Subgroups — C₂₃×#23674 #19171300

Order 5888.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2911	C₂	1	2	yes	maximal
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2090	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2089	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1742	C₂	1	2	yes	maximal
7	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	2	yes	—
13	C₂₃×#428	—	2	4	no	—
14	C₂₃×#428	C₂²	1	8	yes	—
15	C₂₃×#427	—	2	2	no	—
16	C₂₃×#427	—	2	2	no	—
17	C₂₃×#427	—	2	4	no	—
18	C₂×C₂₃×#107	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	C₂²	1	2	yes	—
20	C₂₃×#424	—	2	2	no	—
21	C₂₃×#424	C₂²	1	2	yes	—
22	C₂₃×#424	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
24	C₂₃×#343	C₂²	1	8	yes	—
25	C₂₃×#341	C₂²	1	2	yes	—
26	C₂₃×#341	C₂²	1	2	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	4	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	2	4	no	—
32	C₂×M₁₆×C₂₃	Q₈	1	8	yes	—
33	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂²×C₈×C₂₃	—	2	4	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	D₄	1	2	yes	—
39	C₂²×C₈×C₂₃	D₄	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#117	—	2	1	no	—
43	C₂₃×#117	—	2	1	no	—
44	C₂₃×#117	—	2	2	no	—
45	C₂₃×#117	—	2	2	no	—
46	C₂₃×#117	—	2	2	no	—
47	C₂₃×#117	—	2	2	no	—
48	C₂₃×#117	D₄	1	2	yes	—
49	C₂₃×#117	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄²×C₂₃	—	2	1	no	—
55	C₂×C₄²×C₂₃	—	2	2	no	—
56	C₂₃×#107	—	2	4	no	—
57	C₂₃×#107	—	2	8	no	—
58	C₂₃×#106	—	2	4	no	—
59	C₂₃×#106	—	2	8	no	—
60	C₂₃×C₄⋊C₈	—	2	4	no	—
61	C₂₃×C₄⋊C₈	—	2	4	no	—
62	C₂⁴×C₂₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	#41	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₂₃	—	2	8	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	4	2	no	—
77	M₁₆×C₂₃	C₂×Q₈	1	8	yes	—
78	C₂×C₈×C₂₃	—	2	8	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	4	2	no	—
86	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
92	C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	2	no	—
95	C₂²⋊C₄×C₂₃	—	4	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	—	4	2	no	—
100	#23674	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	#121	1	1	yes	characteristic
103	C₂³×C₂₃	#121	1	1	yes	characteristic
104	C₂³×C₂₃	#125	1	4	yes	—
105	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	8	4	no	—
110	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₂₃	#123	1	2	yes	—
115	C₂×C₄×C₂₃	#123	1	2	yes	—
116	C₂×C₄×C₂₃	#125	1	4	yes	—
117	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
123	C₈×C₂₃	—	2	8	no	—
124	C₈×C₂₃	—	2	2	no	—
125	C₈×C₂₃	—	2	2	no	—
126	C₈×C₂₃	—	4	2	no	—
127	#2911	C₂×C₂₃	1	2	yes	—
128	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
129	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
130	#2090	C₂×C₂₃	1	4	yes	complemented
131	#2089	C₂×C₂₃	1	4	yes	complemented
132	#1742	C₂×C₂₃	1	2	yes	—
133	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	—	4	4	no	—
138	C₂²×C₂₃	#398	1	2	yes	—
139	C₂²×C₂₃	#398	1	2	yes	—
140	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
141	C₂²×C₂₃	C₂×#125	1	2	yes	—
142	C₄×C₂₃	—	4	4	no	—
143	C₄×C₂₃	—	8	4	no	—
144	C₄×C₂₃	#398	1	2	yes	—
145	C₄×C₂₃	#398	1	2	yes	—
146	C₄×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
147	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
148	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
150	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
151	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×#117	C₂²×C₂₃	1	2	yes	—
154	#428	—	2	4	no	—
155	#428	C₂²×C₂₃	1	8	yes	—
156	#427	—	2	2	no	—
157	#427	—	2	2	no	—
158	#427	—	2	4	no	—
159	C₂×#107	C₂²×C₂₃	1	2	yes	—
160	C₂×#106	C₂²×C₂₃	1	2	yes	—
161	#424	—	2	2	no	—
162	#424	C₂²×C₂₃	1	2	yes	—
163	#424	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄⋊C₈	—	2	2	no	—
165	#343	C₂²×C₂₃	1	8	yes	—
166	#341	C₂²×C₂₃	1	2	yes	—
167	#341	C₂²×C₂₃	1	2	yes	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	—	2	2	no	—
171	C₂×C₂₃	C₂×#398	1	1	yes	characteristic
172	C₂×C₂₃	#3067	1	2	yes	—
173	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	Q₈×C₂₃	1	8	yes	—
179	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
180	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	D₄×C₂₃	1	2	yes	—
185	C₂²×C₈	D₄×C₂₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	D₄×C₂₃	1	2	yes	—
195	#117	C₂³×C₂₃	1	2	yes	—
196	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	#107	—	2	4	no	—
203	#107	—	2	8	no	—
204	#106	—	2	4	no	—
205	#106	—	2	8	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	4	no	—
208	C₂₃	#23674	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	#41×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
217	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₂₃	1	4	yes	—
219	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	C₂×Q₈×C₂₃	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂₃×#121	1	1	yes	characteristic
249	C₂³	C₂₃×#121	1	1	yes	characteristic
250	C₂³	C₂₃×#125	1	4	yes	—
251	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
258	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
259	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
260	C₂×C₄	C₂₃×#123	1	2	yes	—
261	C₂×C₄	C₂₃×#123	1	2	yes	—
262	C₂×C₄	C₂₃×#125	1	4	yes	—
263	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
264	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
268	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
269	C₈	—	2	8	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂₃×#398	1	2	yes	—
278	C₂²	C₂₃×#398	1	2	yes	—
279	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
280	C₂²	C₂×C₂₃×#125	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂₃×#398	1	2	yes	—
284	C₄	C₂₃×#398	1	2	yes	—
285	C₄	C₂×C₂₃×C₂²⋊Q₈	1	2	yes	—
286	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
287	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂×C₂₃×#398	1	1	yes	characteristic
292	C₂	C₂₃×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#121
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2090	C₂	semidirect
5. C₂₃×#2089	C₂	semidirect
100. #23674	C₂₃	direct (N × H)
130. #2090	C₂×C₂₃	semidirect
131. #2089	C₂×C₂₃	semidirect
208. C₂₃	#23674	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×#2911	2
C₂×C₂₃×#427	2
C₂×C₂₃×#424	2
C₂₃×#2090	2
C₂₃×#2089	2
C₂₃×#1742	2
C₂×C₂₃×#341	2
#23674	23

Subgroups — C23×#23674 #19171300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23674 #19171300