Subgroups — C₂₃×#22733 #19170359

Order 5888.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1950	C₂	1	4	yes	maximal
4	C₂₃×#1950	C₂	1	2	yes	maximal
5	C₂₃×#1849	C₂	1	2	yes	maximal
6	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1778	C₂	1	2	yes	maximal
8	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#446	—	2	2	no	—
11	C₂₃×#446	—	2	4	no	—
12	C₂₃×#446	C₄	1	4	yes	—
13	C₂₃×#446	C₄	1	4	yes	—
14	C₈×Q₈×C₂₃	—	2	2	no	—
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#108	C₂²	1	2	yes	—
17	C₂×C₂₃×#108	C₂²	1	2	yes	—
18	C₂×C₂₃×#108	C₂²	1	4	yes	—
19	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
20	C₂₃×#423	C₂²	1	2	yes	—
21	C₂₃×#423	C₂²	1	4	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
25	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
26	C₂×C₂₃×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#341	C₂²	1	2	yes	—
29	C₂₃×#341	C₂²	1	4	yes	—
30	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
32	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
33	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₂₃	D₄	1	2	yes	—
40	C₄×Q₈×C₂₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	Q₈	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂₃×#108	—	2	2	no	—
46	C₂₃×#108	—	2	4	no	—
47	C₂₃×#108	—	4	2	no	—
48	C₂₃×#108	—	4	4	no	—
49	C₂₃×#108	Q₈	1	4	yes	complemented
50	C₂₃×C₄⋊C₈	—	2	2	no	—
51	C₂₃×C₄⋊C₈	—	2	4	no	—
52	C₂₃×C₄⋊C₈	—	2	4	no	—
53	C₂₃×C₄⋊C₈	—	2	2	no	—
54	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₂₃×#97	—	2	4	no	—
58	C₂₃×#97	—	2	2	no	—
59	C₂₃×#97	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₂₃	—	2	2	no	—
61	C₄×C₈×C₂₃	—	2	4	no	—
62	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
64	C₂×Q₈×C₂₃	—	4	2	no	—
65	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₂₃	—	2	2	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	1	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₂₃	#41	1	2	yes	—
90	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
91	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
92	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×D₄	1	4	yes	—
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	#22733	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
102	Q₈×C₂₃	—	2	4	no	—
103	Q₈×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₈×C₂₃	—	2	2	no	—
121	C₈×C₂₃	—	2	4	no	—
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	—	4	4	no	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	C₄×Q₈	1	4	yes	—
126	Q₈×M₁₆	C₂×C₂₃	1	2	yes	—
127	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
128	#1950	C₂×C₂₃	1	4	yes	—
129	#1950	C₂×C₂₃	1	2	yes	—
130	#1849	C₂×C₂₃	1	2	yes	—
131	#1848	C₂×C₂₃	1	1	yes	characteristic
132	#1778	C₂×C₂₃	1	2	yes	—
133	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₂₃	#557	1	1	yes	characteristic
138	C₂²×C₂₃	Q₈²	1	1	yes	characteristic
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	4	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	4	no	—
144	C₄×C₂₃	#430	1	2	yes	—
145	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₂₃	#520	1	1	yes	characteristic
147	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
148	C₄×C₂₃	#550	1	1	yes	characteristic
149	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
150	#446	—	2	2	no	—
151	#446	—	2	4	no	—
152	#446	C₄×C₂₃	1	4	yes	—
153	#446	C₄×C₂₃	1	4	yes	—
154	C₈×Q₈	—	2	2	no	—
155	C₂×#108	C₂²×C₂₃	1	2	yes	—
156	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#108	C₂²×C₂₃	1	2	yes	—
158	C₂×#108	C₂²×C₂₃	1	4	yes	—
159	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
160	#423	C₂²×C₂₃	1	2	yes	—
161	#423	C₂²×C₂₃	1	4	yes	—
162	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
164	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
165	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
166	C₂×#97	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
168	#341	C₂²×C₂₃	1	2	yes	—
169	#341	C₂²×C₂₃	1	4	yes	—
170	C₂×C₂₃	—	2	1	no	—
171	C₂×C₂₃	—	2	1	no	—
172	C₂×C₂₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
174	C₂×C₂₃	#2916	1	1	yes	characteristic
175	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
177	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
178	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
182	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
183	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
184	C₄×Q₈	D₄×C₂₃	1	2	yes	—
185	C₄×Q₈	D₄×C₂₃	1	2	yes	—
186	C₂×C₄⋊C₄	Q₈×C₂₃	1	1	yes	characteristic
187	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
188	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
190	#108	—	2	4	no	—
191	#108	—	2	2	no	—
192	#108	—	4	2	no	—
193	#108	—	4	4	no	—
194	#108	Q₈×C₂₃	1	4	yes	complemented
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	4	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
200	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
202	#97	—	2	4	no	—
203	#97	—	2	2	no	—
204	#97	C₂×C₄×C₂₃	1	8	yes	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
208	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
209	C₂₃	#22733	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
212	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	C₂×Q₈×C₂₃	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
233	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
234	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
235	C₂×C₈	#41×C₂₃	1	2	yes	—
236	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
237	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
238	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
239	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
242	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
243	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₂₃	1	4	yes	—
245	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	4	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	C₄×Q₈×C₂₃	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
274	C₂²	C₂₃×#557	1	1	yes	characteristic
275	C₂²	Q₈²×C₂₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	4	no	—
281	C₄	C₂₃×#430	1	2	yes	—
282	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
283	C₄	C₂₃×#520	1	1	yes	characteristic
284	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
285	C₄	C₂₃×#550	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
289	C₂	C₂×C₂₃×#430	1	1	yes	characteristic
290	C₂	C₂₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
49. C₂₃×#108	Q₈	semidirect
100. #22733	C₂₃	direct (N × H)
194. #108	Q₈×C₂₃	semidirect
209. C₂₃	#22733	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₂₃	2
C₂×C₂₃×#446	2
C₂₃×#1950	2
C₂₃×#1849	2
C₂₃×#1848	2
C₂₃×#1778	2
C₄×C₂₃×#108	2
#22733	23

Subgroups — C23×#22733 #19170359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22733 #19170359