Subgroups — C₂₃×#19667 #19167293

Order 5888.

780 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3207	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3169	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1717	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1706	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂²	1	4	yes	—
9	C₂₃×#500	C₂²	1	2	yes	—
10	C₂₃×#493	C₂²	1	2	yes	—
11	C₂₃×#493	C₂²	1	4	yes	—
12	C₂₃×#492	C₂²	1	4	yes	—
13	C₂₃×#487	C₂²	1	4	yes	—
14	C₂₃×#486	C₂²	1	4	yes	—
15	C₂₃×#486	C₂²	1	4	yes	—
16	C₂₃×#466	—	2	4	no	—
17	C₂₃×#463	—	2	8	no	—
18	C₂₃×#449	—	2	8	no	—
19	C₄×D₈×C₂₃	—	2	4	no	—
20	C₂₃×#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#322	C₂²	1	2	yes	—
23	C₈²×C₂₃	C₂²	1	1	yes	characteristic
24	C₈²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×#41×C₂₃	—	2	4	no	—
26	C₂×SD₁₆×C₂₃	—	2	8	no	—
27	C₂×D₈×C₂₃	—	2	4	no	—
28	C₂×D₈×C₂₃	—	2	8	no	—
29	C₂²×C₈×C₂₃	D₄	1	2	yes	—
30	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
31	C₂₃×#127	C₂³	1	4	yes	—
32	C₂₃×#124	—	2	4	no	—
33	C₂₃×#121	—	2	4	no	—
34	C₂₃×#121	—	2	4	no	—
35	C₄×D₄×C₂₃	—	2	4	no	—
36	C₄×D₄×C₂₃	—	2	4	no	—
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#107	—	2	4	no	—
39	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
40	C₂₃×#103	—	2	8	no	—
41	C₂₃×#102	—	2	8	no	—
42	C₂₃×#102	—	2	8	no	—
43	C₂₃×C₂²⋊C₈	D₄	1	4	yes	—
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₂₃	C₂³	1	2	yes	—
47	C₄×C₈×C₂₃	C₂³	1	2	yes	—
48	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	#41×C₂₃	—	8	8	no	—
50	C₂×Q₈×C₂₃	—	2	4	no	—
51	C₂×D₄×C₂₃	—	2	4	no	—
52	C₂×D₄×C₂₃	—	2	4	no	—
53	C₂×D₄×C₂₃	—	2	4	no	—
54	C₂×D₄×C₂₃	—	4	4	no	—
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	SD₁₆×C₂₃	—	8	8	no	—
59	D₈×C₂₃	—	4	8	no	—
60	D₈×C₂₃	—	8	8	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	2	4	no	—
68	C₂²⋊C₄×C₂₃	—	4	4	no	—
69	C₂²⋊C₄×C₂₃	—	4	4	no	—
70	C₄²×C₂₃	C₂×D₄	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
73	#19667	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	4	4	no	—
75	C₂³×C₂₃	#127	1	1	yes	characteristic
76	Q₈×C₂₃	—	4	8	no	—
77	D₄×C₂₃	—	4	8	no	—
78	D₄×C₂₃	—	4	8	no	—
79	D₄×C₂₃	—	4	8	no	—
80	D₄×C₂₃	—	8	8	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	4	4	no	—
84	C₂×C₄×C₂₃	—	8	8	no	—
85	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#127	1	2	yes	—
87	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
89	C₈×C₂₃	—	2	2	no	—
90	C₈×C₂₃	—	2	4	no	—
91	C₈×C₂₃	—	2	4	no	—
92	C₈×C₂₃	#135	1	4	yes	—
93	#3207	C₂×C₂₃	1	4	yes	complemented
94	#3169	C₂×C₂₃	1	2	yes	complemented
95	#1717	C₂×C₂₃	1	4	yes	complemented
96	#1712	C₂×C₂₃	1	1	yes	characteristic, complemented
97	#1710	C₂×C₂₃	1	1	yes	characteristic, complemented
98	#1706	C₂×C₂₃	1	2	yes	complemented
99	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	8	4	no	—
103	C₂²×C₂₃	—	8	4	no	—
104	C₂²×C₂₃	—	8	4	no	—
105	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	8	4	no	—
109	C₄×C₂₃	C₂×#135	1	2	yes	—
110	C₄×C₂₃	#576	1	4	yes	—
111	#532	C₂²×C₂₃	1	4	yes	—
112	#500	C₂²×C₂₃	1	2	yes	—
113	#493	C₂²×C₂₃	1	2	yes	—
114	#493	C₂²×C₂₃	1	4	yes	—
115	#492	C₂²×C₂₃	1	4	yes	—
116	#487	C₂²×C₂₃	1	4	yes	—
117	#486	C₂²×C₂₃	1	4	yes	—
118	#486	C₂²×C₂₃	1	4	yes	—
119	#466	—	2	4	no	—
120	#463	—	2	8	no	—
121	#449	—	2	8	no	—
122	C₄×D₈	—	2	4	no	—
123	#431	C₂²×C₂₃	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	#322	C₂²×C₂₃	1	2	yes	—
126	C₈²	C₂²×C₂₃	1	1	yes	characteristic
127	C₈²	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₂₃	—	4	1	no	—
129	C₂×C₂₃	—	16	4	no	—
130	C₂×C₂₃	#3147	1	1	yes	characteristic
131	C₂×C₂₃	#3155	1	2	yes	—
132	C₂×#41	—	2	4	no	—
133	C₂×SD₁₆	—	2	8	no	—
134	C₂×D₈	—	2	4	no	—
135	C₂×D₈	—	2	8	no	—
136	C₂²×C₈	D₄×C₂₃	1	2	yes	—
137	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
138	#127	C₂³×C₂₃	1	4	yes	—
139	#124	—	2	4	no	—
140	#121	—	2	4	no	—
141	#121	—	2	4	no	—
142	C₄×D₄	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
147	#103	—	2	8	no	—
148	#102	—	2	8	no	—
149	#102	—	2	8	no	—
150	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
151	C₄×C₈	D₄×C₂₃	1	2	yes	—
152	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₂₃	1	2	yes	—
154	C₄×C₈	C₂³×C₂₃	1	2	yes	—
155	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
156	C₂₃	#19667	1	1	yes	characteristic, direct product factor, Sylow
157	#41	—	8	8	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
166	SD₁₆	—	8	8	no	—
167	D₈	—	4	8	no	—
168	D₈	—	8	8	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
173	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	4	4	no	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₂₃	1	2	yes	—
180	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
181	C₂³	—	4	4	no	—
182	C₂³	C₂₃×#127	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	D₄	—	4	8	no	—
185	D₄	—	4	8	no	—
186	D₄	—	4	8	no	—
187	D₄	—	8	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	8	no	—
192	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
193	C₂×C₄	C₂₃×#127	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₈	C₂₃×#135	1	4	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	8	4	no	—
203	C₂²	—	8	4	no	—
204	C₂²	—	8	4	no	—
205	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	C₂×C₂₃×#135	1	2	yes	—
210	C₄	C₂₃×#576	1	4	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	16	4	no	—
213	C₂	C₂₃×#3147	1	1	yes	characteristic
214	C₂	C₂₃×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3207	C₂	semidirect
2. C₂₃×#3169	C₂	semidirect
3. C₂₃×#1717	C₂	semidirect
4. C₂₃×#1712	C₂	semidirect
5. C₂₃×#1710	C₂	semidirect
6. C₂₃×#1706	C₂	semidirect
7. C₂₃×#1450	C₂	semidirect
73. #19667	C₂₃	direct (N × H)
93. #3207	C₂×C₂₃	semidirect
94. #3169	C₂×C₂₃	semidirect
95. #1717	C₂×C₂₃	semidirect
96. #1712	C₂×C₂₃	semidirect
97. #1710	C₂×C₂₃	semidirect
98. #1706	C₂×C₂₃	semidirect
99. #1450	C₂×C₂₃	semidirect
156. C₂₃	#19667	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×#3207	2
C₂₃×#3169	2
C₂₃×#1717	2
C₂₃×#1712	2
C₂₃×#1710	2
C₂₃×#1706	2
C₂₃×#1450	2
#19667	23

Subgroups — C23×#19667 #19167293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19667 #19167293