Subgroups — C₄×#91763 #14322253

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91763	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₆₀	C₂	1	6	yes	maximal, complemented
3	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751040	C₂	1	6	yes	maximal, complemented
5	#11751036	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#4020	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	C₂²	1	6	yes	—
8	C₂²×C₄×D₁₅	C₂²	1	4	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
12	C₂×#6714	C₂²	1	12	yes	—
13	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
14	#91763	C₄	1	8	yes	direct product factor
15	C₄×D₆₀	—	2	24	no	—
16	C₄×#874	—	5	1	no	maximal, Hall
17	C₂×#4020	—	3	1	no	—
18	C₂×C₄×D₂₀	—	3	6	no	—
19	C₄³×C₅	D₃	1	1	yes	characteristic, complemented
20	#65228	—	3	6	no	—
21	#65224	—	3	1	no	—
22	C₂³×D₁₅	C₂³	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₅	C₂³	1	4	yes	—
27	C₂×D₆₀	—	2	12	no	—
28	C₂×D₆₀	—	2	12	no	—
29	C₂×D₆₀	C₂×C₄	1	24	yes	complemented
30	C₂×C₄×D₁₅	—	2	24	no	—
31	C₃×C₄²×C₅	—	2	6	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
33	C₃×C₄²×C₅	D₄	1	12	yes	complemented
34	#6714	—	2	24	no	—
35	C₄⋊Dic₁₅	—	2	12	no	—
36	C₂×#874	—	5	1	no	—
37	C₂×C₄×D₁₂	—	5	6	no	—
38	C₃×C₄³	D₅	1	1	yes	characteristic, complemented
39	#4939	—	5	6	no	—
40	#4935	—	5	1	no	—
41	C₂²×D₂₀	—	3	6	no	—
42	C₂²×C₄×D₅	—	3	4	no	—
43	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
46	C₂×#685	—	3	12	no	—
47	C₂×C₄⋊Dic₅	—	3	6	no	—
48	#4020	—	3	8	no	—
49	C₄×D₂₀	—	6	24	no	—
50	C₄×#127	—	15	1	no	Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×D₁₅	—	2	12	no	—
53	C₂²×D₁₅	—	2	12	no	—
54	C₂²×D₁₅	C₂²×C₄	1	8	yes	—
55	C₂×C₃×C₄×C₅	—	2	6	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
60	C₂×C₃×C₄×C₅	#41	1	6	yes	—
61	C₂×Dic₁₅	—	2	12	no	—
62	D₆₀	—	2	48	no	—
63	C₄×D₁₅	—	4	16	no	—
64	C₂²×D₁₂	—	5	6	no	—
65	C₂²×C₄×D₃	—	5	4	no	—
66	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
67	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
68	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
69	C₂×(C₂×D₃)⋊C₄	—	5	12	no	—
70	C₂×C₄⋊Dic₃	—	5	6	no	—
71	#874	—	5	8	no	—
72	C₄×D₁₂	—	10	24	no	—
73	C₂³×D₅	—	3	4	no	—
74	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
75	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
76	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₂²×Dic₅	—	3	4	no	—
78	C₂×D₂₀	—	3	24	no	—
79	C₂×D₂₀	—	6	12	no	—
80	C₂×D₂₀	—	6	12	no	—
81	C₂×C₄×D₅	—	6	24	no	—
82	C₄²×C₅	—	2	6	no	—
83	C₄²×C₅	C₄×D₃	1	4	yes	complemented
84	C₄²×C₅	D₁₂	1	12	yes	complemented
85	#685	—	6	24	no	—
86	C₄⋊Dic₅	—	6	12	no	—
87	C₂×#127	—	15	1	no	—
88	C₂×C₄×D₄	—	15	6	no	—
89	#390	—	15	6	no	—
90	#384	—	15	1	no	—
91	C₄³	D₁₅	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
94	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
95	C₂×D₁₅	—	2	24	no	—
96	C₂×D₁₅	—	4	4	no	—
97	C₂×D₁₅	—	4	12	no	—
98	C₃×C₄×C₅	—	2	6	no	—
99	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
100	C₃×C₄×C₅	#127	1	4	yes	complemented
101	Dic₁₅	—	4	4	no	—
102	C₂³×D₃	—	5	4	no	—
103	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
104	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
105	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
106	C₂²×Dic₃	—	5	4	no	—
107	C₂×D₁₂	—	5	24	no	—
108	C₂×D₁₂	—	10	12	no	—
109	C₂×D₁₂	—	10	12	no	—
110	C₂×C₄×D₃	—	10	24	no	—
111	C₃×C₄²	—	2	6	no	—
112	C₃×C₄²	C₄×D₅	1	4	yes	complemented
113	C₃×C₄²	D₂₀	1	12	yes	complemented
114	(C₂×D₃)⋊C₄	—	10	24	no	—
115	C₄⋊Dic₃	—	10	12	no	—
116	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
117	C₂²×D₅	—	3	8	no	—
118	C₂²×D₅	—	6	12	no	—
119	C₂²×D₅	—	6	12	no	—
120	C₂×C₄×C₅	—	2	6	no	—
121	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
122	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
123	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
124	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
125	C₂×C₄×C₅	#235	1	6	yes	—
126	C₂×Dic₅	—	6	12	no	—
127	D₂₀	—	6	48	no	—
128	C₄×D₅	—	12	16	no	—
129	C₂²×D₄	—	15	6	no	—
130	C₂³×C₄	—	15	4	no	—
131	#127	—	15	8	no	—
132	C₄×D₄	—	30	24	no	—
133	C₂×C₄⋊C₄	—	15	6	no	—
134	C₂×C₂²⋊C₄	—	15	12	no	—
135	C₂×C₄²	C₂×D₁₅	1	3	yes	—
136	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₁₅	1	3	yes	—
138	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
139	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
140	C₂×C₃×C₅	#532	1	3	yes	—
141	D₁₅	—	4	8	no	—
142	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
143	C₂²×D₃	—	5	8	no	—
144	C₂²×D₃	—	10	12	no	—
145	C₂²×D₃	—	10	12	no	—
146	C₂×C₃×C₄	—	2	6	no	—
147	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
148	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
149	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
150	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
151	C₂×C₃×C₄	#709	1	6	yes	—
152	C₂×Dic₃	—	10	12	no	—
153	D₁₂	—	10	48	no	—
154	C₄×D₃	—	20	16	no	—
155	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₅	C₂²×D₁₂	1	3	yes	—
157	C₂²×C₅	C₂×#235	1	3	yes	—
158	C₂×D₅	—	6	24	no	—
159	C₂×D₅	—	12	4	no	—
160	C₂×D₅	—	12	12	no	—
161	C₄×C₅	—	2	6	no	—
162	C₄×C₅	C₄×D₁₂	1	12	yes	complemented
163	C₄×C₅	#874	1	4	yes	complemented
164	Dic₅	—	12	4	no	—
165	C₂⁴	—	15	4	no	—
166	C₂×D₄	—	15	24	no	—
167	C₂×D₄	—	30	12	no	—
168	C₂×D₄	—	30	12	no	—
169	C₂²×C₄	—	15	4	no	—
170	C₂²×C₄	—	30	24	no	—
171	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
172	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
173	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
174	C₄⋊C₄	—	30	12	no	—
175	C₂²⋊C₄	—	30	24	no	—
176	C₄²	—	2	6	no	—
177	C₄²	C₄×D₁₅	1	4	yes	complemented
178	C₄²	D₆₀	1	12	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
181	C₂²×C₃	C₂²×D₂₀	1	3	yes	—
182	C₂²×C₃	C₂×#709	1	3	yes	—
183	C₂×D₃	—	10	24	no	—
184	C₂×D₃	—	20	4	no	—
185	C₂×D₃	—	20	12	no	—
186	C₃×C₄	—	2	6	no	—
187	C₃×C₄	C₄×D₂₀	1	12	yes	complemented
188	C₃×C₄	#4020	1	4	yes	complemented
189	Dic₃	—	20	4	no	—
190	C₂×C₅	C₂×C₄×D₁₂	1	3	yes	—
191	C₂×C₅	C₂×#874	1	1	yes	characteristic
192	C₂×C₅	#5477	1	3	yes	—
193	D₅	—	12	8	no	—
194	C₂³	—	15	8	no	—
195	C₂³	—	30	12	no	—
196	C₂³	—	30	12	no	—
197	C₂³	C₂³×D₁₅	1	1	yes	characteristic
198	D₄	—	30	48	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	30	12	no	—
201	C₂×C₄	—	60	16	no	—
202	C₂×C₄	C₂×C₄×D₁₅	1	6	yes	—
203	C₂×C₄	C₂×D₆₀	1	6	yes	—
204	C₂×C₄	C₂×D₆₀	1	6	yes	—
205	C₂×C₄	C₂×D₆₀	1	6	yes	—
206	C₂×C₄	#6817	1	6	yes	—
207	C₂×C₃	C₂×C₄×D₂₀	1	3	yes	—
208	C₂×C₃	C₂×#4020	1	1	yes	characteristic
209	C₂×C₃	#65811	1	3	yes	—
210	D₃	—	20	8	no	—
211	C₅	C₄×#874	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	30	24	no	—
213	C₂²	—	60	4	no	—
214	C₂²	—	60	12	no	—
215	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
216	C₂²	C₂²×D₆₀	1	3	yes	—
217	C₂²	C₂×#6817	1	3	yes	—
218	C₄	—	2	6	no	—
219	C₄	—	60	4	no	—
220	C₄	C₄×D₆₀	1	12	yes	complemented
221	C₄	#91763	1	4	yes	direct product factor
222	C₃	C₄×#4020	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	60	8	no	—
224	C₂	C₂×C₄×D₆₀	1	3	yes	—
225	C₂	C₂×#91763	1	1	yes	characteristic
226	C₂	#11756236	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂³×C₄
Frattini	C₂³	C₂³×D₁₅
Fitting	C₃×C₄³×C₅	C₂
socle, cosocle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×#91763	C₂
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	C₄×#127
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₆₀	C₂	semidirect
3. C₃×C₄³×C₅	C₂	semidirect
4. #11751040	C₂	semidirect
5. #11751036	C₂	semidirect
14. #91763	C₄	direct (N × H)
19. C₄³×C₅	D₃	semidirect
29. C₂×D₆₀	C₂×C₄	semidirect
32. C₃×C₄²×C₅	C₂×C₄	semidirect
33. C₃×C₄²×C₅	D₄	semidirect
38. C₃×C₄³	D₅	semidirect
83. C₄²×C₅	C₄×D₃	semidirect
84. C₄²×C₅	D₁₂	semidirect
91. C₄³	D₁₅	semidirect
99. C₃×C₄×C₅	C₄×D₄	semidirect
100. C₃×C₄×C₅	#127	semidirect
112. C₃×C₄²	C₄×D₅	semidirect
113. C₃×C₄²	D₂₀	semidirect
162. C₄×C₅	C₄×D₁₂	semidirect
163. C₄×C₅	#874	semidirect
177. C₄²	C₄×D₁₅	semidirect
178. C₄²	D₆₀	semidirect
179. C₃×C₅	C₄×#127	semidirect
187. C₃×C₄	C₄×D₂₀	semidirect
188. C₃×C₄	#4020	semidirect
211. C₅	C₄×#874	semidirect
220. C₄	C₄×D₆₀	semidirect
221. C₄	#91763	direct (N × H)
222. C₃	C₄×#4020	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91763	2
C₂×C₄×D₆₀	2
C₃×C₄³×C₅	2
#11751040	2
#11751036	2
C₄×#4020	3
C₄×#874	5

Subgroups — C4×#91763 #14322253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91763 #14322253