Subgroups — C₂₃×#2513 #15670132

Order 2944.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#397	C₂	1	2	yes	maximal, complemented
2	C₂₃×#397	C₂	1	2	yes	maximal, complemented
3	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#388	C₂	1	2	yes	maximal, complemented
8	C₂₃×#388	C₂	1	2	yes	maximal, complemented
9	C₂₃×#382	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	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
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×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂⁴×C₂₃	C₂³	1	2	yes	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	2	yes	—
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	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₂²⋊C₄×C₂₃	—	2	4	no	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	4	2	no	—
58	C₂²⋊C₄×C₂₃	—	4	2	no	—
59	C₂²⋊C₄×C₂₃	—	4	2	no	—
60	C₂²⋊C₄×C₂₃	—	4	2	no	—
61	C₄²×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×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	C₂⁴	1	1	yes	characteristic
73	D₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	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	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	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	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	#41	1	2	yes	—
100	C₂×C₄×C₂₃	#41	1	2	yes	—
101	C₂×C₄×C₂₃	#41	1	2	yes	—
102	C₂×C₄×C₂₃	#41	1	2	yes	—
103	C₂×C₄×C₂₃	#41	1	2	yes	—
104	#2513	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	—	4	2	no	—
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	C₂×#41	1	1	yes	characteristic
113	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	2	2	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	1	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	4	2	no	—
128	C₄×C₂₃	#126	1	4	yes	—
129	#397	C₂×C₂₃	1	2	yes	complemented
130	#397	C₂×C₂₃	1	2	yes	complemented
131	#390	C₂×C₂₃	1	1	yes	characteristic
132	#390	C₂×C₂₃	1	1	yes	characteristic
133	#390	C₂×C₂₃	1	1	yes	characteristic
134	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#388	C₂×C₂₃	1	2	yes	complemented
136	#388	C₂×C₂₃	1	2	yes	complemented
137	#382	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂×C₂₃	—	8	2	no	—
141	C₂×C₂₃	C₂×#126	1	1	yes	characteristic
142	C₂×C₂₃	#529	1	1	yes	characteristic
143	C₂×C₂₃	#540	1	1	yes	characteristic
144	C₂×C₂₃	#553	1	1	yes	characteristic
145	C₂×C₂₃	#555	1	1	yes	characteristic
146	C₂×C₂₃	#559	1	1	yes	characteristic
147	C₂×C₂₃	#559	1	1	yes	characteristic
148	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
165	#95	C₂²×C₂₃	1	1	yes	characteristic
166	#95	C₂²×C₂₃	1	1	yes	characteristic
167	#95	C₂²×C₂₃	1	2	yes	—
168	#95	C₂²×C₂₃	1	2	yes	—
169	#95	C₂²×C₂₃	1	2	yes	—
170	#95	C₂²×C₂₃	1	2	yes	—
171	C₂₃	#2513	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₂³×C₂₃	1	2	yes	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	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	1	no	—
236	C₂×C₄	#41×C₂₃	1	2	yes	—
237	C₂×C₄	#41×C₂₃	1	2	yes	—
238	C₂×C₄	#41×C₂₃	1	2	yes	—
239	C₂×C₄	#41×C₂₃	1	2	yes	—
240	C₂×C₄	#41×C₂₃	1	2	yes	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
249	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
250	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	C₂₃×#126	1	4	yes	—
265	C₂	—	8	2	no	—
266	C₂	C₂×C₂₃×#126	1	1	yes	characteristic
267	C₂	C₂₃×#529	1	1	yes	characteristic
268	C₂	C₂₃×#540	1	1	yes	characteristic
269	C₂	C₂₃×#553	1	1	yes	characteristic
270	C₂	C₂₃×#555	1	1	yes	characteristic
271	C₂	C₂₃×#559	1	1	yes	characteristic
272	C₂	C₂₃×#559	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. C₂₃×#397	C₂	semidirect
2. C₂₃×#397	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
7. C₂₃×#388	C₂	semidirect
8. C₂₃×#388	C₂	semidirect
9. C₂₃×#382	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
104. #2513	C₂₃	direct (N × H)
129. #397	C₂×C₂₃	semidirect
130. #397	C₂×C₂₃	semidirect
134. #389	C₂×C₂₃	semidirect
135. #388	C₂×C₂₃	semidirect
136. #388	C₂×C₂₃	semidirect
137. #382	C₂×C₂₃	semidirect
138. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
139. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
171. C₂₃	#2513	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×#397	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#382	2
C₄×C₄⋊C₄×C₂₃	2
#2513	23

Subgroups — C23×#2513 #15670132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2513 #15670132