C₅×D₃	—	2	16	no	—
27	C₂³×C₅×D₃	C₂³	1	12	yes	—
28	C₂²×C₃×C₄×C₅	—	2	8	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	12	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
32	C₂²×C₅×Dic₃	—	2	8	no	—
33	C₂²×C₅×Dic₃	C₂³	1	12	yes	—
34	C₂×C₅×D₁₂	—	2	48	no	—
35	C₂×C₅×D₁₂	C₂³	1	16	yes	—
36	C₂×C₅×Dic₆	—	2	48	no	—
37	C₂×C₅×Dic₆	C₂³	1	16	yes	complemented
38	C₃×C₄²×C₅	C₂³	1	16	yes	complemented
39	C₅×(C₂×D₃)⋊C₄	C₂³	1	64	yes	complemented
40	C₂³×D₁₂	C₂×C₅	1	1	yes	characteristic
41	C₂³×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₃×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂²×(C₂×D₃)⋊C₄	C₂×C₅	1	4	yes	complemented
44	C₂×#875	C₂×C₅	1	24	yes	direct product factor
45	C₂⁵×C₅	—	3	2	no	—
46	C₂²×C₅×Q₈	—	3	12	no	—
47	C₂²×C₅×Q₈	—	6	8	no	—
48	C₂²×C₅×D₄	—	3	12	no	—
49	C₂²×C₅×D₄	—	6	8	no	—
50	C₂³×C₄×C₅	—	3	2	no	—
51	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
52	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
53	C₅×#124	—	3	64	no	—
54	C₂×C₅×C₂²⋊C₄	—	3	48	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	12	yes	complemented
56	C₂²×#124	—	3	1	no	Sylow
57	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
58	C₂³×C₃×C₅	#41	1	8	yes	—
59	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
60	C₂²×C₅×D₃	—	2	24	no	—
61	C₂²×C₅×D₃	—	2	48	no	—
62	C₂²×C₅×D₃	—	4	32	no	—
63	C₂²×C₅×D₃	C₂⁴	1	8	yes	—
64	C₂×C₃×C₄×C₅	—	2	24	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	—
66	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
67	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
68	C₂×C₅×Dic₃	—	2	24	no	—
69	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
70	C₅×D₁₂	—	2	32	no	—
71	C₅×Dic₆	—	2	32	no	—
72	C₂⁴×D₃	C₂²×C₅	1	2	yes	—
73	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
74	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
75	C₂³×Dic₃	C₂²×C₅	1	2	yes	—
76	C₂²×D₁₂	—	2	8	no	—
77	C₂²×D₁₂	C₂²×C₅	1	12	yes	—
78	C₂²×Dic₆	—	2	8	no	—
79	C₂²×Dic₆	C₂²×C₅	1	12	yes	complemented
80	C₂×C₃×C₄²	C₂²×C₅	1	12	yes	complemented
81	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	48	yes	complemented
82	#875	C₂²×C₅	1	64	yes	direct product factor
83	C₂⁴×C₅	—	3	12	no	—
84	C₂⁴×C₅	—	6	8	no	—
85	C₂⁴×C₅	—	6	16	no	—
86	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₅×Q₈	—	3	16	no	—
88	C₂×C₅×Q₈	—	6	48	no	—
89	C₂×C₅×D₄	—	3	16	no	—
90	C₂×C₅×D₄	—	6	48	no	—
91	C₂²×C₄×C₅	—	2	8	no	—
92	C₂²×C₄×C₅	—	3	12	no	—
93	C₂²×C₄×C₅	—	6	8	no	—
94	C₂²×C₄×C₅	D₁₂	1	8	yes	—
95	C₂²×C₄×C₅	C₂²×D₃	1	12	yes	—
96	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
97	C₅×C₂²⋊C₄	—	3	64	no	—
98	C₄²×C₅	C₂²×D₃	1	16	yes	complemented
99	C₂³×Q₈	—	3	1	no	—
100	C₂³×D₄	—	3	1	no	—
101	C₂×#124	—	3	24	no	—
102	C₂²×C₂²⋊C₄	—	3	4	no	—
103	C₂²×C₄²	C₅×D₃	1	1	yes	characteristic, complemented
104	C₂²×C₃×C₅	#124	1	16	yes	complemented
105	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
106	C₂²×C₃×C₅	C₂×#41	1	12	yes	—
107	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	8	no	—
109	C₂×C₅×D₃	—	2	16	no	—
110	C₂×C₅×D₃	—	4	48	no	—
111	C₃×C₄×C₅	—	2	8	no	—
112	C₃×C₄×C₅	C₂²×D₄	1	8	yes	—
113	C₅×Dic₃	—	2	8	no	—
114	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
115	C₂³×D₃	—	2	8	no	—
116	C₂³×D₃	—	2	16	no	—
117	C₂³×D₃	C₂³×C₅	1	12	yes	—
118	C₂²×C₃×C₄	—	2	8	no	—
119	C₂²×C₃×C₄	C₅×D₄	1	8	yes	—
120	C₂²×C₃×C₄	C₂³×C₅	1	12	yes	—
121	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
122	C₂²×Dic₃	—	2	8	no	—
123	C₂²×Dic₃	C₂³×C₅	1	12	yes	—
124	C₂×D₁₂	—	2	48	no	—
125	C₂×D₁₂	C₂³×C₅	1	16	yes	—
126	C₂×Dic₆	—	2	48	no	—
127	C₂×Dic₆	C₂³×C₅	1	16	yes	complemented
128	C₃×C₄²	C₂³×C₅	1	16	yes	complemented
129	(C₂×D₃)⋊C₄	C₂³×C₅	1	64	yes	complemented
130	C₂³×C₅	—	3	8	no	—
131	C₂³×C₅	—	6	24	no	—
132	C₂³×C₅	—	6	48	no	—
133	C₂³×C₅	—	12	32	no	—
134	C₂³×C₅	C₂×D₁₂	1	4	yes	—
135	C₂³×C₅	#235	1	8	yes	—
136	C₂³×C₅	C₂³×D₃	1	3	yes	—
137	C₅×Q₈	—	6	32	no	—
138	C₅×D₄	—	6	32	no	—
139	C₂×C₄×C₅	—	2	24	no	—
140	C₂×C₄×C₅	—	3	8	no	—
141	C₂×C₄×C₅	—	6	24	no	—
142	C₂×C₄×C₅	C₂×D₁₂	1	24	yes	—
143	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
144	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
145	C₂⁵	—	3	2	no	—
146	C₂²×Q₈	—	3	12	no	—
147	C₂²×Q₈	—	6	8	no	—
148	C₂²×D₄	—	3	12	no	—
149	C₂²×D₄	—	6	8	no	—
150	C₂³×C₄	—	3	2	no	—
151	C₂³×C₄	C₂×C₅×D₃	1	2	yes	—
152	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
153	#124	—	3	64	no	—
154	C₂×C₂²⋊C₄	—	3	48	no	—
155	C₂×C₄²	C₂×C₅×D₃	1	12	yes	complemented
156	C₂×C₃×C₅	C₂×#124	1	12	yes	complemented
157	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
158	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
159	C₅×D₃	—	4	8	no	—
160	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
161	C₂³×C₃	C₅×#41	1	8	yes	—
162	C₂³×C₃	C₂⁴×C₅	1	3	yes	—
163	C₂²×D₃	—	2	24	no	—
164	C₂²×D₃	—	2	48	no	—
165	C₂²×D₃	—	4	32	no	—
166	C₂²×D₃	C₂⁴×C₅	1	8	yes	—
167	C₂×C₃×C₄	—	2	24	no	—
168	C₂×C₃×C₄	C₂×C₅×D₄	1	24	yes	—
169	C₂×C₃×C₄	C₂⁴×C₅	1	8	yes	—
170	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
171	C₂×Dic₃	—	2	24	no	—
172	C₂×Dic₃	C₂⁴×C₅	1	8	yes	—
173	D₁₂	—	2	32	no	—
174	Dic₆	—	2	32	no	—
175	C₂²×C₅	—	6	8	no	—
176	C₂²×C₅	—	6	16	no	—
177	C₂²×C₅	—	12	48	no	—
178	C₂²×C₅	#875	1	16	yes	direct product factor
179	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
180	C₂²×C₅	C₂×#235	1	12	yes	—
181	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
182	C₄×C₅	—	2	8	no	—
183	C₄×C₅	—	6	8	no	—
184	C₄×C₅	C₂²×D₁₂	1	8	yes	—
185	C₂⁴	—	3	12	no	—
186	C₂⁴	—	6	8	no	—
187	C₂⁴	—	6	16	no	—
188	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
189	C₂×Q₈	—	3	16	no	—
190	C₂×Q₈	—	6	48	no	—
191	C₂×D₄	—	3	16	no	—
192	C₂×D₄	—	6	48	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	3	12	no	—
195	C₂²×C₄	—	6	8	no	—
196	C₂²×C₄	C₅×D₁₂	1	8	yes	—
197	C₂²×C₄	C₂²×C₅×D₃	1	12	yes	—
198	C₂²×C₄	C₂²×C₅×D₃	1	6	yes	—
199	C₂²⋊C₄	—	3	64	no	—
200	C₄²	C₂²×C₅×D₃	1	16	yes	complemented
201	C₃×C₅	C₂²×#124	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	C₅×#124	1	16	yes	complemented
203	C₂²×C₃	C₂²×C₅×D₄	1	6	yes	—
204	C₂²×C₃	C₂×C₅×#41	1	12	yes	—
205	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
206	C₂×D₃	—	2	8	no	—
207	C₂×D₃	—	2	16	no	—
208	C₂×D₃	—	4	48	no	—
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	C₂²×C₅×D₄	1	8	yes	—
211	Dic₃	—	2	8	no	—
212	C₂×C₅	—	12	8	no	—
213	C₂×C₅	C₂×#875	1	12	yes	direct product factor
214	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
215	C₂×C₅	C₂²×#235	1	2	yes	—
216	C₂³	—	3	8	no	—
217	C₂³	—	6	24	no	—
218	C₂³	—	6	48	no	—
219	C₂³	—	12	32	no	—
220	C₂³	C₂×C₅×D₁₂	1	4	yes	—
221	C₂³	C₅×#235	1	8	yes	—
222	C₂³	C₂³×C₅×D₃	1	3	yes	—
223	Q₈	—	6	32	no	—
224	D₄	—	6	32	no	—
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	3	8	no	—
227	C₂×C₄	—	6	24	no	—
228	C₂×C₄	C₂×C₅×D₁₂	1	24	yes	—
229	C₂×C₄	C₂³×C₅×D₃	1	8	yes	—
230	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
231	C₂×C₃	C₂×C₅×#124	1	12	yes	complemented
232	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
233	C₂×C₃	C₂²×C₅×#41	1	2	yes	—
234	D₃	—	4	8	no	—
235	C₅	C₂²×#875	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	6	8	no	—
237	C₂²	—	6	16	no	—
238	C₂²	—	12	48	no	—
239	C₂²	C₅×#875	1	16	yes	direct product factor
240	C₂²	C₂²×C₅×D₁₂	1	6	yes	—
241	C₂²	C₂×C₅×#235	1	12	yes	—
242	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
243	C₄	—	2	8	no	—
244	C₄	—	6	8	no	—
245	C₄	C₂²×C₅×D₁₂	1	8	yes	—
246	C₃	C₂²×C₅×#124	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	12	8	no	—
248	C₂	C₂×C₅×#875	1	12	yes	direct product factor
249	C₂	C₂³×C₅×D₁₂	1	1	yes	characteristic
250	C₂	C₂²×C₅×#235	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂³×C₅×D₁₂	C₂
hypercenter	C₂²×C₄²×C₅	D₃
hypocenter	C₃	C₂²×C₅×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×Dic₆	C₂	semidirect
3. C₂²×C₃×C₄²×C₅	C₂	semidirect
4. C₂²×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
5. C₂×C₅×#875	C₂	direct (N × H)
14. C₂²×C₅×Dic₆	C₂²	semidirect
15. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₂×C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
17. C₅×#875	C₂²	direct (N × H)
18. C₂²×#875	C₅	direct (N × H)
23. C₂²×C₄²×C₅	D₃	semidirect
37. C₂×C₅×Dic₆	C₂³	semidirect
38. C₃×C₄²×C₅	C₂³	semidirect
39. C₅×(C₂×D₃)⋊C₄	C₂³	semidirect
41. C₂³×Dic₆	C₂×C₅	semidirect
42. C₂²×C₃×C₄²	C₂×C₅	semidirect
43. C₂²×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
44. C₂×#875	C₂×C₅	direct (N × H)
55. C₂×C₄²×C₅	C₂×D₃	semidirect
79. C₂²×Dic₆	C₂²×C₅	semidirect
80. C₂×C₃×C₄²	C₂²×C₅	semidirect
81. C₂×(C₂×D₃)⋊C₄	C₂²×C₅	semidirect
82. #875	C₂²×C₅	direct (N × H)
98. C₄²×C₅	C₂²×D₃	semidirect
103. C₂²×C₄²	C₅×D₃	semidirect
104. C₂²×C₃×C₅	#124	semidirect
127. C₂×Dic₆	C₂³×C₅	semidirect
128. C₃×C₄²	C₂³×C₅	semidirect
129. (C₂×D₃)⋊C₄	C₂³×C₅	semidirect
155. C₂×C₄²	C₂×C₅×D₃	semidirect
156. C₂×C₃×C₅	C₂×#124	semidirect
178. C₂²×C₅	#875	direct (N × H)
200. C₄²	C₂²×C₅×D₃	semidirect
201. C₃×C₅	C₂²×#124	semidirect
202. C₂²×C₃	C₅×#124	semidirect
213. C₂×C₅	C₂×#875	direct (N × H)
231. C₂×C₃	C₂×C₅×#124	semidirect
235. C₅	C₂²×#875	direct (N × H)
239. C₂²	C₅×#875	direct (N × H)
246. C₃	C₂²×C₅×#124	semidirect
248. C₂	C₂×C₅×#875	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×C₅×D₁₂	2
C₂³×C₅×Dic₆	2
C₂²×C₃×C₄²×C₅	2
C₂²×C₅×(C₂×D₃)⋊C₄	2
C₂×C₅×#875	2
C₂²×C₅×#124	3
C₂²×#875	5

Subgroups — C22×C5×#875 #14479659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#875 #14479659