Subgroups — C₂₃×#13137 #19160763

Order 5888.

316 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2148	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1974	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#104	—	2	1	no	—
15	C₂₃×#408	—	2	1	no	—
16	C₄×M₁₆×C₂₃	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#368	—	2	2	no	—
19	C₂₃×#362	C₄	1	2	yes	complemented
20	C₂₃×#361	C₄	1	2	yes	complemented
21	C₂₃×#359	—	2	2	no	—
22	C₂₃×#341	—	2	1	no	—
23	C₂×#41×C₂₃	—	2	1	no	—
24	C₂₃×#135	—	2	1	no	—
25	C₂₃×#135	—	2	1	no	—
26	C₂₃×#135	—	2	1	no	—
27	C₂₃×#135	—	2	1	no	—
28	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂₃×M₃₂	—	2	1	no	—
38	C₂₃×M₃₂	—	2	1	no	—
39	C₂₃×M₃₂	—	2	1	no	—
40	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₁₆×C₂₃	—	2	1	no	—
42	C₂×C₁₆×C₂₃	—	2	1	no	—
43	C₂₃×#108	—	2	1	no	—
44	C₂₃×#108	—	2	1	no	—
45	C₂₃×#108	—	2	1	no	—
46	C₂₃×#108	—	2	1	no	—
47	C₂₃×#108	C₂×C₄	1	2	yes	—
48	C₂₃×#108	D₄	1	2	yes	—
49	C₂₃×#104	—	4	1	no	—
50	C₂₃×#104	—	4	1	no	—
51	C₂₃×C₂²⋊C₈	—	4	2	no	—
52	C₂₃×#97	—	4	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	C₄×C₈×C₂₃	—	4	1	no	—
56	#41×C₂₃	—	4	1	no	—
57	#41×C₂₃	—	4	1	no	—
58	#41×C₂₃	—	4	1	no	—
59	#41×C₂₃	—	4	1	no	—
60	C₂×Q₈×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₂₃	—	2	1	no	—
67	Q₁₆×C₂₃	—	2	1	no	—
68	SD₁₆×C₂₃	—	4	1	no	—
69	SD₁₆×C₂₃	—	4	1	no	—
70	D₈×C₂₃	—	2	1	no	—
71	D₈×C₂₃	—	2	1	no	—
72	M₁₆×C₂₃	—	2	2	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	—	4	1	no	—
75	M₁₆×C₂₃	—	4	1	no	—
76	M₁₆×C₂₃	—	4	1	no	—
77	M₁₆×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	1	no	—
83	C₂×C₈×C₂₃	—	4	1	no	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
90	C₄²×C₂₃	—	2	1	no	—
91	C₄²×C₂₃	D₈	1	1	yes	characteristic
92	C₄²×C₂₃	D₈	1	1	yes	characteristic
93	C₁₆×C₂₃	—	2	2	no	—
94	C₁₆×C₂₃	—	4	1	no	—
95	#13137	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	4	1	no	—
97	C₂³×C₂₃	#137	1	1	yes	characteristic
98	Q₈×C₂₃	—	4	1	no	—
99	Q₈×C₂₃	—	4	1	no	—
100	D₄×C₂₃	—	4	1	no	—
101	D₄×C₂₃	—	4	1	no	—
102	D₄×C₂₃	—	8	1	no	—
103	D₄×C₂₃	—	8	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	8	1	no	—
109	C₂×C₄×C₂₃	—	8	1	no	—
110	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
115	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₈×C₂₃	—	4	2	no	—
117	C₈×C₂₃	—	4	2	no	—
118	C₈×C₂₃	—	4	1	no	—
119	C₈×C₂₃	—	8	1	no	—
120	C₈×C₂₃	#117	1	1	yes	characteristic
121	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
122	C₈×C₂₃	#124	1	1	yes	characteristic
123	C₈×C₂₃	#126	1	1	yes	characteristic
124	#2161	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#2148	C₂×C₂₃	1	1	yes	characteristic
126	#2143	C₂×C₂₃	1	1	yes	characteristic
127	#1974	C₂×C₂₃	1	1	yes	characteristic
128	C₄×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#1388	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1376	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	8	2	no	—
133	C₂²×C₂₃	—	8	1	no	—
134	C₂²×C₂₃	#440	1	1	yes	characteristic
135	C₂²×C₂₃	#464	1	1	yes	characteristic
136	C₂²×C₂₃	#480	1	1	yes	characteristic
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	8	1	no	—
140	C₄×C₂₃	#388	1	1	yes	characteristic
141	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
142	C₄×C₂₃	#459	1	1	yes	characteristic
143	C₄×C₂₃	#482	1	1	yes	characteristic
144	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×#104	—	2	1	no	—
151	#408	—	2	1	no	—
152	C₄×M₁₆	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	#368	—	2	2	no	—
155	#362	C₄×C₂₃	1	2	yes	complemented
156	#361	C₄×C₂₃	1	2	yes	complemented
157	#359	—	2	2	no	—
158	#341	—	2	1	no	—
159	C₂×C₂₃	—	2	1	no	—
160	C₂×C₂₃	—	2	1	no	—
161	C₂×C₂₃	—	2	1	no	—
162	C₂×C₂₃	—	16	1	no	—
163	C₂×C₂₃	#1925	1	1	yes	characteristic
164	C₂×#41	—	2	1	no	—
165	#135	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	—	2	1	no	—
168	#135	—	2	1	no	—
169	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂×SD₁₆	—	2	1	no	—
171	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	C₂×C₄×C₂₃	1	2	yes	—
189	#108	D₄×C₂₃	1	2	yes	—
190	#104	—	4	1	no	—
191	#104	—	4	1	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	#97	—	4	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	4	1	no	—
197	C₂₃	#13137	1	1	yes	characteristic, direct product factor, Sylow
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	Q₁₆	—	2	1	no	—
209	Q₁₆	—	2	1	no	—
210	SD₁₆	—	4	1	no	—
211	SD₁₆	—	4	1	no	—
212	D₈	—	2	1	no	—
213	D₈	—	2	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
229	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
230	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
231	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	D₈×C₂₃	1	1	yes	characteristic
234	C₄²	D₈×C₂₃	1	1	yes	characteristic
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₂₃×#137	1	1	yes	characteristic
239	Q₈	—	4	1	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	1	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
254	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
256	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	1	no	—
260	C₈	—	8	1	no	—
261	C₈	C₂₃×#117	1	1	yes	characteristic
262	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
263	C₈	C₂₃×#124	1	1	yes	characteristic
264	C₈	C₂₃×#126	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₂₃×#440	1	1	yes	characteristic
269	C₂²	C₂₃×#464	1	1	yes	characteristic
270	C₂²	C₂₃×#480	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₂₃×#388	1	1	yes	characteristic
275	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
276	C₄	C₂₃×#459	1	1	yes	characteristic
277	C₄	C₂₃×#482	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	16	1	no	—
282	C₂	C₂₃×#1925	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#459
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1925
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2161	C₂	semidirect
5. C₄×C₂₃×#108	C₂	semidirect
6. C₂₃×#1388	C₂	semidirect
7. C₂₃×#1376	C₂	semidirect
19. C₂₃×#362	C₄	semidirect
20. C₂₃×#361	C₄	semidirect
95. #13137	C₂₃	direct (N × H)
124. #2161	C₂×C₂₃	semidirect
128. C₄×#108	C₂×C₂₃	semidirect
129. #1388	C₂×C₂₃	semidirect
130. #1376	C₂×C₂₃	semidirect
155. #362	C₄×C₂₃	semidirect
156. #361	C₄×C₂₃	semidirect
197. C₂₃	#13137	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×#2161	2
C₂₃×#2148	2
C₂₃×#2143	2
C₂₃×#1974	2
C₄×C₂₃×#108	2
C₂₃×#1388	2
C₂₃×#1376	2
#13137	23

Subgroups — C23×#13137 #19160763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13137 #19160763