Subgroups — C₂₃×#11926 #19159552

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1360	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×SD₁₆×C₂₃	—	2	1	no	—
7	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₂₃×#124	—	2	1	no	—
9	C₂×C₂₃×M₃₂	—	2	1	no	—
10	C₂₃×#495	—	2	1	no	—
11	C₂₃×#495	—	2	1	no	—
12	C₂₃×#495	C₄	1	2	yes	—
13	C₂₃×#495	C₂²	1	2	yes	—
14	C₂₃×#431	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₁₆	—	2	1	no	—
17	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	complemented
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂×Q₁₆×C₂₃	—	2	1	no	—
23	C₂×Q₁₆×C₂₃	—	4	2	no	—
24	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×SD₁₆×C₂₃	—	2	2	no	—
26	C₂×SD₁₆×C₂₃	—	2	2	no	—
27	C₂×SD₁₆×C₂₃	—	4	2	no	—
28	C₂×SD₁₆×C₂₃	—	4	2	no	—
29	C₂×D₈×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	—	4	2	no	—
31	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂₃×#124	—	2	2	no	—
35	C₂₃×#124	—	2	2	no	—
36	C₂₃×#124	—	4	1	no	—
37	C₂₃×#124	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
39	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂₃×M₃₂	—	4	2	no	—
42	C₂×C₁₆×C₂₃	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₂²⋊C₈	—	4	1	no	—
45	C₄×C₈×C₂₃	—	4	1	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂⁴×C₂₃	—	4	1	no	—
51	C₂×Q₈×C₂₃	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	4	1	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
63	Q₁₆×C₂₃	—	4	2	no	—
64	Q₁₆×C₂₃	—	4	2	no	—
65	SD₁₆×C₂₃	—	4	4	no	—
66	SD₁₆×C₂₃	—	4	2	no	—
67	SD₁₆×C₂₃	—	4	2	no	—
68	D₈×C₂₃	—	4	2	no	—
69	D₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	D₈	1	2	yes	—
75	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₃	—	4	2	no	—
79	C₂²⋊C₄×C₂₃	—	4	2	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₂²⋊C₄×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₂₃	D₈	1	1	yes	characteristic
84	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₁₆×C₂₃	—	4	2	no	—
87	#11926	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	8	2	no	—
90	C₂³×C₂₃	—	8	2	no	—
91	C₂³×C₂₃	—	8	2	no	—
92	C₂³×C₂₃	#102	1	1	yes	characteristic
93	Q₈×C₂₃	—	4	2	no	—
94	Q₈×C₂₃	—	4	2	no	—
95	Q₈×C₂₃	—	8	2	no	—
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	—	4	1	no	—
100	C₂×C₄×C₂₃	—	4	1	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	#100	1	2	yes	—
104	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#102	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	1	no	—
113	C₈×C₂₃	—	8	2	no	—
114	C₈×C₂₃	#102	1	2	yes	—
115	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
116	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×#495	C₂×C₂₃	1	1	yes	characteristic
118	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#1360	C₂×C₂₃	1	4	yes	complemented
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	8	2	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	16	4	no	—
127	C₂²×C₂₃	#331	1	2	yes	—
128	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	8	2	no	—
134	C₄×C₂₃	#331	1	2	yes	—
135	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
136	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
137	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
138	C₂²×SD₁₆	—	2	1	no	—
139	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic
140	C₂×#124	—	2	1	no	—
141	C₂×M₃₂	—	2	1	no	—
142	#495	—	2	1	no	—
143	#495	—	2	1	no	—
144	#495	C₄×C₂₃	1	2	yes	—
145	#495	C₂²×C₂₃	1	2	yes	—
146	#431	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
150	#335	—	2	1	no	—
151	#334	C₂²×C₂₃	1	2	yes	complemented
152	C₂×C₂₃	—	2	2	no	—
153	C₂×C₂₃	—	16	2	no	—
154	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
155	C₂×C₂₃	#2143	1	1	yes	characteristic
156	C₂×C₂₃	#2145	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	4	2	no	—
161	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×SD₁₆	—	4	2	no	—
165	C₂×SD₁₆	—	4	2	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	4	2	no	—
168	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	#124	—	2	2	no	—
172	#124	—	2	2	no	—
173	#124	—	4	1	no	—
174	#124	—	4	1	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	M₃₂	—	4	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	4	1	no	—
182	C₄×C₈	—	4	1	no	—
183	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂₃	#11926	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	4	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
201	Q₁₆	—	4	2	no	—
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	4	no	—
204	SD₁₆	—	4	2	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	2	no	—
207	D₈	—	4	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	D₈×C₂₃	1	2	yes	—
213	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₄²	D₈×C₂₃	1	1	yes	characteristic
222	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	C₂₃×#102	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	Q₈	—	8	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₂₃×#100	1	2	yes	—
241	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
242	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
244	C₂×C₄	C₂₃×#102	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	8	2	no	—
251	C₈	C₂₃×#102	1	2	yes	—
252	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
253	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	16	4	no	—
260	C₂²	C₂₃×#331	1	2	yes	—
261	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	2	no	—
267	C₄	C₂₃×#331	1	2	yes	—
268	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
269	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
273	C₂	C₂₃×#2143	1	1	yes	characteristic
274	C₂	C₂₃×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2150	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
4. C₂₃×#1360	C₂	semidirect
17. C₂₃×C₄⋊C₁₆	C₂²	semidirect
19. C₂₃×#334	C₂²	semidirect
87. #11926	C₂₃	direct (N × H)
118. #2150	C₂×C₂₃	semidirect
119. #1564	C₂×C₂₃	semidirect
120. #1360	C₂×C₂₃	semidirect
149. C₄⋊C₁₆	C₂²×C₂₃	semidirect
151. #334	C₂²×C₂₃	semidirect
187. C₂₃	#11926	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×#495	2
C₂₃×#2150	2
C₂₃×#1564	2
C₂₃×#1360	2
#11926	23

Subgroups — C23×#11926 #19159552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11926 #19159552