Subgroups — #19212323

Order 5888.

408 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10625041	C₂	1	1	yes	maximal, characteristic
2	#15673514	C₂	1	1	yes	maximal, characteristic
3	#15671221	C₂	1	2	yes	maximal
4	#15671219	C₂	1	2	yes	maximal
5	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625041	C₄	1	1	yes	characteristic
7	C₂×C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
8	#13092315	C₂²	1	1	yes	characteristic
9	#13092315	C₂²	1	1	yes	characteristic
10	#13092301	—	2	1	no	—
11	#13092301	—	2	1	no	—
12	#13092283	C₄	1	1	yes	characteristic
13	C₄×Dic₂₃⋊C₄	—	2	1	no	—
14	C₄×Dic₂₃⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
17	#13092172	—	2	1	no	—
18	#13092172	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
20	C₄³×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#336	—	2	1	no	—
22	C₂₃×#336	C₂²	1	2	yes	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₂₃⋊C₄	—	2	1	no	—
36	C₂×Dic₂₃⋊C₄	—	2	1	no	—
37	C₂×Dic₂₃⋊C₄	—	2	1	no	—
38	C₂×Dic₂₃⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₂₃	—	2	1	no	—
40	C₂×C₄×Dic₂₃	—	2	1	no	—
41	C₂²×C₂₃⋊C₈	—	2	2	no	—
42	#10625041	—	2	2	no	—
43	#10625041	C₈	1	4	yes	complemented
44	C₂₃×C₄⋊C₈	—	4	2	no	—
45	C₂₃⋊#95	—	2	1	no	—
46	C₂₃⋊#95	—	2	1	no	—
47	C₂₃⋊#95	—	2	1	no	—
48	C₂₃⋊#95	—	2	1	no	—
49	C₂₃⋊#95	—	2	1	no	—
50	C₂₃⋊#95	—	2	1	no	—
51	C₄⋊(C₂₃⋊C₈)	—	4	2	no	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₂₃	—	2	1	no	—
60	C₂²×Dic₂₃	—	2	1	no	—
61	C₂²×Dic₂₃	—	2	1	no	—
62	C₂²×Dic₂₃	—	2	1	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	1	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₄²×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	—	4	1	no	—
72	C₄²×C₂₃	—	4	1	no	—
73	C₄²×C₂₃	—	4	1	no	—
74	C₄²×C₂₃	C₂×C₈	1	2	yes	—
75	C₄²×C₂₃	M₁₆	1	2	yes	—
76	C₄⋊Dic₂₃	—	2	1	no	—
77	C₄⋊Dic₂₃	—	2	1	no	—
78	C₄⋊Dic₂₃	—	4	1	no	—
79	C₄⋊Dic₂₃	—	4	1	no	—
80	C₄⋊Dic₂₃	C₂×C₈	1	2	yes	—
81	C₄⋊Dic₂₃	C₂×C₈	1	2	yes	—
82	Dic₂₃⋊C₄	—	2	4	no	—
83	Dic₂₃⋊C₄	—	2	4	no	—
84	Dic₂₃⋊C₄	—	4	1	no	—
85	Dic₂₃⋊C₄	—	4	1	no	—
86	Dic₂₃⋊C₄	—	4	1	no	—
87	Dic₂₃⋊C₄	—	4	1	no	—
88	C₄×Dic₂₃	—	4	2	no	—
89	C₄×Dic₂₃	—	4	2	no	—
90	C₂×C₂₃⋊C₈	—	4	2	no	—
91	C₂×C₂₃⋊C₈	—	4	2	no	—
92	C₂×C₂₃⋊C₈	—	4	2	no	—
93	#7602	—	23	1	no	maximal, Sylow
94	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₂₃	#101	1	2	yes	—
113	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
119	C₂×Dic₂₃	—	2	2	no	—
120	C₂×Dic₂₃	—	2	2	no	—
121	C₂×Dic₂₃	—	4	1	no	—
122	C₂×Dic₂₃	—	4	1	no	—
123	C₂×Dic₂₃	—	4	1	no	—
124	C₂×Dic₂₃	—	4	1	no	—
125	C₂×Dic₂₃	—	4	1	no	—
126	C₂×Dic₂₃	—	4	1	no	—
127	C₂×Dic₂₃	—	4	1	no	—
128	C₂×Dic₂₃	—	4	1	no	—
129	C₂×Dic₂₃	—	4	1	no	—
130	C₂×Dic₂₃	—	4	1	no	—
131	C₈×C₂₃	—	8	2	no	—
132	C₂₃⋊C₈	—	8	2	no	—
133	C₄×#125	—	23	1	no	—
134	#1831	—	23	1	no	—
135	#1831	D₂₃	1	1	yes	characteristic
136	#1271	—	23	2	no	—
137	#1271	—	23	2	no	—
138	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₂₃	#410	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
141	C₂²×C₂₃	#417	1	1	yes	characteristic
142	C₂²×C₂₃	#417	1	1	yes	characteristic
143	C₂²×C₂₃	#421	1	1	yes	characteristic
144	C₂²×C₂₃	#421	1	1	yes	characteristic
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	2	no	—
154	C₄×C₂₃	—	4	1	no	—
155	Dic₂₃	—	4	2	no	—
156	Dic₂₃	—	4	2	no	—
157	Dic₂₃	—	8	1	no	—
158	Dic₂₃	—	8	1	no	—
159	C₂×#125	—	23	1	no	—
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	—	46	1	no	—
162	#389	—	23	1	no	—
163	#389	—	23	1	no	—
164	#387	—	46	1	no	—
165	#387	—	46	1	no	—
166	#382	—	23	1	no	—
167	C₄×C₄⋊C₄	—	23	1	no	—
168	C₄×C₄⋊C₄	—	23	1	no	—
169	C₄×C₄⋊C₄	—	46	1	no	—
170	C₄×C₄⋊C₄	—	46	1	no	—
171	C₄³	C₂×D₂₃	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#336	—	23	2	no	—
174	#336	—	46	1	no	—
175	#336	C₂×D₂₃	1	2	yes	—
176	C₂×C₂₃	#1461	1	1	yes	characteristic
177	C₂×C₂₃	#1477	1	1	yes	characteristic
178	C₂×C₂₃	#1477	1	1	yes	characteristic
179	C₂×C₂₃	#1512	1	1	yes	characteristic
180	C₂×C₂₃	#1512	1	1	yes	characteristic
181	C₂×C₂₃	#1534	1	1	yes	characteristic
182	C₂×C₂₃	#1561	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	46	2	no	—
185	#125	—	23	4	no	—
186	#125	—	46	2	no	—
187	C₂×C₄⋊C₄	—	23	1	no	—
188	C₂×C₄⋊C₄	—	23	1	no	—
189	C₂×C₄⋊C₄	—	23	1	no	—
190	C₂×C₄⋊C₄	—	23	1	no	—
191	C₂×C₄⋊C₄	—	46	1	no	—
192	C₂×C₄⋊C₄	—	46	1	no	—
193	C₂×C₄⋊C₄	—	46	1	no	—
194	C₂×C₄⋊C₄	—	46	1	no	—
195	C₂×C₄²	—	46	1	no	—
196	C₂×C₄²	—	46	1	no	—
197	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
199	C₂×C₄²	D₉₂	1	1	yes	characteristic
200	C₂×C₄²	D₉₂	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
204	C₄⋊C₈	—	4	2	no	—
205	C₄⋊C₈	—	92	2	no	—
206	#95	—	46	1	no	—
207	#95	—	46	1	no	—
208	#95	—	46	1	no	—
209	#95	—	46	1	no	—
210	#95	—	46	1	no	—
211	#95	—	46	1	no	—
212	C₂₃	#7602	1	1	yes	characteristic, complemented, Sylow
213	C₂²×C₄	—	46	1	no	—
214	C₂²×C₄	—	46	1	no	—
215	C₂²×C₄	—	46	1	no	—
216	C₂²×C₄	—	46	1	no	—
217	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₉₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	92	2	no	—
228	C₂×C₈	—	92	2	no	—
229	C₂×C₈	—	92	2	no	—
230	C₄⋊C₄	—	23	2	no	—
231	C₄⋊C₄	—	23	2	no	—
232	C₄⋊C₄	—	46	1	no	—
233	C₄⋊C₄	—	46	1	no	—
234	C₄⋊C₄	—	46	4	no	—
235	C₄⋊C₄	—	46	4	no	—
236	C₄⋊C₄	—	92	1	no	—
237	C₄⋊C₄	—	92	1	no	—
238	C₄⋊C₄	—	92	1	no	—
239	C₄⋊C₄	—	92	1	no	—
240	C₄⋊C₄	—	92	1	no	—
241	C₄⋊C₄	—	92	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	92	2	no	—
251	C₄²	—	92	2	no	—
252	C₄²	C₈×D₂₃	1	2	yes	—
253	C₄²	#67262	1	2	yes	—
254	C₂³	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	46	2	no	—
271	C₂×C₄	—	46	2	no	—
272	C₂×C₄	—	92	1	no	—
273	C₂×C₄	—	92	1	no	—
274	C₂×C₄	—	92	1	no	—
275	C₂×C₄	—	92	1	no	—
276	C₂×C₄	—	92	1	no	—
277	C₂×C₄	—	92	1	no	—
278	C₂×C₄	—	92	1	no	—
279	C₂×C₄	—	92	1	no	—
280	C₂×C₄	—	92	1	no	—
281	C₂×C₄	—	92	1	no	—
282	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
283	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
284	C₂×C₄	#10625006	1	2	yes	—
285	C₂×C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#67262	1	1	yes	characteristic
287	C₂×C₄	#10625079	1	1	yes	characteristic
288	C₂×C₄	#10625080	1	1	yes	characteristic
289	C₂×C₄	#10625103	1	1	yes	characteristic
290	C₂×C₄	#10625113	1	1	yes	characteristic
291	C₈	—	8	2	no	—
292	C₈	—	184	2	no	—
293	C₂²	C₂×(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
294	C₂²	#13092390	1	1	yes	characteristic
295	C₂²	C₂×#10625006	1	1	yes	characteristic
296	C₂²	#13092416	1	1	yes	characteristic
297	C₂²	#13092419	1	1	yes	characteristic
298	C₂²	#13092433	1	1	yes	characteristic
299	C₂²	#13092435	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	92	2	no	—
311	C₄	—	92	2	no	—
312	C₄	—	184	1	no	—
313	C₄	—	184	1	no	—
314	C₂	#15671720	1	1	yes	characteristic
315	C₂	#15671782	1	1	yes	characteristic
316	C₂	#15671785	1	1	yes	characteristic
317	C₂	#15671933	1	1	yes	characteristic
318	C₂	#15671936	1	1	yes	characteristic
319	C₂	#15672023	1	1	yes	characteristic
320	C₂	#15672134	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₂₃)⋊C₄
commutator	C₂×C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₂₃
Fitting	C₂₃×#1831	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	#1831	D₂₃
hypocenter	C₂₃	#7602
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
43. #10625041	C₈	semidirect
212. C₂₃	#7602	semidirect