Subgroups — C₂²×#91764 #14479661

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×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6714	C₂	1	4	yes	maximal, complemented
5	C₂×#91764	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#4021	—	3	1	no	maximal, Hall
7	C₂⁴×D₁₅	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×Dic₁₅	C₂²	1	2	yes	—
11	C₂²×D₆₀	—	2	8	no	—
12	C₂²×D₆₀	C₂²	1	12	yes	—
13	C₂²×Dic₃₀	—	2	8	no	—
14	C₂²×Dic₃₀	C₂²	1	12	yes	complemented
15	C₂×C₃×C₄²×C₅	C₂²	1	12	yes	complemented
16	C₂×#6714	C₂²	1	48	yes	complemented
17	#91764	C₂²	1	64	yes	direct product factor
18	C₂²×#875	—	5	1	no	maximal, Hall
19	C₂³×D₂₀	—	3	1	no	—
20	C₂³×Dic₁₀	—	3	1	no	—
21	C₂²×C₄²×C₅	D₃	1	1	yes	characteristic, complemented
22	C₂²×#685	—	3	4	no	—
23	C₂×#4021	—	3	24	no	—
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂³×D₁₅	—	2	8	no	—
26	C₂³×D₁₅	—	2	16	no	—
27	C₂³×D₁₅	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×Dic₁₅	—	2	8	no	—
33	C₂²×Dic₁₅	C₂³	1	12	yes	—
34	C₂×D₆₀	—	2	48	no	—
35	C₂×D₆₀	C₂³	1	16	yes	—
36	C₂×Dic₃₀	—	2	48	no	—
37	C₂×Dic₃₀	C₂³	1	16	yes	complemented
38	C₃×C₄²×C₅	C₂³	1	16	yes	complemented
39	#6714	C₂³	1	64	yes	complemented
40	C₂³×D₁₂	—	5	1	no	—
41	C₂³×Dic₆	—	5	1	no	—
42	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic, complemented
43	C₂²×(C₂×D₃)⋊C₄	—	5	4	no	—
44	C₂×#875	—	5	24	no	—
45	C₂⁴×D₅	—	3	2	no	—
46	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
47	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂³×Dic₅	—	3	2	no	—
49	C₂²×D₂₀	—	3	12	no	—
50	C₂²×D₂₀	—	6	8	no	—
51	C₂²×Dic₁₀	—	3	12	no	—
52	C₂²×Dic₁₀	—	6	8	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	12	yes	complemented
54	C₂×#685	—	3	48	no	—
55	#4021	—	3	64	no	—
56	C₂²×#124	—	15	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×D₁₅	—	2	24	no	—
61	C₂²×D₁₅	—	2	48	no	—
62	C₂²×D₁₅	—	4	32	no	—
63	C₂²×D₁₅	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×Dic₁₅	—	2	24	no	—
69	C₂×Dic₁₅	C₂⁴	1	8	yes	—
70	D₆₀	—	2	32	no	—
71	Dic₃₀	—	2	32	no	—
72	C₂⁴×D₃	—	5	2	no	—
73	C₂³×C₃×C₄	C₂×D₅	1	2	yes	—
74	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
75	C₂³×Dic₃	—	5	2	no	—
76	C₂²×D₁₂	—	5	12	no	—
77	C₂²×D₁₂	—	10	8	no	—
78	C₂²×Dic₆	—	5	12	no	—
79	C₂²×Dic₆	—	10	8	no	—
80	C₂×C₃×C₄²	C₂×D₅	1	12	yes	complemented
81	C₂×(C₂×D₃)⋊C₄	—	5	48	no	—
82	#875	—	5	64	no	—
83	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₂³×D₅	—	3	12	no	—
85	C₂³×D₅	—	6	8	no	—
86	C₂³×D₅	—	6	16	no	—
87	C₂²×C₄×C₅	—	2	8	no	—
88	C₂²×C₄×C₅	D₁₂	1	8	yes	—
89	C₂²×C₄×C₅	C₂²×D₃	1	12	yes	—
90	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
91	C₂²×Dic₅	—	3	12	no	—
92	C₂²×Dic₅	—	6	8	no	—
93	C₂×D₂₀	—	3	16	no	—
94	C₂×D₂₀	—	6	48	no	—
95	C₂×Dic₁₀	—	3	16	no	—
96	C₂×Dic₁₀	—	6	48	no	—
97	C₄²×C₅	C₂²×D₃	1	16	yes	complemented
98	#685	—	3	64	no	—
99	C₂³×Q₈	—	15	1	no	—
100	C₂³×D₄	—	15	1	no	—
101	C₂×#124	—	15	24	no	—
102	C₂²×C₂²⋊C₄	—	15	4	no	—
103	C₂²×C₄²	D₁₅	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×D₁₅	—	2	8	no	—
109	C₂×D₁₅	—	2	16	no	—
110	C₂×D₁₅	—	4	48	no	—
111	C₃×C₄×C₅	—	2	8	no	—
112	C₃×C₄×C₅	C₂²×D₄	1	8	yes	—
113	Dic₁₅	—	2	8	no	—
114	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
115	C₂³×D₃	—	5	12	no	—
116	C₂³×D₃	—	10	8	no	—
117	C₂³×D₃	—	10	16	no	—
118	C₂²×C₃×C₄	—	2	8	no	—
119	C₂²×C₃×C₄	D₂₀	1	8	yes	—
120	C₂²×C₃×C₄	C₂²×D₅	1	12	yes	—
121	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
122	C₂²×Dic₃	—	5	12	no	—
123	C₂²×Dic₃	—	10	8	no	—
124	C₂×D₁₂	—	5	16	no	—
125	C₂×D₁₂	—	10	48	no	—
126	C₂×Dic₆	—	5	16	no	—
127	C₂×Dic₆	—	10	48	no	—
128	C₃×C₄²	C₂²×D₅	1	16	yes	complemented
129	(C₂×D₃)⋊C₄	—	5	64	no	—
130	C₂³×C₅	C₂×D₁₂	1	4	yes	—
131	C₂³×C₅	#235	1	8	yes	—
132	C₂³×C₅	C₂³×D₃	1	3	yes	—
133	C₂²×D₅	—	3	8	no	—
134	C₂²×D₅	—	6	24	no	—
135	C₂²×D₅	—	6	48	no	—
136	C₂²×D₅	—	12	32	no	—
137	C₂×C₄×C₅	—	2	24	no	—
138	C₂×C₄×C₅	C₂×D₁₂	1	24	yes	—
139	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
140	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
141	C₂×Dic₅	—	3	8	no	—
142	C₂×Dic₅	—	6	24	no	—
143	D₂₀	—	6	32	no	—
144	Dic₁₀	—	6	32	no	—
145	C₂⁵	—	15	2	no	—
146	C₂²×Q₈	—	15	12	no	—
147	C₂²×Q₈	—	30	8	no	—
148	C₂²×D₄	—	15	12	no	—
149	C₂²×D₄	—	30	8	no	—
150	C₂³×C₄	—	15	2	no	—
151	C₂³×C₄	C₂×D₁₅	1	2	yes	—
152	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
153	#124	—	15	64	no	—
154	C₂×C₂²⋊C₄	—	15	48	no	—
155	C₂×C₄²	C₂×D₁₅	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	D₁₅	—	4	8	no	—
160	C₂³×C₃	C₂×D₂₀	1	4	yes	—
161	C₂³×C₃	#709	1	8	yes	—
162	C₂³×C₃	C₂³×D₅	1	3	yes	—
163	C₂²×D₃	—	5	8	no	—
164	C₂²×D₃	—	10	24	no	—
165	C₂²×D₃	—	10	48	no	—
166	C₂²×D₃	—	20	32	no	—
167	C₂×C₃×C₄	—	2	24	no	—
168	C₂×C₃×C₄	C₂×D₂₀	1	24	yes	—
169	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
170	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
171	C₂×Dic₃	—	5	8	no	—
172	C₂×Dic₃	—	10	24	no	—
173	D₁₂	—	10	32	no	—
174	Dic₆	—	10	32	no	—
175	C₂²×C₅	#875	1	16	yes	complemented
176	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
177	C₂²×C₅	C₂×#235	1	12	yes	—
178	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
179	C₂×D₅	—	6	8	no	—
180	C₂×D₅	—	6	16	no	—
181	C₂×D₅	—	12	48	no	—
182	C₄×C₅	—	2	8	no	—
183	C₄×C₅	C₂²×D₁₂	1	8	yes	—
184	Dic₅	—	6	8	no	—
185	C₂⁴	—	15	12	no	—
186	C₂⁴	—	30	8	no	—
187	C₂⁴	—	30	16	no	—
188	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
189	C₂×Q₈	—	15	16	no	—
190	C₂×Q₈	—	30	48	no	—
191	C₂×D₄	—	15	16	no	—
192	C₂×D₄	—	30	48	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	15	12	no	—
195	C₂²×C₄	—	30	8	no	—
196	C₂²×C₄	D₆₀	1	8	yes	—
197	C₂²×C₄	C₂²×D₁₅	1	12	yes	—
198	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
199	C₂²⋊C₄	—	15	64	no	—
200	C₄²	C₂²×D₁₅	1	16	yes	complemented
201	C₃×C₅	C₂²×#124	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	#4021	1	16	yes	complemented
203	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
204	C₂²×C₃	C₂×#709	1	12	yes	—
205	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
206	C₂×D₃	—	10	8	no	—
207	C₂×D₃	—	10	16	no	—
208	C₂×D₃	—	20	48	no	—
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	C₂²×D₂₀	1	8	yes	—
211	Dic₃	—	10	8	no	—
212	C₂×C₅	C₂×#875	1	12	yes	complemented
213	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
214	C₂×C₅	C₂²×#235	1	2	yes	—
215	D₅	—	12	8	no	—
216	C₂³	—	15	8	no	—
217	C₂³	—	30	24	no	—
218	C₂³	—	30	48	no	—
219	C₂³	—	60	32	no	—
220	C₂³	C₂×D₆₀	1	4	yes	—
221	C₂³	#6817	1	8	yes	—
222	C₂³	C₂³×D₁₅	1	3	yes	—
223	Q₈	—	30	32	no	—
224	D₄	—	30	32	no	—
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	15	8	no	—
227	C₂×C₄	—	30	24	no	—
228	C₂×C₄	C₂×D₆₀	1	24	yes	—
229	C₂×C₄	C₂³×D₁₅	1	8	yes	—
230	C₂×C₄	C₂³×D₁₅	1	4	yes	—
231	C₂×C₃	C₂×#4021	1	12	yes	complemented
232	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
233	C₂×C₃	C₂²×#709	1	2	yes	—
234	D₃	—	20	8	no	—
235	C₅	C₂²×#875	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	30	8	no	—
237	C₂²	—	30	16	no	—
238	C₂²	—	60	48	no	—
239	C₂²	#91764	1	16	yes	direct product factor
240	C₂²	C₂²×D₆₀	1	6	yes	—
241	C₂²	C₂×#6817	1	12	yes	—
242	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
243	C₄	—	2	8	no	—
244	C₄	—	30	8	no	—
245	C₄	C₂²×D₆₀	1	8	yes	—
246	C₃	C₂²×#4021	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	60	8	no	—
248	C₂	C₂×#91764	1	12	yes	direct product factor
249	C₂	C₂³×D₆₀	1	1	yes	characteristic
250	C₂	C₂²×#6817	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂³×D₆₀	C₂
hypercenter	C₂²×C₄²	D₁₅
hypocenter	C₃×C₅	C₂²×#124
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₃₀	C₂	semidirect
3. C₂²×C₃×C₄²×C₅	C₂	semidirect
4. C₂²×#6714	C₂	semidirect
5. C₂×#91764	C₂	direct (N × H)
14. C₂²×Dic₃₀	C₂²	semidirect
15. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₂×#6714	C₂²	semidirect
17. #91764	C₂²	direct (N × H)
21. C₂²×C₄²×C₅	D₃	semidirect
37. C₂×Dic₃₀	C₂³	semidirect
38. C₃×C₄²×C₅	C₂³	semidirect
39. #6714	C₂³	semidirect
42. C₂²×C₃×C₄²	D₅	semidirect
53. C₂×C₄²×C₅	C₂×D₃	semidirect
80. C₂×C₃×C₄²	C₂×D₅	semidirect
97. C₄²×C₅	C₂²×D₃	semidirect
103. C₂²×C₄²	D₁₅	semidirect
104. C₂²×C₃×C₅	#124	semidirect
128. C₃×C₄²	C₂²×D₅	semidirect
155. C₂×C₄²	C₂×D₁₅	semidirect
156. C₂×C₃×C₅	C₂×#124	semidirect
175. C₂²×C₅	#875	semidirect
200. C₄²	C₂²×D₁₅	semidirect
201. C₃×C₅	C₂²×#124	semidirect
202. C₂²×C₃	#4021	semidirect
212. C₂×C₅	C₂×#875	semidirect
231. C₂×C₃	C₂×#4021	semidirect
235. C₅	C₂²×#875	semidirect
239. C₂²	#91764	direct (N × H)
246. C₃	C₂²×#4021	semidirect
248. C₂	C₂×#91764	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×D₆₀	2
C₂³×Dic₃₀	2
C₂²×C₃×C₄²×C₅	2
C₂²×#6714	2
C₂×#91764	2
C₂²×#4021	3
C₂²×#875	5

Subgroups — C22×#91764 #14479661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91764 #14479661