Subgroups — C₂₃×#13628 #19161254

Order 5888.

206 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#405	C₄	1	1	yes	characteristic
8	C₂₃×M₆₄	—	2	2	no	—
9	C₂×C₂₃×C₃₂	—	2	2	no	—
10	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
11	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
12	C₂₃×#346	C₄	1	2	yes	—
13	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂₃×#117	C₈	1	1	yes	characteristic
17	C₂₃×M₃₂	—	2	2	no	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
21	C₂×C₁₆×C₂₃	Q₈	1	2	yes	—
22	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×#97	C₈	1	2	yes	—
24	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂₃×C₃₂	—	2	2	no	—
26	C₂₃×C₃₂	—	2	2	no	—
27	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₂₃	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₂₃	C₁₆	1	2	yes	—
36	C₄²×C₂₃	C₂×C₈	1	2	yes	—
37	C₁₆×C₂₃	—	2	2	no	—
38	C₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	#13628	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×C₂₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×C₁₆	1	4	yes	—
45	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
47	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
48	C₈×C₂₃	—	2	2	no	—
49	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
50	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
52	C₂×M₆₄	C₂×C₂₃	1	1	yes	characteristic
53	C₂²×C₃₂	C₂×C₂₃	1	1	yes	characteristic
54	#2108	C₂×C₂₃	1	1	yes	characteristic
55	C₄⋊C₃₂	C₂×C₂₃	1	4	yes	complemented
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₄⋊C₁₆	1	2	yes	—
58	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₂₃	—	2	4	no	—
60	C₄×C₂₃	C₄⋊C₁₆	1	2	yes	—
61	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
63	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
64	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
65	#405	C₄×C₂₃	1	1	yes	characteristic
66	M₆₄	—	2	2	no	—
67	C₂×C₃₂	—	2	2	no	—
68	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
69	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
70	#346	C₄×C₂₃	1	2	yes	—
71	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
72	C₂×C₂₃	—	2	2	no	—
73	C₂×C₂₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
74	C₂×C₂₃	#2258	1	2	yes	—
75	C₂×M₁₆	C₈×C₂₃	1	1	yes	characteristic
76	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
77	#117	C₈×C₂₃	1	1	yes	characteristic
78	M₃₂	—	2	2	no	—
79	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
80	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
81	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
82	C₂×C₁₆	Q₈×C₂₃	1	2	yes	—
83	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
84	#97	C₈×C₂₃	1	2	yes	—
85	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
86	C₃₂	—	2	2	no	—
87	C₃₂	—	2	2	no	—
88	C₂₃	#13628	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
92	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
93	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
94	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
95	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
96	C₄⋊C₄	C₁₆×C₂₃	1	2	yes	—
97	C₂²⋊C₄	C₁₆×C₂₃	1	2	yes	—
98	C₄²	C₂×C₈×C₂₃	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
101	C₁₆	C₂×D₄×C₂₃	1	1	yes	characteristic
102	C₁₆	C₂×Q₈×C₂₃	1	1	yes	characteristic
103	C₂³	C₂×C₁₆×C₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
105	C₂×C₄	C₂×C₁₆×C₂₃	1	4	yes	—
106	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
107	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
108	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
111	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
112	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₂₃×C₄⋊C₁₆	1	2	yes	—
115	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₂₃×C₄⋊C₁₆	1	2	yes	—
118	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
119	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₂₃×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂	C₂₃×#2258	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×C₄⋊C₃₂	C₂	semidirect
41. #13628	C₂₃	direct (N × H)
55. C₄⋊C₃₂	C₂×C₂₃	semidirect
88. C₂₃	#13628	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×M₆₄	2
C₂²×C₂₃×C₃₂	2
C₂₃×#2108	2
C₂₃×C₄⋊C₃₂	2
#13628	23

Subgroups — C23×#13628 #19161254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13628 #19161254