Subgroups — C₂×#11746755 #14285519

Order 1920.

400 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×C₄⋊(C₃⋊#170)	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#11746755	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64884	—	3	1	no	maximal, Hall
5	C₂²×C₃⋊#170	—	2	2	no	—
6	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×#168	C₄	1	1	yes	characteristic
9	C₄⋊(C₃⋊#170)	—	2	2	no	—
10	C₄⋊(C₃⋊#170)	C₂²	1	4	yes	—
11	C₃×C₈×Dic₅	C₄	1	2	yes	—
12	C₃×C₈×Dic₅	C₂²	1	2	yes	—
13	C₂×#4458	—	5	1	no	maximal, Hall
14	C₂×C₄⋊#170	—	3	2	no	—
15	C₂×C₈×Dic₅	D₃	1	1	yes	characteristic
16	#64884	—	3	4	no	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
19	C₂²×C₃×Dic₅	Q₈	1	1	yes	characteristic
20	C₂×C₃⋊#170	—	2	4	no	—
21	C₂×C₃⋊#170	—	4	4	no	—
22	C₂×C₃×C₅×C₈	—	2	2	no	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
26	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×#168	C₈	1	4	yes	complemented
29	C₂×C₃×#168	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₈	—	5	1	no	—
31	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
32	#4458	—	5	4	no	—
33	C₂²×#170	—	6	2	no	—
34	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
35	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
36	C₂²×#168	Dic₃	1	1	yes	characteristic
37	C₄⋊#170	—	3	4	no	—
38	C₄⋊#170	—	6	2	no	—
39	C₈×Dic₅	Dic₃	1	2	yes	—
40	C₈×Dic₅	C₂×D₃	1	2	yes	—
41	C₂×#335	—	15	1	no	Sylow
42	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
45	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×Dic₅	—	2	2	no	—
48	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×Dic₅	D₈	1	2	yes	—
51	C₂×C₃×Dic₅	Q₁₆	1	2	yes	—
52	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
54	C₃⋊#170	—	4	4	no	—
55	C₃×C₅×C₈	—	2	2	no	—
56	C₃×#168	C₂×C₈	1	4	yes	complemented
57	C₂²×C₃×C₈	—	5	1	no	—
58	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₃×C₄²	—	5	1	no	—
60	C₂²×C₃⋊C₈	—	10	2	no	—
61	C₃×C₄×C₈	—	5	2	no	—
62	C₃×C₄×C₈	—	5	2	no	—
63	C₄⋊(C₃⋊C₈)	—	5	4	no	—
64	C₄⋊(C₃⋊C₈)	—	10	2	no	—
65	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
66	C₂²×Dic₅	Dic₆	1	1	yes	characteristic
67	C₂²×Dic₅	D₁₂	1	1	yes	characteristic
68	C₂×#170	—	6	4	no	—
69	C₂×#170	—	12	4	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
72	C₄×Dic₅	Dic₆	1	1	yes	characteristic
73	C₄×Dic₅	D₁₂	1	1	yes	characteristic
74	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
75	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
76	C₂×#168	C₃⋊C₈	1	4	yes	complemented
77	C₂×#168	C₂×Dic₃	1	2	yes	—
78	C₂×C₄⋊C₈	—	15	2	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	#335	—	15	4	no	—
81	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
82	C₂²×C₃×C₅	#107	1	2	yes	—
83	C₂²×C₃×C₅	#108	1	2	yes	—
84	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
86	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₃×Dic₅	—	2	1	no	—
89	C₃×Dic₅	—	2	1	no	—
90	C₃×Dic₅	#107	1	2	yes	—
91	C₃×Dic₅	C₂×D₈	1	1	yes	characteristic
92	C₃×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	5	1	no	—
94	C₂²×C₃×C₄	—	5	1	no	—
95	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	5	4	no	—
98	C₂×C₃×C₈	—	5	2	no	—
99	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₃×C₄²	—	5	1	no	—
101	C₃×C₄²	—	5	1	no	—
102	C₃×C₄²	—	5	1	no	—
103	C₃×C₄²	—	5	1	no	—
104	C₂×C₃⋊C₈	—	10	4	no	—
105	C₂×C₃⋊C₈	—	20	4	no	—
106	C₂³×C₅	C₄⋊Dic₃	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
108	C₂×C₄×C₅	#208	1	2	yes	—
109	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	D₂₄	1	2	yes	—
113	C₂×Dic₅	Dic₁₂	1	2	yes	—
114	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
115	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
116	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
117	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
118	#170	—	12	4	no	—
119	C₅×C₈	—	2	2	no	—
120	#168	C₂×C₃⋊C₈	1	4	yes	complemented
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	—	30	2	no	—
123	C₂²×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
124	C₂×C₄²	—	5	1	no	—
125	C₄⋊C₈	—	15	4	no	—
126	C₄⋊C₈	—	30	2	no	—
127	C₄×C₈	—	5	2	no	—
128	C₄×C₈	—	5	2	no	—
129	C₂×C₃×C₅	#335	1	4	yes	complemented
130	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
133	C₂³×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₃×C₄	—	5	1	no	—
135	C₂×C₃×C₄	—	5	1	no	—
136	C₂×C₃×C₄	—	5	2	no	—
137	C₂×C₃×C₄	—	5	2	no	—
138	C₂×C₃×C₄	—	5	1	no	—
139	C₂×C₃×C₄	—	5	1	no	—
140	C₂×C₃×C₄	—	10	2	no	—
141	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
142	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₂×C₃×C₄	#704	1	2	yes	—
144	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	—	5	4	no	—
147	C₃⋊C₈	—	20	4	no	—
148	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
149	C₂²×C₅	#818	1	2	yes	—
150	C₂²×C₅	#819	1	2	yes	—
151	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
152	C₄×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
153	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
154	C₄×C₅	C₂×#208	1	1	yes	characteristic
155	Dic₅	—	2	1	no	—
156	Dic₅	—	2	1	no	—
157	Dic₅	#818	1	2	yes	—
158	Dic₅	C₂×D₂₄	1	1	yes	characteristic
159	Dic₅	C₂×Dic₁₂	1	1	yes	characteristic
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	5	4	no	—
165	C₂×C₈	—	5	2	no	—
166	C₂×C₈	—	30	4	no	—
167	C₂×C₈	—	60	4	no	—
168	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
169	C₄²	—	5	1	no	—
170	C₄²	—	5	1	no	—
171	C₄²	—	5	1	no	—
172	C₄²	—	5	1	no	—
173	C₃×C₅	C₂×#335	1	1	yes	characteristic, complemented, Hall
174	C₂²×C₃	#3994	1	2	yes	—
175	C₂²×C₃	#3995	1	2	yes	—
176	C₂²×C₃	#4004	1	2	yes	—
177	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₃×C₄	—	5	2	no	—
179	C₃×C₄	—	5	1	no	—
180	C₃×C₄	—	5	1	no	—
181	C₃×C₄	—	10	1	no	—
182	C₃×C₄	—	10	1	no	—
183	C₃×C₄	#4004	1	2	yes	—
184	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
185	C₃×C₄	C₂×#704	1	1	yes	characteristic
186	C₂×C₅	#4458	1	4	yes	complemented
187	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
188	C₂×C₅	C₂×#818	1	1	yes	characteristic
189	C₂×C₅	C₂×#819	1	1	yes	characteristic
190	C₂³	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	1	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	1	no	—
196	C₂×C₄	—	5	1	no	—
197	C₂×C₄	—	10	2	no	—
198	C₂×C₄	D₅⋊(C₃⋊C₈)	1	2	yes	—
199	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
200	C₂×C₄	#6762	1	2	yes	—
201	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	5	4	no	—
204	C₈	—	60	4	no	—
205	C₂×C₃	#64884	1	4	yes	complemented
206	C₂×C₃	C₂×#3994	1	1	yes	characteristic
207	C₂×C₃	C₂×#3995	1	1	yes	characteristic
208	C₂×C₃	C₂×#4004	1	1	yes	characteristic
209	C₅	C₂×#4458	1	1	yes	characteristic, complemented, Sylow
210	C₂²	#91223	1	2	yes	—
211	C₂²	#91224	1	2	yes	—
212	C₂²	#91233	1	2	yes	—
213	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
214	C₄	—	5	1	no	—
215	C₄	—	5	2	no	—
216	C₄	—	5	1	no	—
217	C₄	—	10	1	no	—
218	C₄	—	10	1	no	—
219	C₄	#91233	1	2	yes	—
220	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
221	C₄	C₂×#6762	1	1	yes	characteristic
222	C₃	C₂×#64884	1	1	yes	characteristic, complemented, Sylow
223	C₂	#11746755	1	4	yes	direct product factor
224	C₂	C₂×#91223	1	1	yes	characteristic
225	C₂	C₂×#91224	1	1	yes	characteristic
226	C₂	C₂×#91233	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄⋊ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle, prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂²×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#335
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #11746755	C₂	direct (N × H)
28. C₂×C₃×#168	C₈	semidirect
56. C₃×#168	C₂×C₈	semidirect
76. C₂×#168	C₃⋊C₈	semidirect
120. #168	C₂×C₃⋊C₈	semidirect
129. C₂×C₃×C₅	#335	semidirect
173. C₃×C₅	C₂×#335	semidirect
186. C₂×C₅	#4458	semidirect
205. C₂×C₃	#64884	semidirect
209. C₅	C₂×#4458	semidirect
222. C₃	C₂×#64884	semidirect
223. C₂	#11746755	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×C₄⋊(C₃⋊#170)	2
C₂×C₃×C₈×Dic₅	2
#11746755	2
C₂×#64884	3
C₂×#4458	5

Subgroups — C2×#11746755 #14285519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746755 #14285519