Subgroups — C₂₃×#2664 #15670283

Order 2944.

434 conjugacy classes of subgroups.

The subgroup lattice has 248 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	1	yes	maximal, characteristic
2	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#399	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#398	C₂	1	2	yes	maximal, complemented
6	C₂₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#397	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#396	C₂	1	2	yes	maximal
9	C₂₃×#389	C₂	1	2	yes	maximal, complemented
10	C₂₃×#388	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#126	—	2	4	no	—
13	C₂₃×#125	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
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	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	2	4	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	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₂²⋊C₄×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	—	4	2	no	—
64	C₄²×C₂₃	D₄	1	4	yes	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	1	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	2	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	2	no	—
81	C₂×C₄×C₂₃	—	2	2	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	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	2	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	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	#41	1	4	yes	—
97	#2664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	4	1	no	—
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	—	4	1	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#41	1	2	yes	—
105	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
107	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	1	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	4	2	no	—
118	C₂×#126	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#400	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#399	C₂×C₂₃	1	1	yes	characteristic
122	#398	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#398	C₂×C₂₃	1	2	yes	complemented
124	#397	C₂×C₂₃	1	1	yes	characteristic
125	#396	C₂×C₂₃	1	2	yes	—
126	#389	C₂×C₂₃	1	2	yes	complemented
127	#388	C₂×C₂₃	1	2	yes	—
128	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×C₂₃	—	8	1	no	—
130	C₂×C₂₃	#532	1	1	yes	characteristic
131	C₂×C₂₃	#536	1	1	yes	characteristic
132	C₂×C₂₃	#537	1	1	yes	characteristic
133	C₂×C₂₃	#539	1	2	yes	—
134	C₂×C₂₃	#541	1	2	yes	—
135	#126	—	2	4	no	—
136	#125	—	2	4	no	—
137	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#95	C₂²×C₂₃	1	1	yes	characteristic
155	#95	C₂²×C₂₃	1	1	yes	characteristic
156	#95	C₂²×C₂₃	1	2	yes	—
157	#95	C₂²×C₂₃	1	2	yes	—
158	#95	C₂²×C₂₃	1	2	yes	—
159	#95	C₂²×C₂₃	1	2	yes	—
160	C₂₃	#2664	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	D₄×C₂₃	1	4	yes	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
220	C₂×C₄	#41×C₂₃	1	4	yes	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₂²	C₂×#41×C₂₃	1	2	yes	—
228	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
230	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₂₃×#532	1	1	yes	characteristic
243	C₂	C₂₃×#536	1	1	yes	characteristic
244	C₂	C₂₃×#537	1	1	yes	characteristic
245	C₂	C₂₃×#539	1	2	yes	—
246	C₂	C₂₃×#541	1	2	yes	—

	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×C₂₃×#125	C₂	semidirect
3. C₂₃×#400	C₂	semidirect
5. C₂₃×#398	C₂	semidirect
6. C₂₃×#398	C₂	semidirect
9. C₂₃×#389	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
97. #2664	C₂₃	direct (N × H)
119. C₂×#125	C₂×C₂₃	semidirect
120. #400	C₂×C₂₃	semidirect
122. #398	C₂×C₂₃	semidirect
123. #398	C₂×C₂₃	semidirect
126. #389	C₂×C₂₃	semidirect
128. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
160. C₂₃	#2664	direct (N × H)

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
C₂×C₂₃×#125	2
C₂₃×#400	2
C₂₃×#399	2
C₂₃×#398	2
C₂₃×#397	2
C₂₃×#396	2
C₂₃×#389	2
C₂₃×#388	2
C₄×C₄⋊C₄×C₂₃	2
#2664	23

Subgroups — C23×#2664 #15670283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2664 #15670283