Subgroups — C₂₃×#13158 #19160784

Order 5888.

316 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1978	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂₃×#430	—	2	1	no	—
12	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂₃×#418	—	2	1	no	—
16	C₂×C₂₃×#100	—	2	1	no	—
17	C₂₃×#405	C₂²	1	1	yes	characteristic
18	C₂₃×#361	—	2	2	no	—
19	C₂₃×#361	C₄	1	4	yes	complemented
20	C₂₃×#347	—	2	2	no	—
21	C₂₃×#344	—	2	1	no	—
22	C₂²×D₄×C₂₃	—	2	1	no	—
23	C₂×D₈×C₂₃	—	2	1	no	—
24	C₂×D₈×C₂₃	—	2	1	no	—
25	C₂×D₈×C₂₃	—	2	1	no	—
26	C₂×D₈×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#117	D₄	1	1	yes	characteristic
35	C₂₃×M₃₂	—	2	1	no	—
36	C₂₃×M₃₂	—	2	1	no	—
37	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
38	C₂₃×M₃₂	D₄	1	2	yes	—
39	C₂×C₁₆×C₂₃	—	2	1	no	—
40	C₂×C₁₆×C₂₃	—	2	1	no	—
41	C₂₃×#108	—	2	1	no	—
42	C₂₃×#108	—	2	1	no	—
43	C₂₃×#108	—	2	1	no	—
44	C₂₃×#108	—	2	1	no	—
45	C₂₃×#108	—	2	1	no	—
46	C₂₃×#108	—	4	2	no	—
47	C₂₃×#108	C₂×C₄	1	2	yes	—
48	C₂₃×#102	—	4	2	no	—
49	C₂₃×#100	—	4	2	no	—
50	C₂₃×#97	—	2	1	no	—
51	C₄×C₈×C₂₃	—	2	1	no	—
52	C₂⁴×C₂₃	—	4	1	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	4	1	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	D₈×C₂₃	—	2	2	no	—
61	D₈×C₂₃	—	2	2	no	—
62	D₈×C₂₃	—	4	1	no	—
63	D₈×C₂₃	—	4	1	no	—
64	M₁₆×C₂₃	—	2	2	no	—
65	M₁₆×C₂₃	—	2	1	no	—
66	M₁₆×C₂₃	—	2	1	no	—
67	M₁₆×C₂₃	—	4	1	no	—
68	M₁₆×C₂₃	—	4	1	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	1	no	—
71	M₁₆×C₂₃	—	4	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₁₆×C₂₃	—	2	2	no	—
86	C₁₆×C₂₃	—	2	2	no	—
87	#13158	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	8	1	no	—
90	C₂³×C₂₃	—	8	1	no	—
91	C₂³×C₂₃	—	16	2	no	—
92	C₂³×C₂₃	#137	1	1	yes	characteristic
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	8	1	no	—
96	D₄×C₂₃	—	8	1	no	—
97	C₂×C₄×C₂₃	—	4	1	no	—
98	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₈×C₂₃	—	4	2	no	—
105	C₈×C₂₃	—	4	1	no	—
106	C₈×C₂₃	—	8	1	no	—
107	C₈×C₂₃	—	8	1	no	—
108	C₈×C₂₃	#117	1	1	yes	characteristic
109	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₂₃	#123	1	1	yes	characteristic
111	C₈×C₂₃	#126	1	1	yes	characteristic
112	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
113	C₂×#361	C₂×C₂₃	1	1	yes	characteristic
114	#2110	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#1978	C₂×C₂₃	1	1	yes	characteristic
116	#1953	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#1388	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#1388	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂²×C₂₃	—	4	1	no	—
120	C₂²×C₂₃	—	8	2	no	—
121	C₂²×C₂₃	—	16	2	no	—
122	C₂²×C₂₃	—	16	2	no	—
123	C₂²×C₂₃	—	16	2	no	—
124	C₂²×C₂₃	#440	1	1	yes	characteristic
125	C₂²×C₂₃	#468	1	1	yes	characteristic
126	C₂²×C₂₃	#489	1	1	yes	characteristic
127	C₄×C₂₃	—	8	1	no	—
128	C₄×C₂₃	#388	1	1	yes	characteristic
129	C₄×C₂₃	#442	1	1	yes	characteristic
130	C₄×C₂₃	#469	1	1	yes	characteristic
131	C₄×C₂₃	#490	1	1	yes	characteristic
132	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
135	#430	—	2	1	no	—
136	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
139	#418	—	2	1	no	—
140	C₂×#100	—	2	1	no	—
141	#405	C₂²×C₂₃	1	1	yes	characteristic
142	#361	—	2	2	no	—
143	#361	C₄×C₂₃	1	4	yes	complemented
144	#347	—	2	2	no	—
145	#344	—	2	1	no	—
146	C₂×C₂₃	—	2	1	no	—
147	C₂×C₂₃	—	2	1	no	—
148	C₂×C₂₃	—	2	1	no	—
149	C₂×C₂₃	—	16	2	no	—
150	C₂×C₂₃	#1936	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
163	#117	D₄×C₂₃	1	1	yes	characteristic
164	M₃₂	—	2	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
167	M₃₂	D₄×C₂₃	1	2	yes	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	#108	—	2	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	—	2	1	no	—
175	#108	—	4	2	no	—
176	#108	C₂×C₄×C₂₃	1	2	yes	—
177	#102	—	4	2	no	—
178	#100	—	4	2	no	—
179	#97	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₂₃	#13158	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	—	4	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
190	D₈	—	2	2	no	—
191	D₈	—	2	2	no	—
192	D₈	—	4	1	no	—
193	D₈	—	4	1	no	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
209	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
210	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
211	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	16	2	no	—
221	C₂³	C₂₃×#137	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	1	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
230	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₈	—	4	2	no	—
234	C₈	—	4	1	no	—
235	C₈	—	8	1	no	—
236	C₈	—	8	1	no	—
237	C₈	C₂₃×#117	1	1	yes	characteristic
238	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
239	C₈	C₂₃×#123	1	1	yes	characteristic
240	C₈	C₂₃×#126	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₂₃×#440	1	1	yes	characteristic
247	C₂²	C₂₃×#468	1	1	yes	characteristic
248	C₂²	C₂₃×#489	1	1	yes	characteristic
249	C₄	—	8	1	no	—
250	C₄	C₂₃×#388	1	1	yes	characteristic
251	C₄	C₂₃×#442	1	1	yes	characteristic
252	C₄	C₂₃×#469	1	1	yes	characteristic
253	C₄	C₂₃×#490	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₂₃×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1936
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2110	C₂	semidirect
5. C₂₃×#1953	C₂	semidirect
6. C₂₃×#1388	C₂	semidirect
7. C₂₃×#1388	C₂	semidirect
19. C₂₃×#361	C₄	semidirect
87. #13158	C₂₃	direct (N × H)
114. #2110	C₂×C₂₃	semidirect
116. #1953	C₂×C₂₃	semidirect
117. #1388	C₂×C₂₃	semidirect
118. #1388	C₂×C₂₃	semidirect
143. #361	C₄×C₂₃	semidirect
181. C₂₃	#13158	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#361	2
C₂₃×#2110	2
C₂₃×#1978	2
C₂₃×#1953	2
C₂₃×#1388	2
#13158	23

Subgroups — C23×#13158 #19160784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13158 #19160784