Subgroups — C₂₃×#20366 #19167992

Order 5888.

1078 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	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1746	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1744	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1729	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
18	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
19	C₂₃×#376	C₂²	1	2	yes	—
20	C₄³×C₂₃	C₂²	1	2	yes	—
21	C₂₃×#336	C₂²	1	16	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	—	2	8	no	—
25	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₂₃	—	2	8	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
30	C₂₃×#117	—	2	8	no	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#97	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
43	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
44	C₂₃×#95	C₂×C₄	1	8	yes	—
45	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	#41	1	8	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
56	M₁₆×C₂₃	—	2	16	no	—
57	C₂×C₈×C₂₃	—	2	16	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₂×C₈×C₂₃	—	2	8	no	—
60	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
61	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
62	C₄⋊C₄×C₂₃	—	2	8	no	—
63	C₂²⋊C₄×C₂₃	—	2	8	no	—
64	C₄²×C₂₃	—	2	16	no	—
65	C₄²×C₂₃	—	2	8	no	—
66	C₄²×C₂₃	C₄²	1	8	yes	—
67	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
71	#20366	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	2	4	no	—
73	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
74	C₂³×C₂₃	#117	1	4	yes	—
75	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	16	no	—
78	C₂×C₄×C₂₃	—	2	8	no	—
79	C₂×C₄×C₂₃	—	2	16	no	—
80	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₂₃	#117	1	8	yes	—
84	C₂×C₄×C₂₃	#117	1	8	yes	—
85	C₂×C₄×C₂₃	#117	1	4	yes	—
86	C₂×C₄×C₂₃	#131	1	8	yes	—
87	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
90	C₈×C₂₃	—	2	8	no	—
91	C₈×C₂₃	—	2	8	no	—
92	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
93	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
94	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
95	#1746	C₂×C₂₃	1	4	yes	complemented
96	#1744	C₂×C₂₃	1	4	yes	complemented
97	#1729	C₂×C₂₃	1	4	yes	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	—	2	8	no	—
100	C₂²×C₂₃	#376	1	4	yes	—
101	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₂₃	C₂×#117	1	2	yes	—
103	C₂²×C₂₃	C₂×#131	1	4	yes	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	4	8	no	—
106	C₄×C₂₃	#376	1	4	yes	—
107	C₄×C₂₃	#405	1	8	yes	—
108	C₄×C₂₃	C₂×#117	1	2	yes	—
109	C₄×C₂₃	C₂×#117	1	2	yes	—
110	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
111	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
112	C₂×#117	C₂²×C₂₃	1	2	yes	—
113	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
114	C₄×M₁₆	C₄×C₂₃	1	8	yes	complemented
115	C₂×#97	C₂²×C₂₃	1	2	yes	—
116	C₂×C₄×C₈	C₄×C₂₃	1	8	yes	complemented
117	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
120	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
121	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
122	#376	C₂²×C₂₃	1	2	yes	—
123	C₄³	C₂²×C₂₃	1	2	yes	—
124	#336	C₂²×C₂₃	1	16	yes	—
125	C₂×C₂₃	—	2	4	no	—
126	C₂×C₂₃	C₂×#376	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#405	1	2	yes	—
128	C₂×C₂₃	#2922	1	4	yes	—
129	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
130	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
131	C₂×M₁₆	—	2	8	no	—
132	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
133	C₂²×C₈	—	2	8	no	—
134	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
135	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
136	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
137	#117	—	2	8	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
140	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
141	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
142	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
145	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
147	#97	C₂×C₄×C₂₃	1	8	yes	—
148	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
149	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
150	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
151	#95	C₂×C₄×C₂₃	1	8	yes	—
152	C₂₃	#20366	1	1	yes	characteristic, direct product factor, Sylow
153	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
156	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
157	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
162	C₂²×C₄	#41×C₂₃	1	8	yes	—
163	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
164	M₁₆	—	2	16	no	—
165	C₂×C₈	—	2	16	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
169	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	2	8	no	—
174	C₄²	C₄²×C₂₃	1	8	yes	—
175	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
176	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
177	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
178	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
181	C₂³	C₂₃×#117	1	4	yes	—
182	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
188	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
190	C₂×C₄	C₂₃×#117	1	4	yes	—
191	C₂×C₄	C₂₃×#117	1	8	yes	—
192	C₂×C₄	C₂₃×#117	1	8	yes	—
193	C₂×C₄	C₂₃×#131	1	8	yes	—
194	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
197	C₈	—	2	8	no	—
198	C₈	—	2	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	4	no	—
201	C₂²	C₂₃×#376	1	4	yes	—
202	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
203	C₂²	C₂×C₂₃×#117	1	2	yes	—
204	C₂²	C₂×C₂₃×#131	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	4	8	no	—
207	C₄	C₂₃×#376	1	4	yes	—
208	C₄	C₂₃×#405	1	8	yes	—
209	C₄	C₂×C₂₃×#117	1	2	yes	—
210	C₄	C₂×C₂₃×#117	1	2	yes	—
211	C₂	—	2	4	no	—
212	C₂	C₂×C₂₃×#376	1	1	yes	characteristic
213	C₂	C₂×C₂₃×#405	1	2	yes	—
214	C₂	C₂₃×#2922	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₂₃	C₂³
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1746	C₂	semidirect
5. C₂₃×#1744	C₂	semidirect
11. C₄×M₁₆×C₂₃	C₄	semidirect
13. C₂×C₄×C₈×C₂₃	C₄	semidirect
71. #20366	C₂₃	direct (N × H)
95. #1746	C₂×C₂₃	semidirect
96. #1744	C₂×C₂₃	semidirect
114. C₄×M₁₆	C₄×C₂₃	semidirect
116. C₂×C₄×C₈	C₄×C₂₃	semidirect
152. C₂₃	#20366	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×C₄×M₁₆×C₂₃	2
C₂²×C₄×C₈×C₂₃	2
C₄×C₂₃×#117	2
C₂₃×#1746	2
C₂₃×#1744	2
C₂₃×#1729	2
#20366	23

Subgroups — C23×#20366 #19167992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20366 #19167992