Subgroups — C₂₃×#9847 #19157473

Order 5888.

498 conjugacy classes of subgroups.

The subgroup lattice has 254 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×#1962	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₂₃×#532	C₂²	1	2	yes	—
7	C₂₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
9	C₂₃×#486	C₄	1	2	yes	complemented
10	C₂₃×#466	—	2	2	no	—
11	C₄×D₈×C₂₃	—	2	2	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂×C₂₃×#104	—	2	4	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	2	no	—
18	C₂₃×#335	—	2	2	no	—
19	C₂₃×D₄⋊C₈	—	2	4	no	—
20	C₂×#41×C₂₃	—	2	2	no	—
21	C₂×D₈×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₂₃×#127	C₂×C₄	1	2	yes	—
27	C₂₃×#124	—	2	2	no	—
28	C₂₃×#121	—	2	2	no	—
29	C₂₃×#121	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₄×D₄×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#107	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×#104	—	4	8	no	—
40	C₂₃×#102	—	2	4	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₄×C₈×C₂₃	D₄	1	2	yes	—
46	#41×C₂₃	—	4	4	no	—
47	#41×C₂₃	—	4	4	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	2	2	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	2	2	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
58	D₈×C₂₃	—	4	4	no	—
59	M₁₆×C₂₃	—	4	4	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	—	4	2	no	—
69	C₂²⋊C₄×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	D₈	1	4	yes	complemented
74	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₂₃	#41	1	1	yes	characteristic
78	#9847	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	—	4	2	no	—
80	C₂³×C₂₃	#121	1	1	yes	characteristic
81	Q₈×C₂₃	—	4	4	no	—
82	D₄×C₂₃	—	4	4	no	—
83	D₄×C₂₃	—	4	4	no	—
84	D₄×C₂₃	—	4	4	no	—
85	D₄×C₂₃	—	8	4	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	8	2	no	—
93	C₂×C₄×C₂₃	—	8	2	no	—
94	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₂₃	#135	1	2	yes	—
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	—	2	2	no	—
104	C₈×C₂₃	—	4	2	no	—
105	C₈×C₂₃	—	8	2	no	—
106	#3169	C₂×C₂₃	1	1	yes	characteristic
107	#1962	C₂×C₂₃	1	2	yes	complemented
108	#1744	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
110	#1492	C₂×C₂₃	1	2	yes	complemented
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	8	2	no	—
114	C₂²×C₂₃	—	8	2	no	—
115	C₂²×C₂₃	—	8	2	no	—
116	C₂²×C₂₃	#390	1	1	yes	characteristic
117	C₂²×C₂₃	#466	1	2	yes	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	4	4	no	—
121	C₄×C₂₃	—	8	2	no	—
122	C₄×C₂₃	#421	1	2	yes	—
123	C₄×C₂₃	C₄×D₈	1	2	yes	—
124	C₄×C₂₃	#444	1	2	yes	—
125	C₄×C₂₃	#493	1	1	yes	characteristic
126	C₄×C₂₃	#495	1	1	yes	characteristic
127	#532	C₂²×C₂₃	1	2	yes	—
128	#500	C₄×C₂₃	1	1	yes	characteristic, complemented
129	#493	C₄×C₂₃	1	1	yes	characteristic, complemented
130	#486	C₄×C₂₃	1	2	yes	complemented
131	#466	—	2	2	no	—
132	C₄×D₈	—	2	2	no	—
133	#423	C₂²×C₂₃	1	2	yes	—
134	C₂×#104	—	2	4	no	—
135	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₄³	C₂²×C₂₃	1	1	yes	characteristic
138	#336	—	2	2	no	—
139	#335	—	2	2	no	—
140	D₄⋊C₈	—	2	4	no	—
141	C₂×C₂₃	—	2	2	no	—
142	C₂×C₂₃	—	16	2	no	—
143	C₂×C₂₃	#1963	1	1	yes	characteristic
144	C₂×C₂₃	#1970	1	1	yes	characteristic
145	C₂×C₂₃	#1975	1	1	yes	characteristic
146	C₂×#41	—	2	2	no	—
147	C₂×D₈	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	D₄×C₂₃	1	2	yes	—
151	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
152	#127	C₂×C₄×C₂₃	1	2	yes	—
153	#124	—	2	2	no	—
154	#121	—	2	2	no	—
155	#121	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	2	2	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	D₄×C₂₃	1	2	yes	—
161	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
162	#107	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	#104	—	4	8	no	—
166	#102	—	2	4	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₄×C₈	D₄×C₂₃	1	2	yes	—
171	C₄×C₈	D₄×C₂₃	1	2	yes	—
172	C₂₃	#9847	1	1	yes	characteristic, direct product factor, Sylow
173	#41	—	4	4	no	—
174	#41	—	4	4	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
185	D₈	—	4	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
192	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	2	no	—
200	C₄²	D₈×C₂₃	1	4	yes	complemented
201	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₄²	#41×C₂₃	1	1	yes	characteristic
205	C₂³	—	4	2	no	—
206	C₂³	C₂₃×#121	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	8	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
224	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
225	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
226	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
227	C₂×C₄	C₂₃×#135	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	8	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	8	2	no	—
235	C₂²	—	8	2	no	—
236	C₂²	—	8	2	no	—
237	C₂²	C₂₃×#390	1	1	yes	characteristic
238	C₂²	C₂₃×#466	1	2	yes	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	4	4	no	—
242	C₄	—	8	2	no	—
243	C₄	C₂₃×#421	1	2	yes	—
244	C₄	C₄×D₈×C₂₃	1	2	yes	—
245	C₄	C₂₃×#444	1	2	yes	—
246	C₄	C₂₃×#493	1	1	yes	characteristic
247	C₄	C₂₃×#495	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	16	2	no	—
250	C₂	C₂₃×#1963	1	1	yes	characteristic
251	C₂	C₂₃×#1970	1	1	yes	characteristic
252	C₂	C₂₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1962	C₂	semidirect
3. C₂₃×#1744	C₂	semidirect
4. C₂₃×#1567	C₂	semidirect
5. C₂₃×#1492	C₂	semidirect
7. C₂₃×#500	C₄	semidirect
8. C₂₃×#493	C₄	semidirect
9. C₂₃×#486	C₄	semidirect
73. C₄²×C₂₃	D₈	semidirect
78. #9847	C₂₃	direct (N × H)
107. #1962	C₂×C₂₃	semidirect
108. #1744	C₂×C₂₃	semidirect
109. #1567	C₂×C₂₃	semidirect
110. #1492	C₂×C₂₃	semidirect
128. #500	C₄×C₂₃	semidirect
129. #493	C₄×C₂₃	semidirect
130. #486	C₄×C₂₃	semidirect
172. C₂₃	#9847	direct (N × H)
200. C₄²	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×#1962	2
C₂₃×#1744	2
C₂₃×#1567	2
C₂₃×#1492	2
#9847	23

Subgroups — C23×#9847 #19157473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9847 #19157473