Subgroups — C₂₃×#2512 #15670131

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#126	C₂	1	2	yes	maximal, complemented
2	C₂₃×#397	C₂	1	2	yes	maximal, complemented
3	C₂₃×#396	C₂	1	2	yes	maximal, complemented
4	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#388	C₂	1	2	yes	maximal, complemented
9	C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#126	—	2	8	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
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	2	yes	—
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	C₂×C₄²×C₂₃	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	1	yes	characteristic
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂⁴×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₄⋊C₄×C₂₃	D₄	1	4	yes	complemented
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	2	4	no	—
61	C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₂²⋊C₄×C₂₃	—	4	2	no	—
63	C₂²⋊C₄×C₂₃	—	4	2	no	—
64	C₄²×C₂₃	—	2	4	no	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	—	2	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
76	D₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	2	1	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	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	2	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	1	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	#41	1	2	yes	—
104	C₂×C₄×C₂₃	#41	1	2	yes	—
105	C₂×C₄×C₂₃	#41	1	2	yes	—
106	C₂×C₄×C₂₃	#41	1	2	yes	—
107	#2512	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	2	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	2	no	—
115	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	#126	1	4	yes	complemented
132	C₂×#126	C₂×C₂₃	1	2	yes	complemented
133	#397	C₂×C₂₃	1	2	yes	complemented
134	#396	C₂×C₂₃	1	2	yes	complemented
135	#390	C₂×C₂₃	1	1	yes	characteristic
136	#390	C₂×C₂₃	1	1	yes	characteristic
137	#390	C₂×C₂₃	1	1	yes	characteristic
138	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#388	C₂×C₂₃	1	2	yes	complemented
140	#384	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×C₂₃	—	8	2	no	—
144	C₂×C₂₃	C₂×#126	1	1	yes	characteristic
145	C₂×C₂₃	#532	1	1	yes	characteristic
146	C₂×C₂₃	#542	1	1	yes	characteristic
147	C₂×C₂₃	#547	1	1	yes	characteristic
148	C₂×C₂₃	#550	1	1	yes	characteristic
149	C₂×C₂₃	#553	1	1	yes	characteristic
150	C₂×C₂₃	#556	1	1	yes	characteristic
151	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
152	#126	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
168	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
171	#95	C₂²×C₂₃	1	1	yes	characteristic
172	#95	C₂²×C₂₃	1	2	yes	—
173	#95	C₂²×C₂₃	1	1	yes	characteristic
174	#95	C₂²×C₂₃	1	2	yes	—
175	C₂₃	#2512	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂³×C₂₃	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	D₄×C₂₃	1	4	yes	complemented
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
243	C₂×C₄	#41×C₂₃	1	2	yes	—
244	C₂×C₄	#41×C₂₃	1	2	yes	—
245	C₂×C₄	#41×C₂₃	1	2	yes	—
246	C₂×C₄	#41×C₂₃	1	2	yes	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
255	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
256	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
257	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
258	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
259	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	C₂₃×#126	1	4	yes	complemented
271	C₂	—	8	2	no	—
272	C₂	C₂×C₂₃×#126	1	1	yes	characteristic
273	C₂	C₂₃×#532	1	1	yes	characteristic
274	C₂	C₂₃×#542	1	1	yes	characteristic
275	C₂	C₂₃×#547	1	1	yes	characteristic
276	C₂	C₂₃×#550	1	1	yes	characteristic
277	C₂	C₂₃×#553	1	1	yes	characteristic
278	C₂	C₂₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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
1. C₂×C₂₃×#126	C₂	semidirect
2. C₂₃×#397	C₂	semidirect
3. C₂₃×#396	C₂	semidirect
7. C₂₃×#389	C₂	semidirect
8. C₂₃×#388	C₂	semidirect
9. C₂₃×#384	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
57. C₄⋊C₄×C₂₃	D₄	semidirect
107. #2512	C₂₃	direct (N × H)
131. C₄×C₂₃	#126	semidirect
132. C₂×#126	C₂×C₂₃	semidirect
133. #397	C₂×C₂₃	semidirect
134. #396	C₂×C₂₃	semidirect
138. #389	C₂×C₂₃	semidirect
139. #388	C₂×C₂₃	semidirect
140. #384	C₂×C₂₃	semidirect
141. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
142. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
175. C₂₃	#2512	direct (N × H)
197. C₄⋊C₄	D₄×C₂₃	semidirect
270. C₄	C₂₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#126	2
C₂₃×#397	2
C₂₃×#396	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#384	2
C₄×C₄⋊C₄×C₂₃	2
#2512	23

Subgroups — C23×#2512 #15670131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2512 #15670131