Subgroups — C₂₃×#33828 #19181454

Order 5888.

488 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	C₂₃×#3175	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2227	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2224	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2215	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2210	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#542	—	2	2	no	—
11	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#495	C₂²	1	2	yes	—
18	C₂₃×#489	—	2	2	no	—
19	C₂₃×#468	—	2	2	no	—
20	C₂₃×#467	C₂²	1	2	yes	—
21	C₂₃×#466	C₂²	1	2	yes	—
22	C₂₃×#440	—	2	2	no	—
23	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
24	C₄×D₈×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#405	C₂²	1	1	yes	characteristic
26	C₂₃×#366	C₂²	1	1	yes	characteristic
27	C₂₃×#366	C₂²	1	1	yes	characteristic
28	C₂₃×#366	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂₃×#358	C₂²	1	2	yes	—
32	C₂₃×#358	C₂²	1	2	yes	—
33	C₂₃×#357	C₂²	1	2	yes	—
34	C₂₃×#357	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	C₂³	1	2	yes	—
38	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊Q₈	—	2	2	no	—
41	C₂₃×#126	—	4	2	no	—
42	C₂₃×#124	—	2	2	no	—
43	C₂₃×#124	—	4	2	no	—
44	C₂₃×#123	—	4	2	no	—
45	C₂₃×C₂²⋊Q₈	—	2	2	no	—
46	C₂₃×C₂²⋊Q₈	—	4	2	no	—
47	C₂₃×#121	—	2	2	no	—
48	C₄×Q₈×C₂₃	—	2	2	no	—
49	C₄×D₄×C₂₃	—	2	2	no	—
50	C₂₃×#117	D₄	1	1	yes	characteristic
51	C₂₃×Q₃₂	—	4	1	no	—
52	C₂₃×SD₃₂	—	4	2	no	—
53	C₂₃×D₁₆	—	4	1	no	—
54	C₂₃×M₃₂	—	4	1	no	—
55	C₂×C₁₆×C₂₃	—	4	1	no	—
56	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
58	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#107	—	2	2	no	—
60	C₂₃×#107	C₂³	1	2	yes	—
61	C₂₃×#107	C₂³	1	2	yes	—
62	C₂₃×#103	—	2	2	no	—
63	C₂₃×#103	—	2	2	no	—
64	C₂₃×#102	—	2	2	no	—
65	C₂₃×#102	—	2	2	no	—
66	C₂₃×#97	D₄	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	4	2	no	—
70	C₂×D₄×C₂₃	—	2	2	no	—
71	C₂×D₄×C₂₃	—	4	2	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₂₃	—	2	2	no	—
75	Q₁₆×C₂₃	—	2	2	no	—
76	SD₁₆×C₂₃	—	4	4	no	—
77	D₈×C₂₃	—	2	2	no	—
78	D₈×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	1	no	—
82	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₂₃	—	4	2	no	—
93	C₂²⋊C₄×C₂₃	—	4	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	2	no	—
95	C₂²⋊C₄×C₂₃	—	4	2	no	—
96	C₂²⋊C₄×C₂₃	—	4	2	no	—
97	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	C₂×D₄	1	2	yes	—
100	C₁₆×C₂₃	—	2	1	no	—
101	C₁₆×C₂₃	—	2	1	no	—
102	C₁₆×C₂₃	—	2	2	no	—
103	#33828	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	4	2	no	—
105	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₂₃	—	4	2	no	—
107	Q₈×C₂₃	—	4	2	no	—
108	Q₈×C₂₃	—	16	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	16	2	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	16	2	no	—
118	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	2	no	—
124	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
126	#3175	C₂×C₂₃	1	2	yes	complemented
127	#2227	C₂×C₂₃	1	2	yes	complemented
128	#2224	C₂×C₂₃	1	2	yes	complemented
129	#2220	C₂×C₂₃	1	2	yes	complemented
130	#2216	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#2215	C₂×C₂₃	1	1	yes	characteristic
132	#2211	C₂×C₂₃	1	2	yes	complemented
133	#2210	C₂×C₂₃	1	2	yes	complemented
134	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	8	2	no	—
139	C₂²×C₂₃	—	8	2	no	—
140	C₂²×C₂₃	—	8	2	no	—
141	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	8	2	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	—	8	2	no	—
148	C₄×C₂₃	#537	1	1	yes	characteristic
149	C₄×C₂₃	#575	1	1	yes	characteristic
150	#542	—	2	2	no	—
151	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
153	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
156	#500	C₂²×C₂₃	1	2	yes	—
157	#495	C₂²×C₂₃	1	2	yes	—
158	#489	—	2	2	no	—
159	#468	—	2	2	no	—
160	#467	C₂²×C₂₃	1	2	yes	—
161	#466	C₂²×C₂₃	1	2	yes	—
162	#440	—	2	2	no	—
163	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
164	C₄×D₈	C₂²×C₂₃	1	2	yes	—
165	#405	C₂²×C₂₃	1	1	yes	characteristic
166	#366	C₂²×C₂₃	1	1	yes	characteristic
167	#366	C₂²×C₂₃	1	1	yes	characteristic
168	#366	C₂²×C₂₃	1	2	yes	—
169	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
170	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
171	#358	C₂²×C₂₃	1	2	yes	—
172	#358	C₂²×C₂₃	1	2	yes	—
173	#357	C₂²×C₂₃	1	2	yes	—
174	#357	C₂²×C₂₃	1	2	yes	—
175	C₂×C₂₃	—	4	1	no	—
176	C₂×C₂₃	—	16	2	no	—
177	C₂×C₂₃	#3245	1	1	yes	characteristic
178	C₂×C₂₃	#3415	1	1	yes	characteristic
179	C₂×C₂₃	#3417	1	1	yes	characteristic
180	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
181	C₂×SD₁₆	—	2	2	no	—
182	C₂×D₈	C₂³×C₂₃	1	2	yes	—
183	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	2	no	—
186	#126	—	4	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	#123	—	4	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	—	4	2	no	—
192	#121	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	#117	D₄×C₂₃	1	1	yes	characteristic
196	Q₃₂	—	4	1	no	—
197	SD₃₂	—	4	2	no	—
198	D₁₆	—	4	1	no	—
199	M₃₂	—	4	1	no	—
200	C₂×C₁₆	—	4	1	no	—
201	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
202	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
203	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
204	#107	—	2	2	no	—
205	#107	C₂³×C₂₃	1	2	yes	—
206	#107	C₂³×C₂₃	1	2	yes	—
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	#102	—	2	2	no	—
211	#97	D₄×C₂₃	1	2	yes	—
212	C₄×C₈	C₂³×C₂₃	1	2	yes	—
213	C₂₃	#33828	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	D₈	—	2	2	no	—
224	D₈	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
237	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
244	C₄²	—	4	2	no	—
245	C₄²	C₂×D₄×C₂₃	1	2	yes	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	16	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	16	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	16	2	no	—
263	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
264	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
265	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	1	no	—
269	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₂₃×#537	1	1	yes	characteristic
285	C₄	C₂₃×#575	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	16	2	no	—
288	C₂	C₂₃×#3245	1	1	yes	characteristic
289	C₂	C₂₃×#3415	1	1	yes	characteristic
290	C₂	C₂₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3175	C₂	semidirect
2. C₂₃×#2227	C₂	semidirect
3. C₂₃×#2224	C₂	semidirect
4. C₂₃×#2220	C₂	semidirect
5. C₂₃×#2216	C₂	semidirect
7. C₂₃×#2211	C₂	semidirect
8. C₂₃×#2210	C₂	semidirect
9. C₂₃×#2151	C₂	semidirect
103. #33828	C₂₃	direct (N × H)
126. #3175	C₂×C₂₃	semidirect
127. #2227	C₂×C₂₃	semidirect
128. #2224	C₂×C₂₃	semidirect
129. #2220	C₂×C₂₃	semidirect
130. #2216	C₂×C₂₃	semidirect
132. #2211	C₂×C₂₃	semidirect
133. #2210	C₂×C₂₃	semidirect
134. #2151	C₂×C₂₃	semidirect
213. C₂₃	#33828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3175	2
C₂₃×#2227	2
C₂₃×#2224	2
C₂₃×#2220	2
C₂₃×#2216	2
C₂₃×#2215	2
C₂₃×#2211	2
C₂₃×#2210	2
C₂₃×#2151	2
#33828	23

Subgroups — C23×#33828 #19181454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33828 #19181454