Subgroups — C₂₃×#2433 #15670052

Order 2944.

1348 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#127	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#120	C₂	1	4	yes	maximal, complemented
5	C₂₃×#392	C₂	1	4	yes	maximal, complemented
6	C₂₃×#390	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×C₂₃	C₂²	1	2	yes	—
9	C₂²×D₄×C₂₃	—	2	8	no	—
10	C₂²×D₄×C₂₃	—	2	4	no	—
11	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂²×D₄×C₂₃	C₂²	1	8	yes	complemented
16	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#127	—	2	8	no	—
18	C₂₃×#121	—	2	4	no	—
19	C₂₃×#120	—	2	16	no	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#95	C₂²	1	2	yes	complemented
26	C₂⁴×C₂₃	—	2	8	no	—
27	C₂⁴×C₂₃	—	2	4	no	—
28	C₂⁴×C₂₃	—	2	8	no	—
29	C₂⁴×C₂₃	—	4	4	no	—
30	C₂⁴×C₂₃	C₂³	1	2	yes	—
31	C₂⁴×C₂₃	C₂³	1	4	yes	—
32	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₂₃	—	2	32	no	—
34	C₂×D₄×C₂₃	—	2	8	no	—
35	C₂×D₄×C₂₃	—	2	8	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	—	2	16	no	—
38	C₂×D₄×C₂₃	—	2	4	no	—
39	C₂×D₄×C₂₃	—	2	4	no	—
40	C₂×D₄×C₂₃	—	2	4	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	—	2	4	no	—
43	C₂×D₄×C₂₃	—	2	16	no	—
44	C₂×D₄×C₂₃	—	4	8	no	—
45	C₂×D₄×C₂₃	D₄	1	8	yes	complemented
46	C₂²×C₄×C₂₃	—	2	4	no	—
47	C₂²×C₄×C₂₃	D₄	1	4	yes	complemented
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂²⋊C₄×C₂₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃	—	2	16	no	—
55	C₂²⋊C₄×C₂₃	D₄	1	8	yes	complemented
56	C₄²×C₂₃	—	2	4	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	16	no	—
60	C₂³×C₂₃	—	2	8	no	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	—	2	4	no	—
63	C₂³×C₂₃	—	2	8	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	4	no	—
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	—	4	32	no	—
69	C₂³×C₂₃	—	4	8	no	—
70	C₂³×C₂₃	—	4	8	no	—
71	C₂³×C₂₃	C₂×D₄	1	8	yes	—
72	C₂³×C₂₃	C₂×D₄	1	4	yes	—
73	C₂³×C₂₃	C₂×D₄	1	2	yes	—
74	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
75	D₄×C₂₃	—	2	16	no	—
76	D₄×C₂₃	—	2	16	no	—
77	D₄×C₂₃	—	4	32	no	—
78	D₄×C₂₃	—	4	8	no	—
79	D₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	complemented
90	#2433	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	8	no	—
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	2	16	no	—
96	C₂²×C₂₃	—	4	32	no	—
97	C₂²×C₂₃	—	4	4	no	—
98	C₂²×C₂₃	—	4	4	no	—
99	C₂²×C₂₃	—	4	8	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	8	no	—
104	C₂²×C₂₃	#127	1	4	yes	complemented
105	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
107	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
108	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	8	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	#120	1	4	yes	complemented
113	C₂³×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
114	C₂×#127	C₂×C₂₃	1	2	yes	complemented
115	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×#120	C₂×C₂₃	1	4	yes	complemented
117	#392	C₂×C₂₃	1	4	yes	complemented
118	#390	C₂×C₂₃	1	2	yes	complemented
119	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×C₂₃	—	2	4	no	—
121	C₂×C₂₃	—	4	8	no	—
122	C₂×C₂₃	—	8	2	no	—
123	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
124	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
125	C₂×C₂₃	#535	1	1	yes	characteristic
126	C₂×C₂₃	D₄²	1	4	yes	—
127	C₂⁵	C₂²×C₂₃	1	2	yes	—
128	C₂²×D₄	—	2	8	no	—
129	C₂²×D₄	—	2	4	no	—
130	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
131	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
132	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
133	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
134	C₂²×D₄	C₂²×C₂₃	1	8	yes	complemented
135	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
136	#127	—	2	8	no	—
137	#121	—	2	4	no	—
138	#120	—	2	16	no	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
142	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
143	C₂×C₄²	C₂²×C₂₃	1	2	yes	complemented
144	#95	C₂²×C₂₃	1	2	yes	complemented
145	C₂₃	#2433	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	—	2	8	no	—
147	C₂⁴	—	2	4	no	—
148	C₂⁴	—	2	8	no	—
149	C₂⁴	—	4	4	no	—
150	C₂⁴	C₂³×C₂₃	1	2	yes	—
151	C₂⁴	C₂³×C₂₃	1	4	yes	—
152	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
153	C₂×D₄	—	2	32	no	—
154	C₂×D₄	—	2	8	no	—
155	C₂×D₄	—	2	8	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	2	16	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	2	16	no	—
164	C₂×D₄	—	4	8	no	—
165	C₂×D₄	D₄×C₂₃	1	8	yes	complemented
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	D₄×C₂₃	1	4	yes	complemented
168	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	16	no	—
175	C₂²⋊C₄	D₄×C₂₃	1	8	yes	complemented
176	C₄²	—	2	4	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	16	no	—
180	C₂³	—	2	8	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	8	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	4	no	—
188	C₂³	—	4	32	no	—
189	C₂³	—	4	8	no	—
190	C₂³	—	4	8	no	—
191	C₂³	C₂×D₄×C₂₃	1	8	yes	—
192	C₂³	C₂×D₄×C₂₃	1	4	yes	—
193	C₂³	C₂×D₄×C₂₃	1	2	yes	—
194	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
195	D₄	—	2	16	no	—
196	D₄	—	2	16	no	—
197	D₄	—	4	32	no	—
198	D₄	—	4	8	no	—
199	D₄	—	4	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₂₃	1	8	yes	complemented
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	8	no	—
213	C₂²	—	2	8	no	—
214	C₂²	—	2	16	no	—
215	C₂²	—	4	32	no	—
216	C₂²	—	4	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	8	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	8	no	—
223	C₂²	C₂₃×#127	1	4	yes	complemented
224	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
226	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
227	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	8	no	—
230	C₄	—	4	2	no	—
231	C₄	C₂₃×#120	1	4	yes	complemented
232	C₂	—	2	4	no	—
233	C₂	—	4	8	no	—
234	C₂	—	8	2	no	—
235	C₂	C₂×C₂₃×#120	1	1	yes	characteristic
236	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
237	C₂	C₂₃×#535	1	1	yes	characteristic
238	C₂	D₄²×C₂₃	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×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
1. C₂³×D₄×C₂₃	C₂	semidirect
2. C₂×C₂₃×#127	C₂	semidirect
3. C₂×C₂₃×#121	C₂	semidirect
4. C₂×C₂₃×#120	C₂	semidirect
5. C₂₃×#392	C₂	semidirect
6. C₂₃×#390	C₂	semidirect
7. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
14. C₂²×D₄×C₂₃	C₂²	semidirect
15. C₂²×D₄×C₂₃	C₂²	semidirect
20. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
22. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
23. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
24. C₂×C₄²×C₂₃	C₂²	semidirect
25. C₂₃×#95	C₂²	semidirect
45. C₂×D₄×C₂₃	D₄	semidirect
47. C₂²×C₄×C₂₃	D₄	semidirect
55. C₂²⋊C₄×C₂₃	D₄	semidirect
89. C₂×C₄×C₂₃	C₂×D₄	semidirect
90. #2433	C₂₃	direct (N × H)
104. C₂²×C₂₃	#127	semidirect
112. C₄×C₂₃	#120	semidirect
113. C₂³×D₄	C₂×C₂₃	semidirect
114. C₂×#127	C₂×C₂₃	semidirect
115. C₂×#121	C₂×C₂₃	semidirect
116. C₂×#120	C₂×C₂₃	semidirect
117. #392	C₂×C₂₃	semidirect
118. #390	C₂×C₂₃	semidirect
119. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
133. C₂²×D₄	C₂²×C₂₃	semidirect
134. C₂²×D₄	C₂²×C₂₃	semidirect
139. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
141. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
142. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
143. C₂×C₄²	C₂²×C₂₃	semidirect
144. #95	C₂²×C₂₃	semidirect
145. C₂₃	#2433	direct (N × H)
165. C₂×D₄	D₄×C₂₃	semidirect
167. C₂²×C₄	D₄×C₂₃	semidirect
175. C₂²⋊C₄	D₄×C₂₃	semidirect
209. C₂×C₄	C₂×D₄×C₂₃	semidirect
223. C₂²	C₂₃×#127	semidirect
231. C₄	C₂₃×#120	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₃	2
C₂×C₂₃×#127	2
C₂×C₂₃×#121	2
C₂×C₂₃×#120	2
C₂₃×#392	2
C₂₃×#390	2
C₄×C₂²⋊C₄×C₂₃	2
#2433	23

Subgroups — C23×#2433 #15670052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2433 #15670052