Subgroups — C₂₃×#33486 #19181112

Order 5888.

662 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	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	D₄×C₁₆×C₂₃	C₂	1	8	yes	maximal, complemented
3	C₂₃×#2162	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	C₂²	1	8	yes	complemented
8	C₈×D₄×C₂₃	C₄	1	4	yes	—
9	C₈×D₄×C₂₃	C₂²	1	4	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
14	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
15	C₂₃×C₂²⋊C₁₆	C₂²	1	8	yes	complemented
16	C₄×C₁₆×C₂₃	C₂²	1	4	yes	complemented
17	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₂₃	—	4	4	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
25	C₄×D₄×C₂₃	C₈	1	4	yes	—
26	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₂₃	—	4	8	no	—
31	C₂×C₁₆×C₂₃	C₂³	1	8	yes	—
32	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
33	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂⁴×C₂₃	C₂×C₈	1	2	yes	—
37	C₂×D₄×C₂₃	C₁₆	1	8	yes	complemented
38	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	4	4	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
58	C₄²×C₂₃	C₂×C₈	1	2	yes	—
59	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
60	C₁₆×C₂₃	—	2	8	no	—
61	#33486	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	—	4	2	no	—
63	C₂³×C₂₃	—	4	2	no	—
64	C₂³×C₂₃	C₂×C₁₆	1	8	yes	complemented
65	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
66	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
67	D₄×C₂₃	C₂×C₁₆	1	16	yes	complemented
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	1	no	—
75	C₂×C₄×C₂₃	C₂×C₁₆	1	4	yes	complemented
76	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	4	no	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
85	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
86	D₄×C₁₆	C₂×C₂₃	1	8	yes	complemented
87	#2162	C₂×C₂₃	1	2	yes	complemented
88	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	4	yes	complemented
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	—	4	8	no	—
94	C₂²×C₂₃	C₂²×C₁₆	1	2	yes	—
95	C₂²×C₂₃	C₂²×C₁₆	1	8	yes	—
96	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	4	no	—
100	C₄×C₂₃	C₂²×C₁₆	1	4	yes	—
101	C₄×C₂₃	#518	1	1	yes	characteristic
102	C₄×C₂₃	#568	1	1	yes	characteristic
103	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
104	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₂₃	1	8	yes	complemented
106	C₈×D₄	C₄×C₂₃	1	4	yes	—
107	C₈×D₄	C₂²×C₂₃	1	4	yes	—
108	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
109	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
111	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	complemented
112	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	—
113	C₂²⋊C₁₆	C₂²×C₂₃	1	8	yes	complemented
114	C₄×C₁₆	C₂²×C₂₃	1	4	yes	complemented
115	C₂×C₂₃	—	2	2	no	—
116	C₂×C₂₃	—	2	8	no	—
117	C₂×C₂₃	#2972	1	1	yes	characteristic
118	C₂×C₂₃	C₂³×C₁₆	1	1	yes	characteristic
119	C₂×C₂₃	#3407	1	1	yes	characteristic
120	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
121	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
122	C₂²×C₈	—	4	4	no	—
123	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
124	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
125	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
128	C₄×D₄	C₈×C₂₃	1	4	yes	—
129	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₈×C₂₃	1	2	yes	—
132	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₁₆	—	4	8	no	—
134	C₂×C₁₆	C₂³×C₂₃	1	8	yes	—
135	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
136	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
137	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
138	C₄×C₈	C₂³×C₂₃	1	2	yes	—
139	C₂₃	#33486	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁴	C₂×C₈×C₂₃	1	2	yes	—
141	C₂×D₄	C₁₆×C₂₃	1	8	yes	complemented
142	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
148	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
158	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
159	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
161	C₂²⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
162	C₄²	C₂×C₈×C₂₃	1	2	yes	—
163	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
164	C₁₆	—	2	8	no	—
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂×C₁₆×C₂₃	1	8	yes	complemented
168	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
169	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
170	D₄	C₂×C₁₆×C₂₃	1	16	yes	complemented
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₁₆×C₂₃	1	4	yes	complemented
179	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
181	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
182	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
183	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	4	no	—
187	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	C₂²×C₁₆×C₂₃	1	2	yes	—
194	C₂²	C₂²×C₁₆×C₂₃	1	8	yes	—
195	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	C₂²×C₁₆×C₂₃	1	4	yes	—
200	C₄	C₂₃×#518	1	1	yes	characteristic
201	C₄	C₂₃×#568	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	C₂₃×#2972	1	1	yes	characteristic
205	C₂	C₂³×C₁₆×C₂₃	1	1	yes	characteristic
206	C₂	C₂₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂	C₂³×C₁₆×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂²×D₄×C₂₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×C₁₆×C₂₃	C₂	semidirect
3. C₂₃×#2162	C₂	semidirect
4. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
7. C₂²×C₁₆×C₂₃	C₂²	semidirect
13. C₂₃×C₄⋊C₁₆	C₂²	semidirect
15. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
16. C₄×C₁₆×C₂₃	C₂²	semidirect
37. C₂×D₄×C₂₃	C₁₆	semidirect
61. #33486	C₂₃	direct (N × H)
64. C₂³×C₂₃	C₂×C₁₆	semidirect
67. D₄×C₂₃	C₂×C₁₆	semidirect
75. C₂×C₄×C₂₃	C₂×C₁₆	semidirect
86. D₄×C₁₆	C₂×C₂₃	semidirect
87. #2162	C₂×C₂₃	semidirect
88. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
105. C₂²×C₁₆	C₂²×C₂₃	semidirect
111. C₄⋊C₁₆	C₂²×C₂₃	semidirect
113. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
114. C₄×C₁₆	C₂²×C₂₃	semidirect
139. C₂₃	#33486	direct (N × H)
141. C₂×D₄	C₁₆×C₂₃	semidirect
167. C₂³	C₂×C₁₆×C₂₃	semidirect
170. D₄	C₂×C₁₆×C₂₃	semidirect
178. C₂×C₄	C₂×C₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₂₃	2
D₄×C₁₆×C₂₃	2
C₂₃×#2162	2
C₂×C₂₃×C₂²⋊C₁₆	2
#33486	23

Subgroups — C23×#33486 #19181112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33486 #19181112