Subgroups — D₂₃×#344 #15673137

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092189	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092188	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#117	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	1	yes	characteristic
15	M₁₆×D₂₃	—	2	1	no	—
16	M₁₆×D₂₃	—	2	1	no	—
17	M₁₆×D₂₃	—	2	1	no	—
18	M₁₆×D₂₃	—	2	1	no	—
19	M₁₆×D₂₃	—	2	1	no	—
20	M₁₆×D₂₃	—	2	1	no	—
21	M₁₆×D₂₃	C₄	1	2	yes	complemented
22	M₁₆×D₂₃	C₄	1	2	yes	complemented
23	C₂×#67262	—	2	1	no	—
24	C₂×#67262	C₄	1	2	yes	complemented
25	C₂×C₈×D₂₃	—	2	1	no	—
26	C₂×C₈×D₂₃	C₄	1	2	yes	complemented
27	#10625063	—	2	1	no	—
28	C₄⋊C₄×D₂₃	—	2	1	no	—
29	C₂²⋊C₄×D₂₃	C₄	1	2	yes	—
30	#10625048	C₄	1	2	yes	—
31	#10625043	—	2	1	no	—
32	C₄²×D₂₃	—	2	1	no	—
33	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₃	—	2	2	no	—
37	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
42	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
43	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
44	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
45	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄×D₂₃	Q₈	1	1	yes	characteristic
48	C₂×C₄×D₂₃	Q₈	1	1	yes	characteristic
49	M₁₆×C₂₃	—	2	1	no	—
50	M₁₆×C₂₃	—	2	1	no	—
51	M₁₆×C₂₃	—	2	1	no	—
52	M₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
53	C₂×C₈×C₂₃	—	2	1	no	—
54	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
57	C₄²×C₂₃	—	2	1	no	—
58	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
59	(C₂×D₂₃)⋊C₄	—	2	2	no	—
60	C₄⋊Dic₂₃	—	2	1	no	—
61	Dic₂₃⋊C₄	—	2	2	no	—
62	C₄×Dic₂₃	—	2	1	no	—
63	C₄×Dic₂₃	—	2	2	no	—
64	#67267	—	2	1	no	—
65	#67267	—	2	1	no	—
66	#67267	—	2	1	no	—
67	#67267	C₂×C₄	1	2	yes	complemented
68	C₂×C₂₃⋊C₈	—	2	1	no	—
69	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	complemented
70	#67262	—	2	2	no	—
71	#67262	—	2	2	no	—
72	#67262	—	2	2	no	—
73	#67262	—	2	2	no	—
74	C₈×D₂₃	—	2	2	no	—
75	C₈×D₂₃	—	2	2	no	—
76	C₈×D₂₃	—	2	2	no	—
77	C₈×D₂₃	—	2	2	no	—
78	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₂₃	—	4	2	no	—
80	C₂²×D₂₃	C₄²	1	1	yes	characteristic
81	C₂²×D₂₃	C₄²	1	1	yes	characteristic
82	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×D₂₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×D₂₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×Dic₂₃	—	2	2	no	—
94	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
95	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
96	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×Dic₂₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₂₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₂₃	—	4	4	no	—
101	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
116	C₄×D₂₃	C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	—	2	2	no	—
119	C₂₃⋊C₈	—	2	2	no	—
120	C₂₃⋊C₈	—	2	2	no	—
121	C₂×#344	—	23	1	no	maximal, Sylow
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×D₂₃	—	2	2	no	—
127	C₂×D₂₃	—	2	2	no	—
128	C₂×D₂₃	—	2	2	no	—
129	C₂×D₂₃	—	4	1	no	—
130	C₂×D₂₃	—	4	1	no	—
131	C₂×D₂₃	—	4	1	no	—
132	C₂×D₂₃	#95	1	1	yes	characteristic
133	C₂×D₂₃	#95	1	1	yes	characteristic
134	C₂×D₂₃	#95	1	1	yes	characteristic
135	C₂×D₂₃	#95	1	1	yes	characteristic
136	C₄×C₂₃	—	4	2	no	—
137	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	Dic₂₃	—	4	2	no	—
142	Dic₂₃	#95	1	1	yes	characteristic
143	Dic₂₃	#95	1	1	yes	characteristic
144	Dic₂₃	#95	1	1	yes	characteristic
145	Dic₂₃	#95	1	1	yes	characteristic
146	C₂²×M₁₆	—	23	1	no	—
147	C₂²×M₁₆	—	23	1	no	—
148	C₂×#117	—	23	1	no	—
149	#344	—	23	1	no	—
150	#344	—	23	1	no	—
151	#344	—	23	1	no	—
152	#344	D₂₃	1	1	yes	characteristic, direct product factor
153	C₂×C₂₃	—	2	1	no	—
154	C₂×C₂₃	—	2	1	no	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
157	D₂₃	—	2	1	no	—
158	D₂₃	—	2	1	no	—
159	D₂₃	—	2	1	no	—
160	D₂₃	#344	1	2	yes	direct product factor
161	C₂³×C₄	—	23	1	no	—
162	C₂×M₁₆	—	23	1	no	—
163	C₂×M₁₆	—	23	2	no	—
164	C₂×M₁₆	—	23	1	no	—
165	C₂×M₁₆	—	23	2	no	—
166	C₂×M₁₆	—	23	2	no	—
167	C₂×M₁₆	—	46	1	no	—
168	C₂×M₁₆	—	46	1	no	—
169	C₂×M₁₆	—	46	1	no	—
170	C₂×M₁₆	—	46	1	no	—
171	C₂×M₁₆	—	46	1	no	—
172	C₂×M₁₆	—	46	1	no	—
173	C₂×M₁₆	—	46	1	no	—
174	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
175	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
176	C₂²×C₈	—	23	2	no	—
177	C₂²×C₈	—	46	1	no	—
178	#117	—	23	1	no	—
179	#117	—	23	2	no	—
180	#117	—	46	1	no	—
181	#117	—	46	1	no	—
182	#117	C₂×D₂₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	46	1	no	—
184	C₂×C₂²⋊C₄	—	23	2	no	—
185	C₂×C₄²	—	46	1	no	—
186	C₂₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
187	C₂⁴	—	23	1	no	—
188	C₂²×C₄	—	23	1	no	—
189	C₂²×C₄	—	23	1	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	1	no	—
193	C₂²×C₄	—	23	1	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	1	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	23	1	no	—
201	C₂²×C₄	—	46	2	no	—
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	23	2	no	—
207	M₁₆	—	46	2	no	—
208	M₁₆	—	46	2	no	—
209	M₁₆	—	46	2	no	—
210	M₁₆	—	46	2	no	—
211	M₁₆	—	46	1	no	—
212	M₁₆	—	46	1	no	—
213	M₁₆	—	46	1	no	—
214	M₁₆	C₄×D₂₃	1	2	yes	complemented
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	23	2	no	—
217	C₂×C₈	—	46	2	no	—
218	C₂×C₈	—	46	2	no	—
219	C₂×C₈	—	46	2	no	—
220	C₂×C₈	—	46	2	no	—
221	C₂×C₈	—	46	1	no	—
222	C₂×C₈	C₄×D₂₃	1	2	yes	complemented
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	46	2	no	—
225	C₄⋊C₄	—	46	1	no	—
226	C₂²⋊C₄	—	23	2	no	—
227	C₂²⋊C₄	—	46	2	no	—
228	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
229	C₄²	—	2	1	no	—
230	C₄²	—	46	2	no	—
231	C₄²	—	46	1	no	—
232	C₂³	—	23	1	no	—
233	C₂³	—	23	1	no	—
234	C₂³	—	23	1	no	—
235	C₂³	—	23	1	no	—
236	C₂³	—	23	1	no	—
237	C₂³	—	23	1	no	—
238	C₂³	—	92	2	no	—
239	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	23	1	no	—
243	C₂×C₄	—	23	1	no	—
244	C₂×C₄	—	23	1	no	—
245	C₂×C₄	—	23	1	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	1	no	—
248	C₂×C₄	—	23	1	no	—
249	C₂×C₄	—	23	1	no	—
250	C₂×C₄	—	23	1	no	—
251	C₂×C₄	—	23	1	no	—
252	C₂×C₄	—	23	1	no	—
253	C₂×C₄	—	23	1	no	—
254	C₂×C₄	—	23	1	no	—
255	C₂×C₄	—	23	1	no	—
256	C₂×C₄	—	23	1	no	—
257	C₂×C₄	—	23	1	no	—
258	C₂×C₄	—	23	1	no	—
259	C₂×C₄	—	23	1	no	—
260	C₂×C₄	—	23	1	no	—
261	C₂×C₄	—	23	1	no	—
262	C₂×C₄	—	23	1	no	—
263	C₂×C₄	—	46	2	no	—
264	C₂×C₄	—	92	4	no	—
265	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
267	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
268	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
269	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
270	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	46	2	no	—
274	C₈	—	46	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	23	1	no	—
277	C₂²	—	23	1	no	—
278	C₂²	—	23	1	no	—
279	C₂²	—	23	1	no	—
280	C₂²	—	46	2	no	—
281	C₂²	—	46	2	no	—
282	C₂²	—	46	2	no	—
283	C₂²	—	92	1	no	—
284	C₂²	—	92	1	no	—
285	C₂²	—	92	1	no	—
286	C₂²	C₄²×D₂₃	1	1	yes	characteristic
287	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
288	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	23	1	no	—
291	C₄	—	23	1	no	—
292	C₄	—	23	1	no	—
293	C₄	—	23	1	no	—
294	C₄	—	92	2	no	—
295	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
296	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
297	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
298	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	23	2	no	—
303	C₂	—	46	1	no	—
304	C₂	—	46	1	no	—
305	C₂	—	46	1	no	—
306	C₂	#95×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#344	C₂
socle	C₂×C₂₃	C₂×#95
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#344	D₂₃
hypocenter	C₂₃	C₂×#344
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. #13092189	C₂	semidirect
5. #13092188	C₂	semidirect
6. #13092187	C₂	semidirect
7. C₂₃×#344	C₂	semidirect
21. M₁₆×D₂₃	C₄	semidirect
22. M₁₆×D₂₃	C₄	semidirect
24. C₂×#67262	C₄	semidirect
26. C₂×C₈×D₂₃	C₄	semidirect
52. M₁₆×C₂₃	C₂×C₄	semidirect
54. C₂×C₈×C₂₃	C₂×C₄	semidirect
67. #67267	C₂×C₄	semidirect
69. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
152. #344	D₂₃	direct (N × H)
160. D₂₃	#344	direct (N × H)
186. C₂₃	C₂×#344	semidirect
214. M₁₆	C₄×D₂₃	semidirect
222. C₂×C₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₃	2
#117×D₂₃	2
#13092189	2
#13092188	2
#13092187	2
C₂₃×#344	2
C₂×#344	23

Subgroups — D23×#344 #15673137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#344 #15673137