Subgroups — C₂₃×#13757 #19161383

Order 5888.

1598 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#376	C₂	1	2	yes	maximal
2	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1434	C₂	1	8	yes	maximal
4	C₄×C₂₃×#95	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#376	C₄	1	16	yes	complemented
10	C₂×C₂₃×#95	C₂²	1	16	yes	—
11	C₂×C₂₃×#95	C₂²	1	8	yes	—
12	C₄³×C₂₃	C₄	1	8	yes	complemented
13	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
14	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	—	2	16	no	—
18	C₂×C₄²×C₂₃	—	2	8	no	—
19	C₂×C₄²×C₂₃	—	2	16	no	—
20	C₂×C₄²×C₂₃	C₂×C₄	1	16	yes	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	16	yes	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
25	C₂₃×#95	—	2	32	no	—
26	C₂₃×#95	C₂×C₄	1	32	yes	—
27	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	—	2	8	no	—
29	C₂²×C₄×C₂₃	—	2	8	no	—
30	C₂²×C₄×C₂₃	—	2	8	no	—
31	C₂²×C₄×C₂₃	—	2	16	no	—
32	C₂²×C₄×C₂₃	—	2	32	no	—
33	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	#41	1	16	yes	—
40	C₂²×C₄×C₂₃	#41	1	8	yes	—
41	C₄²×C₂₃	—	2	32	no	—
42	C₄²×C₂₃	—	2	16	no	—
43	C₄²×C₂₃	—	2	32	no	—
44	C₄²×C₂₃	C₄²	1	16	yes	—
45	#13757	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
48	C₂³×C₂₃	C₂×#41	1	4	yes	—
49	C₂³×C₂₃	C₂×#41	1	8	yes	—
50	C₂×C₄×C₂₃	—	2	8	no	—
51	C₂×C₄×C₂₃	—	2	8	no	—
52	C₂×C₄×C₂₃	—	2	8	no	—
53	C₂×C₄×C₂₃	—	2	16	no	—
54	C₂×C₄×C₂₃	—	2	32	no	—
55	C₂×C₄×C₂₃	—	2	16	no	—
56	C₂×C₄×C₂₃	—	2	32	no	—
57	C₂×C₄×C₂₃	—	4	32	no	—
58	C₂×C₄×C₂₃	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
60	C₂×C₄×C₂₃	#117	1	16	yes	—
61	C₂×C₄×C₂₃	#117	1	8	yes	—
62	C₂×C₄×C₂₃	#117	1	16	yes	—
63	C₂×#376	C₂×C₂₃	1	2	yes	—
64	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
65	#1434	C₂×C₂₃	1	8	yes	—
66	C₄×#95	C₂×C₂₃	1	4	yes	—
67	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
68	C₂²×C₂₃	C₂×#117	1	2	yes	—
69	C₂²×C₂₃	C₂×#117	1	4	yes	—
70	C₂²×C₂₃	C₂×#117	1	4	yes	—
71	C₂²×C₂₃	C₄×#41	1	8	yes	—
72	C₂²×C₂₃	#529	1	16	yes	—
73	C₄×C₂₃	—	2	8	no	—
74	C₄×C₂₃	—	2	16	no	—
75	C₄×C₂₃	—	4	16	no	—
76	C₄×C₂₃	#376	1	8	yes	—
77	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
78	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
79	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
80	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
81	#376	C₄×C₂₃	1	16	yes	complemented
82	C₂×#95	C₂²×C₂₃	1	16	yes	—
83	C₂×#95	C₂²×C₂₃	1	8	yes	—
84	C₄³	C₄×C₂₃	1	8	yes	complemented
85	C₂×C₂₃	C₂×#376	1	1	yes	characteristic
86	C₂×C₂₃	C₄×#117	1	2	yes	—
87	C₂×C₂₃	#2283	1	4	yes	—
88	C₂×C₂₃	#2298	1	8	yes	—
89	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
90	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
91	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
92	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
93	C₂×C₄²	—	2	16	no	—
94	C₂×C₄²	—	2	8	no	—
95	C₂×C₄²	—	2	16	no	—
96	C₂×C₄²	C₂×C₄×C₂₃	1	16	yes	—
97	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
98	C₂×C₄²	C₂×C₄×C₂₃	1	16	yes	—
99	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
100	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
101	#95	—	2	32	no	—
102	#95	C₂×C₄×C₂₃	1	32	yes	—
103	C₂₃	#13757	1	1	yes	characteristic, direct product factor, Sylow
104	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	8	no	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	2	8	no	—
108	C₂²×C₄	—	2	32	no	—
109	C₂²×C₄	—	2	16	no	—
110	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
111	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
112	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
113	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
114	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
115	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
116	C₂²×C₄	#41×C₂₃	1	16	yes	—
117	C₂²×C₄	#41×C₂₃	1	8	yes	—
118	C₄²	—	2	32	no	—
119	C₄²	—	2	16	no	—
120	C₄²	—	2	32	no	—
121	C₄²	C₄²×C₂₃	1	16	yes	—
122	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
123	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
124	C₂³	C₂×#41×C₂₃	1	4	yes	—
125	C₂³	C₂×#41×C₂₃	1	8	yes	—
126	C₂×C₄	—	2	8	no	—
127	C₂×C₄	—	2	8	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	2	32	no	—
131	C₂×C₄	—	2	16	no	—
132	C₂×C₄	—	2	32	no	—
133	C₂×C₄	—	4	32	no	—
134	C₂×C₄	C₂×C₄²×C₂₃	1	8	yes	—
135	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
136	C₂×C₄	C₂₃×#117	1	16	yes	—
137	C₂×C₄	C₂₃×#117	1	8	yes	—
138	C₂×C₄	C₂₃×#117	1	16	yes	—
139	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
140	C₂²	C₂×C₂₃×#117	1	2	yes	—
141	C₂²	C₂×C₂₃×#117	1	4	yes	—
142	C₂²	C₂×C₂₃×#117	1	4	yes	—
143	C₂²	C₄×#41×C₂₃	1	8	yes	—
144	C₂²	C₂₃×#529	1	16	yes	—
145	C₄	—	2	8	no	—
146	C₄	—	2	16	no	—
147	C₄	—	4	16	no	—
148	C₄	C₂₃×#376	1	8	yes	—
149	C₂	C₂×C₂₃×#376	1	1	yes	characteristic
150	C₂	C₄×C₂₃×#117	1	2	yes	—
151	C₂	C₂₃×#2283	1	4	yes	—
152	C₂	C₂₃×#2298	1	8	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂₃×#376	C₄	semidirect
12. C₄³×C₂₃	C₄	semidirect
45. #13757	C₂₃	direct (N × H)
81. #376	C₄×C₂₃	semidirect
84. C₄³	C₄×C₂₃	semidirect
103. C₂₃	#13757	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×#376	2
C₂×C₄³×C₂₃	2
C₂₃×#1434	2
C₄×C₂₃×#95	2
#13757	23

Subgroups — C23×#13757 #19161383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13757 #19161383