Subgroups — C₄×Q₁₆×D₂₃ #15682158

Order 2944.

488 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₂₃	C₂	1	2	yes	maximal
3	C₄×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092511	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#67275	C₂	1	2	yes	maximal, complemented
7	#13092509	C₂	1	2	yes	maximal, complemented
8	#107×D₂₃	C₂	1	1	yes	maximal, characteristic
9	#103×D₂₃	C₂	1	2	yes	maximal
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
15	Q₈×Dic₂₃	C₂²	1	2	yes	—
16	C₂×#67275	C₂²	1	2	yes	—
17	Q₁₆×D₂₃	C₄	1	8	yes	direct product factor
18	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₂₃	—	2	2	no	—
22	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
23	#10625058	—	2	2	no	—
24	#10625058	C₂²	1	2	yes	—
25	C₄²×D₂₃	—	2	2	no	—
26	C₄²×D₂₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₄₆	—	2	2	no	—
28	C₄×Dic₄₆	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	#10625017	C₂²	1	2	yes	—
33	#10625000	C₂²	1	1	yes	characteristic
34	#10624998	C₂²	1	2	yes	—
35	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
37	#10624992	C₂²	1	2	yes	—
38	#10624989	C₂²	1	2	yes	—
39	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
41	Q₈×D₂₃	—	2	4	no	—
42	Q₈×D₂₃	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₂₃	—	2	2	no	—
44	C₂×C₄×D₂₃	—	2	2	no	—
45	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₆	—	2	2	no	—
49	C₂×Dic₄₆	C₂³	1	2	yes	—
50	Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	#67275	C₂×C₄	1	8	yes	complemented
58	C₄⋊Dic₂₃	—	2	2	no	—
59	C₄⋊Dic₂₃	C₂³	1	2	yes	—
60	Dic₂₃⋊C₄	—	2	2	no	—
61	Dic₂₃⋊C₄	—	2	2	no	—
62	Dic₂₃⋊C₄	—	2	2	no	—
63	C₄×Dic₂₃	—	2	2	no	—
64	C₄×Dic₂₃	—	2	2	no	—
65	C₄×Dic₂₃	—	2	2	no	—
66	C₄×Dic₂₃	D₄	1	1	yes	characteristic
67	C₄×Dic₂₃	D₄	1	1	yes	characteristic
68	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	Dic₉₂	C₂×C₄	1	4	yes	complemented
72	C₈×D₂₃	—	2	2	no	—
73	C₈×D₂₃	C₂×C₄	1	4	yes	—
74	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₂₃	—	2	2	no	—
76	Q₈×C₂₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₂₃	—	2	2	no	—
83	C₂×Dic₂₃	—	2	2	no	—
84	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄×D₂₃	—	2	8	no	—
88	C₄×D₂₃	—	2	2	no	—
89	C₄×D₂₃	—	4	4	no	—
90	C₄×D₂₃	Q₁₆	1	4	yes	direct product factor
91	C₄×D₂₃	C₂²×C₄	1	2	yes	—
92	C₄×D₂₃	#41	1	2	yes	—
93	Dic₄₆	—	2	2	no	—
94	Dic₄₆	—	2	8	no	—
95	Dic₄₆	C₂²×C₄	1	4	yes	—
96	C₈×C₂₃	—	2	1	no	—
97	C₈×C₂₃	C₂²×C₄	1	2	yes	—
98	C₂₃⋊C₈	—	2	1	no	—
99	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄×Q₁₆	—	23	1	no	maximal, Sylow
101	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₂₃	C₄×D₄	1	2	yes	—
103	C₂×D₂₃	C₂×Q₁₆	1	2	yes	—
104	C₂×D₂₃	#135	1	2	yes	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	C₂×Q₁₆	1	2	yes	complemented
109	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
110	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
111	Dic₂₃	—	2	4	no	—
112	Dic₂₃	—	2	1	no	—
113	Dic₂₃	—	4	2	no	—
114	Dic₂₃	C₄×D₄	1	2	yes	—
115	Dic₂₃	C₂×Q₁₆	1	2	yes	complemented
116	C₂²×Q₁₆	—	23	1	no	—
117	C₂×C₄×Q₈	—	23	2	no	—
118	C₄×Q₁₆	—	23	1	no	—
119	C₄×Q₁₆	—	23	2	no	—
120	C₄×Q₁₆	—	23	1	no	—
121	C₄×Q₁₆	—	23	2	no	—
122	C₄×Q₁₆	—	23	1	no	—
123	C₄×Q₁₆	D₂₃	1	1	yes	characteristic, direct product factor
124	C₂×#107	—	23	1	no	—
125	C₂×#103	—	23	2	no	—
126	C₂×C₄×C₈	—	23	1	no	—
127	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
129	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
130	D₂₃	C₄×Q₁₆	1	4	yes	direct product factor
131	C₂²×Q₈	—	23	2	no	—
132	C₂×Q₁₆	—	23	2	no	—
133	C₂×Q₁₆	—	23	8	no	—
134	C₂×Q₁₆	—	23	1	no	—
135	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
136	C₂²×C₈	—	23	1	no	—
137	C₂²×C₈	—	23	1	no	—
138	C₄×Q₈	—	23	2	no	—
139	C₄×Q₈	—	23	2	no	—
140	C₄×Q₈	—	23	2	no	—
141	C₄×Q₈	—	46	2	no	—
142	C₄×Q₈	—	46	2	no	—
143	C₄×Q₈	C₂×D₂₃	1	2	yes	—
144	C₂×C₄⋊C₄	—	23	2	no	—
145	C₂×C₄⋊C₄	—	46	2	no	—
146	C₂×C₄²	—	23	1	no	—
147	C₂×C₄²	—	46	2	no	—
148	#107	—	23	1	no	—
149	#107	—	23	2	no	—
150	#107	C₂×D₂₃	1	1	yes	characteristic
151	#103	—	23	2	no	—
152	#103	—	23	2	no	—
153	#103	—	23	2	no	—
154	#103	C₂×D₂₃	1	2	yes	—
155	C₄×C₈	—	23	1	no	—
156	C₄×C₈	—	23	1	no	—
157	C₄×C₈	—	23	1	no	—
158	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
159	C₂₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₈	—	23	8	no	—
161	C₂×Q₈	—	23	2	no	—
162	C₂×Q₈	—	46	2	no	—
163	C₂×Q₈	—	46	4	no	—
164	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
165	C₂²×C₄	—	23	1	no	—
166	C₂²×C₄	—	23	1	no	—
167	C₂²×C₄	—	23	1	no	—
168	C₂²×C₄	—	46	2	no	—
169	C₂²×C₄	—	46	2	no	—
170	Q₁₆	—	23	8	no	—
171	Q₁₆	—	23	4	no	—
172	Q₁₆	C₄×D₂₃	1	4	yes	direct product factor
173	C₂×C₈	—	23	1	no	—
174	C₂×C₈	—	23	4	no	—
175	C₂×C₈	—	23	1	no	—
176	C₂×C₈	—	46	2	no	—
177	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	23	2	no	—
181	C₄⋊C₄	—	46	2	no	—
182	C₄⋊C₄	—	46	2	no	—
183	C₄⋊C₄	—	46	2	no	—
184	C₄⋊C₄	—	46	2	no	—
185	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	23	1	no	—
188	C₄²	—	23	1	no	—
189	C₄²	—	23	1	no	—
190	C₄²	—	46	2	no	—
191	C₄²	—	46	2	no	—
192	C₄²	—	46	2	no	—
193	C₄²	C₂²×D₂₃	1	1	yes	characteristic
194	C₂³	—	23	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	23	4	no	—
197	Q₈	—	46	8	no	—
198	Q₈	—	46	2	no	—
199	Q₈	C₂×C₄×D₂₃	1	4	yes	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	23	1	no	—
203	C₂×C₄	—	23	1	no	—
204	C₂×C₄	—	23	4	no	—
205	C₂×C₄	—	23	2	no	—
206	C₂×C₄	—	23	2	no	—
207	C₂×C₄	—	23	1	no	—
208	C₂×C₄	—	46	2	no	—
209	C₂×C₄	—	46	8	no	—
210	C₂×C₄	—	46	2	no	—
211	C₂×C₄	—	46	2	no	—
212	C₂×C₄	—	92	4	no	—
213	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
214	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	23	2	no	—
218	C₈	—	46	1	no	—
219	C₈	C₂×C₄×D₂₃	1	2	yes	—
220	C₂²	—	23	2	no	—
221	C₂²	—	23	2	no	—
222	C₂²	—	23	2	no	—
223	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	23	2	no	—
228	C₄	—	23	2	no	—
229	C₄	—	46	4	no	—
230	C₄	—	46	1	no	—
231	C₄	—	92	2	no	—
232	C₄	Q₁₆×D₂₃	1	2	yes	direct product factor
233	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
234	C₄	#41×D₂₃	1	1	yes	characteristic
235	C₂	—	23	4	no	—
236	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
237	C₂	C₂×Q₁₆×D₂₃	1	1	yes	characteristic
238	C₂	#135×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×Q₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂²×D₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×Q₁₆	D₂₃
hypocenter	C₂₃	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₉₂	C₂	semidirect
4. Q₁₆×Dic₂₃	C₂	semidirect
5. #13092511	C₂	semidirect
6. C₄×#67275	C₂	semidirect
7. #13092509	C₂	semidirect
11. C₄×Q₁₆×C₂₃	C₂	semidirect
17. Q₁₆×D₂₃	C₄	direct (N × H)
50. Q₁₆×C₂₃	C₂×C₄	semidirect
57. #67275	C₂×C₄	semidirect
71. Dic₉₂	C₂×C₄	semidirect
90. C₄×D₂₃	Q₁₆	direct (N × H)
108. C₄×C₂₃	C₂×Q₁₆	semidirect
115. Dic₂₃	C₂×Q₁₆	semidirect
123. C₄×Q₁₆	D₂₃	direct (N × H)
130. D₂₃	C₄×Q₁₆	direct (N × H)
159. C₂₃	C₂×C₄×Q₁₆	semidirect
172. Q₁₆	C₄×D₂₃	direct (N × H)
232. C₄	Q₁₆×D₂₃	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×Q₁₆×D₂₃	2
C₄×Q₈×D₂₃	2
C₄×Dic₉₂	2
Q₁₆×Dic₂₃	2
#13092511	2
C₄×#67275	2
#13092509	2
#107×D₂₃	2
#103×D₂₃	2
C₄×C₈×D₂₃	2
C₄×Q₁₆×C₂₃	2
C₂×C₄×Q₁₆	23

Subgroups — C4×Q16×D23 #15682158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₁₆×D₂₃ #15682158