Subgroups — C₂₃×#3288 #15670907

Order 2944.

464 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#577	C₂	1	2	yes	maximal, complemented
2	C₂₃×#576	C₂	1	2	yes	maximal, complemented
3	C₂₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#472	C₂	1	2	yes	maximal, complemented
5	C₂₃×#471	C₂	1	2	yes	maximal, complemented
6	C₂₃×#455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#453	C₂	1	2	yes	maximal, complemented
9	C₂₃×#444	C₂	1	2	yes	maximal, complemented
10	C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
11	C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
12	C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	C₂₃×#137	—	2	2	no	—
14	C₂₃×#137	C₂²	1	2	yes	complemented
15	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
16	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
17	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
18	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
19	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
20	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₂₃×#135	—	2	2	no	—
22	C₂₃×#135	—	2	2	no	—
23	C₂₃×#135	C₂²	1	2	yes	—
24	C₂×SD₁₆×C₂₃	—	2	2	no	—
25	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
26	C₂×D₈×C₂₃	—	2	2	no	—
27	C₂×D₈×C₂₃	C₂²	1	2	yes	complemented
28	C₂₃×#131	C₂²	1	2	yes	complemented
29	C₂₃×#131	C₂²	1	2	yes	complemented
30	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#108	C₂²	1	2	yes	complemented
33	C₂₃×#104	C₂²	1	2	yes	complemented
34	C₂₃×#104	C₂²	1	2	yes	complemented
35	C₂₃×#101	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#100	C₂²	1	2	yes	complemented
37	C₂₃×#100	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#97	C₂²	1	1	yes	characteristic, complemented
39	#41×C₂₃	—	2	2	no	—
40	#41×C₂₃	—	2	2	no	—
41	#41×C₂₃	—	2	2	no	—
42	#41×C₂₃	—	2	2	no	—
43	#41×C₂₃	—	4	1	no	—
44	#41×C₂₃	C₂³	1	2	yes	—
45	#41×C₂₃	C₂³	1	2	yes	—
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	2	2	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	4	1	no	—
52	C₂×D₄×C₂₃	—	4	1	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₂₃	C₂³	1	2	yes	—
56	Q₁₆×C₂₃	—	2	2	no	—
57	Q₁₆×C₂₃	D₄	1	2	yes	complemented
58	SD₁₆×C₂₃	—	2	2	no	—
59	SD₁₆×C₂₃	—	2	2	no	—
60	SD₁₆×C₂₃	—	2	2	no	—
61	SD₁₆×C₂₃	—	2	2	no	—
62	SD₁₆×C₂₃	—	2	2	no	—
63	SD₁₆×C₂₃	—	2	2	no	—
64	SD₁₆×C₂₃	D₄	1	2	yes	complemented
65	SD₁₆×C₂₃	D₄	1	2	yes	complemented
66	D₈×C₂₃	—	2	2	no	—
67	D₈×C₂₃	—	2	2	no	—
68	D₈×C₂₃	—	2	2	no	—
69	D₈×C₂₃	—	2	2	no	—
70	D₈×C₂₃	—	2	2	no	—
71	D₈×C₂₃	D₄	1	2	yes	complemented
72	M₁₆×C₂₃	—	2	2	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	C₂³	1	2	yes	—
75	M₁₆×C₂₃	C₂³	1	2	yes	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	C₂³	1	2	yes	—
79	C₂²⋊C₄×C₂₃	—	4	1	no	—
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂³×C₂₃	—	2	1	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	4	2	no	—
84	C₂³×C₂₃	—	4	1	no	—
85	Q₈×C₂₃	—	2	2	no	—
86	Q₈×C₂₃	—	4	1	no	—
87	Q₈×C₂₃	C₂×D₄	1	2	yes	—
88	Q₈×C₂₃	C₂×D₄	1	2	yes	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	2	2	no	—
94	D₄×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	4	1	no	—
98	D₄×C₂₃	—	8	2	no	—
99	D₄×C₂₃	C₂×D₄	1	2	yes	—
100	D₄×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₂₃	—	2	2	no	—
110	C₈×C₂₃	—	2	2	no	—
111	C₈×C₂₃	C₂×D₄	1	2	yes	—
112	C₈×C₂₃	C₂×D₄	1	2	yes	—
113	#3288	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	4	1	no	—
117	C₂²×C₂₃	—	4	2	no	—
118	C₂²×C₂₃	—	8	2	no	—
119	C₂²×C₂₃	—	8	1	no	—
120	C₂²×C₂₃	—	16	2	no	—
121	C₂²×C₂₃	—	16	1	no	—
122	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	C₂²×D₄	1	2	yes	—
128	#577	C₂×C₂₃	1	2	yes	complemented
129	#576	C₂×C₂₃	1	2	yes	complemented
130	#496	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#472	C₂×C₂₃	1	2	yes	complemented
132	#471	C₂×C₂₃	1	2	yes	complemented
133	#455	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#454	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#453	C₂×C₂₃	1	2	yes	complemented
136	#444	C₂×C₂₃	1	2	yes	complemented
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	4	2	no	—
139	C₂×C₂₃	—	4	2	no	—
140	C₂×C₂₃	—	8	2	no	—
141	C₂×C₂₃	—	8	1	no	—
142	C₂×C₂₃	D₄²	1	1	yes	characteristic
143	2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic, complemented
144	H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
145	H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	complemented
146	#137	—	2	2	no	—
147	#137	C₂²×C₂₃	1	2	yes	complemented
148	AGL(1,ℤ/8)	—	2	2	no	—
149	AGL(1,ℤ/8)	—	2	2	no	—
150	AGL(1,ℤ/8)	—	2	2	no	—
151	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
152	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	complemented
153	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	complemented
154	#135	—	2	2	no	—
155	#135	—	2	2	no	—
156	#135	C₂²×C₂₃	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	C₂²×C₂₃	1	2	yes	complemented
161	#131	C₂²×C₂₃	1	2	yes	complemented
162	#131	C₂²×C₂₃	1	2	yes	complemented
163	#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
164	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
165	#108	C₂²×C₂₃	1	2	yes	complemented
166	#104	C₂²×C₂₃	1	2	yes	complemented
167	#104	C₂²×C₂₃	1	2	yes	complemented
168	#101	C₂²×C₂₃	1	1	yes	characteristic, complemented
169	#100	C₂²×C₂₃	1	2	yes	complemented
170	#100	C₂²×C₂₃	1	1	yes	characteristic, complemented
171	#97	C₂²×C₂₃	1	1	yes	characteristic, complemented
172	C₂₃	#3288	1	1	yes	characteristic, direct product factor, Sylow
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	4	1	no	—
178	#41	C₂³×C₂₃	1	2	yes	—
179	#41	C₂³×C₂₃	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×D₄	C₂³×C₂₃	1	2	yes	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	D₄×C₂₃	1	2	yes	complemented
192	SD₁₆	—	2	2	no	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	D₄×C₂₃	1	2	yes	complemented
199	SD₁₆	D₄×C₂₃	1	2	yes	complemented
200	D₈	—	2	2	no	—
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	2	2	no	—
204	D₈	—	2	2	no	—
205	D₈	D₄×C₂₃	1	2	yes	complemented
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₂³×C₂₃	1	2	yes	—
209	M₁₆	C₂³×C₂₃	1	2	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂³×C₂₃	1	2	yes	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₄²	C₂³×C₂₃	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	1	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	4	1	no	—
221	Q₈	C₂×D₄×C₂₃	1	2	yes	—
222	Q₈	C₂×D₄×C₂₃	1	2	yes	—
223	D₄	—	2	2	no	—
224	D₄	—	2	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	1	no	—
232	D₄	—	8	2	no	—
233	D₄	C₂×D₄×C₂₃	1	2	yes	—
234	D₄	C₂×D₄×C₂₃	1	2	yes	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	C₂×D₄×C₂₃	1	2	yes	—
246	C₈	C₂×D₄×C₂₃	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	16	1	no	—
254	C₂²	—	16	2	no	—
255	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	C₂²×D₄×C₂₃	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	—	8	1	no	—
266	C₂	D₄²×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#577	C₂	semidirect
2. C₂₃×#576	C₂	semidirect
3. C₂₃×#496	C₂	semidirect
4. C₂₃×#472	C₂	semidirect
5. C₂₃×#471	C₂	semidirect
6. C₂₃×#455	C₂	semidirect
7. C₂₃×#454	C₂	semidirect
8. C₂₃×#453	C₂	semidirect
9. C₂₃×#444	C₂	semidirect
10. C₂₃×2₋¹⁺⁴	C₂²	semidirect
12. C₂₃×H(5,𝔽₂)	C₂²	semidirect
14. C₂₃×#137	C₂²	semidirect
19. C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
20. C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×SD₁₆×C₂₃	C₂²	semidirect
27. C₂×D₈×C₂₃	C₂²	semidirect
28. C₂₃×#131	C₂²	semidirect
29. C₂₃×#131	C₂²	semidirect
30. C₂₃×#127	C₂²	semidirect
31. C₂₃×#124	C₂²	semidirect
32. C₂₃×#108	C₂²	semidirect
33. C₂₃×#104	C₂²	semidirect
34. C₂₃×#104	C₂²	semidirect
35. C₂₃×#101	C₂²	semidirect
36. C₂₃×#100	C₂²	semidirect
37. C₂₃×#100	C₂²	semidirect
38. C₂₃×#97	C₂²	semidirect
57. Q₁₆×C₂₃	D₄	semidirect
64. SD₁₆×C₂₃	D₄	semidirect
65. SD₁₆×C₂₃	D₄	semidirect
71. D₈×C₂₃	D₄	semidirect
113. #3288	C₂₃	direct (N × H)
128. #577	C₂×C₂₃	semidirect
129. #576	C₂×C₂₃	semidirect
130. #496	C₂×C₂₃	semidirect
131. #472	C₂×C₂₃	semidirect
132. #471	C₂×C₂₃	semidirect
133. #455	C₂×C₂₃	semidirect
134. #454	C₂×C₂₃	semidirect
135. #453	C₂×C₂₃	semidirect
136. #444	C₂×C₂₃	semidirect
143. 2₋¹⁺⁴	C₂²×C₂₃	semidirect
145. H(5,𝔽₂)	C₂²×C₂₃	semidirect
147. #137	C₂²×C₂₃	semidirect
152. AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
153. AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
158. C₂×SD₁₆	C₂²×C₂₃	semidirect
160. C₂×D₈	C₂²×C₂₃	semidirect
161. #131	C₂²×C₂₃	semidirect
162. #131	C₂²×C₂₃	semidirect
163. #127	C₂²×C₂₃	semidirect
164. #124	C₂²×C₂₃	semidirect
165. #108	C₂²×C₂₃	semidirect
166. #104	C₂²×C₂₃	semidirect
167. #104	C₂²×C₂₃	semidirect
168. #101	C₂²×C₂₃	semidirect
169. #100	C₂²×C₂₃	semidirect
170. #100	C₂²×C₂₃	semidirect
171. #97	C₂²×C₂₃	semidirect
172. C₂₃	#3288	direct (N × H)
191. Q₁₆	D₄×C₂₃	semidirect
198. SD₁₆	D₄×C₂₃	semidirect
199. SD₁₆	D₄×C₂₃	semidirect
205. D₈	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#577	2
C₂₃×#576	2
C₂₃×#496	2
C₂₃×#472	2
C₂₃×#471	2
C₂₃×#455	2
C₂₃×#454	2
C₂₃×#453	2
C₂₃×#444	2
#3288	23

Subgroups — C23×#3288 #15670907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3288 #15670907