Subgroups — C₂₃×#12557 #19160183

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×#117	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
8	C₂×C₂₃×M₃₂	—	2	2	no	—
9	C₂×C₂₃×M₃₂	—	2	2	no	—
10	C₂×C₂₃×M₃₂	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#427	—	2	2	no	—
13	C₂₃×#427	C₄	1	4	yes	complemented
14	C₂×C₂₃×#107	—	2	1	no	—
15	C₂×C₂₃×#107	—	2	2	no	—
16	C₂×C₂₃×#107	C₄	1	4	yes	complemented
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×#107	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	—	2	1	no	—
20	C₂³×C₄×C₂₃	—	4	1	no	—
21	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂²×C₈×C₂₃	D₄	1	2	yes	—
28	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×#117	—	2	4	no	—
31	C₂₃×#117	—	4	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
36	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
37	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
38	C₂×C₄²×C₂₃	—	4	1	no	—
39	C₂₃×M₃₂	—	2	8	no	—
40	C₂×C₁₆×C₂₃	—	2	2	no	—
41	C₂×C₁₆×C₂₃	—	2	2	no	—
42	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
43	C₂₃×#107	—	2	4	no	—
44	C₂₃×#107	—	2	2	no	—
45	C₂₃×#107	—	2	2	no	—
46	C₂₃×#107	C₂×C₄	1	4	yes	—
47	C₂₃×#107	C₂×C₄	1	4	yes	complemented
48	C₂₃×#106	—	2	4	no	—
49	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	4	4	no	—
51	C₂²×C₄×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	8	4	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₄²	1	4	yes	—
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	4	4	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₂²⋊C₄×C₂₃	—	4	4	no	—
81	C₄²×C₂₃	—	4	4	no	—
82	C₁₆×C₂₃	—	2	4	no	—
83	#12557	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	#102	1	2	yes	—
87	C₂³×C₂₃	#102	1	2	yes	—
88	C₂³×C₂₃	#107	1	2	yes	—
89	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	—	4	4	no	—
91	C₂×C₄×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	8	8	no	—
93	C₂×C₄×C₂₃	—	8	2	no	—
94	C₂×C₄×C₂₃	—	8	4	no	—
95	C₂×C₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	#95	1	4	yes	—
97	C₂×C₄×C₂₃	#102	1	2	yes	—
98	C₂×C₄×C₂₃	#102	1	2	yes	—
99	C₂×C₄×C₂₃	#103	1	4	yes	—
100	C₂×C₄×C₂₃	#103	1	4	yes	—
101	C₂×C₄×C₂₃	#106	1	4	yes	—
102	C₂×C₄×C₂₃	#107	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₈×C₂₃	#95	1	4	yes	—
110	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
111	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
115	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
116	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic
117	#1380	C₂×C₂₃	1	4	yes	complemented
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	2	4	no	—
122	C₂²×C₂₃	#340	1	4	yes	—
123	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
126	C₂²×C₂₃	#509	1	2	yes	—
127	C₂²×C₂₃	#510	1	2	yes	—
128	C₄×C₂₃	—	8	4	no	—
129	C₄×C₂₃	—	8	4	no	—
130	C₄×C₂₃	#340	1	4	yes	—
131	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
132	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
133	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
134	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×#117	—	2	1	no	—
137	C₂²×C₄⋊C₄	—	2	1	no	—
138	C₂×M₃₂	—	2	2	no	—
139	C₂×M₃₂	—	2	2	no	—
140	C₂×M₃₂	C₄×C₂₃	1	4	yes	complemented
141	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
142	#427	—	2	2	no	—
143	#427	C₄×C₂₃	1	4	yes	complemented
144	C₂×#107	—	2	1	no	—
145	C₂×#107	—	2	2	no	—
146	C₂×#107	C₄×C₂₃	1	4	yes	complemented
147	C₂×#107	C₂²×C₂₃	1	2	yes	—
148	C₂×#107	C₂²×C₂₃	1	2	yes	—
149	C₂×#106	—	2	1	no	—
150	C₂×C₂₃	—	2	4	no	—
151	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
152	C₂×C₂₃	#2139	1	2	yes	—
153	C₂×C₂₃	#2140	1	1	yes	characteristic
154	C₂×C₂₃	#2141	1	1	yes	characteristic
155	C₂×C₂₃	#2159	1	2	yes	—
156	C₂³×C₄	—	4	1	no	—
157	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
160	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
161	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₂₃	1	2	yes	—
163	C₂²×C₈	D₄×C₂₃	1	2	yes	—
164	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
165	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
166	#117	—	2	4	no	—
167	#117	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	4	2	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	M₃₂	—	2	8	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
179	#107	—	2	4	no	—
180	#107	—	2	2	no	—
181	#107	—	2	2	no	—
182	#107	C₂×C₄×C₂₃	1	4	yes	—
183	#107	C₂×C₄×C₂₃	1	4	yes	complemented
184	#106	—	2	4	no	—
185	C₂₃	#12557	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	8	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
193	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
194	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
196	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
197	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂×C₈	C₄²×C₂₃	1	4	yes	—
200	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
201	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
202	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
203	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
204	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
205	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂₃×#102	1	2	yes	—
223	C₂³	C₂₃×#102	1	2	yes	—
224	C₂³	C₂₃×#107	1	2	yes	—
225	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	8	8	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	C₂₃×#95	1	4	yes	—
233	C₂×C₄	C₂₃×#102	1	2	yes	—
234	C₂×C₄	C₂₃×#102	1	2	yes	—
235	C₂×C₄	C₂₃×#103	1	4	yes	—
236	C₂×C₄	C₂₃×#103	1	4	yes	—
237	C₂×C₄	C₂₃×#106	1	4	yes	—
238	C₂×C₄	C₂₃×#107	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
245	C₈	C₂₃×#95	1	4	yes	—
246	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
247	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
249	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	C₂₃×#340	1	4	yes	—
255	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
256	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
257	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
258	C₂²	C₂₃×#509	1	2	yes	—
259	C₂²	C₂₃×#510	1	2	yes	—
260	C₄	—	8	4	no	—
261	C₄	—	8	4	no	—
262	C₄	C₂₃×#340	1	4	yes	—
263	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
264	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
265	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
266	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
269	C₂	C₂₃×#2139	1	2	yes	—
270	C₂	C₂₃×#2140	1	1	yes	characteristic
271	C₂	C₂₃×#2141	1	1	yes	characteristic
272	C₂	C₂₃×#2159	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1380	C₂	semidirect
10. C₂×C₂₃×M₃₂	C₄	semidirect
13. C₂₃×#427	C₄	semidirect
16. C₂×C₂₃×#107	C₄	semidirect
42. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
47. C₂₃×#107	C₂×C₄	semidirect
83. #12557	C₂₃	direct (N × H)
117. #1380	C₂×C₂₃	semidirect
140. C₂×M₃₂	C₄×C₂₃	semidirect
143. #427	C₄×C₂₃	semidirect
146. C₂×#107	C₄×C₂₃	semidirect
178. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
183. #107	C₂×C₄×C₂₃	semidirect
185. C₂₃	#12557	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×M₃₂	2
C₂×C₂₃×#427	2
C₂²×C₂₃×#107	2
C₂₃×#1380	2
#12557	23

Subgroups — C23×#12557 #19160183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12557 #19160183