Subgroups — C₂×C₂₃×D₄⋊C₁₆ #19159207

Order 5888.

470 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×D₄⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₈×C₂₃	—	2	1	no	—
6	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₂²×C₁₆×C₂₃	—	2	1	no	—
9	C₈×D₄×C₂₃	—	2	1	no	—
10	C₈×D₄×C₂₃	—	2	1	no	—
11	C₈×D₄×C₂₃	C₄	1	2	yes	—
12	C₈×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₁₆	—	2	1	no	—
17	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₄×C₁₆×C₂₃	—	2	1	no	—
19	C₄×C₁₆×C₂₃	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	4	2	no	—
26	C₂²×C₈×C₂₃	—	4	2	no	—
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₂₃	—	2	1	no	—
30	C₄×D₄×C₂₃	—	2	1	no	—
31	C₄×D₄×C₂₃	C₈	1	2	yes	—
32	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₂₃	—	2	4	no	—
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂×C₁₆×C₂₃	—	2	2	no	—
39	C₂×C₁₆×C₂₃	—	4	2	no	—
40	C₂₃×C₄⋊C₈	—	2	1	no	—
41	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂₃×C₂²⋊C₈	—	2	2	no	—
43	C₂₃×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	C₁₆	1	4	yes	complemented
52	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	4	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	D₈	1	2	yes	—
70	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
71	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
80	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₁₆×C₂₃	—	2	4	no	—
82	C₁₆×C₂₃	—	4	2	no	—
83	C₂×D₄⋊C₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	2	4	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	C₂×C₁₆	1	4	yes	complemented
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₂₃	#102	1	2	yes	—
102	C₂×C₄×C₂₃	#104	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
104	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
108	C₈×C₂₃	—	2	1	no	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	4	2	no	—
111	C₈×C₂₃	#102	1	2	yes	—
112	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
113	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
115	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
117	D₄⋊C₁₆	C₂×C₂₃	1	4	yes	direct product factor
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	4	8	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
123	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
124	C₂²×C₂₃	#350	1	2	yes	—
125	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
130	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
131	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
132	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
133	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
134	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
135	C₂³×C₈	—	2	1	no	—
136	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
137	C₂²×C₁₆	—	2	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	C₄×C₂₃	1	2	yes	—
142	C₈×D₄	C₂²×C₂₃	1	2	yes	—
143	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₈	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₄⋊C₁₆	—	2	1	no	—
147	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
148	C₄×C₁₆	—	2	1	no	—
149	C₄×C₁₆	C₂²×C₂₃	1	2	yes	complemented
150	C₂×C₂₃	—	4	4	no	—
151	C₂×C₂₃	D₄⋊C₁₆	1	4	yes	direct product factor
152	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
153	C₂×C₂₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
154	C₂×C₂₃	C₂×#350	1	1	yes	characteristic
155	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
156	C₂³×C₄	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	4	2	no	—
161	C₂²×C₈	—	4	2	no	—
162	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₈×C₂₃	1	2	yes	—
167	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	2	1	no	—
170	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	4	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
181	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
182	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂₃	C₂×D₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₁₆×C₂₃	1	4	yes	complemented
188	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₈	D₈×C₂₃	1	2	yes	—
206	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
215	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
216	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₁₆	—	2	4	no	—
218	C₁₆	—	4	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂×C₁₆×C₂₃	1	4	yes	complemented
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
234	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂₃×#102	1	2	yes	—
237	C₂×C₄	C₂₃×#104	1	2	yes	—
238	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
239	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	C₂₃×#102	1	2	yes	—
247	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
248	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
249	C₂²	—	2	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
254	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
255	C₂²	C₂₃×#350	1	2	yes	—
256	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
261	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
262	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
263	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
264	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
265	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
266	C₂	—	4	4	no	—
267	C₂	C₂₃×D₄⋊C₁₆	1	4	yes	direct product factor
268	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
269	C₂	C₂×C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
270	C₂	C₂×C₂₃×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₃	D₄
commutator	C₄	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂³×C₂₃	C₂²⋊C₈
prime radical	C₂²×D₄×C₂₃	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₄×C₁₆×C₂₃	C₂	semidirect
4. C₂₃×D₄⋊C₁₆	C₂	direct (N × H)
17. C₂₃×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₆×C₂₃	C₂²	semidirect
51. C₂×D₄×C₂₃	C₁₆	semidirect
83. C₂×D₄⋊C₁₆	C₂₃	direct (N × H)
91. D₄×C₂₃	C₂×C₁₆	semidirect
115. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
116. C₂×C₄×C₁₆	C₂×C₂₃	semidirect
117. D₄⋊C₁₆	C₂×C₂₃	direct (N × H)
147. C₄⋊C₁₆	C₂²×C₂₃	semidirect
149. C₄×C₁₆	C₂²×C₂₃	semidirect
151. C₂×C₂₃	D₄⋊C₁₆	direct (N × H)
183. C₂₃	C₂×D₄⋊C₁₆	direct (N × H)
187. C₂×D₄	C₁₆×C₂₃	semidirect
226. D₄	C₂×C₁₆×C₂₃	semidirect
267. C₂	C₂₃×D₄⋊C₁₆	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×D₄×C₂₃	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×D₄⋊C₁₆	2
C₂×D₄⋊C₁₆	23

Subgroups — C2×C23×D4⋊C16 #19159207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×D₄⋊C₁₆ #19159207