Subgroups — C₂₃×#32872 #19180498

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 282 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
3	C₂₃×#2082	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2070	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2049	C₂	1	2	yes	maximal
6	C₂₃×#2047	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2020	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1902	C₂	1	2	yes	maximal
9	C₂₃×#1858	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#137	C₂²	1	2	yes	—
11	C₂²×Q₁₆×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×#419	C₂²	1	2	yes	—
19	C₂₃×#419	C₂²	1	2	yes	—
20	C₂₃×#413	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#101	C₂²	1	2	yes	—
22	C₂×C₂₃×#101	C₂²	1	2	yes	—
23	C₂×C₂₃×#101	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×Q₈×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#137	—	4	2	no	—
34	C₂₃×#137	—	4	2	no	—
35	C₂×Q₁₆×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	—	4	2	no	—
37	C₂×Q₁₆×C₂₃	—	4	1	no	—
38	C₂×SD₁₆×C₂₃	—	2	2	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×#103	—	4	2	no	—
54	C₂₃×#103	—	4	2	no	—
55	C₂₃×#101	—	2	1	no	—
56	C₂₃×#101	—	4	2	no	—
57	C₂₃×#101	—	4	2	no	—
58	C₂₃×#101	—	4	2	no	—
59	C₂₃×#100	—	2	1	no	—
60	C₂₃×C₂²⋊C₈	—	4	1	no	—
61	#41×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	2	2	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	2	2	no	—
65	C₂×Q₈×C₂₃	—	2	2	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
70	C₂×D₄×C₂₃	—	2	1	no	—
71	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	—	2	1	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₂₃	—	4	4	no	—
76	Q₁₆×C₂₃	—	4	4	no	—
77	SD₁₆×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	#32872	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	1	no	—
101	C₂³×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
102	Q₈×C₂₃	—	2	2	no	—
103	Q₈×C₂₃	—	4	2	no	—
104	Q₈×C₂₃	—	4	2	no	—
105	Q₈×C₂₃	—	4	2	no	—
106	Q₈×C₂₃	—	4	2	no	—
107	D₄×C₂₃	—	2	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	4	1	no	—
123	C₈×C₂₃	—	4	2	no	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	—	4	1	no	—
126	#2103	C₂×C₂₃	1	2	yes	complemented
127	#2098	C₂×C₂₃	1	1	yes	characteristic
128	#2082	C₂×C₂₃	1	2	yes	complemented
129	#2070	C₂×C₂₃	1	2	yes	complemented
130	#2049	C₂×C₂₃	1	2	yes	—
131	#2047	C₂×C₂₃	1	2	yes	complemented
132	#2020	C₂×C₂₃	1	1	yes	characteristic
133	#1902	C₂×C₂₃	1	2	yes	—
134	#1858	C₂×C₂₃	1	1	yes	characteristic
135	C₂²×C₂₃	—	4	1	no	—
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	8	1	no	—
138	C₂²×C₂₃	#539	1	1	yes	characteristic
139	C₂²×C₂₃	#562	1	2	yes	—
140	C₄×C₂₃	—	4	1	no	—
141	C₄×C₂₃	—	8	2	no	—
142	C₄×C₂₃	—	8	2	no	—
143	C₄×C₂₃	#538	1	2	yes	—
144	C₄×C₂₃	#560	1	1	yes	characteristic
145	C₄×C₂₃	#560	1	1	yes	characteristic
146	C₂×#137	C₂²×C₂₃	1	2	yes	—
147	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
148	#568	C₂²×C₂₃	1	1	yes	characteristic
149	#430	C₂²×C₂₃	1	2	yes	—
150	#430	C₂²×C₂₃	1	1	yes	characteristic
151	#430	C₂²×C₂₃	1	2	yes	—
152	#424	C₂²×C₂₃	1	2	yes	—
153	#421	C₂²×C₂₃	1	2	yes	—
154	#419	C₂²×C₂₃	1	2	yes	—
155	#419	C₂²×C₂₃	1	2	yes	—
156	#413	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#101	C₂²×C₂₃	1	2	yes	—
158	C₂×#101	C₂²×C₂₃	1	2	yes	—
159	C₂×#101	C₂²×C₂₃	1	2	yes	—
160	#408	C₂²×C₂₃	1	1	yes	characteristic
161	#344	C₂²×C₂₃	1	2	yes	—
162	#344	C₂²×C₂₃	1	2	yes	—
163	#341	C₂²×C₂₃	1	1	yes	characteristic
164	#341	C₂²×C₂₃	1	2	yes	—
165	#338	C₂²×C₂₃	1	2	yes	—
166	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	—	8	1	no	—
170	C₂×C₂₃	#2826	1	1	yes	characteristic
171	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
173	#137	—	4	2	no	—
174	#137	—	4	2	no	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	4	1	no	—
177	C₂×Q₁₆	—	4	2	no	—
178	C₂×SD₁₆	—	2	2	no	—
179	#131	—	4	2	no	—
180	C₂×M₁₆	—	4	1	no	—
181	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
182	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
183	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
185	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	#117	C₂³×C₂₃	1	2	yes	—
188	#108	—	4	1	no	—
189	#108	—	4	2	no	—
190	#108	—	4	2	no	—
191	C₄⋊C₈	—	4	2	no	—
192	#104	—	4	2	no	—
193	#103	—	4	2	no	—
194	#103	—	4	2	no	—
195	#101	—	2	1	no	—
196	#101	—	4	2	no	—
197	#101	—	4	2	no	—
198	#101	—	4	2	no	—
199	#100	—	2	1	no	—
200	C₂²⋊C₈	—	4	1	no	—
201	C₂₃	#32872	1	1	yes	characteristic, direct product factor, Sylow
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	4	no	—
218	SD₁₆	—	4	4	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
259	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
260	C₈	—	4	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₂₃×#539	1	1	yes	characteristic
270	C₂²	C₂₃×#562	1	2	yes	—
271	C₄	—	4	1	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂₃×#538	1	2	yes	—
275	C₄	C₂₃×#560	1	1	yes	characteristic
276	C₄	C₂₃×#560	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₂₃×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂³
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
3. C₂₃×#2082	C₂	semidirect
4. C₂₃×#2070	C₂	semidirect
6. C₂₃×#2047	C₂	semidirect
99. #32872	C₂₃	direct (N × H)
126. #2103	C₂×C₂₃	semidirect
128. #2082	C₂×C₂₃	semidirect
129. #2070	C₂×C₂₃	semidirect
131. #2047	C₂×C₂₃	semidirect
201. C₂₃	#32872	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×#2082	2
C₂₃×#2070	2
C₂₃×#2049	2
C₂₃×#2047	2
C₂₃×#2020	2
C₂₃×#1902	2
C₂₃×#1858	2
#32872	23

Subgroups — C23×#32872 #19180498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32872 #19180498