Subgroups — C₂₃×#3200 #15670819

Order 2944.

410 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#485	C₂	1	2	yes	maximal, complemented
4	C₂₃×#482	C₂	1	2	yes	maximal, complemented
5	C₂₃×#468	C₂	1	2	yes	maximal, complemented
6	C₂₃×#465	C₂	1	2	yes	maximal, complemented
7	C₂₃×#450	C₂	1	2	yes	maximal, complemented
8	C₂₃×#448	C₂	1	2	yes	maximal, complemented
9	C₂₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×C₄⋊Q₈	—	2	1	no	—
18	C₂₃×#127	—	2	1	no	—
19	C₂₃×#124	—	2	1	no	—
20	C₂₃×#124	—	2	1	no	—
21	C₂₃×#123	—	2	1	no	—
22	C₂₃×#123	—	2	1	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
25	C₂₃×#121	C₂²	1	2	yes	—
26	C₂₃×#121	C₂²	1	2	yes	—
27	C₂₃×#120	—	2	1	no	—
28	C₂₃×#120	—	2	1	no	—
29	C₂₃×#117	C₂²	1	1	yes	characteristic
30	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#107	C₂²	1	2	yes	complemented
32	C₂₃×#106	C₂²	1	2	yes	complemented
33	C₂₃×#103	C₂²	1	2	yes	complemented
34	C₂₃×#103	C₂²	1	2	yes	complemented
35	C₂₃×#102	C₂²	1	2	yes	—
36	C₂₃×#102	C₂²	1	2	yes	—
37	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
39	C₂⁴×C₂₃	—	2	1	no	—
40	#41×C₂₃	—	4	1	no	—
41	#41×C₂₃	—	4	1	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	—	4	1	no	—
44	C₂×Q₈×C₂₃	—	4	1	no	—
45	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	4	1	no	—
51	C₂×D₄×C₂₃	—	4	1	no	—
52	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	—	2	1	no	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	SD₁₆×C₂₃	—	4	2	no	—
61	SD₁₆×C₂₃	—	4	2	no	—
62	M₁₆×C₂₃	—	4	1	no	—
63	C₂×C₈×C₂₃	—	4	1	no	—
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	2	2	no	—
68	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	4	2	no	—
80	C₂³×C₂₃	—	8	2	no	—
81	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₂₃	—	2	2	no	—
85	Q₈×C₂₃	—	2	2	no	—
86	Q₈×C₂₃	—	4	1	no	—
87	Q₈×C₂₃	—	4	2	no	—
88	D₄×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	4	1	no	—
91	D₄×C₂₃	—	4	1	no	—
92	D₄×C₂₃	—	4	1	no	—
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	8	2	no	—
95	D₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₈×C₂₃	—	2	2	no	—
111	C₈×C₂₃	—	2	2	no	—
112	#3200	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	8	4	no	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	4	2	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
132	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#535	C₂×C₂₃	1	1	yes	characteristic
134	#485	C₂×C₂₃	1	2	yes	complemented
135	#482	C₂×C₂₃	1	2	yes	complemented
136	#468	C₂×C₂₃	1	2	yes	complemented
137	#465	C₂×C₂₃	1	2	yes	complemented
138	#450	C₂×C₂₃	1	2	yes	complemented
139	#448	C₂×C₂₃	1	2	yes	complemented
140	#408	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×C₂₃	—	4	1	no	—
142	C₂×C₂₃	—	4	1	no	—
143	C₂×C₂₃	—	4	1	no	—
144	C₂×C₂₃	—	8	2	no	—
145	C₂×C₂₃	#534	1	1	yes	characteristic
146	C₂×C₂₃	#577	1	2	yes	—
147	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
148	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
149	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
151	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
152	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
153	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
154	C₄⋊Q₈	—	2	1	no	—
155	#127	—	2	1	no	—
156	#124	—	2	1	no	—
157	#124	—	2	1	no	—
158	#123	—	2	1	no	—
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
161	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
162	#121	C₂²×C₂₃	1	2	yes	—
163	#121	C₂²×C₂₃	1	2	yes	—
164	#120	—	2	1	no	—
165	#120	—	2	1	no	—
166	#117	C₂²×C₂₃	1	1	yes	characteristic
167	#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
168	#107	C₂²×C₂₃	1	2	yes	complemented
169	#106	C₂²×C₂₃	1	2	yes	complemented
170	#103	C₂²×C₂₃	1	2	yes	complemented
171	#103	C₂²×C₂₃	1	2	yes	complemented
172	#102	C₂²×C₂₃	1	2	yes	—
173	#102	C₂²×C₂₃	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
175	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
176	C₂₃	#3200	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	—	2	1	no	—
178	#41	—	4	1	no	—
179	#41	—	4	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
184	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
191	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
192	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₂₃	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
198	SD₁₆	—	4	2	no	—
199	SD₁₆	—	4	2	no	—
200	M₁₆	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	C₂³×C₂₃	1	2	yes	—
203	C₂×C₈	C₂³×C₂₃	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
207	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	2	no	—
219	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	4	1	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	1	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	1	no	—
272	C₂	—	8	2	no	—
273	C₂	C₂₃×#534	1	1	yes	characteristic
274	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#536	C₂	semidirect
3. C₂₃×#485	C₂	semidirect
4. C₂₃×#482	C₂	semidirect
5. C₂₃×#468	C₂	semidirect
6. C₂₃×#465	C₂	semidirect
7. C₂₃×#450	C₂	semidirect
8. C₂₃×#448	C₂	semidirect
9. C₂₃×#408	C₂	semidirect
11. C₂²×Q₈×C₂₃	C₂²	semidirect
15. C₂×M₁₆×C₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₃	C₂²	semidirect
24. C₂₃×C₂²⋊Q₈	C₂²	semidirect
30. C₂₃×#117	C₂²	semidirect
31. C₂₃×#107	C₂²	semidirect
32. C₂₃×#106	C₂²	semidirect
33. C₂₃×#103	C₂²	semidirect
34. C₂₃×#103	C₂²	semidirect
38. C₂₃×C₂²⋊C₈	C₂²	semidirect
112. #3200	C₂₃	direct (N × H)
132. #536	C₂×C₂₃	semidirect
134. #485	C₂×C₂₃	semidirect
135. #482	C₂×C₂₃	semidirect
136. #468	C₂×C₂₃	semidirect
137. #465	C₂×C₂₃	semidirect
138. #450	C₂×C₂₃	semidirect
139. #448	C₂×C₂₃	semidirect
140. #408	C₂×C₂₃	semidirect
148. C₂²×Q₈	C₂²×C₂₃	semidirect
152. C₂×M₁₆	C₂²×C₂₃	semidirect
153. C₂²×C₈	C₂²×C₂₃	semidirect
161. C₂²⋊Q₈	C₂²×C₂₃	semidirect
167. #117	C₂²×C₂₃	semidirect
168. #107	C₂²×C₂₃	semidirect
169. #106	C₂²×C₂₃	semidirect
170. #103	C₂²×C₂₃	semidirect
171. #103	C₂²×C₂₃	semidirect
175. C₂²⋊C₈	C₂²×C₂₃	semidirect
176. C₂₃	#3200	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×#536	2
C₂₃×#535	2
C₂₃×#485	2
C₂₃×#482	2
C₂₃×#468	2
C₂₃×#465	2
C₂₃×#450	2
C₂₃×#448	2
C₂₃×#408	2
#3200	23

Subgroups — C23×#3200 #15670819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3200 #15670819