Subgroups — C₂₃×#2423 #15670042

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#399	C₂	1	2	yes	maximal
4	C₂₃×#398	C₂	1	2	yes	maximal, complemented
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#388	C₂	1	2	yes	maximal
8	C₂₃×#387	C₂	1	2	yes	maximal, complemented
9	C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₂²⋊Q₈	—	2	4	no	—
15	C₄×Q₈×C₂₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂₃×#95	C₂²	1	1	yes	characteristic
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₈×C₂₃	D₄	1	4	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₂²⋊C₄×C₂₃	—	2	4	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	—	4	2	no	—
65	C₂²⋊C₄×C₂₃	Q₈	1	4	yes	complemented
66	C₄²×C₂₃	—	2	4	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	—	2	1	no	—
71	C₂³×C₂₃	—	2	1	no	—
72	C₂³×C₂₃	—	2	1	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	1	no	—
75	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
76	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
77	Q₈×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₂₃	#41	1	2	yes	—
107	C₂×C₄×C₂₃	#41	1	2	yes	—
108	C₂×C₄×C₂₃	#41	1	2	yes	—
109	#2423	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	4	1	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	—	4	1	no	—
117	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
122	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	2	no	—
133	C₄×C₂₃	#123	1	4	yes	—
134	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#399	C₂×C₂₃	1	2	yes	—
137	#398	C₂×C₂₃	1	2	yes	complemented
138	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#388	C₂×C₂₃	1	2	yes	—
141	#387	C₂×C₂₃	1	2	yes	complemented
142	#384	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#380	C₂×C₂₃	1	1	yes	characteristic
144	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
145	C₂×C₂₃	—	4	2	no	—
146	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
147	C₂×C₂₃	#533	1	1	yes	characteristic
148	C₂×C₂₃	#542	1	1	yes	characteristic
149	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
150	C₂×C₂₃	#550	1	1	yes	characteristic
151	C₂×C₂₃	#552	1	1	yes	characteristic
152	C₂×C₂₃	#554	1	1	yes	characteristic
153	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²⋊Q₈	—	2	4	no	—
156	C₄×Q₈	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
171	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
172	#95	C₂²×C₂₃	1	1	yes	characteristic
173	#95	C₂²×C₂₃	1	1	yes	characteristic
174	#95	C₂²×C₂₃	1	2	yes	—
175	#95	C₂²×C₂₃	1	2	yes	—
176	#95	C₂²×C₂₃	1	2	yes	—
177	#95	C₂²×C₂₃	1	2	yes	—
178	C₂₃	#2423	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	D₄×C₂₃	1	4	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	Q₈×C₂₃	1	4	yes	complemented
208	C₄²	—	2	4	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
218	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	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	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
245	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
246	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
247	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
248	C₂×C₄	#41×C₂₃	1	2	yes	—
249	C₂×C₄	#41×C₂₃	1	2	yes	—
250	C₂×C₄	#41×C₂₃	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
259	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
260	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
261	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
262	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
263	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	C₂₃×#123	1	4	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
277	C₂	C₂₃×#533	1	1	yes	characteristic
278	C₂	C₂₃×#542	1	1	yes	characteristic
279	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
280	C₂	C₂₃×#550	1	1	yes	characteristic
281	C₂	C₂₃×#552	1	1	yes	characteristic
282	C₂	C₂₃×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#398	C₂	semidirect
5. C₂₃×#391	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
8. C₂₃×#387	C₂	semidirect
9. C₂₃×#384	C₂	semidirect
65. C₂²⋊C₄×C₂₃	Q₈	semidirect
109. #2423	C₂₃	direct (N × H)
135. C₂×C₄×Q₈	C₂×C₂₃	semidirect
137. #398	C₂×C₂₃	semidirect
138. #391	C₂×C₂₃	semidirect
139. #389	C₂×C₂₃	semidirect
141. #387	C₂×C₂₃	semidirect
142. #384	C₂×C₂₃	semidirect
178. C₂₃	#2423	direct (N × H)
207. C₂²⋊C₄	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₂²⋊Q₈	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#399	2
C₂₃×#398	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#387	2
C₂₃×#384	2
C₂₃×#380	2
C₄×C₂²⋊C₄×C₂₃	2
#2423	23

Subgroups — C23×#2423 #15670042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2423 #15670042