Subgroups — C₂₃×#3067 #15670686

Order 2944.

452 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#575	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#473	C₂	1	2	yes	maximal, complemented
5	C₂₃×#472	C₂	1	2	yes	maximal, complemented
6	C₂₃×#471	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#413	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#41×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#137	—	2	2	no	—
12	C₂₃×#137	C₂²	1	4	yes	—
13	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
14	C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
15	C₂₃×#135	—	2	4	no	—
16	C₂₃×#135	—	2	1	no	—
17	C₂₃×#135	—	2	1	no	—
18	C₂₃×#135	—	2	1	no	—
19	C₂₃×#135	—	2	1	no	—
20	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
22	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#131	—	2	1	no	—
24	C₂₃×#131	—	2	1	no	—
25	C₂₃×#131	C₂²	1	2	yes	—
26	C₂₃×#131	C₂²	1	2	yes	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#108	C₂²	1	2	yes	complemented
33	C₂₃×#108	C₂²	1	2	yes	complemented
34	C₂₃×#101	C₂²	1	4	yes	complemented
35	C₂₃×#100	C₂²	1	4	yes	complemented
36	#41×C₂₃	—	2	1	no	—
37	#41×C₂₃	—	2	1	no	—
38	#41×C₂₃	—	2	2	no	—
39	#41×C₂₃	—	2	2	no	—
40	#41×C₂₃	—	2	2	no	—
41	#41×C₂₃	—	2	2	no	—
42	#41×C₂₃	D₄	1	2	yes	—
43	#41×C₂₃	C₂³	1	2	yes	—
44	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	Q₁₆×C₂₃	—	2	4	no	—
54	Q₁₆×C₂₃	—	2	1	no	—
55	Q₁₆×C₂₃	—	2	1	no	—
56	SD₁₆×C₂₃	—	2	4	no	—
57	SD₁₆×C₂₃	—	2	4	no	—
58	SD₁₆×C₂₃	—	2	2	no	—
59	SD₁₆×C₂₃	—	2	2	no	—
60	D₈×C₂₃	—	2	4	no	—
61	D₈×C₂₃	—	2	1	no	—
62	D₈×C₂₃	—	2	1	no	—
63	M₁₆×C₂₃	—	2	1	no	—
64	M₁₆×C₂₃	—	2	2	no	—
65	M₁₆×C₂₃	—	2	2	no	—
66	M₁₆×C₂₃	—	2	2	no	—
67	M₁₆×C₂₃	C₂³	1	2	yes	—
68	M₁₆×C₂₃	C₂³	1	4	yes	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂³×C₂₃	—	2	1	no	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	#41	1	1	yes	characteristic
82	Q₈×C₂₃	—	2	1	no	—
83	Q₈×C₂₃	—	2	2	no	—
84	Q₈×C₂₃	—	2	2	no	—
85	Q₈×C₂₃	C₂×D₄	1	2	yes	—
86	D₄×C₂₃	—	2	2	no	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	—	2	4	no	—
108	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
112	#3067	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	8	2	no	—
118	C₂²×C₂₃	#121	1	1	yes	characteristic
119	C₂²×C₂₃	#121	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	#121	1	1	yes	characteristic
124	C₄×C₂₃	#121	1	1	yes	characteristic
125	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	#575	C₂×C₂₃	1	2	yes	complemented
128	C₂×#135	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#473	C₂×C₂₃	1	2	yes	complemented
131	#472	C₂×C₂₃	1	2	yes	complemented
132	#471	C₂×C₂₃	1	4	yes	complemented
133	C₂×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#413	C₂×C₂₃	1	2	yes	complemented
135	C₂×C₂₃	—	2	1	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	4	2	no	—
139	C₂×C₂₃	—	8	2	no	—
140	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
141	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#41	C₂²×C₂₃	1	2	yes	—
143	#137	—	2	2	no	—
144	#137	C₂²×C₂₃	1	4	yes	—
145	AGL(1,ℤ/8)	—	2	2	no	—
146	AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
147	#135	—	2	4	no	—
148	#135	—	2	1	no	—
149	#135	—	2	1	no	—
150	#135	—	2	1	no	—
151	#135	—	2	1	no	—
152	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
153	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
154	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
155	#131	—	2	1	no	—
156	#131	—	2	1	no	—
157	#131	C₂²×C₂₃	1	2	yes	—
158	#131	C₂²×C₂₃	1	2	yes	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
164	#108	C₂²×C₂₃	1	2	yes	complemented
165	#108	C₂²×C₂₃	1	2	yes	complemented
166	#101	C₂²×C₂₃	1	4	yes	complemented
167	#100	C₂²×C₂₃	1	4	yes	complemented
168	C₂₃	#3067	1	1	yes	characteristic, direct product factor, Sylow
169	#41	—	2	1	no	—
170	#41	—	2	1	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	#41	D₄×C₂₃	1	2	yes	—
176	#41	C₂³×C₂₃	1	2	yes	—
177	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	2	1	no	—
188	Q₁₆	—	2	1	no	—
189	SD₁₆	—	2	4	no	—
190	SD₁₆	—	2	4	no	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	2	2	no	—
193	D₈	—	2	4	no	—
194	D₈	—	2	1	no	—
195	D₈	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	C₂³×C₂₃	1	2	yes	—
201	M₁₆	C₂³×C₂₃	1	4	yes	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
209	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	#41×C₂₃	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	C₂×D₄×C₂₃	1	2	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	C₂×D₄×C₂₃	1	2	yes	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	4	no	—
241	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₂₃×#121	1	1	yes	characteristic
251	C₂²	C₂₃×#121	1	1	yes	characteristic
252	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	4	2	no	—
255	C₄	C₂₃×#121	1	1	yes	characteristic
256	C₄	C₂₃×#121	1	1	yes	characteristic
257	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
258	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₂×C₂₃×#121	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#575	C₂	semidirect
2. C₂×C₂₃×#135	C₂	semidirect
3. C₂×C₂₃×#131	C₂	semidirect
4. C₂₃×#473	C₂	semidirect
5. C₂₃×#472	C₂	semidirect
6. C₂₃×#471	C₂	semidirect
7. C₂×C₂₃×#108	C₂	semidirect
8. C₂₃×#413	C₂	semidirect
20. C₂×Q₁₆×C₂₃	C₂²	semidirect
21. C₂×SD₁₆×C₂₃	C₂²	semidirect
22. C₂×D₈×C₂₃	C₂²	semidirect
32. C₂₃×#108	C₂²	semidirect
33. C₂₃×#108	C₂²	semidirect
34. C₂₃×#101	C₂²	semidirect
35. C₂₃×#100	C₂²	semidirect
112. #3067	C₂₃	direct (N × H)
127. #575	C₂×C₂₃	semidirect
128. C₂×#135	C₂×C₂₃	semidirect
129. C₂×#131	C₂×C₂₃	semidirect
130. #473	C₂×C₂₃	semidirect
131. #472	C₂×C₂₃	semidirect
132. #471	C₂×C₂₃	semidirect
133. C₂×#108	C₂×C₂₃	semidirect
134. #413	C₂×C₂₃	semidirect
152. C₂×Q₁₆	C₂²×C₂₃	semidirect
153. C₂×SD₁₆	C₂²×C₂₃	semidirect
154. C₂×D₈	C₂²×C₂₃	semidirect
164. #108	C₂²×C₂₃	semidirect
165. #108	C₂²×C₂₃	semidirect
166. #101	C₂²×C₂₃	semidirect
167. #100	C₂²×C₂₃	semidirect
168. C₂₃	#3067	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×#575	2
C₂×C₂₃×#135	2
C₂×C₂₃×#131	2
C₂₃×#473	2
C₂₃×#472	2
C₂₃×#471	2
C₂×C₂₃×#108	2
C₂₃×#413	2
#3067	23

Subgroups — C23×#3067 #15670686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3067 #15670686