Subgroups — D₅⋊(C₄⋊(C₃⋊C₁₆)) #14314305

Order 1920.

368 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×D₅⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11746759	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₃⋊#674)	C₂	1	2	yes	maximal, complemented
5	D₅⋊(C₄⋊C₁₆)	—	3	1	no	maximal, Hall
6	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₃×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂×C₃⋊#674	C₂²	1	4	yes	—
10	D₅⋊(C₃⋊C₁₆)	—	2	4	no	—
11	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
13	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×#168	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₁₆)	—	5	1	no	maximal, Hall
16	C₂×D₅⋊C₁₆	—	3	2	no	—
17	C₄×C₈×D₅	D₃	1	1	yes	characteristic
18	#64888	—	3	2	no	—
19	C₄⋊#674	—	3	2	no	—
20	C₂×C₃×C₄×D₅	C₈	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₈	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄×Dic₅	C₈	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₈	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×#168	C₂³	1	1	yes	characteristic
31	C₃×C₈×D₅	—	2	2	no	—
32	C₃×C₈×D₅	D₄	1	2	yes	—
33	C₃×C₈×D₅	Q₈	1	2	yes	—
34	C₃⋊#674	—	2	4	no	—
35	C₂×C₃×C₄×C₈	—	5	1	no	—
36	C₂²×C₃⋊C₁₆	—	5	2	no	—
37	C₄⋊(C₃⋊C₁₆)	—	5	2	no	—
38	C₄⋊(C₃⋊C₁₆)	—	5	2	no	—
39	C₂×C₈×D₅	Dic₃	1	1	yes	characteristic
40	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
41	C₄²×D₅	Dic₃	1	1	yes	characteristic
42	C₂×#674	—	3	4	no	—
43	D₅⋊C₁₆	—	6	4	no	—
44	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic
45	C₈×Dic₅	Dic₃	1	1	yes	characteristic
46	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
47	C₄×#168	C₂×D₃	1	1	yes	characteristic
48	C₂×C₄⋊C₁₆	—	15	1	no	Sylow
49	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
56	C₃×C₄×D₅	—	2	2	no	—
57	C₃×C₄×D₅	C₁₆	1	4	yes	complemented
58	C₃×C₄×D₅	C₄⋊C₄	1	2	yes	—
59	C₃×C₄×D₅	M₁₆	1	2	yes	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
62	C₃×#168	—	2	1	no	—
63	C₃×#168	C₂×D₄	1	1	yes	characteristic
64	C₃×#168	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₃×C₈	—	5	1	no	—
66	C₂²×C₃×C₈	—	5	1	no	—
67	C₂×C₃×C₄²	—	5	1	no	—
68	C₃×C₄×C₈	—	5	1	no	—
69	C₃×C₄×C₈	—	5	1	no	—
70	C₃×C₄×C₈	—	5	1	no	—
71	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×C₃⋊C₁₆	—	5	4	no	—
73	C₂×C₃⋊C₁₆	—	10	4	no	—
74	C₂×C₄×D₅	C₃⋊C₈	1	1	yes	characteristic
75	C₂×C₄×D₅	C₃⋊C₈	1	1	yes	characteristic
76	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
80	C₄×Dic₅	C₃⋊C₈	1	1	yes	characteristic
81	C₄×Dic₅	C₃⋊C₈	1	1	yes	characteristic
82	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
83	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
84	C₂×#168	C₂²×D₃	1	1	yes	characteristic
85	C₈×D₅	—	2	2	no	—
86	C₈×D₅	Dic₆	1	2	yes	—
87	C₈×D₅	D₁₂	1	2	yes	—
88	#674	—	6	4	no	—
89	C₂²×C₁₆	—	15	2	no	—
90	C₂×C₄×C₈	—	5	1	no	—
91	C₄⋊C₁₆	—	15	2	no	—
92	C₄⋊C₁₆	—	15	2	no	—
93	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
95	C₂×C₃×D₅	C₂×C₁₆	1	2	yes	—
96	C₂×C₃×D₅	M₃₂	1	2	yes	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	C₂×C₁₆	1	2	yes	complemented
99	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₃×Dic₅	—	2	1	no	—
102	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
103	C₃×Dic₅	C₂×C₁₆	1	2	yes	complemented
104	C₂²×C₃×C₄	—	5	1	no	—
105	C₂²×C₃×C₄	—	5	1	no	—
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂×C₃×C₈	—	5	1	no	—
108	C₂×C₃×C₈	—	5	2	no	—
109	C₂×C₃×C₈	—	5	2	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	10	2	no	—
112	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₃×C₄²	—	5	1	no	—
115	C₃×C₄²	—	5	1	no	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₃⋊C₁₆	—	10	4	no	—
119	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
123	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
124	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
125	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
126	C₄×D₅	—	2	2	no	—
127	C₄×D₅	C₃⋊C₁₆	1	4	yes	complemented
128	C₄×D₅	#208	1	2	yes	—
129	C₄×D₅	C₄⋊Dic₃	1	2	yes	—
130	C₅×C₈	—	2	1	no	—
131	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
132	#168	—	2	1	no	—
133	#168	C₂×Dic₆	1	1	yes	characteristic
134	#168	C₂×D₁₂	1	1	yes	characteristic
135	C₂²×C₈	—	5	1	no	—
136	C₂²×C₈	—	5	1	no	—
137	C₂×C₄²	—	5	1	no	—
138	C₂×C₁₆	—	15	4	no	—
139	C₂×C₁₆	—	30	4	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	—	5	1	no	—
143	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
144	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
147	C₃×D₅	C₄⋊C₁₆	1	4	yes	complemented
148	C₂³×C₃	—	5	1	no	—
149	C₂×C₃×C₄	—	5	1	no	—
150	C₂×C₃×C₄	—	5	1	no	—
151	C₂×C₃×C₄	—	5	2	no	—
152	C₂×C₃×C₄	—	5	2	no	—
153	C₂×C₃×C₄	—	5	4	no	—
154	C₂×C₃×C₄	—	5	1	no	—
155	C₂×C₃×C₄	—	10	2	no	—
156	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
157	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	5	1	no	—
161	C₃×C₈	—	5	1	no	—
162	C₃×C₈	—	10	1	no	—
163	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
164	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
165	C₂×D₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
166	C₂×D₅	C₂×C₃⋊C₁₆	1	2	yes	—
167	C₂×D₅	#812	1	2	yes	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	C₂×C₃⋊C₁₆	1	2	yes	complemented
170	C₄×C₅	C₂×#208	1	1	yes	characteristic
171	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
172	Dic₅	—	2	1	no	—
173	Dic₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
174	Dic₅	C₂×C₃⋊C₁₆	1	2	yes	complemented
175	C₂²×C₄	—	5	1	no	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	1	no	—
178	C₂×C₈	—	5	1	no	—
179	C₂×C₈	—	5	2	no	—
180	C₂×C₈	—	5	2	no	—
181	C₂×C₈	—	5	1	no	—
182	C₂×C₈	—	10	2	no	—
183	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
184	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
185	C₄²	—	5	1	no	—
186	C₄²	—	5	1	no	—
187	C₄²	—	5	1	no	—
188	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
189	C₁₆	—	30	4	no	—
190	C₃×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	—	5	2	no	—
192	C₂²×C₃	—	5	2	no	—
193	C₂²×C₃	—	5	2	no	—
194	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	5	2	no	—
197	C₃×C₄	—	5	2	no	—
198	C₃×C₄	—	10	1	no	—
199	C₃×C₄	D₅⋊C₁₆	1	2	yes	complemented
200	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
201	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
203	C₂×C₅	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
204	C₂×C₅	C₂×#812	1	1	yes	characteristic
205	D₅	C₄⋊(C₃⋊C₁₆)	1	4	yes	complemented
206	C₂³	—	5	1	no	—
207	C₂×C₄	—	5	1	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	4	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	5	1	no	—
213	C₂×C₄	—	10	2	no	—
214	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
215	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
216	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	5	1	no	—
219	C₈	—	5	1	no	—
220	C₈	—	10	1	no	—
221	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
222	C₂×C₃	—	5	4	no	—
223	C₂×C₃	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
224	C₂×C₃	C₂×D₅⋊C₁₆	1	1	yes	characteristic
225	C₂×C₃	D₅⋊M₃₂	1	1	yes	characteristic
226	C₅	C₂×C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	5	2	no	—
228	C₂²	—	5	2	no	—
229	C₂²	—	5	2	no	—
230	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	5	2	no	—
233	C₄	—	5	2	no	—
234	C₄	—	10	1	no	—
235	C₄	D₅⋊(C₃⋊C₁₆)	1	2	yes	complemented
236	C₄	D₅⋊#208	1	1	yes	characteristic
237	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
238	C₃	D₅⋊(C₄⋊C₁₆)	1	1	yes	characteristic, complemented, Sylow
239	C₂	—	5	4	no	—
240	C₂	D₅⋊(C₄⋊(C₃⋊C₈))	1	1	yes	characteristic
241	C₂	C₂×D₅⋊(C₃⋊C₁₆)	1	1	yes	characteristic
242	C₂	D₅⋊#812	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746759	C₂	semidirect
4. C₄⋊(C₃⋊#674)	C₂	semidirect
57. C₃×C₄×D₅	C₁₆	semidirect
98. C₃×C₄×C₅	C₂×C₁₆	semidirect
103. C₃×Dic₅	C₂×C₁₆	semidirect
127. C₄×D₅	C₃⋊C₁₆	semidirect
147. C₃×D₅	C₄⋊C₁₆	semidirect
169. C₄×C₅	C₂×C₃⋊C₁₆	semidirect
174. Dic₅	C₂×C₃⋊C₁₆	semidirect
190. C₃×C₅	C₂×C₄⋊C₁₆	semidirect
199. C₃×C₄	D₅⋊C₁₆	semidirect
205. D₅	C₄⋊(C₃⋊C₁₆)	semidirect
226. C₅	C₂×C₄⋊(C₃⋊C₁₆)	semidirect
235. C₄	D₅⋊(C₃⋊C₁₆)	semidirect
238. C₃	D₅⋊(C₄⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅⋊(C₃⋊C₁₆)	2
C₃×C₄×C₈×D₅	2
#11746759	2
C₄⋊(C₃⋊#674)	2
D₅⋊(C₄⋊C₁₆)	3
C₂×C₄⋊(C₃⋊C₁₆)	5

Subgroups — D5⋊(C4⋊(C3⋊C16)) #14314305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅⋊(C₄⋊(C₃⋊C₁₆)) #14314305