Subgroups — C₂₃×#11410 #19159036

Order 5888.

306 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	C₂₃×#2162	C₂	1	2	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1312	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₂₃×C₂²⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₁₆×C₂₃	C₂²	1	4	yes	—
11	C₈²×C₂₃	C₄	1	1	yes	characteristic
12	C₈²×C₂₃	C₄	1	1	yes	characteristic
13	C₈²×C₂₃	C₂²	1	1	yes	characteristic
14	C₈²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₈	1	2	yes	—
16	C₂²×C₈×C₂₃	C₈	1	2	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₂₃	C₈	1	2	yes	—
21	C₄×C₈×C₂₃	C₈	1	2	yes	—
22	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	—	2	4	no	—
34	C₂×C₈×C₂₃	C₁₆	1	8	yes	complemented
35	C₂×C₈×C₂₃	C₄²	1	2	yes	—
36	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
41	C₄²×C₂₃	C₄²	1	1	yes	characteristic
42	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₂₃	—	2	8	no	—
46	#11410	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×C₁₆	1	4	yes	—
54	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₂₃	—	2	2	no	—
58	C₈×C₂₃	—	2	2	no	—
59	C₈×C₂₃	C₂×C₁₆	1	4	yes	—
60	C₈×C₂₃	C₂×C₁₆	1	4	yes	—
61	C₈×C₂₃	M₃₂	1	4	yes	—
62	C₈×C₂₃	#131	1	4	yes	—
63	#2162	C₂×C₂₃	1	2	yes	—
64	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
65	#1312	C₂×C₂₃	1	2	yes	complemented
66	C₈×C₁₆	C₂×C₂₃	1	2	yes	complemented
67	C₂²×C₂₃	—	2	1	no	—
68	C₂²×C₂₃	—	2	1	no	—
69	C₂²×C₂₃	C₄×C₁₆	1	2	yes	—
70	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₂₃	—	2	1	no	—
72	C₄×C₂₃	—	2	1	no	—
73	C₄×C₂₃	C₄×C₁₆	1	2	yes	—
74	C₄×C₂₃	#405	1	2	yes	—
75	C₄×C₂₃	C₂²×C₁₆	1	2	yes	—
76	C₄×C₂₃	C₂×M₃₂	1	2	yes	—
77	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
78	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
80	C₄⋊C₁₆	C₄×C₂₃	1	4	yes	—
81	C₂²⋊C₁₆	C₄×C₂₃	1	4	yes	—
82	C₄×C₁₆	C₂²×C₂₃	1	4	yes	—
83	C₈²	C₄×C₂₃	1	1	yes	characteristic
84	C₈²	C₄×C₂₃	1	1	yes	characteristic
85	C₈²	C₂²×C₂₃	1	1	yes	characteristic
86	C₈²	C₂²×C₂₃	1	1	yes	characteristic
87	C₂×C₂₃	—	2	2	no	—
88	C₂×C₂₃	#1450	1	1	yes	characteristic
89	C₂×C₂₃	C₂×C₄×C₁₆	1	1	yes	characteristic
90	C₂×C₂₃	C₄×M₃₂	1	1	yes	characteristic
91	C₂²×C₈	C₈×C₂₃	1	2	yes	—
92	C₂²×C₈	C₈×C₂₃	1	2	yes	—
93	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
94	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₁₆	C₂×C₄×C₂₃	1	8	yes	—
96	C₄×C₈	C₈×C₂₃	1	2	yes	—
97	C₄×C₈	C₈×C₂₃	1	2	yes	—
98	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
99	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
100	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
101	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
105	C₂₃	#11410	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	C₁₆×C₂₃	1	8	yes	complemented
112	C₂×C₈	C₄²×C₂₃	1	2	yes	—
113	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
114	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
115	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
116	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
117	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
118	C₄²	C₄²×C₂₃	1	1	yes	characteristic
119	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
120	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
121	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
122	C₁₆	—	2	8	no	—
123	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
127	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
128	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₁₆×C₂₃	1	4	yes	—
130	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	C₂×C₁₆×C₂₃	1	4	yes	—
136	C₈	C₂×C₁₆×C₂₃	1	4	yes	—
137	C₈	C₂₃×M₃₂	1	4	yes	—
138	C₈	C₂₃×#131	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×C₁₆×C₂₃	1	2	yes	—
142	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×C₁₆×C₂₃	1	2	yes	—
146	C₄	C₂₃×#405	1	2	yes	—
147	C₄	C₂²×C₁₆×C₂₃	1	2	yes	—
148	C₄	C₂×C₂₃×M₃₂	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	C₂₃×#1450	1	1	yes	characteristic
151	C₂	C₂×C₄×C₁₆×C₂₃	1	1	yes	characteristic
152	C₂	C₄×C₂₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈²×C₂₃	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄×C₈
prime radical	C₂³×C₂₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1312	C₂	semidirect
4. C₈×C₁₆×C₂₃	C₂	semidirect
34. C₂×C₈×C₂₃	C₁₆	semidirect
46. #11410	C₂₃	direct (N × H)
65. #1312	C₂×C₂₃	semidirect
66. C₈×C₁₆	C₂×C₂₃	semidirect
105. C₂₃	#11410	direct (N × H)
111. C₂×C₈	C₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2162	2
C₂×C₈²×C₂₃	2
C₂₃×#1312	2
C₈×C₁₆×C₂₃	2
#11410	23

Subgroups — C23×#11410 #19159036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11410 #19159036