Subgroups — C₂₃×#20536 #19168162

Order 5888.

1026 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1770	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1749	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#431	C₄	1	16	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₃×#97	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
13	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂×C₂₃×#95	C₄	1	2	yes	—
16	C₂₃×#336	C₂²	1	8	yes	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
22	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
24	C₂×C₄²×C₂₃	—	4	4	no	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
31	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
32	C₂₃×#97	—	4	8	no	—
33	C₄×C₈×C₂₃	C₂×C₄	1	16	yes	complemented
34	C₂₃×#95	C₂×C₄	1	8	yes	—
35	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	8	no	—
39	C₂²×C₄×C₂₃	—	4	4	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	—	4	4	no	—
42	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
43	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
44	C₂²×C₄×C₂₃	M₁₆	1	8	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₂₃	—	2	8	no	—
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	C₂²×C₄	1	16	yes	—
54	C₄⋊C₄×C₂₃	—	2	16	no	—
55	C₄²×C₂₃	—	2	8	no	—
56	C₄²×C₂₃	—	4	4	no	—
57	C₄²×C₂₃	C₄²	1	8	yes	—
58	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
59	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
60	#20536	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	—	2	1	no	—
62	C₂³×C₂₃	—	2	1	no	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
65	C₂³×C₂₃	C₂×M₁₆	1	4	yes	—
66	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	—	2	8	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	2	8	no	—
74	C₂×C₄×C₂₃	—	4	4	no	—
75	C₂×C₄×C₂₃	—	4	16	no	—
76	C₂×C₄×C₂₃	#97	1	16	yes	complemented
77	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₂₃	C₂×M₁₆	1	8	yes	—
81	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
82	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
83	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
84	C₈×C₂₃	—	2	16	no	—
85	C₂×#431	C₂×C₂₃	1	2	yes	—
86	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
87	#1770	C₂×C₂₃	1	8	yes	complemented
88	#1749	C₂×C₂₃	1	4	yes	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	4	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	C₂×#97	1	4	yes	—
94	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₂₃	#518	1	1	yes	characteristic
96	C₂²×C₂₃	#519	1	1	yes	characteristic
97	C₂²×C₂₃	C₂²×M₁₆	1	2	yes	—
98	C₂²×C₂₃	#568	1	2	yes	—
99	C₄×C₂₃	—	2	4	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	4	8	no	—
102	C₄×C₂₃	C₂×#97	1	8	yes	—
103	C₄×C₂₃	#520	1	4	yes	—
104	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
105	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
106	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
107	#431	C₄×C₂₃	1	16	yes	complemented
108	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
109	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
110	C₂×#97	C₂²×C₂₃	1	8	yes	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
112	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
113	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
114	C₂×#95	C₄×C₂₃	1	2	yes	—
115	#336	C₂²×C₂₃	1	8	yes	—
116	C₂×C₂₃	—	2	4	no	—
117	C₂×C₂₃	#2274	1	1	yes	characteristic
118	C₂×C₂₃	C₂²×#97	1	1	yes	characteristic
119	C₂×C₂₃	#2873	1	1	yes	characteristic
120	C₂×C₂₃	#2975	1	2	yes	—
121	C₂×C₂₃	#2983	1	2	yes	—
122	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
123	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
127	C₂×C₄⋊C₄	—	2	8	no	—
128	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
129	C₂×C₄²	—	4	4	no	—
130	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
132	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
134	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
135	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	complemented
136	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
137	#97	—	4	8	no	—
138	C₄×C₈	C₂×C₄×C₂₃	1	16	yes	complemented
139	#95	C₂×C₄×C₂₃	1	8	yes	—
140	C₂₃	#20536	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
149	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
150	C₂²×C₄	M₁₆×C₂₃	1	8	yes	—
151	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	C₂²×C₄×C₂₃	1	16	yes	—
160	C₄⋊C₄	—	2	16	no	—
161	C₄²	—	2	8	no	—
162	C₄²	—	4	4	no	—
163	C₄²	C₄²×C₂₃	1	8	yes	—
164	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
165	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	1	no	—
169	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
170	C₂³	C₂×M₁₆×C₂₃	1	4	yes	—
171	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	C₂₃×#97	1	16	yes	complemented
182	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
183	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
184	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
185	C₂×C₄	C₂×M₁₆×C₂₃	1	8	yes	—
186	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
187	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
188	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
189	C₈	—	2	16	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	C₂×C₂₃×#97	1	4	yes	—
195	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
196	C₂²	C₂₃×#518	1	1	yes	characteristic
197	C₂²	C₂₃×#519	1	1	yes	characteristic
198	C₂²	C₂²×M₁₆×C₂₃	1	2	yes	—
199	C₂²	C₂₃×#568	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	4	8	no	—
203	C₄	C₂×C₂₃×#97	1	8	yes	—
204	C₄	C₂₃×#520	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂₃×#2274	1	1	yes	characteristic
207	C₂	C₂²×C₂₃×#97	1	1	yes	characteristic
208	C₂	C₂₃×#2873	1	1	yes	characteristic
209	C₂	C₂₃×#2975	1	2	yes	—
210	C₂	C₂₃×#2983	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
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 ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1770	C₂	semidirect
8. C₂₃×#431	C₄	semidirect
30. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
33. C₄×C₈×C₂₃	C₂×C₄	semidirect
60. #20536	C₂₃	direct (N × H)
76. C₂×C₄×C₂₃	#97	semidirect
87. #1770	C₂×C₂₃	semidirect
107. #431	C₄×C₂₃	semidirect
135. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
138. C₄×C₈	C₂×C₄×C₂₃	semidirect
140. C₂₃	#20536	direct (N × H)
181. C₂×C₄	C₂₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#431	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂₃×#1770	2
C₂₃×#1749	2
#20536	23

Subgroups — C23×#20536 #19168162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20536 #19168162