Subgroups — #19212253

Order 5888.

426 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15673540	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#15671220	C₂	1	2	yes	maximal
4	#15671220	C₂	1	2	yes	maximal
5	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₂₃⋊C₈	—	2	1	no	—
8	#13092172	—	2	1	no	—
9	#13092172	—	2	1	no	—
10	#13092172	C₂²	1	2	yes	—
11	#13092172	C₂²	1	2	yes	—
12	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
13	C₂₃×#389	C₂²	1	1	yes	characteristic
14	C₂₃×#389	C₂²	1	1	yes	characteristic
15	C₂₃×#387	—	2	1	no	—
16	C₂₃×#387	—	2	1	no	—
17	C₂₃×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#125	—	2	2	no	—
24	C₂₃×#125	C₈	1	4	yes	complemented
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₂₃⋊C₈	—	2	2	no	—
43	C₂²×C₂₃⋊C₈	—	2	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	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
51	C₄×C₂₃⋊C₈	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	—	2	4	no	—
64	C₄⋊C₄×C₂₃	—	2	4	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₄⋊C₄×C₂₃	—	4	1	no	—
68	C₄⋊C₄×C₂₃	—	4	1	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	—	4	1	no	—
71	C₄⋊C₄×C₂₃	—	4	1	no	—
72	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
73	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	—	4	1	no	—
80	C₄²×C₂₃	—	4	1	no	—
81	C₄²×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	C₂×C₈	1	2	yes	—
84	C₄²×C₂₃	M₁₆	1	2	yes	—
85	C₂×C₂₃⋊C₈	—	2	4	no	—
86	C₂×C₂₃⋊C₈	—	4	2	no	—
87	C₂×C₂₃⋊C₈	—	4	2	no	—
88	C₂×C₂₃⋊C₈	—	4	2	no	—
89	C₂×C₂₃⋊C₈	—	4	2	no	—
90	C₂×C₂₃⋊C₈	—	4	2	no	—
91	#7585	—	23	1	no	maximal, Sylow
92	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
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	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₂₃	#101	1	2	yes	—
122	C₂×C₄×C₂₃	#104	1	2	yes	—
123	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#135	1	2	yes	—
126	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
128	C₂₃⋊C₈	—	4	4	no	—
129	C₂₃⋊C₈	—	8	2	no	—
130	C₄×#125	D₂₃	1	1	yes	characteristic
131	#1839	—	23	1	no	—
132	C₄×C₄⋊C₈	—	23	1	no	—
133	#1271	—	23	2	no	—
134	#1271	—	23	2	no	—
135	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₂₃	#410	1	1	yes	characteristic
137	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
138	C₂²×C₂₃	#416	1	1	yes	characteristic
139	C₂²×C₂₃	#417	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
141	C₂²×C₂₃	#421	1	1	yes	characteristic
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	4	2	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	—	8	1	no	—
154	C₄×C₂₃	—	8	1	no	—
155	C₂×#125	Dic₂₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	46	1	no	—
157	C₂×C₄×C₈	—	46	1	no	—
158	#389	C₂×D₂₃	1	1	yes	characteristic
159	#389	C₂×D₂₃	1	1	yes	characteristic
160	#387	—	2	1	no	—
161	#387	—	2	1	no	—
162	#382	Dic₂₃	1	1	yes	characteristic
163	C₄×C₄⋊C₄	—	2	1	no	—
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
167	C₄³	C₂×D₂₃	1	1	yes	characteristic
168	#336	—	23	2	no	—
169	#336	—	23	2	no	—
170	#336	—	46	1	no	—
171	#336	—	46	1	no	—
172	C₂×C₂₃	#1461	1	1	yes	characteristic
173	C₂×C₂₃	#1476	1	1	yes	characteristic
174	C₂×C₂₃	#1477	1	1	yes	characteristic
175	C₂×C₂₃	#1509	1	1	yes	characteristic
176	C₂×C₂₃	#1521	1	1	yes	characteristic
177	C₂×C₂₃	#1528	1	1	yes	characteristic
178	C₂×C₂₃	#1552	1	1	yes	characteristic
179	C₂²×C₈	—	46	2	no	—
180	C₂²×C₈	—	46	2	no	—
181	#125	—	2	2	no	—
182	#125	C₂₃⋊C₈	1	4	yes	complemented
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
200	C₄⋊C₈	—	46	4	no	—
201	C₄×C₈	—	92	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	C₂₃	#7585	1	1	yes	characteristic, complemented, Sylow
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
220	C₂×C₈	—	46	4	no	—
221	C₂×C₈	—	92	2	no	—
222	C₂×C₈	—	92	2	no	—
223	C₂×C₈	—	92	2	no	—
224	C₂×C₈	—	92	2	no	—
225	C₂×C₈	—	92	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	C₂×C₂₃⋊C₈	1	2	yes	—
236	C₄⋊C₄	C₂×C₂₃⋊C₈	1	2	yes	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₂×C₂₃⋊C₈	1	2	yes	—
247	C₄²	#67267	1	2	yes	—
248	C₂³	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
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	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
276	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
277	C₂×C₄	#10625018	1	2	yes	—
278	C₂×C₄	#10625019	1	2	yes	—
279	C₂×C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×#67267	1	1	yes	characteristic
281	C₂×C₄	#10625120	1	1	yes	characteristic
282	C₂×C₄	#10625121	1	2	yes	—
283	C₂×C₄	#10625122	1	1	yes	characteristic
284	C₈	—	92	4	no	—
285	C₈	—	184	2	no	—
286	C₂²	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
287	C₂²	#13092391	1	1	yes	characteristic
288	C₂²	C₂×#10625018	1	1	yes	characteristic
289	C₂²	#13092413	1	1	yes	characteristic
290	C₂²	#13092418	1	1	yes	characteristic
291	C₂²	C₂×#10625019	1	1	yes	characteristic
292	C₂²	#13092434	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₂	#15671721	1	1	yes	characteristic
307	C₂	#15671779	1	1	yes	characteristic
308	C₂	#15671784	1	1	yes	characteristic
309	C₂	#15671922	1	1	yes	characteristic
310	C₂	#15671974	1	1	yes	characteristic
311	C₂	#15671999	1	1	yes	characteristic
312	C₂	#15672097	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₂₃
Fitting	C₄×C₂₃×#125	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	C₄×#125	D₂₃
hypocenter	C₂₃	#7585
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
24. C₂₃×#125	C₈	semidirect
182. #125	C₂₃⋊C₈	semidirect
208. C₂₃	#7585	semidirect