Subgroups — C₂₃×#20811 #19168437

Order 5888.

834 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	2	yes	maximal
2	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1843	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1832	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1798	C₂	1	4	yes	maximal
6	C₂₃×#1767	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#432	—	2	8	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
12	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₂₃×#97	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
16	C₂₃×#376	C₂²	1	1	yes	characteristic
17	C₂₃×#376	C₂²	1	1	yes	characteristic
18	C₄³×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#336	C₂²	1	8	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
23	C₂₃×#117	—	2	8	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	4	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊C₈	—	2	8	no	—
35	C₂₃×C₄⋊C₈	—	2	16	no	—
36	C₂₃×C₂²⋊C₈	—	2	8	no	—
37	C₂₃×#97	—	2	8	no	—
38	C₂₃×#95	C₂×C₄	1	4	yes	—
39	C₂₃×#95	C₂×C₄	1	4	yes	—
40	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	4	4	no	—
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	2	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	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	2	16	no	—
54	C₄⋊C₄×C₂₃	—	2	8	no	—
55	C₂²⋊C₄×C₂₃	—	2	8	no	—
56	C₄²×C₂₃	—	2	4	no	—
57	C₄²×C₂₃	—	4	8	no	—
58	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
59	C₄²×C₂₃	C₂×D₄	1	2	yes	—
60	C₄²×C₂₃	C₂×D₄	1	2	yes	—
61	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
62	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
63	C₄²×C₂₃	#41	1	8	yes	—
64	#20811	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	8	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	8	no	—
77	C₂×C₄×C₂₃	—	4	8	no	—
78	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₂₃	#117	1	8	yes	—
82	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
85	C₈×C₂₃	—	4	8	no	—
86	C₂×#432	C₂×C₂₃	1	2	yes	—
87	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
88	#1843	C₂×C₂₃	1	2	yes	complemented
89	#1832	C₂×C₂₃	1	2	yes	complemented
90	#1798	C₂×C₂₃	1	4	yes	—
91	#1767	C₂×C₂₃	1	4	yes	complemented
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	4	4	no	—
95	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₂₃	C₂×#117	1	2	yes	—
97	C₂²×C₂₃	#568	1	4	yes	—
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	—	2	4	no	—
100	C₄×C₂₃	—	2	8	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	4	4	no	—
103	C₄×C₂₃	#532	1	2	yes	—
104	C₄×C₂₃	#533	1	2	yes	—
105	C₂×#117	C₂²×C₂₃	1	2	yes	—
106	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
107	#432	—	2	8	no	—
108	C₂×C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
109	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
111	C₂×#97	C₂²×C₂₃	1	4	yes	—
112	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
113	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
114	#376	C₂²×C₂₃	1	1	yes	characteristic
115	#376	C₂²×C₂₃	1	1	yes	characteristic
116	C₄³	C₂²×C₂₃	1	2	yes	—
117	#336	C₂²×C₂₃	1	8	yes	—
118	C₂×C₂₃	—	4	2	no	—
119	C₂×C₂₃	#2285	1	1	yes	characteristic
120	C₂×C₂₃	#2905	1	2	yes	—
121	C₂×C₂₃	#2923	1	4	yes	—
122	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
123	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
125	#117	—	2	8	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
129	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
130	C₂×C₄²	D₄×C₂₃	1	4	yes	—
131	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
136	C₄⋊C₈	—	2	8	no	—
137	C₄⋊C₈	—	2	16	no	—
138	C₂²⋊C₈	—	2	8	no	—
139	#97	—	2	8	no	—
140	#95	C₂×C₄×C₂₃	1	4	yes	—
141	#95	C₂×C₄×C₂₃	1	4	yes	—
142	C₂₃	#20811	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
147	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	16	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	4	8	no	—
161	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
162	C₄²	C₂×D₄×C₂₃	1	2	yes	—
163	C₄²	C₂×D₄×C₂₃	1	2	yes	—
164	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
165	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
166	C₄²	#41×C₂₃	1	8	yes	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
170	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
183	C₂×C₄	C₂₃×#117	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
187	C₈	—	4	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂²	C₂×C₂₃×#117	1	2	yes	—
193	C₂²	C₂₃×#568	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	C₂₃×#532	1	2	yes	—
200	C₄	C₂₃×#533	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	C₂₃×#2285	1	1	yes	characteristic
203	C₂	C₂₃×#2905	1	2	yes	—
204	C₂	C₂₃×#2923	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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×#1843	C₂	semidirect
4. C₂₃×#1832	C₂	semidirect
6. C₂₃×#1767	C₂	semidirect
64. #20811	C₂₃	direct (N × H)
88. #1843	C₂×C₂₃	semidirect
89. #1832	C₂×C₂₃	semidirect
91. #1767	C₂×C₂₃	semidirect
142. C₂₃	#20811	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×C₂₃×#432	2
C₄×C₂₃×#117	2
C₂₃×#1843	2
C₂₃×#1832	2
C₂₃×#1798	2
C₂₃×#1767	2
#20811	23

Subgroups — C23×#20811 #19168437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20811 #19168437