Subgroups — C₂₃×#2949 #15670568

Order 2944.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#518	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#440	C₂	1	4	yes	maximal, complemented
5	C₄×SD₁₆×C₂₃	C₂	1	4	yes	maximal, complemented
6	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#418	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₃	C₄	1	8	yes	complemented
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₂₃	—	2	2	no	—
18	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
19	C₄×D₄×C₂₃	—	2	2	no	—
20	C₄×D₄×C₂₃	C₂²	1	4	yes	—
21	C₂₃×#117	—	2	2	no	—
22	C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂₃×#117	C₂²	1	1	yes	characteristic
24	C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
26	C₂₃×#107	C₂²	1	2	yes	complemented
27	C₂₃×#106	C₂²	1	2	yes	complemented
28	C₂₃×#103	C₂²	1	4	yes	complemented
29	C₂₃×#102	C₂²	1	4	yes	—
30	C₂₃×#97	C₂²	1	2	yes	complemented
31	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
32	C₂⁴×C₂₃	—	2	1	no	—
33	C₂×Q₈×C₂₃	—	2	1	no	—
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	1	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
45	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
46	M₁₆×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
48	C₂×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
53	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂²⋊C₄×C₂₃	—	2	1	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	2	1	no	—
61	C₄²×C₂₃	—	2	1	no	—
62	C₄²×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	—	8	2	no	—
66	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₂₃	—	2	4	no	—
68	Q₈×C₂₃	—	2	2	no	—
69	Q₈×C₂₃	C₂²×C₄	1	4	yes	—
70	D₄×C₂₃	—	2	4	no	—
71	D₄×C₂₃	—	2	2	no	—
72	D₄×C₂₃	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	8	4	no	—
78	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#41	1	2	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	C₂²×C₄	1	2	yes	—
85	C₈×C₂₃	C₂²×C₄	1	2	yes	—
86	#2949	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	4	4	no	—
90	C₂²×C₂₃	—	8	4	no	—
91	C₂²×C₂₃	C₄×D₄	1	2	yes	—
92	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₂₃	—	2	4	no	—
94	C₄×C₂₃	—	2	4	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	C₄×D₄	1	2	yes	—
98	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
100	C₂²×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
101	#519	C₂×C₂₃	1	1	yes	characteristic, complemented
102	#518	C₂×C₂₃	1	1	yes	characteristic
103	#440	C₂×C₂₃	1	4	yes	complemented
104	C₄×SD₁₆	C₂×C₂₃	1	4	yes	complemented
105	#427	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#419	C₂×C₂₃	1	1	yes	characteristic, complemented
107	#418	C₂×C₂₃	1	1	yes	characteristic
108	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
109	C₂×C₂₃	—	2	2	no	—
110	C₂×C₂₃	—	4	4	no	—
111	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₂₃	#577	1	2	yes	—
113	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×SD₁₆	C₄×C₂₃	1	8	yes	complemented
116	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
117	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
118	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₄×Q₈	—	2	2	no	—
121	C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
122	C₄×D₄	—	2	2	no	—
123	C₄×D₄	C₂²×C₂₃	1	4	yes	—
124	#117	—	2	2	no	—
125	#117	C₂²×C₂₃	1	1	yes	characteristic
126	#117	C₂²×C₂₃	1	1	yes	characteristic
127	#117	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	—	2	2	no	—
129	#107	C₂²×C₂₃	1	2	yes	complemented
130	#106	C₂²×C₂₃	1	2	yes	complemented
131	#103	C₂²×C₂₃	1	4	yes	complemented
132	#102	C₂²×C₂₃	1	4	yes	—
133	#97	C₂²×C₂₃	1	2	yes	complemented
134	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
135	C₂₃	#2949	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
140	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
144	C₂×D₄	C₂³×C₂₃	1	2	yes	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
148	SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
149	SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
150	M₁₆	—	2	2	no	—
151	C₂×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
152	C₂×C₈	C₂³×C₂₃	1	2	yes	—
153	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
154	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
157	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
158	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	C₂³×C₂₃	1	2	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	8	2	no	—
170	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
171	Q₈	—	2	4	no	—
172	Q₈	—	2	2	no	—
173	Q₈	C₂²×C₄×C₂₃	1	4	yes	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	D₄	C₂²×C₄×C₂₃	1	4	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
184	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	#41×C₂₃	1	2	yes	—
186	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	C₂²×C₄×C₂₃	1	2	yes	—
189	C₈	C₂²×C₄×C₂₃	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	8	4	no	—
194	C₂²	C₄×D₄×C₂₃	1	2	yes	—
195	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₄×D₄×C₂₃	1	2	yes	—
201	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
202	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	4	4	no	—
205	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
206	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#519	C₂	semidirect
4. C₂₃×#440	C₂	semidirect
5. C₄×SD₁₆×C₂₃	C₂	semidirect
6. C₂₃×#427	C₂	semidirect
7. C₂₃×#419	C₂	semidirect
9. C₂₃×#405	C₂	semidirect
12. C₂×SD₁₆×C₂₃	C₄	semidirect
18. C₄×Q₈×C₂₃	C₂²	semidirect
26. C₂₃×#107	C₂²	semidirect
27. C₂₃×#106	C₂²	semidirect
28. C₂₃×#103	C₂²	semidirect
30. C₂₃×#97	C₂²	semidirect
31. C₄×C₈×C₂₃	C₂²	semidirect
35. C₂×Q₈×C₂₃	C₂×C₄	semidirect
44. SD₁₆×C₂₃	C₂×C₄	semidirect
45. SD₁₆×C₂₃	C₂×C₄	semidirect
47. C₂×C₈×C₂₃	C₂×C₄	semidirect
86. #2949	C₂₃	direct (N × H)
101. #519	C₂×C₂₃	semidirect
103. #440	C₂×C₂₃	semidirect
104. C₄×SD₁₆	C₂×C₂₃	semidirect
105. #427	C₂×C₂₃	semidirect
106. #419	C₂×C₂₃	semidirect
108. #405	C₂×C₂₃	semidirect
115. C₂×SD₁₆	C₄×C₂₃	semidirect
121. C₄×Q₈	C₂²×C₂₃	semidirect
129. #107	C₂²×C₂₃	semidirect
130. #106	C₂²×C₂₃	semidirect
131. #103	C₂²×C₂₃	semidirect
133. #97	C₂²×C₂₃	semidirect
134. C₄×C₈	C₂²×C₂₃	semidirect
135. C₂₃	#2949	direct (N × H)
139. C₂×Q₈	C₂×C₄×C₂₃	semidirect
148. SD₁₆	C₂×C₄×C₂₃	semidirect
149. SD₁₆	C₂×C₄×C₂₃	semidirect
151. C₂×C₈	C₂×C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₃	2
C₂₃×#519	2
C₂₃×#518	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
C₂₃×#427	2
C₂₃×#419	2
C₂₃×#418	2
C₂₃×#405	2
#2949	23

Subgroups — C23×#2949 #15670568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2949 #15670568