Subgroups — C₄×#91231 #14294214

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91231	C₂	1	4	yes	maximal
2	C₂×C₄×C₃⋊#170	C₂	1	2	yes	maximal
3	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₄×#4002	—	3	1	no	maximal, Hall
5	C₂²×C₃⋊#170	C₂²	1	4	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
11	#91231	C₄	1	16	yes	direct product factor
12	C₄×C₃⋊#170	C₄	1	8	yes	complemented
13	C₄×#803	—	5	1	no	maximal, Hall
14	C₂×#4002	—	3	4	no	—
15	C₂×C₄×#170	—	3	2	no	—
16	C₄²×Dic₅	D₃	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊#170	C₂×C₄	1	8	yes	—
23	C₂×C₃⋊#170	C₂×C₄	1	16	yes	—
24	C₃×C₄²×C₅	—	2	2	no	—
25	C₃×C₄×Dic₅	—	2	2	no	—
26	C₃×C₄×Dic₅	—	2	4	no	—
27	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	—
28	C₃×C₄×Dic₅	C₂×C₄	1	8	yes	—
29	C₃×C₄³	—	5	1	no	—
30	C₂×#803	—	5	4	no	—
31	C₂×C₄×C₃⋊C₈	—	5	2	no	—
32	C₂²×#170	—	3	4	no	—
33	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
34	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
35	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
36	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
37	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
38	#4002	—	3	16	no	—
39	C₄×#170	—	3	8	no	—
40	C₄×#97	—	15	1	no	Sylow
41	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
45	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
46	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
47	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
48	C₂×C₃×Dic₅	—	2	2	no	—
49	C₂×C₃×Dic₅	—	2	2	no	—
50	C₂×C₃×Dic₅	C₄²	1	4	yes	—
51	C₂×C₃×Dic₅	C₄²	1	2	yes	—
52	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
53	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
54	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
55	C₃⋊#170	C₄²	1	16	yes	complemented
56	C₂×C₃×C₄²	—	5	1	no	—
57	C₂×C₃×C₄²	—	5	2	no	—
58	C₂×C₃×C₄²	—	5	2	no	—
59	C₂×C₃×C₄²	—	5	1	no	—
60	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	—	5	4	no	—
62	#803	—	5	16	no	—
63	C₄×C₃⋊C₈	—	5	8	no	—
64	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
65	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
66	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
67	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
68	C₂²×Dic₅	C₂²×D₃	1	1	yes	characteristic
69	C₂×#170	—	3	8	no	—
70	C₂×#170	—	3	16	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄×Dic₅	—	2	2	no	—
73	C₄×Dic₅	—	2	4	no	—
74	C₄×Dic₅	C₄×D₃	1	4	yes	—
75	C₄×Dic₅	C₄×D₃	1	8	yes	—
76	C₂×#97	—	15	4	no	—
77	C₂×C₄×C₈	—	15	2	no	—
78	C₄³	—	5	1	no	—
79	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
80	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
81	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
83	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	4	no	—
85	C₃×C₄×C₅	#97	1	4	yes	complemented
86	C₃×Dic₅	—	2	4	no	—
87	C₃×Dic₅	#97	1	4	yes	complemented
88	C₃×Dic₅	C₂×C₄²	1	4	yes	—
89	C₂²×C₃×C₄	—	5	1	no	—
90	C₂²×C₃×C₄	—	5	1	no	—
91	C₂²×C₃×C₄	—	5	2	no	—
92	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
93	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₃×C₄²	—	2	2	no	—
95	C₃×C₄²	—	5	4	no	—
96	C₃×C₄²	—	5	8	no	—
97	C₃×C₄²	—	10	4	no	—
98	C₃×C₄²	—	10	2	no	—
99	C₂×C₃⋊C₈	—	5	8	no	—
100	C₂×C₃⋊C₈	—	5	16	no	—
101	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	#208	1	2	yes	—
105	C₂×C₄×C₅	#208	1	2	yes	—
106	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
107	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	2	2	no	—
110	C₂×Dic₅	#208	1	4	yes	—
111	C₂×Dic₅	C₄×Dic₃	1	4	yes	—
112	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
113	C₂×Dic₅	C₂×C₄×D₃	1	2	yes	—
114	C₂×Dic₅	C₂×C₄×D₃	1	4	yes	—
115	#170	—	3	16	no	—
116	C₂²×C₈	—	15	4	no	—
117	C₂×C₄²	—	5	2	no	—
118	C₂×C₄²	—	5	1	no	—
119	C₂×C₄²	—	5	2	no	—
120	C₂×C₄²	—	5	1	no	—
121	C₂×C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
122	#97	—	15	16	no	—
123	C₄×C₈	—	15	8	no	—
124	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
125	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
126	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
127	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
128	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
129	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	5	4	no	—
133	C₂×C₃×C₄	—	5	2	no	—
134	C₂×C₃×C₄	—	5	2	no	—
135	C₂×C₃×C₄	—	5	4	no	—
136	C₂×C₃×C₄	—	5	4	no	—
137	C₂×C₃×C₄	—	10	2	no	—
138	C₂×C₃×C₄	—	10	2	no	—
139	C₂×C₃×C₄	#700	1	2	yes	—
140	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	4	yes	—
141	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
142	C₂×C₃×C₄	#704	1	2	yes	—
143	C₃⋊C₈	—	5	16	no	—
144	C₂²×C₅	C₂×#208	1	1	yes	characteristic
145	C₂²×C₅	C₂×#208	1	2	yes	—
146	C₂²×C₅	C₂×#208	1	1	yes	characteristic
147	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₄×Dic₃	1	2	yes	—
149	C₄×C₅	—	2	4	no	—
150	C₄×C₅	#803	1	4	yes	complemented
151	Dic₅	—	2	4	no	—
152	Dic₅	#803	1	4	yes	complemented
153	Dic₅	C₄²×D₃	1	4	yes	—
154	C₂²×C₄	—	5	1	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
158	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
159	C₂×C₈	—	15	8	no	—
160	C₂×C₈	—	15	16	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	5	4	no	—
163	C₄²	—	5	8	no	—
164	C₄²	—	10	4	no	—
165	C₄²	—	10	2	no	—
166	C₃×C₅	C₄×#97	1	1	yes	characteristic, complemented, Hall
167	C₂²×C₃	C₂×#700	1	1	yes	characteristic
168	C₂²×C₃	D₅⋊M₁₆	1	2	yes	—
169	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
171	C₂²×C₃	C₂×#704	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	5	4	no	—
174	C₃×C₄	—	5	4	no	—
175	C₃×C₄	—	10	4	no	—
176	C₃×C₄	#4002	1	4	yes	complemented
177	C₂×C₅	C₂×#803	1	1	yes	characteristic
178	C₂×C₅	C₂×#803	1	1	yes	characteristic
179	C₂×C₅	C₄×#208	1	2	yes	—
180	C₂×C₅	C₄×#208	1	2	yes	—
181	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
182	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	5	4	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	—	5	4	no	—
188	C₂×C₄	—	5	2	no	—
189	C₂×C₄	—	5	4	no	—
190	C₂×C₄	—	10	2	no	—
191	C₂×C₄	—	10	2	no	—
192	C₂×C₄	#6758	1	2	yes	—
193	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	4	yes	—
194	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
195	C₂×C₄	#6762	1	2	yes	—
196	C₈	—	15	16	no	—
197	C₂×C₃	#65676	1	1	yes	characteristic
198	C₂×C₃	C₄×#700	1	2	yes	—
199	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×C₃	C₂×#4002	1	1	yes	characteristic
201	C₂×C₃	C₄×#704	1	2	yes	—
202	C₅	C₄×#803	1	1	yes	characteristic, complemented, Sylow
203	C₂²	C₂×#6758	1	1	yes	characteristic
204	C₂²	D₅⋊#208	1	2	yes	—
205	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
206	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
207	C₂²	C₂×#6762	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	5	4	no	—
210	C₄	—	5	4	no	—
211	C₄	—	10	4	no	—
212	C₄	#91231	1	4	yes	direct product factor
213	C₃	C₄×#4002	1	1	yes	characteristic, complemented, Sylow
214	C₂	#11752335	1	1	yes	characteristic
215	C₂	C₄×#6758	1	2	yes	—
216	C₂	C₄²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
217	C₂	C₂×#91231	1	1	yes	characteristic
218	C₂	C₄×#6762	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₄³
Frattini	C₂³	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄²×C₅	C₄
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×Dic₅	C₂³
hypercenter	C₂×C₄²	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄×#97
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. #91231	C₄	direct (N × H)
12. C₄×C₃⋊#170	C₄	semidirect
55. C₃⋊#170	C₄²	semidirect
85. C₃×C₄×C₅	#97	semidirect
87. C₃×Dic₅	#97	semidirect
150. C₄×C₅	#803	semidirect
152. Dic₅	#803	semidirect
166. C₃×C₅	C₄×#97	semidirect
176. C₃×C₄	#4002	semidirect
202. C₅	C₄×#803	semidirect
212. C₄	#91231	direct (N × H)
213. C₃	C₄×#4002	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91231	2
C₂×C₄×C₃⋊#170	2
C₃×C₄²×Dic₅	2
C₄×#4002	3
C₄×#803	5

Subgroups — C4×#91231 #14294214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91231 #14294214