Subgroups — C₂₃×#1869 #15669488

Order 2944.

432 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂₃×#336	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	—	2	2	no	—
10	C₂²×C₈×C₂₃	—	2	4	no	—
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
13	C₄×D₄×C₂₃	—	2	2	no	—
14	C₄×D₄×C₂₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₃	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₂²⋊C₈	—	2	2	no	—
20	C₂₃×C₂²⋊C₈	—	2	4	no	—
21	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₂₃	—	2	2	no	—
24	C₂×D₄×C₂₃	—	2	2	no	—
25	C₂×D₄×C₂₃	C₈	1	4	yes	complemented
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	—	2	2	no	—
28	C₂²×C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄×C₂₃	—	2	1	no	—
30	C₂²×C₄×C₂₃	—	2	2	no	—
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	D₄	1	2	yes	—
37	C₂²×C₄×C₂₃	D₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₃	—	2	8	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂²⋊C₄×C₂₃	—	2	2	no	—
48	C₂²⋊C₄×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	—	2	1	no	—
50	C₄²×C₂₃	—	2	1	no	—
51	C₂³×C₂₃	—	2	2	no	—
52	C₂³×C₂₃	—	2	2	no	—
53	C₂³×C₂₃	—	2	1	no	—
54	C₂³×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂³×C₂₃	C₂×C₈	1	2	yes	—
59	C₂³×C₂₃	C₂×C₈	1	2	yes	complemented
60	C₂³×C₂₃	M₁₆	1	2	yes	—
61	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	D₄×C₂₃	—	2	8	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	1	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	4	1	no	—
76	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	complemented
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#41	1	2	yes	—
82	C₈×C₂₃	—	2	4	no	—
83	C₈×C₂₃	—	4	2	no	—
84	#1869	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	2	2	no	—
87	C₂²×C₂₃	—	2	4	no	—
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	4	1	no	—
92	C₂²×C₂₃	—	4	1	no	—
93	C₂²×C₂₃	—	4	2	no	—
94	C₂²×C₂₃	C₂²⋊C₈	1	4	yes	complemented
95	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₂₃	C₄×D₄	1	2	yes	—
97	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
98	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×C₂₃	#131	1	2	yes	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	4	1	no	—
103	C₄×C₂₃	—	4	1	no	—
104	C₄×C₂₃	#120	1	1	yes	characteristic
105	C₄×C₂₃	#121	1	2	yes	—
106	C₄×C₂₃	#123	1	1	yes	characteristic
107	C₂³×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
108	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
109	C₂×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂×C₂²⋊C₈	C₂×C₂₃	1	2	yes	complemented
111	#336	C₂×C₂₃	1	2	yes	complemented
112	C₂×C₂₃	—	2	4	no	—
113	C₂×C₂₃	—	4	2	no	—
114	C₂×C₂₃	#386	1	1	yes	characteristic
115	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₂₃	#408	1	1	yes	characteristic
117	C₂×C₂₃	C₈×D₄	1	2	yes	—
118	C₂×C₂₃	#435	1	2	yes	—
119	C₂²×D₄	C₄×C₂₃	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
121	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
125	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	4	no	—
134	C₂₃	#1869	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₈×C₂₃	1	4	yes	complemented
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	D₄×C₂₃	1	2	yes	—
151	C₂²×C₄	D₄×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
160	C₄⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂²⋊C₄×C₂₃	1	2	yes	—
172	C₂³	C₂×C₈×C₂₃	1	2	yes	—
173	C₂³	C₂×C₈×C₂₃	1	2	yes	complemented
174	C₂³	M₁₆×C₂₃	1	2	yes	—
175	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
176	D₄	—	2	8	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
191	C₂×C₄	C₂×C₈×C₂₃	1	2	yes	complemented
192	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	#41×C₂₃	1	2	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	4	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	C₂₃×C₂²⋊C₈	1	4	yes	complemented
208	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂²	C₄×D₄×C₂₃	1	2	yes	—
210	C₂²	C₂²×C₈×C₂₃	1	1	yes	characteristic
211	C₂²	C₂×M₁₆×C₂₃	1	1	yes	characteristic
212	C₂²	C₂₃×#131	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	4	1	no	—
217	C₄	C₂₃×#120	1	1	yes	characteristic
218	C₄	C₂₃×#121	1	2	yes	—
219	C₄	C₂₃×#123	1	1	yes	characteristic
220	C₂	—	2	4	no	—
221	C₂	—	4	2	no	—
222	C₂	C₂₃×#386	1	1	yes	characteristic
223	C₂	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
224	C₂	C₂₃×#408	1	1	yes	characteristic
225	C₂	C₈×D₄×C₂₃	1	2	yes	—
226	C₂	C₂₃×#435	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂³
commutator	C₂²	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂²×D₄×C₂₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×C₂₃	C₂	semidirect
3. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
4. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
5. C₂₃×#336	C₂	semidirect
25. C₂×D₄×C₂₃	C₈	semidirect
45. C₂×C₈×C₂₃	D₄	semidirect
59. C₂³×C₂₃	C₂×C₈	semidirect
77. C₂×C₄×C₂₃	C₂×C₈	semidirect
84. #1869	C₂₃	direct (N × H)
94. C₂²×C₂₃	C₂²⋊C₈	semidirect
107. C₂³×C₈	C₂×C₂₃	semidirect
109. C₂×C₂²⋊C₈	C₂×C₂₃	semidirect
110. C₂×C₂²⋊C₈	C₂×C₂₃	semidirect
111. #336	C₂×C₂₃	semidirect
134. C₂₃	#1869	direct (N × H)
139. C₂×D₄	C₈×C₂₃	semidirect
159. C₂×C₈	D₄×C₂₃	semidirect
173. C₂³	C₂×C₈×C₂₃	semidirect
191. C₂×C₄	C₂×C₈×C₂₃	semidirect
207. C₂²	C₂₃×C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂×C₂₃×C₂²⋊C₈	2
C₂₃×#336	2
#1869	23

Subgroups — C23×#1869 #15669488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1869 #15669488