Subgroups — C₂₃×#32703 #19180329

Order 5888.

476 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2080	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2050	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1938	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1881	C₂	1	2	yes	maximal
7	C₂₃×#1861	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1858	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#575	C₂²	1	1	yes	characteristic
11	C₂₃×#568	C₂²	1	2	yes	—
12	C₂₃×#430	C₂²	1	1	yes	characteristic
13	C₂₃×#430	C₂²	1	2	yes	—
14	C₂₃×#424	C₂²	1	2	yes	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂₃×#421	C₂²	1	2	yes	—
18	C₂×C₂₃×#104	C₂²	1	2	yes	—
19	C₂₃×#416	C₂²	1	2	yes	—
20	C₂₃×#413	C₂²	1	2	yes	—
21	C₂₃×#408	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
23	C₂₃×#344	C₂²	1	2	yes	—
24	C₂₃×#344	C₂²	1	2	yes	—
25	C₂₃×#341	C₂²	1	2	yes	—
26	C₂₃×#341	C₂²	1	2	yes	—
27	C₂₃×#341	C₂²	1	2	yes	—
28	C₂₃×#339	C₂²	1	2	yes	—
29	C₂₃×#338	C₂²	1	1	yes	characteristic
30	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₃	C₂³	1	2	yes	—
32	C₂₃×#137	—	2	2	no	—
33	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
34	C₂₃×#135	—	4	1	no	—
35	C₂₃×#131	—	4	2	no	—
36	C₂₃×#131	—	4	2	no	—
37	C₂×M₁₆×C₂₃	—	4	2	no	—
38	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂₃×#117	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#108	—	4	2	no	—
47	C₂₃×#108	—	4	1	no	—
48	C₂₃×C₄⋊C₈	—	4	1	no	—
49	C₂₃×C₄⋊C₈	—	4	1	no	—
50	C₂₃×C₄⋊C₈	—	4	2	no	—
51	C₂₃×#104	—	2	4	no	—
52	C₂₃×#104	—	4	2	no	—
53	C₂₃×#103	—	2	2	no	—
54	C₂₃×#102	—	2	2	no	—
55	C₂₃×#101	—	2	2	no	—
56	C₂₃×#100	—	2	2	no	—
57	C₂₃×C₂²⋊C₈	—	4	2	no	—
58	C₂₃×#97	—	2	2	no	—
59	#41×C₂₃	—	2	2	no	—
60	#41×C₂₃	—	2	2	no	—
61	#41×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	2	2	no	—
63	C₂×Q₈×C₂₃	#41	1	2	yes	—
64	C₂×D₄×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	#41	1	2	yes	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₂₃	—	4	1	no	—
70	SD₁₆×C₂₃	—	4	2	no	—
71	D₈×C₂₃	—	4	1	no	—
72	M₁₆×C₂₃	—	2	2	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	—	2	2	no	—
75	M₁₆×C₂₃	—	2	2	no	—
76	M₁₆×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	—	2	2	no	—
78	M₁₆×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	2	2	no	—
80	M₁₆×C₂₃	—	4	4	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	4	4	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	1	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	#41	1	2	yes	—
95	#32703	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
98	Q₈×C₂₃	—	2	2	no	—
99	Q₈×C₂₃	—	2	2	no	—
100	D₄×C₂₃	—	2	2	no	—
101	D₄×C₂₃	—	2	2	no	—
102	D₄×C₂₃	—	4	2	no	—
103	D₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	8	1	no	—
110	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₂₃	—	4	1	no	—
115	C₈×C₂₃	—	4	1	no	—
116	C₈×C₂₃	—	4	2	no	—
117	C₈×C₂₃	—	4	2	no	—
118	C₈×C₂₃	—	4	2	no	—
119	C₈×C₂₃	—	4	2	no	—
120	#2103	C₂×C₂₃	1	2	yes	complemented
121	#2102	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#2080	C₂×C₂₃	1	2	yes	complemented
123	#2050	C₂×C₂₃	1	1	yes	characteristic
124	#1938	C₂×C₂₃	1	2	yes	complemented
125	#1881	C₂×C₂₃	1	2	yes	—
126	#1861	C₂×C₂₃	1	2	yes	complemented
127	#1858	C₂×C₂₃	1	2	yes	complemented
128	#1838	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	8	2	no	—
132	C₂²×C₂₃	#539	1	1	yes	characteristic
133	C₂²×C₂₃	#539	1	2	yes	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	8	2	no	—
137	C₄×C₂₃	#551	1	1	yes	characteristic
138	C₄×C₂₃	#553	1	2	yes	—
139	C₄×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
140	#575	C₂²×C₂₃	1	1	yes	characteristic
141	#568	C₂²×C₂₃	1	2	yes	—
142	#430	C₂²×C₂₃	1	1	yes	characteristic
143	#430	C₂²×C₂₃	1	2	yes	—
144	#424	C₂²×C₂₃	1	2	yes	—
145	#423	C₂²×C₂₃	1	1	yes	characteristic
146	#423	C₂²×C₂₃	1	1	yes	characteristic
147	#421	C₂²×C₂₃	1	2	yes	—
148	C₂×#104	C₂²×C₂₃	1	2	yes	—
149	#416	C₂²×C₂₃	1	2	yes	—
150	#413	C₂²×C₂₃	1	2	yes	—
151	#408	C₂²×C₂₃	1	2	yes	—
152	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
153	#344	C₂²×C₂₃	1	2	yes	—
154	#344	C₂²×C₂₃	1	2	yes	—
155	#341	C₂²×C₂₃	1	2	yes	—
156	#341	C₂²×C₂₃	1	2	yes	—
157	#341	C₂²×C₂₃	1	2	yes	—
158	#339	C₂²×C₂₃	1	2	yes	—
159	#338	C₂²×C₂₃	1	1	yes	characteristic
160	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×C₂₃	—	2	1	no	—
162	C₂×C₂₃	—	2	2	no	—
163	C₂×C₂₃	—	8	2	no	—
164	C₂×C₂₃	#2782	1	1	yes	characteristic
165	C₂×#41	C₂³×C₂₃	1	2	yes	—
166	#137	—	2	2	no	—
167	AGL(1,ℤ/8)	—	2	2	no	—
168	#135	—	4	1	no	—
169	#131	—	4	2	no	—
170	#131	—	4	2	no	—
171	C₂×M₁₆	—	4	2	no	—
172	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
173	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
174	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
177	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
178	#117	C₂³×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	#108	—	4	2	no	—
181	#108	—	4	1	no	—
182	C₄⋊C₈	—	4	1	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄⋊C₈	—	4	2	no	—
185	#104	—	2	4	no	—
186	#104	—	4	2	no	—
187	#103	—	2	2	no	—
188	#102	—	2	2	no	—
189	#101	—	2	2	no	—
190	#100	—	2	2	no	—
191	C₂²⋊C₈	—	4	2	no	—
192	#97	—	2	2	no	—
193	C₂₃	#32703	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#41×C₂₃	1	2	yes	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	#41×C₂₃	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
204	Q₁₆	—	4	1	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	1	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	2	no	—
229	C₄²	#41×C₂₃	1	2	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
245	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	C₂₃×#539	1	1	yes	characteristic
258	C₂²	C₂₃×#539	1	2	yes	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	2	no	—
262	C₄	C₂₃×#551	1	1	yes	characteristic
263	C₄	C₂₃×#553	1	2	yes	—
264	C₄	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₂₃×#2782	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	#2782
prime radical	C₂₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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×#2102	C₂	semidirect
3. C₂₃×#2080	C₂	semidirect
5. C₂₃×#1938	C₂	semidirect
7. C₂₃×#1861	C₂	semidirect
8. C₂₃×#1858	C₂	semidirect
9. C₂₃×#1838	C₂	semidirect
95. #32703	C₂₃	direct (N × H)
120. #2103	C₂×C₂₃	semidirect
121. #2102	C₂×C₂₃	semidirect
122. #2080	C₂×C₂₃	semidirect
124. #1938	C₂×C₂₃	semidirect
126. #1861	C₂×C₂₃	semidirect
127. #1858	C₂×C₂₃	semidirect
128. #1838	C₂×C₂₃	semidirect
193. C₂₃	#32703	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×#2102	2
C₂₃×#2080	2
C₂₃×#2050	2
C₂₃×#1938	2
C₂₃×#1881	2
C₂₃×#1861	2
C₂₃×#1858	2
C₂₃×#1838	2
#32703	23

Subgroups — C23×#32703 #19180329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32703 #19180329