Subgroups — #19212187

Order 5888.

516 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#15671222	C₂	1	4	yes	maximal
4	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625040	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₄₆	C₄	1	2	yes	—
7	C₂×C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₂₃⋊C₈	—	2	1	no	—
9	#13092323	C₂²	1	2	yes	—
10	#13092315	—	2	1	no	—
11	#13092315	—	2	1	no	—
12	#13092287	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₂₃	—	2	1	no	—
14	C₄×C₄⋊Dic₂₃	—	2	1	no	—
15	#13092172	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	—	2	1	no	—
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#336	C₂²	1	2	yes	—
20	C₂²×Dic₄₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	D₄	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	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₃	—	2	1	no	—
30	C₂×C₄⋊Dic₂₃	—	2	1	no	—
31	C₂×C₄⋊Dic₂₃	—	2	1	no	—
32	C₂×C₄⋊Dic₂₃	—	2	1	no	—
33	C₂×C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₃⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₂₃	—	2	2	no	—
36	C₂²×C₂₃⋊C₈	—	2	2	no	—
37	#10625040	—	2	2	no	—
38	#10625040	C₈	1	4	yes	complemented
39	C₄×Dic₄₆	—	4	4	no	—
40	C₂₃×C₄⋊C₈	—	2	4	no	—
41	C₄×C₈×C₂₃	—	4	2	no	—
42	C₂₃⋊#95	—	2	2	no	—
43	C₂₃⋊#95	—	2	2	no	—
44	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
45	C₄×C₂₃⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₂₃	—	2	2	no	—
54	C₂²×Dic₂₃	—	2	2	no	—
55	C₂×Dic₄₆	—	2	2	no	—
56	C₂×Dic₄₆	—	4	2	no	—
57	C₂×Dic₄₆	—	4	2	no	—
58	C₂×Dic₄₆	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	—	4	1	no	—
66	C₄²×C₂₃	C₂×C₈	1	2	yes	—
67	C₄²×C₂₃	M₁₆	1	2	yes	—
68	C₄²×C₂₃	SD₁₆	1	2	yes	—
69	C₄²×C₂₃	SD₁₆	1	2	yes	—
70	C₄²×C₂₃	Q₁₆	1	2	yes	—
71	C₄²×C₂₃	Q₁₆	1	2	yes	—
72	C₄⋊Dic₂₃	—	2	4	no	—
73	C₄⋊Dic₂₃	—	2	4	no	—
74	C₄⋊Dic₂₃	—	4	2	no	—
75	C₄⋊Dic₂₃	—	4	2	no	—
76	C₄⋊Dic₂₃	—	4	2	no	—
77	Dic₂₃⋊C₄	—	4	4	no	—
78	C₄×Dic₂₃	—	4	4	no	—
79	C₂×C₂₃⋊C₈	—	2	4	no	—
80	C₂×C₂₃⋊C₈	—	4	2	no	—
81	C₂×C₂₃⋊C₈	—	4	2	no	—
82	#7571	—	23	1	no	maximal, Sylow
83	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₂₃	#100	1	2	yes	—
96	C₂×C₄×C₂₃	#103	1	2	yes	—
97	C₂×C₄×C₂₃	#103	1	2	yes	—
98	C₂×C₄×C₂₃	#103	1	2	yes	—
99	C₂×C₄×C₂₃	#103	1	2	yes	—
100	C₂×C₄×C₂₃	#104	1	2	yes	—
101	C₂×C₄×C₂₃	#104	1	2	yes	—
102	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×Dic₂₃	—	2	4	no	—
109	C₂×Dic₂₃	—	4	2	no	—
110	C₂×Dic₂₃	—	4	2	no	—
111	C₂×Dic₂₃	—	4	2	no	—
112	C₂×Dic₂₃	—	4	2	no	—
113	C₂×Dic₂₃	—	4	2	no	—
114	Dic₄₆	—	4	4	no	—
115	Dic₄₆	—	4	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₂₃⋊C₈	—	4	4	no	—
118	C₄×C₄⋊Q₈	—	23	1	no	—
119	C₄×C₄⋊C₈	—	23	1	no	—
120	C₄×C₄⋊C₈	D₂₃	1	1	yes	characteristic
121	#1272	—	23	4	no	—
122	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₂₃	#410	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
126	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
127	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	4	1	no	—
136	C₄×C₂₃	Q₈⋊C₈	1	4	yes	—
137	C₄×C₂₃	Q₈⋊C₈	1	4	yes	—
138	C₄×C₂₃	#333	1	4	yes	—
139	Dic₂₃	—	4	4	no	—
140	Dic₂₃	—	8	2	no	—
141	C₂×C₄⋊Q₈	—	23	1	no	—
142	C₂×C₄×Q₈	—	23	2	no	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄⋊C₈	—	46	1	no	—
145	C₂×C₄×C₈	—	2	1	no	—
146	C₂×C₄×C₈	—	46	1	no	—
147	#391	—	23	2	no	—
148	#389	—	46	1	no	—
149	#389	—	46	1	no	—
150	#384	—	23	1	no	—
151	C₄×C₄⋊C₄	—	46	1	no	—
152	C₄×C₄⋊C₄	—	46	1	no	—
153	C₄³	C₂×D₂₃	1	1	yes	characteristic
154	#336	—	23	2	no	—
155	#336	C₂×D₂₃	1	2	yes	—
156	C₂×C₂₃	#1460	1	1	yes	characteristic
157	C₂×C₂₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
158	C₂×C₂₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
159	C₂×C₂₃	#1503	1	1	yes	characteristic
160	C₂×C₂₃	#1503	1	1	yes	characteristic
161	C₂×C₂₃	#1525	1	1	yes	characteristic
162	C₂×C₂₃	C₂×#333	1	1	yes	characteristic
163	C₂²×Q₈	—	23	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	46	2	no	—
166	C₄⋊Q₈	—	23	4	no	—
167	C₄⋊Q₈	—	46	2	no	—
168	C₄×Q₈	—	92	4	no	—
169	C₂×C₄⋊C₄	—	23	2	no	—
170	C₂×C₄⋊C₄	—	46	1	no	—
171	C₂×C₄⋊C₄	—	46	1	no	—
172	C₂×C₄⋊C₄	—	46	2	no	—
173	C₂×C₄⋊C₄	—	46	1	no	—
174	C₂×C₄⋊C₄	—	46	1	no	—
175	C₂×C₄²	—	46	2	no	—
176	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₂₃	1	1	yes	characteristic
178	C₂×C₄²	D₉₂	1	1	yes	characteristic
179	C₂×C₄²	D₉₂	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	46	4	no	—
185	C₄×C₈	—	4	2	no	—
186	C₄×C₈	—	92	2	no	—
187	#95	—	46	2	no	—
188	#95	—	46	2	no	—
189	C₂₃	#7571	1	1	yes	characteristic, complemented, Sylow
190	C₂×Q₈	—	23	4	no	—
191	C₂×Q₈	—	46	2	no	—
192	C₂×Q₈	—	92	2	no	—
193	C₂×Q₈	—	92	2	no	—
194	C₂²×C₄	—	46	2	no	—
195	C₂²×C₄	—	46	2	no	—
196	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₉₂	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	46	4	no	—
207	C₂×C₈	—	92	2	no	—
208	C₂×C₈	—	92	2	no	—
209	C₄⋊C₄	—	46	4	no	—
210	C₄⋊C₄	—	46	4	no	—
211	C₄⋊C₄	—	92	2	no	—
212	C₄⋊C₄	—	92	4	no	—
213	C₄⋊C₄	—	92	2	no	—
214	C₄⋊C₄	—	92	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	92	4	no	—
220	C₄²	C₈×D₂₃	1	2	yes	—
221	C₄²	#67262	1	2	yes	—
222	C₄²	#67263	1	2	yes	—
223	C₄²	Dic₉₂	1	2	yes	—
224	C₄²	#67273	1	2	yes	—
225	C₄²	#67275	1	2	yes	—
226	C₂³	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
227	Q₈	—	92	4	no	—
228	Q₈	—	92	4	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	46	4	no	—
239	C₂×C₄	—	92	2	no	—
240	C₂×C₄	—	92	2	no	—
241	C₂×C₄	—	92	2	no	—
242	C₂×C₄	—	92	2	no	—
243	C₂×C₄	—	92	2	no	—
244	C₂×C₄	#10624987	1	2	yes	—
245	C₂×C₄	#10624992	1	2	yes	—
246	C₂×C₄	#10624992	1	2	yes	—
247	C₂×C₄	#10624998	1	2	yes	—
248	C₂×C₄	#10624998	1	2	yes	—
249	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
250	C₂×C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
251	C₂×C₄	#10625005	1	2	yes	—
252	C₂×C₄	#10625007	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#67262	1	1	yes	characteristic
255	C₂×C₄	C₂×#67263	1	1	yes	characteristic
256	C₂×C₄	C₂×Dic₉₂	1	1	yes	characteristic
257	C₂×C₄	C₂×#67273	1	1	yes	characteristic
258	C₂×C₄	C₂×#67275	1	1	yes	characteristic
259	C₈	—	4	4	no	—
260	C₈	—	92	4	no	—
261	C₂²	C₂×(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
262	C₂²	#13092390	1	1	yes	characteristic
263	C₂²	C₂×#10625005	1	1	yes	characteristic
264	C₂²	C₂×#10624992	1	1	yes	characteristic
265	C₂²	C₂×#10624998	1	1	yes	characteristic
266	C₂²	C₂×#10625007	1	1	yes	characteristic
267	C₂²	C₂×#10624987	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	92	4	no	—
276	C₄	—	184	2	no	—
277	C₄	#13092147	1	4	yes	—
278	C₄	#13092149	1	4	yes	—
279	C₄	#13092165	1	4	yes	—
280	C₂	#15671717	1	1	yes	characteristic
281	C₂	C₂×#13092147	1	1	yes	characteristic
282	C₂	C₂×#13092149	1	1	yes	characteristic
283	C₂	#15671895	1	1	yes	characteristic
284	C₂	#15671897	1	1	yes	characteristic
285	C₂	#15671988	1	1	yes	characteristic
286	C₂	C₂×#13092165	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₂₃)⋊C₄
commutator	C₂×C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₂₃
Fitting	C₄×C₂₃×C₄⋊C₈	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	C₄×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#7571
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. #10625040	C₈	semidirect
189. C₂₃	#7571	semidirect