Subgroups — C₂₃×#23361 #19170987

Order 5888.

756 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3151	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1904	C₂	1	4	yes	maximal, complemented
3	C₂₃×#1903	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1887	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1884	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
8	C₂₃×#568	C₂²	1	2	yes	—
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂₃×#518	C₂²	1	2	yes	—
12	C₂₃×#496	C₄	1	4	yes	complemented
13	C₂₃×#493	C₄	1	2	yes	complemented
14	C₂₃×#492	C₄	1	2	yes	complemented
15	C₂₃×#421	C₂²	1	4	yes	—
16	C₂₃×#418	C₂²	1	4	yes	—
17	C₂×C₂₃×#100	C₂²	1	2	yes	—
18	C₂₃×#410	C₂²	1	2	yes	—
19	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#344	C₂²	1	4	yes	—
21	C₂₃×#339	C₂²	1	4	yes	—
22	C₂₃×#338	C₂²	1	2	yes	—
23	C₂₃×H(3,ℤ/4)	C₂²	1	2	yes	—
24	C₂×#41×C₂₃	C₂³	1	2	yes	—
25	C₂²×D₄×C₂₃	—	4	1	no	—
26	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
27	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
28	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
29	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
30	C₂×D₈×C₂₃	C₂×C₄	1	8	yes	complemented
31	C₂₃×#131	—	4	4	no	—
32	C₂×M₁₆×C₂₃	—	4	2	no	—
33	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
34	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂₃×#127	—	4	1	no	—
37	C₂₃×#127	C₂×C₄	1	2	yes	—
38	C₂₃×#127	C₂×C₄	1	2	yes	—
39	C₂₃×#127	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂₃×#124	C₂×C₄	1	2	yes	complemented
41	C₂₃×#121	—	4	1	no	—
42	C₄×D₄×C₂₃	—	4	4	no	—
43	C₄×D₄×C₂₃	—	4	1	no	—
44	C₂₃×#117	C₂³	1	4	yes	—
45	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#104	—	4	4	no	—
48	C₂₃×#102	—	4	4	no	—
49	C₂₃×#100	—	4	2	no	—
50	C₂₃×#99	—	4	2	no	—
51	C₂₃×#97	C₂×C₄	1	2	yes	complemented
52	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
53	C₂⁴×C₂₃	—	2	2	no	—
54	#41×C₂₃	—	2	4	no	—
55	#41×C₂₃	—	2	4	no	—
56	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	4	no	—
59	C₂×D₄×C₂₃	—	2	4	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂×D₄×C₂₃	—	4	2	no	—
64	C₂×D₄×C₂₃	—	8	4	no	—
65	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	—	2	2	no	—
69	C₂²×C₄×C₂₃	—	4	4	no	—
70	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	SD₁₆×C₂₃	—	4	8	no	—
72	D₈×C₂₃	—	4	8	no	—
73	M₁₆×C₂₃	—	2	4	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	4	4	no	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	4	1	no	—
82	C₂²⋊C₄×C₂₃	—	2	4	no	—
83	C₂²⋊C₄×C₂₃	—	4	2	no	—
84	C₂²⋊C₄×C₂₃	—	4	4	no	—
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	2	4	no	—
87	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	#23361	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	4	no	—
92	C₂³×C₂₃	—	4	2	no	—
93	C₂³×C₂₃	—	4	2	no	—
94	C₂³×C₂₃	—	8	4	no	—
95	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	4	no	—
97	D₄×C₂₃	—	2	4	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	4	4	no	—
101	D₄×C₂₃	—	8	2	no	—
102	D₄×C₂₃	—	8	4	no	—
103	D₄×C₂₃	—	8	4	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	16	4	no	—
112	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
114	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	#3151	C₂×C₂₃	1	1	yes	characteristic
118	#1904	C₂×C₂₃	1	4	yes	complemented
119	#1903	C₂×C₂₃	1	4	yes	complemented
120	#1887	C₂×C₂₃	1	2	yes	complemented
121	#1884	C₂×C₂₃	1	2	yes	complemented
122	#1841	C₂×C₂₃	1	2	yes	complemented
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	—	8	4	no	—
127	C₂²×C₂₃	—	8	2	no	—
128	C₂²×C₂₃	—	8	4	no	—
129	C₂²×C₂₃	—	16	2	no	—
130	C₂²×C₂₃	#518	1	1	yes	characteristic
131	C₂²×C₂₃	#560	1	2	yes	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	4	1	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	8	4	no	—
136	C₄×C₂₃	#518	1	2	yes	—
137	C₄×C₂₃	#560	1	2	yes	—
138	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
139	#568	C₂²×C₂₃	1	2	yes	—
140	#532	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
142	#518	C₂²×C₂₃	1	2	yes	—
143	#496	C₄×C₂₃	1	4	yes	complemented
144	#493	C₄×C₂₃	1	2	yes	complemented
145	#492	C₄×C₂₃	1	2	yes	complemented
146	#421	C₂²×C₂₃	1	4	yes	—
147	#418	C₂²×C₂₃	1	4	yes	—
148	C₂×#100	C₂²×C₂₃	1	2	yes	—
149	#410	C₂²×C₂₃	1	2	yes	—
150	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
151	#344	C₂²×C₂₃	1	4	yes	—
152	#339	C₂²×C₂₃	1	4	yes	—
153	#338	C₂²×C₂₃	1	2	yes	—
154	H(3,ℤ/4)	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	8	2	no	—
158	C₂×C₂₃	—	8	4	no	—
159	C₂×C₂₃	#2380	1	1	yes	characteristic
160	C₂×#41	C₂³×C₂₃	1	2	yes	—
161	C₂²×D₄	—	4	1	no	—
162	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
163	AGL(1,ℤ/8)	—	4	4	no	—
164	AGL(1,ℤ/8)	—	4	4	no	—
165	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
166	C₂×D₈	C₂×C₄×C₂₃	1	8	yes	complemented
167	#131	—	4	4	no	—
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
170	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
171	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
172	#127	—	4	1	no	—
173	#127	C₂×C₄×C₂₃	1	2	yes	—
174	#127	C₂×C₄×C₂₃	1	2	yes	—
175	#127	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
176	#124	C₂×C₄×C₂₃	1	2	yes	complemented
177	#121	—	4	1	no	—
178	C₄×D₄	—	4	4	no	—
179	C₄×D₄	—	4	1	no	—
180	#117	C₂³×C₂₃	1	4	yes	—
181	C₂×C₂²⋊C₄	—	4	2	no	—
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	#104	—	4	4	no	—
184	#102	—	4	4	no	—
185	#100	—	4	2	no	—
186	#99	—	4	2	no	—
187	#97	C₂×C₄×C₂₃	1	2	yes	complemented
188	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
189	C₂₃	#23361	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	—	2	2	no	—
191	#41	—	2	4	no	—
192	#41	—	2	4	no	—
193	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	8	4	no	—
202	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
203	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
208	SD₁₆	—	4	8	no	—
209	D₈	—	4	8	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	8	4	no	—
231	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	8	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	8	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	16	4	no	—
248	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	4	4	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	16	2	no	—
260	C₂²	C₂₃×#518	1	1	yes	characteristic
261	C₂²	C₂₃×#560	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₂₃×#518	1	2	yes	—
267	C₄	C₂₃×#560	1	2	yes	—
268	C₂	—	2	1	no	—
269	C₂	—	2	2	no	—
270	C₂	—	8	4	no	—
271	C₂	—	8	2	no	—
272	C₂	C₂₃×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2380
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3151	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1904	C₂	semidirect
3. C₂₃×#1903	C₂	semidirect
4. C₂₃×#1887	C₂	semidirect
5. C₂₃×#1884	C₂	semidirect
6. C₂₃×#1841	C₂	semidirect
12. C₂₃×#496	C₄	semidirect
13. C₂₃×#493	C₄	semidirect
14. C₂₃×#492	C₄	semidirect
29. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
30. C₂×D₈×C₂₃	C₂×C₄	semidirect
35. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
39. C₂₃×#127	C₂×C₄	semidirect
40. C₂₃×#124	C₂×C₄	semidirect
51. C₂₃×#97	C₂×C₄	semidirect
52. C₄×C₈×C₂₃	C₂×C₄	semidirect
89. #23361	C₂₃	direct (N × H)
118. #1904	C₂×C₂₃	semidirect
119. #1903	C₂×C₂₃	semidirect
120. #1887	C₂×C₂₃	semidirect
121. #1884	C₂×C₂₃	semidirect
122. #1841	C₂×C₂₃	semidirect
143. #496	C₄×C₂₃	semidirect
144. #493	C₄×C₂₃	semidirect
145. #492	C₄×C₂₃	semidirect
165. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
166. C₂×D₈	C₂×C₄×C₂₃	semidirect
171. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
175. #127	C₂×C₄×C₂₃	semidirect
176. #124	C₂×C₄×C₂₃	semidirect
187. #97	C₂×C₄×C₂₃	semidirect
188. C₄×C₈	C₂×C₄×C₂₃	semidirect
189. C₂₃	#23361	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×#3151	2
C₂₃×#1904	2
C₂₃×#1903	2
C₂₃×#1887	2
C₂₃×#1884	2
C₂₃×#1841	2
#23361	23

Subgroups — C23×#23361 #19170987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23361 #19170987