Subgroups — C₂×#11746715 #14312549

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11746715	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64844	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	1	yes	characteristic
7	C₂×D₅⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊#674	C₄	1	2	yes	complemented
10	C₈×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
11	C₈×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
12	D₅⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
13	D₅⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×#672	C₄	1	1	yes	characteristic, complemented
16	C₃×D₅×C₁₆	C₄	1	2	yes	complemented
17	C₃×D₅×C₁₆	C₂²	1	2	yes	—
18	C₂×#4502	—	5	1	no	maximal, Hall
19	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
20	C₂×D₅⋊C₁₆	—	3	1	no	—
21	C₂×D₅×C₁₆	D₃	1	1	yes	characteristic
22	#64844	—	3	4	no	—
23	C₂²×ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃⋊#170	C₈	1	2	yes	—
26	C₄×ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
27	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
28	D₅⋊(C₃⋊C₈)	C₈	1	2	yes	—
29	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
36	C₃⋊#674	C₂×C₄	1	4	yes	complemented
37	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
38	C₃×#672	C₂×C₄	1	2	yes	complemented
39	C₂²×C₃×C₁₆	—	5	1	no	—
40	C₂×C₈×Dic₃	—	5	1	no	—
41	C₂²×C₃⋊C₁₆	—	5	1	no	—
42	#4502	—	5	4	no	—
43	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
44	C₂×D₅⋊C₈	—	3	1	no	—
45	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
46	C₂×#674	—	3	2	no	—
47	C₈×AGL(1,𝔽₅)	—	3	2	no	—
48	C₈×AGL(1,𝔽₅)	—	3	2	no	—
49	D₅⋊C₁₆	—	3	2	no	—
50	D₅⋊C₁₆	—	3	2	no	—
51	C₂×C₅×C₁₆	Dic₃	1	1	yes	characteristic, complemented
52	C₂×#672	Dic₃	1	1	yes	characteristic, complemented
53	D₅×C₁₆	Dic₃	1	2	yes	complemented
54	D₅×C₁₆	C₂×D₃	1	2	yes	—
55	C₂×#346	—	15	1	no	Sylow
56	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
57	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
58	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
61	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
62	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
63	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
64	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
65	C₃⋊#170	C₂×C₈	1	4	yes	—
66	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
67	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₃×#168	C₄²	1	1	yes	characteristic
69	C₃×#168	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₈	—	5	1	no	—
71	C₂×C₄×Dic₃	—	5	1	no	—
72	C₂²×C₃⋊C₈	—	5	1	no	—
73	C₂×C₃×C₁₆	—	5	1	no	—
74	C₂×C₃×C₁₆	—	5	2	no	—
75	C₂×C₃×C₁₆	—	5	2	no	—
76	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
77	C₈×Dic₃	—	5	2	no	—
78	C₈×Dic₃	—	5	2	no	—
79	C₂×C₃⋊C₁₆	—	5	2	no	—
80	C₂×C₃⋊C₁₆	—	5	2	no	—
81	C₂×C₃⋊C₁₆	—	5	2	no	—
82	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
83	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
84	C₂×#170	—	3	2	no	—
85	C₄×AGL(1,𝔽₅)	—	3	2	no	—
86	C₄×AGL(1,𝔽₅)	—	3	2	no	—
87	D₅⋊C₈	—	3	2	no	—
88	D₅⋊C₈	—	3	2	no	—
89	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂×#168	C₄×D₃	1	1	yes	characteristic
91	C₈×D₅	C₄×D₃	1	1	yes	characteristic
92	C₈×D₅	C₄×D₃	1	1	yes	characteristic
93	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
94	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
95	#674	—	3	4	no	—
96	C₅×C₁₆	C₂×Dic₃	1	2	yes	complemented
97	#672	C₂×Dic₃	1	2	yes	complemented
98	C₂²×C₁₆	—	5	1	no	—
99	C₂²×C₁₆	—	15	1	no	—
100	C₂×C₄×C₈	—	15	1	no	—
101	#346	—	15	4	no	—
102	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
103	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
104	C₂×C₃×D₅	M₃₂	1	4	yes	—
105	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
106	ΓL(1,𝔽₁₆)	—	2	4	no	—
107	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
109	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
110	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	5	1	no	—
112	C₂²×Dic₃	—	5	2	no	—
113	C₂×C₃×C₈	—	5	1	no	—
114	C₂×C₃×C₈	—	5	1	no	—
115	C₂×C₃×C₈	—	5	1	no	—
116	C₂×C₃×C₈	—	5	1	no	—
117	C₂×C₃×C₈	—	5	1	no	—
118	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₄×Dic₃	—	5	2	no	—
120	C₄×Dic₃	—	5	2	no	—
121	C₂×C₃⋊C₈	—	5	2	no	—
122	C₂×C₃⋊C₈	—	5	2	no	—
123	C₂×C₃⋊C₈	—	5	2	no	—
124	C₃×C₁₆	—	5	2	no	—
125	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
126	C₃⋊C₁₆	—	5	4	no	—
127	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
128	C₂×AGL(1,𝔽₅)	—	3	4	no	—
129	C₂×AGL(1,𝔽₅)	—	6	4	no	—
130	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
131	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
132	C₄×D₅	D₃×C₈	1	1	yes	characteristic
133	C₄×D₅	D₃×C₈	1	1	yes	characteristic
134	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
135	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
136	#170	—	3	4	no	—
137	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
138	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
139	#168	C₄×Dic₃	1	1	yes	characteristic
140	#168	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	15	1	no	—
143	C₂×C₄²	—	15	1	no	—
144	C₂×C₁₆	—	5	1	no	—
145	C₂×C₁₆	—	5	2	no	—
146	C₂×C₁₆	—	5	2	no	—
147	C₂×C₁₆	—	15	2	no	—
148	C₂×C₁₆	—	15	2	no	—
149	C₂×C₁₆	—	15	2	no	—
150	C₂×C₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
151	C₄×C₈	—	15	2	no	—
152	C₄×C₈	—	15	2	no	—
153	C₂×C₃×C₅	#346	1	2	yes	complemented
154	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
155	C₃×D₅	#346	1	2	yes	complemented
156	C₃×D₅	C₂×M₃₂	1	2	yes	—
157	C₂³×C₃	—	5	1	no	—
158	C₂×C₃×C₄	—	5	1	no	—
159	C₂×C₃×C₄	—	5	1	no	—
160	C₂×C₃×C₄	—	5	1	no	—
161	C₂×C₃×C₄	—	5	1	no	—
162	C₂×C₃×C₄	—	5	1	no	—
163	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂×Dic₃	—	5	4	no	—
165	C₂×Dic₃	—	10	4	no	—
166	C₃×C₈	—	5	1	no	—
167	C₃×C₈	—	5	1	no	—
168	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₃⋊C₈	—	5	4	no	—
171	C₂²×C₅	C₈×Dic₃	1	1	yes	characteristic
172	C₂×D₅	#798	1	4	yes	—
173	C₂×D₅	C₈×Dic₃	1	1	yes	characteristic
174	C₂×D₅	C₂×D₃×C₈	1	1	yes	characteristic
175	AGL(1,𝔽₅)	—	6	4	no	—
176	C₄×C₅	C₈×Dic₃	1	1	yes	characteristic
177	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
178	Dic₅	C₈×Dic₃	1	1	yes	characteristic
179	Dic₅	C₂×D₃×C₈	1	1	yes	characteristic
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	15	2	no	—
182	C₂×C₈	—	5	1	no	—
183	C₂×C₈	—	5	1	no	—
184	C₂×C₈	—	5	1	no	—
185	C₂×C₈	—	5	1	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	15	2	no	—
188	C₂×C₈	—	15	2	no	—
189	C₂×C₈	—	15	2	no	—
190	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
191	C₄²	—	15	2	no	—
192	C₄²	—	15	2	no	—
193	C₁₆	—	5	2	no	—
194	C₁₆	—	15	4	no	—
195	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
196	C₃×C₅	C₂×#346	1	1	yes	characteristic, complemented, Hall
197	C₂²×C₃	—	5	4	no	—
198	C₂²×C₃	—	5	1	no	—
199	C₂²×C₃	—	5	1	no	—
200	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₃×C₄	—	5	1	no	—
202	C₃×C₄	—	5	1	no	—
203	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
205	Dic₃	—	10	4	no	—
206	C₂×C₅	#4502	1	2	yes	complemented
207	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
208	D₅	#4502	1	2	yes	complemented
209	D₅	C₂×#798	1	2	yes	—
210	C₂³	—	5	1	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	5	1	no	—
213	C₂×C₄	—	5	1	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	15	4	no	—
217	C₂×C₄	—	30	4	no	—
218	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
219	C₈	—	5	1	no	—
220	C₈	—	5	1	no	—
221	C₈	—	15	4	no	—
222	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
223	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
224	C₂×C₃	—	5	2	no	—
225	C₂×C₃	—	5	2	no	—
226	C₂×C₃	#64844	1	2	yes	complemented
227	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₅	C₂×#4502	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	5	4	no	—
230	C₂²	—	5	1	no	—
231	C₂²	—	5	1	no	—
232	C₂²	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
233	C₄	—	5	1	no	—
234	C₄	—	5	1	no	—
235	C₄	—	30	4	no	—
236	C₄	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
237	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
238	C₃	C₂×#64844	1	1	yes	characteristic, complemented, Sylow
239	C₂	—	5	2	no	—
240	C₂	—	5	2	no	—
241	C₂	#11746715	1	2	yes	direct product factor
242	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂²×C₃×C₅	C₄×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×C₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#346
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. #11746715	C₂	direct (N × H)
9. C₂×C₃⋊#674	C₄	semidirect
12. D₅⋊(C₃⋊C₁₆)	C₄	semidirect
14. C₂×C₃×C₅×C₁₆	C₄	semidirect
15. C₂×C₃×#672	C₄	semidirect
16. C₃×D₅×C₁₆	C₄	semidirect
36. C₃⋊#674	C₂×C₄	semidirect
37. C₃×C₅×C₁₆	C₂×C₄	semidirect
38. C₃×#672	C₂×C₄	semidirect
51. C₂×C₅×C₁₆	Dic₃	semidirect
52. C₂×#672	Dic₃	semidirect
53. D₅×C₁₆	Dic₃	semidirect
76. C₂×C₃×C₁₆	AGL(1,𝔽₅)	semidirect
96. C₅×C₁₆	C₂×Dic₃	semidirect
97. #672	C₂×Dic₃	semidirect
125. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
150. C₂×C₁₆	ΓL(1,𝔽₁₆)	semidirect
153. C₂×C₃×C₅	#346	semidirect
155. C₃×D₅	#346	semidirect
195. C₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
196. C₃×C₅	C₂×#346	semidirect
206. C₂×C₅	#4502	semidirect
208. D₅	#4502	semidirect
226. C₂×C₃	#64844	semidirect
228. C₅	C₂×#4502	semidirect
238. C₃	C₂×#64844	semidirect
241. C₂	#11746715	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×C₈×ΓL(1,𝔽₁₆)	2
C₂×D₅⋊(C₃⋊C₁₆)	2
C₂×C₃×D₅×C₁₆	2
#11746715	2
C₂×#64844	3
C₂×#4502	5

Subgroups — C2×#11746715 #14312549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746715 #14312549