Subgroups — D₄×M₁₆×C₂₃ #15670553

Order 2944.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#436	C₂	1	2	yes	maximal, complemented
4	C₂₃×#435	C₂	1	4	yes	maximal, complemented
5	C₈×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#407	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×M₁₆×C₂₃	—	2	8	no	—
12	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	C₂²	1	4	yes	complemented
16	C₂²×C₈×C₂₃	C₂²	1	4	yes	complemented
17	C₄×D₄×C₂₃	C₄	1	2	yes	—
18	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
19	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
20	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₂₃	C₂²	1	2	yes	—
22	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
29	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
30	C₂₃×#97	C₂²	1	2	yes	complemented
31	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
32	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₃	—	2	4	no	—
34	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₃	—	2	4	no	—
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	8	no	—
50	M₁₆×C₂₃	D₄	1	4	yes	direct product factor
51	C₂×C₈×C₂₃	—	2	8	no	—
52	C₂×C₈×C₂₃	C₂³	1	4	yes	—
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₄²×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	M₁₆	1	4	yes	direct product factor
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	4	no	—
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	C₂×D₄	1	4	yes	complemented
97	D₄×M₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	4	8	no	—
106	C₂²×C₂₃	C₄×D₄	1	2	yes	—
107	C₂²×C₂₃	C₂×M₁₆	1	4	yes	complemented
108	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	C₄×D₄	1	2	yes	—
115	C₄×C₂₃	C₂×M₁₆	1	2	yes	complemented
116	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂²×M₁₆	C₂×C₂₃	1	2	yes	complemented
119	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
120	#436	C₂×C₂₃	1	2	yes	complemented
121	#435	C₂×C₂₃	1	4	yes	complemented
122	C₈×D₄	C₂×C₂₃	1	2	yes	complemented
123	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#407	C₂×C₂₃	1	2	yes	complemented
125	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×C₂₃	—	2	2	no	—
127	C₂×C₂₃	—	2	4	no	—
128	C₂×C₂₃	—	4	2	no	—
129	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
131	C₂×C₂₃	#568	1	1	yes	characteristic
132	C₂²×D₄	C₄×C₂₃	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
136	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×M₁₆	C₂²×C₂₃	1	4	yes	complemented
139	C₂²×C₈	C₂²×C₂₃	1	4	yes	complemented
140	C₄×D₄	C₄×C₂₃	1	2	yes	—
141	C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
142	C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₂₃	1	2	yes	—
145	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
148	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
150	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
151	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	complemented
152	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	complemented
153	#97	C₂²×C₂₃	1	2	yes	complemented
154	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
155	C₂₃	D₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	C₂×C₄×C₂₃	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
159	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	D₄×C₂₃	1	4	yes	direct product factor
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	C₂³×C₂₃	1	2	yes	—
177	C₂×C₈	C₂³×C₂₃	1	4	yes	—
178	C₂×C₈	C₂³×C₂₃	1	2	yes	—
179	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
180	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
183	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
184	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
185	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
186	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
194	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
195	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	M₁₆×C₂₃	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂×C₄	#41×C₂₃	1	2	yes	—
217	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	2	2	no	—
220	C₈	C₂×D₄×C₂₃	1	4	yes	complemented
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₄×D₄×C₂₃	1	2	yes	—
230	C₂²	C₂×M₁₆×C₂₃	1	4	yes	complemented
231	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	C₄×D₄×C₂₃	1	2	yes	—
238	C₄	C₂×M₁₆×C₂₃	1	2	yes	complemented
239	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
240	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
245	C₂	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
246	C₂	C₂₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂²×D₄×C₂₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₁₆×C₂₃	C₂	semidirect
3. C₂₃×#436	C₂	semidirect
4. C₂₃×#435	C₂	semidirect
5. C₈×D₄×C₂₃	C₂	semidirect
6. C₂₃×#423	C₂	semidirect
7. C₂₃×#407	C₂	semidirect
8. C₄×M₁₆×C₂₃	C₂	semidirect
15. C₂×M₁₆×C₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₃	C₂²	semidirect
26. C₂₃×C₄⋊C₈	C₂²	semidirect
27. C₂₃×C₄⋊C₈	C₂²	semidirect
28. C₂₃×C₂²⋊C₈	C₂²	semidirect
29. C₂₃×C₂²⋊C₈	C₂²	semidirect
30. C₂₃×#97	C₂²	semidirect
31. C₄×C₈×C₂₃	C₂²	semidirect
50. M₁₆×C₂₃	D₄	direct (N × H)
74. D₄×C₂₃	M₁₆	direct (N × H)
96. C₈×C₂₃	C₂×D₄	semidirect
97. D₄×M₁₆	C₂₃	direct (N × H)
107. C₂²×C₂₃	C₂×M₁₆	semidirect
115. C₄×C₂₃	C₂×M₁₆	semidirect
118. C₂²×M₁₆	C₂×C₂₃	semidirect
120. #436	C₂×C₂₃	semidirect
121. #435	C₂×C₂₃	semidirect
122. C₈×D₄	C₂×C₂₃	semidirect
123. #423	C₂×C₂₃	semidirect
124. #407	C₂×C₂₃	semidirect
125. C₄×M₁₆	C₂×C₂₃	semidirect
138. C₂×M₁₆	C₂²×C₂₃	semidirect
139. C₂²×C₈	C₂²×C₂₃	semidirect
149. C₄⋊C₈	C₂²×C₂₃	semidirect
150. C₄⋊C₈	C₂²×C₂₃	semidirect
151. C₂²⋊C₈	C₂²×C₂₃	semidirect
152. C₂²⋊C₈	C₂²×C₂₃	semidirect
153. #97	C₂²×C₂₃	semidirect
154. C₄×C₈	C₂²×C₂₃	semidirect
155. C₂₃	D₄×M₁₆	direct (N × H)
174. M₁₆	D₄×C₂₃	direct (N × H)
198. D₄	M₁₆×C₂₃	direct (N × H)
220. C₈	C₂×D₄×C₂₃	semidirect
230. C₂²	C₂×M₁₆×C₂₃	semidirect
238. C₄	C₂×M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#436	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#423	2
C₂₃×#407	2
C₄×M₁₆×C₂₃	2
D₄×M₁₆	23

Subgroups — D4×M16×C23 #15670553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×M₁₆×C₂₃ #15670553