Subgroups — C₂₃×#8132 #19155758

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1273	C₂	1	4	yes	maximal
5	C₂×C₂₃×#97	C₄	1	2	yes	—
6	C₂×C₂₃×#97	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₄³×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#336	C₄	1	4	yes	—
13	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
14	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×#97	—	2	4	no	—
22	C₂₃×#97	C₈	1	8	yes	complemented
23	C₄×C₈×C₂₃	—	2	8	no	—
24	C₄×C₈×C₂₃	—	2	4	no	—
25	C₄×C₈×C₂₃	—	2	4	no	—
26	C₄×C₈×C₂₃	—	2	4	no	—
27	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	—	2	4	no	—
34	C₂×C₈×C₂₃	—	2	4	no	—
35	C₂×C₈×C₂₃	—	2	4	no	—
36	C₂×C₈×C₂₃	—	2	4	no	—
37	C₂×C₈×C₂₃	—	2	8	no	—
38	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
39	C₂×C₈×C₂₃	M₁₆	1	8	yes	—
40	C₄²×C₂₃	—	2	4	no	—
41	C₄²×C₂₃	—	2	4	no	—
42	C₄²×C₂₃	C₂×C₈	1	4	yes	—
43	C₄²×C₂₃	M₁₆	1	4	yes	—
44	C₄²×C₂₃	#41	1	4	yes	—
45	#8132	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
52	C₂×C₄×C₂₃	#97	1	4	yes	—
53	C₂×C₄×C₂₃	#117	1	2	yes	—
54	C₂×C₄×C₂₃	#117	1	2	yes	—
55	C₂×C₄×C₂₃	#117	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₂₃	#131	1	4	yes	—
60	C₈×C₂₃	—	2	8	no	—
61	C₈×C₂₃	—	2	8	no	—
62	C₈×C₂₃	—	4	4	no	—
63	#1744	C₂×C₂₃	1	1	yes	characteristic
64	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
65	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
66	#1273	C₂×C₂₃	1	4	yes	—
67	C₂²×C₂₃	#376	1	1	yes	characteristic
68	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
70	C₂²×C₂₃	C₄×M₁₆	1	2	yes	—
71	C₂²×C₂₃	#405	1	2	yes	—
72	C₄×C₂₃	—	2	4	no	—
73	C₄×C₂₃	—	2	4	no	—
74	C₄×C₂₃	#431	1	4	yes	—
75	C₄×C₂₃	#432	1	4	yes	—
76	C₄×C₂₃	#433	1	4	yes	—
77	C₂×#97	C₄×C₂₃	1	2	yes	—
78	C₂×#97	C₄×C₂₃	1	2	yes	—
79	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	—
80	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
83	C₄³	C₂²×C₂₃	1	1	yes	characteristic
84	#336	C₄×C₂₃	1	4	yes	—
85	C₂×C₂₃	C₈×M₁₆	1	2	yes	—
86	C₂×C₂₃	#1451	1	2	yes	—
87	C₂×C₂₃	#1744	1	1	yes	characteristic
88	C₂×C₂₃	#1745	1	1	yes	characteristic
89	C₂×C₂₃	#1746	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
91	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
92	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
93	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
94	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
95	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
98	#97	—	2	4	no	—
99	#97	C₈×C₂₃	1	8	yes	complemented
100	C₄×C₈	—	2	8	no	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	2	4	no	—
104	C₂₃	#8132	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
106	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
107	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
117	C₂×C₈	M₁₆×C₂₃	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×C₈×C₂₃	1	4	yes	—
121	C₄²	M₁₆×C₂₃	1	4	yes	—
122	C₄²	#41×C₂₃	1	4	yes	—
123	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
129	C₂×C₄	C₂₃×#97	1	4	yes	—
130	C₂×C₄	C₂₃×#117	1	2	yes	—
131	C₂×C₄	C₂₃×#117	1	2	yes	—
132	C₂×C₄	C₂₃×#117	1	2	yes	—
133	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
134	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
135	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
136	C₂×C₄	C₂₃×#131	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	2	8	no	—
139	C₈	—	4	4	no	—
140	C₂²	C₂₃×#376	1	1	yes	characteristic
141	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
142	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
143	C₂²	C₄×M₁₆×C₂₃	1	2	yes	—
144	C₂²	C₂₃×#405	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	C₂₃×#431	1	4	yes	—
148	C₄	C₂₃×#432	1	4	yes	—
149	C₄	C₂₃×#433	1	4	yes	—
150	C₂	C₈×M₁₆×C₂₃	1	2	yes	—
151	C₂	C₂₃×#1451	1	2	yes	—
152	C₂	C₂₃×#1744	1	1	yes	characteristic
153	C₂	C₂₃×#1745	1	1	yes	characteristic
154	C₂	C₂₃×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄²
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
22. C₂₃×#97	C₈	semidirect
45. #8132	C₂₃	direct (N × H)
99. #97	C₈×C₂₃	semidirect
104. C₂₃	#8132	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#1744	2
C₄×C₂₃×#97	2
C₄²×C₈×C₂₃	2
C₂₃×#1273	2
#8132	23

Subgroups — C23×#8132 #19155758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8132 #19155758