Subgroups — C₄×C₂₃×#499 #19169347

Order 5888.

774 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	2	yes	maximal
3	C₂₃×#1991	C₂	1	2	yes	maximal
4	C₂₃×#1942	C₂	1	2	yes	maximal
5	C₂₃×#1941	C₂	1	2	yes	maximal
6	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₃×#107	C₂	1	2	yes	maximal
8	C₄×C₂₃×#106	C₂	1	2	yes	maximal
9	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂₃×#499	C₄	1	8	yes	direct product factor
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#389	C₂²	1	4	yes	—
23	C₂₃×#387	—	2	4	no	—
24	C₂₃×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
27	C₄³×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#125	—	2	4	no	—
35	C₂₃×#125	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₂₃	—	2	4	no	—
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	D₄	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#107	—	2	4	no	—
47	C₂₃×#107	C₂×C₄	1	8	yes	—
48	C₂₃×#106	—	2	4	no	—
49	C₂₃×#106	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₂₃	—	2	2	no	—
51	C₄×C₈×C₂₃	—	2	2	no	—
52	C₄×C₈×C₂₃	—	2	2	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₂₃	Q₈	1	4	yes	—
55	C₄×C₈×C₂₃	Q₈	1	4	yes	—
56	C₂₃×#95	—	2	4	no	—
57	C₂₃×#95	—	2	4	no	—
58	C₂₃×#95	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₂₃	—	2	1	no	—
67	C₂×C₈×C₂₃	—	2	1	no	—
68	C₂×C₈×C₂₃	—	2	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₂₃	#41	1	2	yes	—
77	C₂×C₈×C₂₃	#41	1	2	yes	—
78	C₄⋊C₄×C₂₃	—	2	16	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	4	8	no	—
83	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
84	C₄²×C₂₃	—	2	4	no	—
85	C₄²×C₂₃	—	2	4	no	—
86	C₄²×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	—	4	8	no	—
89	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
92	C₄²×C₂₃	#41	1	2	yes	—
93	C₄×#499	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	8	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	#135	1	4	yes	—
110	C₂×C₄×C₂₃	#135	1	4	yes	—
111	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	C₄×Q₈	1	4	yes	—
120	C₈×C₂₃	C₄×Q₈	1	4	yes	—
121	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
122	C₄×#125	C₂×C₂₃	1	2	yes	—
123	#1991	C₂×C₂₃	1	2	yes	—
124	#1942	C₂×C₂₃	1	2	yes	—
125	#1941	C₂×C₂₃	1	2	yes	—
126	#1846	C₂×C₂₃	1	1	yes	characteristic
127	C₄×#107	C₂×C₂₃	1	2	yes	—
128	C₄×#106	C₂×C₂₃	1	2	yes	—
129	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₂₃	#532	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×#135	1	2	yes	—
134	C₂²×C₂₃	C₂×#135	1	2	yes	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	4	no	—
137	C₄×C₂₃	—	2	4	no	—
138	C₄×C₂₃	—	4	8	no	—
139	C₄×C₂₃	—	8	4	no	—
140	C₄×C₂₃	#499	1	4	yes	direct product factor
141	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₂₃	#533	1	1	yes	characteristic
144	C₄×C₂₃	#533	1	1	yes	characteristic
145	C₂×#125	C₂²×C₂₃	1	2	yes	—
146	#499	C₄×C₂₃	1	8	yes	direct product factor
147	C₂×#107	C₂²×C₂₃	1	2	yes	—
148	C₂×#107	C₂²×C₂₃	1	2	yes	—
149	C₂×#106	C₂²×C₂₃	1	2	yes	—
150	C₂×#106	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	#389	C₂²×C₂₃	1	4	yes	—
158	#387	—	2	4	no	—
159	#382	C₂²×C₂₃	1	2	yes	—
160	C₄×C₄⋊C₄	—	2	4	no	—
161	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
162	C₄³	C₂²×C₂₃	1	1	yes	characteristic
163	#340	C₂²×C₂₃	1	4	yes	—
164	#340	C₂²×C₂₃	1	4	yes	—
165	C₂×C₂₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₂₃	C₄×#135	1	2	yes	—
167	C₂×C₂₃	C₂×#499	1	1	yes	characteristic
168	C₂×C₂₃	#3160	1	1	yes	characteristic
169	C₂×C₂₃	#3168	1	2	yes	—
170	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	#125	—	2	4	no	—
175	#125	C₂×C₄×C₂₃	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	D₄×C₂₃	1	2	yes	—
182	C₂×C₄²	D₄×C₂₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	C₂×C₄×C₂₃	1	8	yes	—
188	#106	—	2	4	no	—
189	#106	C₂×C₄×C₂₃	1	8	yes	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
194	C₄×C₈	Q₈×C₂₃	1	4	yes	—
195	C₄×C₈	Q₈×C₂₃	1	4	yes	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	4	no	—
199	C₂₃	C₄×#499	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
215	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
216	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
217	C₂×C₈	#41×C₂₃	1	2	yes	—
218	C₂×C₈	#41×C₂₃	1	2	yes	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	4	8	no	—
230	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
231	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
232	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
233	C₄²	#41×C₂₃	1	2	yes	—
234	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
247	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
248	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
249	C₂×C₄	C₂₃×#135	1	4	yes	—
250	C₂×C₄	C₂₃×#135	1	4	yes	—
251	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	C₄×Q₈×C₂₃	1	4	yes	—
260	C₈	C₄×Q₈×C₂₃	1	4	yes	—
261	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
262	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
263	C₂²	C₂₃×#532	1	1	yes	characteristic
264	C₂²	C₂×C₂₃×#135	1	2	yes	—
265	C₂²	C₂×C₂₃×#135	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	8	no	—
270	C₄	—	8	4	no	—
271	C₄	C₂₃×#499	1	4	yes	direct product factor
272	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
273	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
274	C₄	C₂₃×#533	1	1	yes	characteristic
275	C₄	C₂₃×#533	1	1	yes	characteristic
276	C₂	C₄×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
277	C₂	C₄×C₂₃×#135	1	2	yes	—
278	C₂	C₂×C₂₃×#499	1	1	yes	characteristic
279	C₂	C₂₃×#3160	1	1	yes	characteristic
280	C₂	C₂₃×#3168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#499	C₄	direct (N × H)
93. C₄×#499	C₂₃	direct (N × H)
140. C₄×C₂₃	#499	direct (N × H)
146. #499	C₄×C₂₃	direct (N × H)
199. C₂₃	C₄×#499	direct (N × H)
271. C₄	C₂₃×#499	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#499	2
C₄×C₂₃×#125	2
C₂₃×#1991	2
C₂₃×#1942	2
C₂₃×#1941	2
C₂₃×#1846	2
C₄×C₂₃×#107	2
C₄×C₂₃×#106	2
C₄²×C₈×C₂₃	2
C₄×#499	23

Subgroups — C4×C23×#499 #19169347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#499 #19169347