Subgroups — C₂₃×#20438 #19168064

Order 5888.

1110 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#405	C₂	1	2	yes	maximal
3	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1771	C₂	1	8	yes	maximal, complemented
6	C₂₃×#1729	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#423	C₄	1	8	yes	complemented
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂₃×#405	—	2	8	no	—
15	C₂×C₂₃×#97	C₄	1	8	yes	complemented
16	C₂×C₂₃×#97	C₂²	1	4	yes	—
17	C₂×C₂₃×#97	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#384	C₄	1	2	yes	—
20	C₂₃×#382	C₄	1	2	yes	—
21	C₂₃×#380	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₂₃×#336	C₂²	1	8	yes	—
24	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₂₃	D₄	1	4	yes	—
34	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂₃×#117	—	2	4	no	—
39	C₂₃×#117	—	2	4	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
42	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
49	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
50	C₂₃×#97	—	2	8	no	—
51	C₂₃×#97	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₂₃	—	2	8	no	—
53	C₂₃×#95	C₂×C₄	1	8	yes	—
54	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₂₃	#41	1	2	yes	—
66	C₂²×C₄×C₂₃	#41	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₂₃	—	2	16	no	—
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
80	C₄⋊C₄×C₂₃	—	2	8	no	—
81	C₄⋊C₄×C₂₃	—	4	8	no	—
82	C₂²⋊C₄×C₂₃	—	2	8	no	—
83	C₄²×C₂₃	—	2	4	no	—
84	C₄²×C₂₃	—	2	4	no	—
85	C₄²×C₂₃	C₄²	1	4	yes	—
86	C₄²×C₂₃	C₄²	1	4	yes	—
87	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
89	#20438	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	4	no	—
91	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
92	C₂³×C₂₃	C₄×D₄	1	2	yes	—
93	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
94	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	2	16	no	—
99	C₂×C₄×C₂₃	—	4	8	no	—
100	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₂₃	#117	1	4	yes	—
105	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	2	8	no	—
117	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
118	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#405	C₂×C₂₃	1	2	yes	—
120	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
121	#2285	C₂×C₂₃	1	1	yes	characteristic
122	#1771	C₂×C₂₃	1	8	yes	complemented
123	#1729	C₂×C₂₃	1	2	yes	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	—	2	4	no	—
126	C₂²×C₂₃	—	2	4	no	—
127	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
128	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂²×C₂₃	#568	1	4	yes	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	—	4	8	no	—
135	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
136	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
138	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
141	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×#117	C₂²×C₂₃	1	2	yes	—
144	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
145	#423	C₄×C₂₃	1	8	yes	complemented
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
147	#405	—	2	8	no	—
148	C₂×#97	C₄×C₂₃	1	8	yes	complemented
149	C₂×#97	C₂²×C₂₃	1	4	yes	—
150	C₂×#97	C₂²×C₂₃	1	4	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
152	#384	C₄×C₂₃	1	2	yes	—
153	#382	C₄×C₂₃	1	2	yes	—
154	#380	C₄×C₂₃	1	4	yes	—
155	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
156	#336	C₂²×C₂₃	1	8	yes	—
157	C₂×C₂₃	—	2	4	no	—
158	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₂₃	#2873	1	2	yes	—
160	C₂×C₂₃	#2932	1	2	yes	—
161	C₂×C₂₃	#2961	1	2	yes	—
162	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
170	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
171	C₂²×C₈	D₄×C₂₃	1	4	yes	—
172	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
173	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
175	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
176	#117	—	2	4	no	—
177	#117	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
186	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
187	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
188	#97	—	2	8	no	—
189	#97	C₂×C₄×C₂₃	1	16	yes	—
190	C₄×C₈	—	2	8	no	—
191	#95	C₂×C₄×C₂₃	1	8	yes	—
192	C₂₃	#20438	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
204	C₂²×C₄	#41×C₂₃	1	2	yes	—
205	C₂²×C₄	#41×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
207	M₁₆	—	2	16	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	complemented
214	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
215	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
216	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
218	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₄²×C₂₃	1	4	yes	—
225	C₄²	C₄²×C₂₃	1	4	yes	—
226	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
227	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
228	C₂³	—	2	4	no	—
229	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
230	C₂³	C₄×D₄×C₂₃	1	2	yes	—
231	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
232	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
239	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
242	C₂×C₄	C₂₃×#117	1	4	yes	—
243	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
244	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
245	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
246	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
247	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
260	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
261	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
262	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
263	C₂²	C₂₃×#568	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	8	no	—
267	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
268	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
269	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
270	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
271	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
274	C₂	C₂₃×#2873	1	2	yes	—
275	C₂	C₂₃×#2932	1	2	yes	—
276	C₂	C₂₃×#2961	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1771	C₂	semidirect
12. C₂₃×#423	C₄	semidirect
15. C₂×C₂₃×#97	C₄	semidirect
74. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
89. #20438	C₂₃	direct (N × H)
122. #1771	C₂×C₂₃	semidirect
145. #423	C₄×C₂₃	semidirect
148. C₂×#97	C₄×C₂₃	semidirect
192. C₂₃	#20438	direct (N × H)
213. C₂×C₈	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#423	2
C₂×C₂₃×#405	2
C₂²×C₂₃×#97	2
C₂₃×#2285	2
C₂₃×#1771	2
C₂₃×#1729	2
#20438	23

Subgroups — C23×#20438 #19168064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20438 #19168064