Subgroups — C₂₃×#17975 #19165601

Order 5888.

726 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	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2931	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1581	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#131	—	2	2	no	—
8	C₂₃×#532	C₂²	1	2	yes	—
9	C₂₃×#500	C₄	1	1	yes	characteristic
10	C₂₃×#493	C₄	1	1	yes	characteristic
11	C₂₃×#486	C₄	1	2	yes	—
12	C₂₃×#466	C₄	1	4	yes	—
13	C₄×D₈×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#436	—	2	2	no	—
15	C₂₃×#436	C₂²	1	4	yes	—
16	C₈×D₄×C₂₃	—	2	2	no	—
17	C₈×D₄×C₂₃	C₂²	1	4	yes	—
18	C₂₃×#423	C₂²	1	2	yes	—
19	C₂₃×#408	—	2	2	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#335	C₂²	1	4	yes	—
24	C₂₃×D₄⋊C₈	C₂²	1	8	yes	—
25	C₂₃×#322	C₂²	1	4	yes	—
26	C₂×#41×C₂₃	—	2	2	no	—
27	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂₃×#131	—	4	8	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	—	2	4	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₂₃×#127	C₂×C₄	1	2	yes	—
38	C₂₃×#124	—	2	2	no	—
39	C₂₃×#121	—	2	2	no	—
40	C₂₃×#121	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₂₃	—	2	2	no	—
42	C₄×D₄×C₂₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×#107	C₂×C₄	1	4	yes	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
47	C₂₃×#102	C₂×C₄	1	8	yes	—
48	C₂₃×C₂²⋊C₈	—	2	4	no	—
49	C₂₃×C₂²⋊C₈	—	2	4	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄×C₈×C₂₃	C₂³	1	2	yes	—
56	C₄×C₈×C₂₃	C₂³	1	2	yes	—
57	#41×C₂₃	—	4	4	no	—
58	#41×C₂₃	—	4	4	no	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	4	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
68	D₈×C₂₃	—	4	4	no	—
69	M₁₆×C₂₃	—	4	4	no	—
70	M₁₆×C₂₃	—	4	8	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	8	no	—
77	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
84	C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	#41	1	1	yes	characteristic
89	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
90	#17975	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	4	no	—
92	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₂₃	—	4	4	no	—
94	D₄×C₂₃	—	2	8	no	—
95	D₄×C₂₃	—	4	8	no	—
96	D₄×C₂₃	—	4	4	no	—
97	D₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₂₃	#135	1	2	yes	—
108	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	—	4	2	no	—
115	C₈×C₂₃	—	4	4	no	—
116	C₈×C₂₃	C₂×D₈	1	4	yes	—
117	#3169	C₂×C₂₃	1	1	yes	characteristic
118	#2931	C₂×C₂₃	1	2	yes	complemented
119	#1581	C₂×C₂₃	1	8	yes	complemented
120	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#1492	C₂×C₂₃	1	2	yes	complemented
122	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	—	4	4	no	—
127	C₂²×C₂₃	—	4	4	no	—
128	C₂²×C₂₃	C₄×D₈	1	2	yes	—
129	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	4	4	no	—
133	C₄×C₂₃	C₄×D₈	1	2	yes	—
134	C₄×C₂₃	#444	1	2	yes	—
135	C₄×C₂₃	#568	1	2	yes	—
136	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
138	C₂×#131	—	2	2	no	—
139	#532	C₂²×C₂₃	1	2	yes	—
140	#500	C₄×C₂₃	1	1	yes	characteristic
141	#493	C₄×C₂₃	1	1	yes	characteristic
142	#486	C₄×C₂₃	1	2	yes	—
143	#466	C₄×C₂₃	1	4	yes	—
144	C₄×D₈	C₂²×C₂₃	1	4	yes	—
145	#436	—	2	2	no	—
146	#436	C₂²×C₂₃	1	4	yes	—
147	C₈×D₄	—	2	2	no	—
148	C₈×D₄	C₂²×C₂₃	1	4	yes	—
149	#423	C₂²×C₂₃	1	2	yes	—
150	#408	—	2	2	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	#335	C₂²×C₂₃	1	4	yes	—
155	D₄⋊C₈	C₂²×C₂₃	1	8	yes	—
156	#322	C₂²×C₂₃	1	4	yes	—
157	C₂×C₂₃	—	2	2	no	—
158	C₂×C₂₃	—	8	4	no	—
159	C₂×C₂₃	#2932	1	1	yes	characteristic
160	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
161	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
162	C₂×#41	—	2	2	no	—
163	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
164	#131	—	4	8	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
173	#127	C₂×C₄×C₂₃	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	C₂×C₄×C₂₃	1	4	yes	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	C₂³×C₂₃	1	4	yes	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	#107	C₂×C₄×C₂₃	1	4	yes	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
183	#102	C₂×C₄×C₂₃	1	8	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₄×C₈	D₄×C₂₃	1	2	yes	—
190	C₄×C₈	C₂³×C₂₃	1	2	yes	—
191	C₄×C₈	C₂³×C₂₃	1	2	yes	—
192	C₄×C₈	C₂³×C₂₃	1	2	yes	—
193	C₂₃	#17975	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
205	D₈	—	4	4	no	—
206	M₁₆	—	4	4	no	—
207	M₁₆	—	4	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
215	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₄²	#41×C₂₃	1	1	yes	characteristic
226	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
243	C₂×C₄	C₂₃×#135	1	2	yes	—
244	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₈	C₂×D₈×C₂₃	1	4	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₄×D₈×C₂₃	1	2	yes	—
259	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	C₄×D₈×C₂₃	1	2	yes	—
264	C₄	C₂₃×#444	1	2	yes	—
265	C₄	C₂₃×#568	1	2	yes	—
266	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
267	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂₃×#2932	1	1	yes	characteristic
271	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
272	C₂	C₂×C₂₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#2931	C₂	semidirect
3. C₂₃×#1581	C₂	semidirect
4. C₂₃×#1567	C₂	semidirect
5. C₂₃×#1492	C₂	semidirect
6. C₂×C₂₃×#322	C₂	semidirect
77. C₂×C₈×C₂₃	D₈	semidirect
90. #17975	C₂₃	direct (N × H)
118. #2931	C₂×C₂₃	semidirect
119. #1581	C₂×C₂₃	semidirect
120. #1567	C₂×C₂₃	semidirect
121. #1492	C₂×C₂₃	semidirect
122. C₂×#322	C₂×C₂₃	semidirect
193. C₂₃	#17975	direct (N × H)
214. C₂×C₈	D₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3169	2
C₂₃×#2931	2
C₂₃×#1581	2
C₂₃×#1567	2
C₂₃×#1492	2
C₂×C₂₃×#322	2
#17975	23

Subgroups — C23×#17975 #19165601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17975 #19165601