Subgroups — C₂₃×#32870 #19180496

Order 5888.

502 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×#2103	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2098	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2081	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2069	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2048	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2045	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2018	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1903	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1858	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
11	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#568	C₂²	1	1	yes	characteristic
13	C₂₃×#430	C₂²	1	2	yes	—
14	C₂₃×#430	C₂²	1	1	yes	characteristic
15	C₂₃×#430	C₂²	1	2	yes	—
16	C₂₃×#424	C₂²	1	2	yes	—
17	C₂₃×#421	C₂²	1	2	yes	—
18	C₂₃×#418	C₂²	1	2	yes	—
19	C₂₃×#418	C₂²	1	2	yes	—
20	C₂₃×#413	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#100	C₂²	1	2	yes	—
22	C₂×C₂₃×#100	C₂²	1	2	yes	—
23	C₂×C₂₃×#100	C₂²	1	2	yes	—
24	C₂₃×#408	C₂²	1	1	yes	characteristic
25	C₂₃×#344	C₂²	1	2	yes	—
26	C₂₃×#344	C₂²	1	2	yes	—
27	C₂₃×#341	C₂²	1	1	yes	characteristic
28	C₂₃×#341	C₂²	1	2	yes	—
29	C₂₃×#338	C₂²	1	2	yes	—
30	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
33	C₂₃×AGL(1,ℤ/8)	—	4	2	no	—
34	C₂₃×AGL(1,ℤ/8)	—	4	2	no	—
35	C₂×SD₁₆×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	—	4	2	no	—
38	C₂×D₈×C₂₃	—	4	1	no	—
39	C₂₃×#131	—	4	2	no	—
40	C₂×M₁₆×C₂₃	—	4	1	no	—
41	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
42	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
43	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
45	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#117	C₂³	1	2	yes	—
48	C₂₃×#108	—	4	2	no	—
49	C₂₃×#108	—	4	2	no	—
50	C₂₃×#108	—	4	1	no	—
51	C₂₃×C₄⋊C₈	—	4	2	no	—
52	C₂₃×#104	—	4	2	no	—
53	C₂₃×#102	—	4	2	no	—
54	C₂₃×#102	—	4	2	no	—
55	C₂₃×#101	—	2	1	no	—
56	C₂₃×#100	—	2	1	no	—
57	C₂₃×#100	—	4	2	no	—
58	C₂₃×#100	—	4	2	no	—
59	C₂₃×#100	—	4	2	no	—
60	C₂₃×C₂²⋊C₈	—	4	1	no	—
61	C₂⁴×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	2	2	no	—
63	#41×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	2	2	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	2	2	no	—
71	C₂×D₄×C₂₃	—	2	2	no	—
72	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	—	2	1	no	—
74	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	SD₁₆×C₂₃	—	4	4	no	—
76	D₈×C₂₃	—	4	4	no	—
77	D₈×C₂₃	—	4	4	no	—
78	M₁₆×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	2	2	no	—
80	M₁₆×C₂₃	—	2	2	no	—
81	M₁₆×C₂₃	—	2	2	no	—
82	M₁₆×C₂₃	—	2	2	no	—
83	M₁₆×C₂₃	—	2	2	no	—
84	M₁₆×C₂₃	—	2	2	no	—
85	M₁₆×C₂₃	—	2	2	no	—
86	M₁₆×C₂₃	—	2	2	no	—
87	M₁₆×C₂₃	—	4	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	1	no	—
94	C₂×C₈×C₂₃	—	2	1	no	—
95	C₂×C₈×C₂₃	—	4	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	2	2	no	—
99	#32870	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	1	no	—
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	—	4	2	no	—
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	—	16	2	no	—
105	C₂³×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
106	Q₈×C₂₃	—	2	2	no	—
107	D₄×C₂₃	—	2	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	4	2	no	—
123	C₈×C₂₃	—	4	1	no	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	—	4	2	no	—
126	C₈×C₂₃	—	4	1	no	—
127	#2103	C₂×C₂₃	1	2	yes	complemented
128	#2098	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#2081	C₂×C₂₃	1	2	yes	complemented
130	#2069	C₂×C₂₃	1	2	yes	complemented
131	#2048	C₂×C₂₃	1	2	yes	complemented
132	#2045	C₂×C₂₃	1	2	yes	complemented
133	#2018	C₂×C₂₃	1	1	yes	characteristic
134	#1903	C₂×C₂₃	1	2	yes	complemented
135	#1858	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	—	8	1	no	—
139	C₂²×C₂₃	—	8	2	no	—
140	C₂²×C₂₃	—	16	2	no	—
141	C₂²×C₂₃	—	16	2	no	—
142	C₂²×C₂₃	—	16	2	no	—
143	C₂²×C₂₃	#539	1	1	yes	characteristic
144	C₂²×C₂₃	#562	1	2	yes	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	#538	1	2	yes	—
148	C₄×C₂₃	#560	1	1	yes	characteristic
149	C₄×C₂₃	#560	1	1	yes	characteristic
150	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
151	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
152	#568	C₂²×C₂₃	1	1	yes	characteristic
153	#430	C₂²×C₂₃	1	2	yes	—
154	#430	C₂²×C₂₃	1	1	yes	characteristic
155	#430	C₂²×C₂₃	1	2	yes	—
156	#424	C₂²×C₂₃	1	2	yes	—
157	#421	C₂²×C₂₃	1	2	yes	—
158	#418	C₂²×C₂₃	1	2	yes	—
159	#418	C₂²×C₂₃	1	2	yes	—
160	#413	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×#100	C₂²×C₂₃	1	2	yes	—
162	C₂×#100	C₂²×C₂₃	1	2	yes	—
163	C₂×#100	C₂²×C₂₃	1	2	yes	—
164	#408	C₂²×C₂₃	1	1	yes	characteristic
165	#344	C₂²×C₂₃	1	2	yes	—
166	#344	C₂²×C₂₃	1	2	yes	—
167	#341	C₂²×C₂₃	1	1	yes	characteristic
168	#341	C₂²×C₂₃	1	2	yes	—
169	#338	C₂²×C₂₃	1	2	yes	—
170	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₂₃	—	2	2	no	—
172	C₂×C₂₃	—	2	1	no	—
173	C₂×C₂₃	—	8	1	no	—
174	C₂×C₂₃	—	16	2	no	—
175	C₂×C₂₃	#2826	1	1	yes	characteristic
176	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
177	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
178	AGL(1,ℤ/8)	—	4	2	no	—
179	AGL(1,ℤ/8)	—	4	2	no	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	4	1	no	—
183	C₂×D₈	—	4	2	no	—
184	#131	—	4	2	no	—
185	C₂×M₁₆	—	4	1	no	—
186	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
187	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
188	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
190	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	#117	C₂³×C₂₃	1	2	yes	—
193	#108	—	4	1	no	—
194	#108	—	4	2	no	—
195	#108	—	4	2	no	—
196	C₄⋊C₈	—	4	2	no	—
197	#104	—	4	2	no	—
198	#102	—	4	2	no	—
199	#102	—	4	2	no	—
200	#101	—	2	1	no	—
201	#100	—	2	1	no	—
202	#100	—	4	2	no	—
203	#100	—	4	2	no	—
204	#100	—	4	2	no	—
205	C₂²⋊C₈	—	4	1	no	—
206	C₂₃	#32870	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	2	2	no	—
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
221	SD₁₆	—	4	4	no	—
222	D₈	—	4	4	no	—
223	D₈	—	4	4	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	16	2	no	—
250	C₂³	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
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×H(5,𝔽₂)	1	2	yes	—
265	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
266	C₈	—	4	1	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	2	no	—
278	C₂²	—	16	2	no	—
279	C₂²	C₂₃×#539	1	1	yes	characteristic
280	C₂²	C₂₃×#562	1	2	yes	—
281	C₄	—	4	1	no	—
282	C₄	—	8	2	no	—
283	C₄	C₂₃×#538	1	2	yes	—
284	C₄	C₂₃×#560	1	1	yes	characteristic
285	C₄	C₂₃×#560	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	1	no	—
289	C₂	—	16	2	no	—
290	C₂	C₂₃×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#2018	C₂
LCS	G ▷ C₂²×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
1. C₂₃×#2103	C₂	semidirect
2. C₂₃×#2098	C₂	semidirect
3. C₂₃×#2081	C₂	semidirect
4. C₂₃×#2069	C₂	semidirect
5. C₂₃×#2048	C₂	semidirect
6. C₂₃×#2045	C₂	semidirect
8. C₂₃×#1903	C₂	semidirect
9. C₂₃×#1858	C₂	semidirect
99. #32870	C₂₃	direct (N × H)
127. #2103	C₂×C₂₃	semidirect
128. #2098	C₂×C₂₃	semidirect
129. #2081	C₂×C₂₃	semidirect
130. #2069	C₂×C₂₃	semidirect
131. #2048	C₂×C₂₃	semidirect
132. #2045	C₂×C₂₃	semidirect
134. #1903	C₂×C₂₃	semidirect
135. #1858	C₂×C₂₃	semidirect
206. C₂₃	#32870	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×#2103	2
C₂₃×#2098	2
C₂₃×#2081	2
C₂₃×#2069	2
C₂₃×#2048	2
C₂₃×#2045	2
C₂₃×#2018	2
C₂₃×#1903	2
C₂₃×#1858	2
#32870	23

Subgroups — C23×#32870 #19180496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32870 #19180496