Subgroups — C₂₃×#20467 #19168093

Order 5888.

930 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#1768	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1750	C₂	1	4	yes	maximal
5	C₂₃×#1745	C₂	1	2	yes	maximal, complemented
6	C₄×C₂₃×#97	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#432	C₄	1	16	yes	complemented
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₂₃×#97	C₂²	1	4	yes	—
13	C₂×C₂₃×#97	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
16	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
17	C₂₃×#376	C₂²	1	2	yes	—
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₄³×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#336	C₂²	1	8	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
24	C₂₃×#117	—	2	8	no	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
36	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
37	C₂₃×#97	—	2	8	no	—
38	C₂₃×#97	C₂×C₄	1	16	yes	—
39	C₄×C₈×C₂₃	—	2	8	no	—
40	C₂₃×#95	C₂×C₄	1	8	yes	—
41	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	2	8	no	—
54	C₂×C₈×C₂₃	C₂²×C₄	1	16	yes	—
55	C₄⋊C₄×C₂₃	—	2	8	no	—
56	C₂²⋊C₄×C₂₃	—	2	8	no	—
57	C₄²×C₂₃	—	2	4	no	—
58	C₄²×C₂₃	—	4	4	no	—
59	C₄²×C₂₃	—	4	4	no	—
60	C₄²×C₂₃	C₄²	1	8	yes	—
61	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
62	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
63	C₄²×C₂₃	#41	1	8	yes	—
64	#20467	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×C₄²	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	8	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	8	no	—
76	C₂×C₄×C₂₃	—	4	8	no	—
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	#117	1	8	yes	—
81	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
83	C₈×C₂₃	—	2	16	no	—
84	C₂×#432	C₂×C₂₃	1	2	yes	—
85	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
86	#1768	C₂×C₂₃	1	4	yes	complemented
87	#1750	C₂×C₂₃	1	4	yes	—
88	#1745	C₂×C₂₃	1	2	yes	complemented
89	C₄×#97	C₂×C₂₃	1	2	yes	complemented
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×#117	1	2	yes	—
95	C₂²×C₂₃	#568	1	4	yes	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	4	no	—
98	C₄×C₂₃	—	2	8	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	4	4	no	—
101	C₄×C₂₃	C₄×#41	1	4	yes	—
102	C₂×#117	C₂²×C₂₃	1	2	yes	—
103	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
104	#432	C₄×C₂₃	1	16	yes	complemented
105	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
106	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
107	C₂×#97	C₂²×C₂₃	1	4	yes	—
108	C₂×#97	C₂²×C₂₃	1	4	yes	—
109	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
110	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
111	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
112	#376	C₂²×C₂₃	1	2	yes	—
113	C₄³	C₂²×C₂₃	1	1	yes	characteristic
114	C₄³	C₂²×C₂₃	1	1	yes	characteristic
115	#336	C₂²×C₂₃	1	8	yes	—
116	C₂×C₂₃	—	4	2	no	—
117	C₂×C₂₃	C₄×#117	1	1	yes	characteristic
118	C₂×C₂₃	#2873	1	2	yes	—
119	C₂×C₂₃	#2923	1	4	yes	—
120	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
121	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
123	#117	—	2	8	no	—
124	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
125	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
126	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
130	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
131	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
134	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
135	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
136	#97	—	2	8	no	—
137	#97	C₂×C₄×C₂₃	1	16	yes	—
138	C₄×C₈	—	2	8	no	—
139	#95	C₂×C₄×C₂₃	1	8	yes	—
140	C₂₃	#20467	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
145	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
151	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	C₂²×C₄×C₂₃	1	16	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₄²×C₂₃	1	8	yes	—
161	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
162	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
163	C₄²	#41×C₂₃	1	8	yes	—
164	C₂³	—	2	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
167	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
177	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
178	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
179	C₂×C₄	C₂₃×#117	1	8	yes	—
180	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
181	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
182	C₈	—	2	16	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
187	C₂²	C₂×C₂₃×#117	1	2	yes	—
188	C₂²	C₂₃×#568	1	4	yes	—
189	C₄	—	2	8	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	4	4	no	—
194	C₄	C₄×#41×C₂₃	1	4	yes	—
195	C₂	—	4	2	no	—
196	C₂	C₄×C₂₃×#117	1	1	yes	characteristic
197	C₂	C₂₃×#2873	1	2	yes	—
198	C₂	C₂₃×#2923	1	4	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×#1768	C₂	semidirect
5. C₂₃×#1745	C₂	semidirect
6. C₄×C₂₃×#97	C₂	semidirect
9. C₂₃×#432	C₄	semidirect
64. #20467	C₂₃	direct (N × H)
86. #1768	C₂×C₂₃	semidirect
88. #1745	C₂×C₂₃	semidirect
89. C₄×#97	C₂×C₂₃	semidirect
104. #432	C₄×C₂₃	semidirect
140. C₂₃	#20467	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×#1768	2
C₂₃×#1750	2
C₂₃×#1745	2
C₄×C₂₃×#97	2
#20467	23

Subgroups — C23×#20467 #19168093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20467 #19168093