Subgroups — #19212330

Order 5888.

440 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#15673540	C₂	1	1	yes	maximal, characteristic
3	#15671220	C₂	1	4	yes	maximal
4	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	#13092172	—	2	1	no	—
6	#13092172	—	2	1	no	—
7	#13092172	—	2	1	no	—
8	#13092172	—	2	1	no	—
9	#13092172	C₂²	1	2	yes	—
10	#13092172	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
13	C₂₃×#391	—	2	1	no	—
14	C₂₃×#389	—	2	1	no	—
15	C₂₃×#389	C₂²	1	2	yes	—
16	C₂₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
19	C₄³×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×C₄⋊Q₈	C₈	1	4	yes	complemented
23	C₄×Q₈×C₂₃	—	4	4	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₂₃⋊C₈	—	2	2	no	—
39	C₂²×C₂₃⋊C₈	—	2	2	no	—
40	C₂₃×#95	—	2	2	no	—
41	C₂₃×#95	—	2	2	no	—
42	C₂×Q₈×C₂₃	—	2	4	no	—
43	C₂×Q₈×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₄⋊C₄×C₂₃	—	4	4	no	—
60	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
61	C₄²×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	4	1	no	—
65	C₄²×C₂₃	—	4	1	no	—
66	C₄²×C₂₃	—	4	1	no	—
67	C₄²×C₂₃	—	4	4	no	—
68	C₄²×C₂₃	C₂×C₈	1	2	yes	—
69	C₄²×C₂₃	M₁₆	1	2	yes	—
70	C₂×C₂₃⋊C₈	—	4	2	no	—
71	C₂×C₂₃⋊C₈	—	4	2	no	—
72	C₂×C₂₃⋊C₈	—	4	2	no	—
73	C₂×C₂₃⋊C₈	—	4	2	no	—
74	C₂×C₂₃⋊C₈	—	4	2	no	—
75	C₂×C₂₃⋊C₈	—	4	2	no	—
76	#7604	—	23	1	no	maximal, Sylow
77	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₂₃	—	4	8	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
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	4	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₂₃	#101	1	2	yes	—
103	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₂₃	#137	1	2	yes	—
107	C₂₃⋊C₈	—	8	2	no	—
108	C₂₃⋊C₈	—	8	2	no	—
109	C₄×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
110	#1839	—	23	1	no	—
111	#1839	—	23	1	no	—
112	#1271	—	23	4	no	—
113	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₂₃	#410	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
116	C₂²×C₂₃	#418	1	1	yes	characteristic
117	C₂²×C₂₃	#419	1	1	yes	characteristic
118	C₂²×C₂₃	#421	1	1	yes	characteristic
119	C₂²×C₂₃	#421	1	1	yes	characteristic
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	2	1	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	4	no	—
129	C₄×C₂₃	—	8	2	no	—
130	C₂×C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic
131	C₂×C₄×Q₈	—	2	1	no	—
132	#391	—	2	1	no	—
133	#389	—	2	1	no	—
134	#389	C₂×D₂₃	1	2	yes	—
135	#384	Dic₂₃	1	1	yes	characteristic
136	C₄×C₄⋊C₄	—	2	1	no	—
137	C₄×C₄⋊C₄	Dic₂₃	1	2	yes	—
138	C₄³	C₂×D₂₃	1	1	yes	characteristic
139	#336	—	23	2	no	—
140	#336	—	23	2	no	—
141	#336	—	46	1	no	—
142	#336	—	46	1	no	—
143	#336	—	46	1	no	—
144	#336	—	46	1	no	—
145	C₂×C₂₃	#1461	1	1	yes	characteristic
146	C₂×C₂₃	#1478	1	1	yes	characteristic
147	C₂×C₂₃	#1479	1	1	yes	characteristic
148	C₂×C₂₃	#1514	1	1	yes	characteristic
149	C₂×C₂₃	#1515	1	1	yes	characteristic
150	C₂×C₂₃	#1537	1	1	yes	characteristic
151	C₂×C₂₃	#1556	1	1	yes	characteristic
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×C₈	—	46	2	no	—
154	C₂²×C₈	—	46	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	C₂₃⋊C₈	1	4	yes	complemented
157	C₄×Q₈	—	4	4	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₂₃	#7604	1	1	yes	characteristic, complemented, Sylow
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
187	C₂×C₈	—	92	2	no	—
188	C₂×C₈	—	92	2	no	—
189	C₂×C₈	—	92	2	no	—
190	C₂×C₈	—	92	2	no	—
191	C₂×C₈	—	92	2	no	—
192	C₂×C₈	—	92	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	C₂×C₂₃⋊C₈	1	4	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₂×C₂₃⋊C₈	1	2	yes	—
208	C₄²	#67267	1	2	yes	—
209	C₂³	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
210	Q₈	—	4	8	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
233	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
234	C₂×C₄	#10625018	1	2	yes	—
235	C₂×C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×#67267	1	1	yes	characteristic
237	C₂×C₄	#10625103	1	2	yes	—
238	C₂×C₄	#10625113	1	2	yes	—
239	C₈	—	184	2	no	—
240	C₈	—	184	2	no	—
241	C₂²	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
242	C₂²	#13092391	1	1	yes	characteristic
243	C₂²	C₂×#10625018	1	1	yes	characteristic
244	C₂²	#13092422	1	1	yes	characteristic
245	C₂²	#13092426	1	1	yes	characteristic
246	C₂²	#13092434	1	1	yes	characteristic
247	C₂²	#13092434	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₂	#15671721	1	1	yes	characteristic
259	C₂	#15671788	1	1	yes	characteristic
260	C₂	#15671792	1	1	yes	characteristic
261	C₂	#15671944	1	1	yes	characteristic
262	C₂	#15671948	1	1	yes	characteristic
263	C₂	#15672035	1	1	yes	characteristic
264	C₂	#15672114	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×C₄⋊Q₈	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	C₄×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#7604
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. C₂₃×C₄⋊Q₈	C₈	semidirect
156. C₄⋊Q₈	C₂₃⋊C₈	semidirect
174. C₂₃	#7604	semidirect