Subgroups — C₂₃×#11619 #19159245

Order 5888.

336 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×#2965	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1338	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1330	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₂₃×#446	C₄	1	1	yes	characteristic
9	C₂₃×#446	C₂²	1	1	yes	characteristic
10	C₂₃×#436	C₄	1	1	yes	characteristic
11	C₂₃×#436	C₂²	1	1	yes	characteristic
12	C₂₃×#435	—	2	1	no	—
13	C₂₃×#435	—	2	1	no	—
14	C₂₃×#431	—	2	1	no	—
15	C₂₃×#431	C₄	1	1	yes	characteristic
16	C₄×M₁₆×C₂₃	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
20	C₂₃×#346	—	2	1	no	—
21	C₂₃×#346	—	2	1	no	—
22	C₂₃×#346	C₂²	1	2	yes	—
23	C₂×#41×C₂₃	C₈	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₂₃	—	2	1	no	—
31	C₄×D₄×C₂₃	—	2	1	no	—
32	C₄×D₄×C₂₃	C₈	1	1	yes	characteristic
33	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂₃×#117	—	2	1	no	—
35	C₂₃×#117	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	—	2	1	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₂₃	—	2	4	no	—
39	C₂×C₁₆×C₂₃	—	2	2	no	—
40	C₂×C₁₆×C₂₃	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂₃×C₂²⋊C₈	—	2	1	no	—
46	C₂₃×C₂²⋊C₈	—	2	1	no	—
47	C₂₃×C₂²⋊C₈	—	2	1	no	—
48	C₂₃×#97	—	2	1	no	—
49	C₂₃×#97	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	#41×C₂₃	—	4	2	no	—
56	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
65	M₁₆×C₂₃	—	4	2	no	—
66	M₁₆×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₂²⋊C₄×C₂₃	—	2	1	no	—
81	C₂²⋊C₄×C₂₃	—	2	1	no	—
82	C₄²×C₂₃	—	2	1	no	—
83	C₄²×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₁₆×C₂₃	—	2	4	no	—
90	C₁₆×C₂₃	—	4	2	no	—
91	#11619	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
94	Q₈×C₂₃	—	4	1	no	—
95	D₄×C₂₃	—	4	1	no	—
96	D₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
114	C₈×C₂₃	—	2	1	no	—
115	C₈×C₂₃	—	2	1	no	—
116	C₈×C₂₃	—	4	1	no	—
117	C₈×C₂₃	—	4	1	no	—
118	C₈×C₂₃	#104	1	2	yes	—
119	C₈×C₂₃	#135	1	2	yes	—
120	#2965	C₂×C₂₃	1	1	yes	characteristic
121	#2163	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#1338	C₂×C₂₃	1	2	yes	complemented
124	#1330	C₂×C₂₃	1	2	yes	complemented
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	4	1	no	—
128	C₂²×C₂₃	—	4	1	no	—
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	#349	1	2	yes	—
131	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	4	1	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	#416	1	1	yes	characteristic
141	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
142	C₄×#41	C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₁₆	—	2	1	no	—
144	#446	C₄×C₂₃	1	1	yes	characteristic
145	#446	C₂²×C₂₃	1	1	yes	characteristic
146	#436	C₄×C₂₃	1	1	yes	characteristic
147	#436	C₂²×C₂₃	1	1	yes	characteristic
148	#435	—	2	1	no	—
149	#435	—	2	1	no	—
150	#431	—	2	1	no	—
151	#431	C₄×C₂₃	1	1	yes	characteristic
152	C₄×M₁₆	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂²⋊C₁₆	—	2	1	no	—
156	#346	—	2	1	no	—
157	#346	—	2	1	no	—
158	#346	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	—	8	1	no	—
161	C₂×C₂₃	#1476	1	1	yes	characteristic
162	C₂×C₂₃	C₂×#349	1	1	yes	characteristic
163	C₂×C₂₃	#2117	1	1	yes	characteristic
164	C₂×#41	C₈×C₂₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
169	C₄×Q₈	C₈×C₂₃	1	1	yes	characteristic
170	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₈×C₂₃	1	1	yes	characteristic
174	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
175	#117	—	2	1	no	—
176	#117	C₈×C₂₃	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	#97	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
194	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
196	C₂₃	#11619	1	1	yes	characteristic, direct product factor, Sylow
197	#41	—	4	2	no	—
198	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
229	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
230	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₁₆	—	2	4	no	—
232	C₁₆	—	4	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	C₂₃×#104	1	2	yes	—
260	C₈	C₂₃×#135	1	2	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₂₃×#349	1	2	yes	—
267	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	C₂₃×#416	1	1	yes	characteristic
277	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
278	C₂	—	2	2	no	—
279	C₂	—	8	1	no	—
280	C₂	C₂₃×#1476	1	1	yes	characteristic
281	C₂	C₂×C₂₃×#349	1	1	yes	characteristic
282	C₂	C₂₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₃	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2163	C₂	semidirect
3. C₂×C₂₃×#346	C₂	semidirect
4. C₂₃×#1338	C₂	semidirect
5. C₂₃×#1330	C₂	semidirect
91. #11619	C₂₃	direct (N × H)
121. #2163	C₂×C₂₃	semidirect
122. C₂×#346	C₂×C₂₃	semidirect
123. #1338	C₂×C₂₃	semidirect
124. #1330	C₂×C₂₃	semidirect
196. C₂₃	#11619	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×#2965	2
C₂₃×#2163	2
C₂×C₂₃×#346	2
C₂₃×#1338	2
C₂₃×#1330	2
#11619	23

Subgroups — C23×#11619 #19159245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11619 #19159245