Subgroups — C₂×#11746476 #14314561

Order 1920.

336 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	C₂×#91223	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3994	C₂	1	1	yes	maximal, characteristic
3	C₂×D₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11746476	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64849	—	3	1	no	maximal, Hall
6	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×C₃×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	#91223	—	2	1	no	—
10	#91223	C₂²	1	2	yes	—
11	C₃×#3994	—	2	1	no	—
12	C₃×#3994	C₂²	1	2	yes	—
13	C₂×C₃⋊#672	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic, complemented
15	D₅×C₃⋊C₁₆	C₄	1	2	yes	complemented
16	D₅×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×#4489	—	5	1	no	maximal, Hall
18	C₂×#3994	—	3	1	no	—
19	C₂×#3994	D₃	1	1	yes	characteristic
20	C₂×D₅×C₁₆	—	3	1	no	—
21	#64849	—	3	4	no	—
22	C₂²×ΓL(1,𝔽₁₆)	—	4	1	no	—
23	C₂²×C₃×AGL(1,𝔽₅)	—	4	1	no	—
24	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
25	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
26	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
27	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
28	C₃×C₄⋊AGL(1,𝔽₅)	—	4	1	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×#168	Q₈	1	1	yes	characteristic
31	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₈×D₅	D₄	1	1	yes	characteristic
33	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
34	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
35	C₃⋊#672	C₂×C₄	1	2	yes	complemented
36	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
37	C₂×C₃×#107	—	5	1	no	—
38	C₂×#818	—	5	1	no	—
39	C₂²×C₃⋊C₁₆	—	5	1	no	—
40	#4489	—	5	4	no	—
41	C₂×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
42	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
43	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
44	#3994	—	2	1	no	—
45	#3994	—	3	2	no	—
46	#3994	—	6	1	no	—
47	#3994	C₂×D₃	1	2	yes	—
48	C₂×C₅×C₁₆	—	3	1	no	—
49	C₂×#672	—	3	1	no	—
50	D₅×C₁₆	—	3	2	no	—
51	D₅×C₁₆	—	3	2	no	—
52	C₂×#366	—	15	1	no	Sylow
53	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
54	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
55	C₂×ΓL(1,𝔽₁₆)	—	8	2	no	—
56	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
57	C₂×C₃×AGL(1,𝔽₅)	—	8	2	no	—
58	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×Dic₅	Q₁₆	1	1	yes	characteristic
60	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₄×D₅	D₈	1	1	yes	characteristic
62	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
63	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
64	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
67	C₃×#168	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₃×C₈	—	5	1	no	—
69	C₂×C₃×C₄⋊C₄	—	10	1	no	—
70	C₂×C₄⋊Dic₃	—	10	1	no	—
71	C₃×#107	—	5	2	no	—
72	C₃×#107	—	10	1	no	—
73	#818	—	5	2	no	—
74	#818	—	10	1	no	—
75	C₂×C₃⋊C₁₆	—	5	1	no	—
76	C₂×C₃⋊C₁₆	—	5	2	no	—
77	C₂×C₃⋊C₁₆	—	5	2	no	—
78	C₂×C₃⋊C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
79	C₂²×AGL(1,𝔽₅)	—	4	1	no	—
80	C₂²×AGL(1,𝔽₅)	—	12	1	no	—
81	C₂×C₄×D₅	#67	1	1	yes	characteristic
82	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
83	C₄⋊AGL(1,𝔽₅)	—	4	1	no	—
84	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
85	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
86	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
87	C₂×#168	Dic₆	1	1	yes	characteristic
88	C₈×D₅	Dic₆	1	1	yes	characteristic
89	C₈×D₅	C₄×D₃	1	1	yes	characteristic
90	C₈×D₅	#67	1	1	yes	characteristic
91	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
92	C₅×C₁₆	—	3	2	no	—
93	#672	—	3	2	no	—
94	C₂²×C₁₆	—	15	1	no	—
95	C₂×#107	—	5	1	no	—
96	C₂×#107	—	15	1	no	—
97	#366	—	15	4	no	—
98	C₂²×C₃×C₅	#107	1	1	yes	characteristic
99	C₂×C₃×D₅	#107	1	1	yes	characteristic
100	C₂×C₃×D₅	D₁₆	1	2	yes	—
101	C₂×C₃×D₅	Q₃₂	1	2	yes	—
102	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
103	ΓL(1,𝔽₁₆)	—	8	2	no	—
104	C₃×AGL(1,𝔽₅)	—	8	2	no	—
105	C₃×C₄×C₅	#107	1	1	yes	characteristic
106	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₃×Dic₅	#107	1	1	yes	characteristic
108	C₃×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	5	1	no	—
110	C₂²×C₃×C₄	—	20	1	no	—
111	C₂²×Dic₃	—	20	1	no	—
112	C₂×C₃×C₈	—	5	1	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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	10	2	no	—
119	C₃×C₄⋊C₄	—	20	1	no	—
120	C₄⋊Dic₃	—	10	2	no	—
121	C₄⋊Dic₃	—	20	1	no	—
122	C₃⋊C₁₆	—	5	2	no	—
123	C₃⋊C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
124	C₂²×D₅	#213	1	1	yes	characteristic
125	C₂×AGL(1,𝔽₅)	—	4	2	no	—
126	C₂×AGL(1,𝔽₅)	—	8	2	no	—
127	C₂×AGL(1,𝔽₅)	—	12	2	no	—
128	C₂×AGL(1,𝔽₅)	—	24	2	no	—
129	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
130	C₂×Dic₅	#216	1	1	yes	characteristic
131	C₄×D₅	Dic₃⋊C₄	1	1	yes	characteristic
132	C₄×D₅	#213	1	1	yes	characteristic
133	C₄×D₅	#216	1	1	yes	characteristic
134	C₄×D₅	C₂×#67	1	1	yes	characteristic
135	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
136	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
137	#168	Dic₃⋊C₄	1	1	yes	characteristic
138	#168	C₂×Dic₆	1	1	yes	characteristic
139	C₂²×C₈	—	5	1	no	—
140	C₂×C₄⋊C₄	—	10	1	no	—
141	C₂×C₄⋊C₄	—	30	1	no	—
142	C₂×C₁₆	—	3	1	no	—
143	C₂×C₁₆	—	15	2	no	—
144	C₂×C₁₆	—	15	2	no	—
145	C₂×C₁₆	—	15	1	no	—
146	#107	—	5	2	no	—
147	#107	—	10	1	no	—
148	#107	—	15	2	no	—
149	#107	—	30	1	no	—
150	C₂×C₃×C₅	#366	1	2	yes	complemented
151	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
152	C₃×D₅	#366	1	2	yes	complemented
153	C₃×D₅	C₂×D₁₆	1	1	yes	characteristic
154	C₃×D₅	C₂×Q₃₂	1	1	yes	characteristic
155	C₂³×C₃	—	5	1	no	—
156	C₂×C₃×C₄	—	5	1	no	—
157	C₂×C₃×C₄	—	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	—	20	2	no	—
162	C₂×C₃×C₄	—	40	2	no	—
163	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂×Dic₃	—	20	2	no	—
165	C₂×Dic₃	—	40	2	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	C₂²×C₅	#807	1	1	yes	characteristic
171	C₂×D₅	#807	1	1	yes	characteristic
172	C₂×D₅	#826	1	2	yes	—
173	C₂×D₅	#829	1	2	yes	—
174	C₂×D₅	C₂×#213	1	1	yes	characteristic
175	AGL(1,𝔽₅)	—	8	2	no	—
176	AGL(1,𝔽₅)	—	24	2	no	—
177	C₄×C₅	#807	1	1	yes	characteristic
178	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
179	Dic₅	#807	1	1	yes	characteristic
180	Dic₅	C₂×#216	1	1	yes	characteristic
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	20	1	no	—
183	C₂²×C₄	—	60	1	no	—
184	C₂×C₈	—	5	1	no	—
185	C₂×C₈	—	5	1	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	5	1	no	—
188	C₂×C₈	—	5	1	no	—
189	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₄⋊C₄	—	10	2	no	—
191	C₄⋊C₄	—	20	1	no	—
192	C₄⋊C₄	—	30	2	no	—
193	C₄⋊C₄	—	60	1	no	—
194	C₁₆	—	3	2	no	—
195	C₁₆	—	15	2	no	—
196	C₃×C₅	C₂×#366	1	1	yes	characteristic, complemented, Hall
197	C₂²×C₃	—	5	2	no	—
198	C₂²×C₃	—	5	2	no	—
199	C₂²×C₃	—	5	1	no	—
200	C₂²×C₃	—	5	1	no	—
201	C₂²×C₃	#3994	1	1	yes	characteristic
202	C₃×C₄	—	5	1	no	—
203	C₃×C₄	—	5	1	no	—
204	C₃×C₄	—	40	2	no	—
205	C₃×C₄	#3994	1	1	yes	characteristic
206	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
207	Dic₃	—	40	2	no	—
208	C₂×C₅	#4489	1	2	yes	complemented
209	C₂×C₅	C₂×#807	1	1	yes	characteristic
210	D₅	#4489	1	2	yes	complemented
211	D₅	C₂×#826	1	1	yes	characteristic
212	D₅	C₂×#829	1	1	yes	characteristic
213	C₂³	—	5	1	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	5	1	no	—
217	C₂×C₄	—	5	1	no	—
218	C₂×C₄	—	5	1	no	—
219	C₂×C₄	—	20	2	no	—
220	C₂×C₄	—	40	2	no	—
221	C₂×C₄	—	60	2	no	—
222	C₂×C₄	—	120	2	no	—
223	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₈	—	5	1	no	—
225	C₈	—	5	1	no	—
226	C₈	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₈	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂×C₃	—	5	1	no	—
229	C₂×C₃	—	5	2	no	—
230	C₂×C₃	—	5	1	no	—
231	C₂×C₃	#64849	1	2	yes	complemented
232	C₂×C₃	C₂×#3994	1	1	yes	characteristic
233	C₅	C₂×#4489	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	5	1	no	—
235	C₂²	—	5	1	no	—
236	C₂²	—	5	2	no	—
237	C₂²	—	5	2	no	—
238	C₂²	#91160	1	1	yes	characteristic
239	C₄	—	5	1	no	—
240	C₄	—	5	1	no	—
241	C₄	—	40	2	no	—
242	C₄	—	120	2	no	—
243	C₄	#91160	1	1	yes	characteristic
244	C₄	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₃	C₂×#64849	1	1	yes	characteristic, complemented, Sylow
246	C₂	—	5	1	no	—
247	C₂	—	5	1	no	—
248	C₂	—	5	2	no	—
249	C₂	#11746476	1	2	yes	direct product factor
250	C₂	C₂×#91160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91160
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₈	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	#107
prime radical	C₂²×C₃×D₅	D₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₂×#366
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11746476	C₂	direct (N × H)
13. C₂×C₃⋊#672	C₄	semidirect
14. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
15. D₅×C₃⋊C₁₆	C₄	semidirect
35. C₃⋊#672	C₂×C₄	semidirect
36. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
78. C₂×C₃⋊C₁₆	AGL(1,𝔽₅)	semidirect
123. C₃⋊C₁₆	C₂×AGL(1,𝔽₅)	semidirect
150. C₂×C₃×C₅	#366	semidirect
152. C₃×D₅	#366	semidirect
196. C₃×C₅	C₂×#366	semidirect
208. C₂×C₅	#4489	semidirect
210. D₅	#4489	semidirect
231. C₂×C₃	#64849	semidirect
233. C₅	C₂×#4489	semidirect
245. C₃	C₂×#64849	semidirect
249. C₂	#11746476	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×#91223	2
C₂×C₃×#3994	2
C₂×D₅×C₃⋊C₁₆	2
#11746476	2
C₂×#64849	3
C₂×#4489	5

Subgroups — C2×#11746476 #14314561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746476 #14314561