Subgroups — C₂₃×#3252 #15670871

Order 2944.

348 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×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#479	C₂	1	2	yes	maximal, complemented
4	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#470	C₂	1	2	yes	maximal, complemented
7	C₂₃×#468	C₂	1	2	yes	maximal
8	C₂₃×#464	C₂	1	2	yes	maximal, complemented
9	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#126	—	2	1	no	—
18	C₂₃×#125	—	2	1	no	—
19	C₂₃×#125	C₂²	1	1	yes	characteristic
20	C₂₃×#124	C₂²	1	1	yes	characteristic
21	C₂₃×#123	—	2	1	no	—
22	C₂₃×C₂²⋊Q₈	—	2	1	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#121	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₂₃	—	2	1	no	—
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
29	C₄×D₄×C₂₃	—	2	1	no	—
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#117	—	2	1	no	—
32	C₂₃×#117	—	2	1	no	—
33	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#107	C₂²	1	2	yes	—
35	C₂₃×#106	C₂²	1	2	yes	—
36	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
39	C₂₃×#103	C₂²	1	2	yes	—
40	C₂₃×#103	C₂²	1	2	yes	—
41	C₂₃×#103	C₂²	1	2	yes	—
42	C₂₃×#102	C₂²	1	2	yes	—
43	C₂×Q₈×C₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	Q₁₆×C₂₃	—	4	2	no	—
53	SD₁₆×C₂₃	—	4	2	no	—
54	M₁₆×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₄²×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	D₄	1	1	yes	characteristic
78	C₄²×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂³×C₂₃	—	2	1	no	—
82	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₂₃	—	2	2	no	—
84	Q₈×C₂₃	—	2	1	no	—
85	Q₈×C₂₃	—	2	1	no	—
86	Q₈×C₂₃	—	2	2	no	—
87	Q₈×C₂₃	—	4	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	2	1	no	—
90	D₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	4	1	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₈×C₂₃	—	2	2	no	—
108	C₈×C₂₃	—	2	2	no	—
109	#3252	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	4	1	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	4	1	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	#137	1	2	yes	—
127	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
129	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#529	C₂×C₂₃	1	1	yes	characteristic
131	#479	C₂×C₂₃	1	2	yes	complemented
132	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#470	C₂×C₂₃	1	2	yes	complemented
135	#468	C₂×C₂₃	1	2	yes	—
136	#464	C₂×C₂₃	1	2	yes	complemented
137	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#461	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂×C₂₃	—	4	1	no	—
141	C₂×C₂₃	—	8	1	no	—
142	C₂×C₂₃	#537	1	1	yes	characteristic
143	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
144	C₂×C₂₃	#575	1	1	yes	characteristic
145	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
146	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
147	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
148	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
150	#126	—	2	1	no	—
151	#125	—	2	1	no	—
152	#125	C₂²×C₂₃	1	1	yes	characteristic
153	#124	C₂²×C₂₃	1	1	yes	characteristic
154	#123	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
158	#121	C₂²×C₂₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#117	—	2	1	no	—
166	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
167	#107	C₂²×C₂₃	1	2	yes	—
168	#106	C₂²×C₂₃	1	2	yes	—
169	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
170	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
171	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
172	#103	C₂²×C₂₃	1	2	yes	—
173	#103	C₂²×C₂₃	1	2	yes	—
174	#103	C₂²×C₂₃	1	2	yes	—
175	#102	C₂²×C₂₃	1	2	yes	—
176	C₂₃	#3252	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	SD₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	C₂×C₈	C₂³×C₂₃	1	2	yes	—
190	C₂×C₈	C₂³×C₂₃	1	2	yes	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
201	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	D₄×C₂₃	1	1	yes	characteristic
212	C₄²	D₄×C₂₃	1	1	yes	characteristic
213	C₄²	C₂³×C₂₃	1	1	yes	characteristic
214	C₄²	C₂³×C₂₃	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	C₂₃×#137	1	2	yes	—
260	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
261	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂₃×#537	1	1	yes	characteristic
265	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
266	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#533	C₂	semidirect
3. C₂₃×#479	C₂	semidirect
4. C₂₃×#477	C₂	semidirect
5. C₂₃×#476	C₂	semidirect
6. C₂₃×#470	C₂	semidirect
8. C₂₃×#464	C₂	semidirect
9. C₂₃×#462	C₂	semidirect
10. C₂₃×#461	C₂	semidirect
11. C₂₃×#423	C₂	semidirect
109. #3252	C₂₃	direct (N × H)
129. #533	C₂×C₂₃	semidirect
131. #479	C₂×C₂₃	semidirect
132. #477	C₂×C₂₃	semidirect
133. #476	C₂×C₂₃	semidirect
134. #470	C₂×C₂₃	semidirect
136. #464	C₂×C₂₃	semidirect
137. #462	C₂×C₂₃	semidirect
138. #461	C₂×C₂₃	semidirect
139. #423	C₂×C₂₃	semidirect
176. C₂₃	#3252	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×#533	2
C₂₃×#529	2
C₂₃×#479	2
C₂₃×#477	2
C₂₃×#476	2
C₂₃×#470	2
C₂₃×#468	2
C₂₃×#464	2
C₂₃×#462	2
C₂₃×#461	2
C₂₃×#423	2
#3252	23

Subgroups — C23×#3252 #15670871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3252 #15670871