Subgroups — #19212214

Order 5888.

444 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15675874	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#15671222	C₂	1	2	yes	maximal
4	#15671219	C₂	1	2	yes	maximal
5	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₄₆	C₄	1	1	yes	characteristic
7	C₂×C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₂₃⋊C₈	—	2	1	no	—
9	#13092323	C₄	1	1	yes	characteristic
10	#13092323	C₂²	1	1	yes	characteristic
11	#13092315	C₄	1	1	yes	characteristic
12	#13092315	C₂²	1	1	yes	characteristic
13	#13092301	—	2	1	no	—
14	#13092301	—	2	1	no	—
15	C₄×Dic₂₃⋊C₄	—	2	1	no	—
16	C₄×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
17	#13092263	—	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×Dic₄₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂×C₄	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	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂³	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×C₄×Dic₂₃	—	2	1	no	—
39	C₂×C₄×Dic₂₃	—	2	1	no	—
40	C₂²×C₂₃⋊C₈	—	2	2	no	—
41	C₄×Dic₄₆	—	2	2	no	—
42	C₄×Dic₄₆	C₈	1	4	yes	complemented
43	C₂₃×C₄⋊C₈	—	4	2	no	—
44	C₂₃⋊#95	—	2	1	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)	—	2	4	no	—
52	C₄×C₂₃⋊C₈	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₂₃	—	2	1	no	—
61	C₂²×Dic₂₃	—	2	1	no	—
62	C₂²×Dic₂₃	—	2	1	no	—
63	C₂²×Dic₂₃	—	2	1	no	—
64	C₂×Dic₄₆	—	2	1	no	—
65	C₂×Dic₄₆	—	4	1	no	—
66	C₂×Dic₄₆	—	4	1	no	—
67	C₂×Dic₄₆	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	4	1	no	—
75	C₄²×C₂₃	—	4	1	no	—
76	C₄²×C₂₃	—	4	1	no	—
77	C₄²×C₂₃	C₂×C₈	1	2	yes	—
78	C₄²×C₂₃	M₁₆	1	2	yes	—
79	C₄²×C₂₃	SD₁₆	1	2	yes	—
80	C₄²×C₂₃	Q₁₆	1	2	yes	—
81	C₄⋊Dic₂₃	—	2	1	no	—
82	C₄⋊Dic₂₃	—	4	1	no	—
83	C₄⋊Dic₂₃	—	4	1	no	—
84	C₄⋊Dic₂₃	C₂×C₈	1	2	yes	—
85	Dic₂₃⋊C₄	—	2	4	no	—
86	Dic₂₃⋊C₄	—	4	2	no	—
87	Dic₂₃⋊C₄	—	4	1	no	—
88	Dic₂₃⋊C₄	—	4	1	no	—
89	C₄×Dic₂₃	—	2	4	no	—
90	C₄×Dic₂₃	—	4	2	no	—
91	C₂×C₂₃⋊C₈	—	2	4	no	—
92	C₂×C₂₃⋊C₈	—	4	2	no	—
93	C₂×C₂₃⋊C₈	—	4	2	no	—
94	#7577	—	23	1	no	maximal, Sylow
95	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₂₃	#103	1	2	yes	—
112	C₂×C₄×C₂₃	#103	1	2	yes	—
113	C₂×C₄×C₂₃	#104	1	2	yes	—
114	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
120	C₂×Dic₂₃	—	2	2	no	—
121	C₂×Dic₂₃	—	2	2	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	C₂×Dic₂₃	—	4	1	no	—
132	Dic₄₆	—	4	2	no	—
133	Dic₄₆	—	4	2	no	—
134	C₈×C₂₃	—	8	2	no	—
135	C₂₃⋊C₈	—	4	4	no	—
136	#2295	—	23	1	no	—
137	#1831	D₂₃	1	1	yes	characteristic
138	C₄×C₄⋊C₈	—	23	1	no	—
139	#1272	—	23	2	no	—
140	#1271	—	23	2	no	—
141	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂²×C₂₃	#410	1	1	yes	characteristic
143	C₂²×C₂₃	#413	1	1	yes	characteristic
144	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
145	C₂²×C₂₃	#417	1	1	yes	characteristic
146	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
147	C₂²×C₂₃	#421	1	1	yes	characteristic
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	4	1	no	—
157	C₄×C₂₃	Q₈⋊C₈	1	4	yes	—
158	Dic₂₃	—	4	2	no	—
159	Dic₂₃	—	4	2	no	—
160	Dic₂₃	—	8	1	no	—
161	Dic₂₃	—	8	1	no	—
162	C₂×C₄×Q₈	—	23	1	no	—
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	46	1	no	—
165	C₂×C₄×C₈	—	46	1	no	—
166	#391	—	23	1	no	—
167	#391	—	23	1	no	—
168	#389	—	23	1	no	—
169	#389	—	23	1	no	—
170	#387	—	46	1	no	—
171	#387	—	46	1	no	—
172	C₄×C₄⋊C₄	—	23	1	no	—
173	C₄×C₄⋊C₄	—	46	1	no	—
174	#376	—	46	1	no	—
175	C₄³	C₂×D₂₃	1	1	yes	characteristic
176	#336	—	2	1	no	—
177	#336	—	23	2	no	—
178	#336	C₂×D₂₃	1	2	yes	—
179	C₂×C₂₃	#1458	1	1	yes	characteristic
180	C₂×C₂₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
181	C₂×C₂₃	#1477	1	1	yes	characteristic
182	C₂×C₂₃	#1509	1	1	yes	characteristic
183	C₂×C₂₃	#1518	1	1	yes	characteristic
184	C₂×C₂₃	#1529	1	1	yes	characteristic
185	C₂×C₂₃	#1550	1	1	yes	characteristic
186	C₂²×Q₈	—	23	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	46	2	no	—
189	C₄×Q₈	—	23	4	no	—
190	C₄×Q₈	—	46	2	no	—
191	C₂×C₄⋊C₄	—	23	1	no	—
192	C₂×C₄⋊C₄	—	23	1	no	—
193	C₂×C₄⋊C₄	—	23	1	no	—
194	C₂×C₄⋊C₄	—	46	1	no	—
195	C₂×C₄⋊C₄	—	46	1	no	—
196	C₂×C₄⋊C₄	—	46	1	no	—
197	C₂×C₄²	—	46	1	no	—
198	C₂×C₄²	—	46	1	no	—
199	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
201	C₂×C₄²	D₉₂	1	1	yes	characteristic
202	C₂×C₄²	D₉₂	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
206	C₄⋊C₈	—	4	2	no	—
207	C₄⋊C₈	—	46	4	no	—
208	C₄×C₈	—	92	2	no	—
209	#95	—	46	1	no	—
210	#95	—	46	1	no	—
211	#95	—	46	1	no	—
212	#95	—	46	1	no	—
213	#95	—	46	1	no	—
214	#95	—	46	1	no	—
215	#95	—	46	1	no	—
216	C₂₃	#7577	1	1	yes	characteristic, complemented, Sylow
217	C₂×Q₈	—	23	2	no	—
218	C₂×Q₈	—	46	1	no	—
219	C₂×Q₈	—	92	1	no	—
220	C₂×Q₈	—	92	1	no	—
221	C₂²×C₄	—	46	1	no	—
222	C₂²×C₄	—	46	1	no	—
223	C₂²×C₄	—	46	1	no	—
224	C₂²×C₄	—	46	1	no	—
225	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₉₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	46	4	no	—
236	C₂×C₈	—	92	2	no	—
237	C₂×C₈	—	92	2	no	—
238	C₄⋊C₄	—	23	2	no	—
239	C₄⋊C₄	—	46	1	no	—
240	C₄⋊C₄	—	46	4	no	—
241	C₄⋊C₄	—	92	1	no	—
242	C₄⋊C₄	—	92	1	no	—
243	C₄⋊C₄	—	92	2	no	—
244	C₄⋊C₄	—	92	1	no	—
245	C₄⋊C₄	—	92	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	46	4	no	—
253	C₄²	—	92	2	no	—
254	C₄²	C₈×D₂₃	1	2	yes	—
255	C₄²	#67262	1	2	yes	—
256	C₄²	#67273	1	2	yes	—
257	C₄²	#67275	1	2	yes	—
258	C₂³	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
259	Q₈	—	92	2	no	—
260	Q₈	—	92	2	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	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	46	2	no	—
275	C₂×C₄	—	46	2	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	—	92	1	no	—
283	C₂×C₄	—	92	1	no	—
284	C₂×C₄	—	92	1	no	—
285	C₂×C₄	—	92	1	no	—
286	C₂×C₄	#10624992	1	2	yes	—
287	C₂×C₄	#10624992	1	2	yes	—
288	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
289	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
290	C₂×C₄	#10625007	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#67262	1	1	yes	characteristic
293	C₂×C₄	#10625079	1	1	yes	characteristic
294	C₂×C₄	#10625080	1	1	yes	characteristic
295	C₂×C₄	C₂×#67273	1	1	yes	characteristic
296	C₂×C₄	C₂×#67275	1	1	yes	characteristic
297	C₈	—	8	2	no	—
298	C₈	—	92	4	no	—
299	C₂²	C₂×(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
300	C₂²	#13092390	1	1	yes	characteristic
301	C₂²	#13092399	1	1	yes	characteristic
302	C₂²	C₂×#10624992	1	1	yes	characteristic
303	C₂²	#13092419	1	1	yes	characteristic
304	C₂²	C₂×#10625007	1	1	yes	characteristic
305	C₂²	#13092435	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	92	2	no	—
316	C₄	—	92	2	no	—
317	C₄	—	184	1	no	—
318	C₄	—	184	1	no	—
319	C₄	#13092147	1	4	yes	—
320	C₂	#15671709	1	1	yes	characteristic
321	C₂	C₂×#13092147	1	1	yes	characteristic
322	C₂	#15671785	1	1	yes	characteristic
323	C₂	#15671923	1	1	yes	characteristic
324	C₂	#15671960	1	1	yes	characteristic
325	C₂	#15672005	1	1	yes	characteristic
326	C₂	#15672088	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	#7577
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
42. C₄×Dic₄₆	C₈	semidirect
216. C₂₃	#7577	semidirect