Subgroups — C₂₃×#12002 #19159628

Order 5888.

220 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1369	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×M₃₂	—	2	2	no	—
5	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#364	—	2	2	no	—
9	C₂₃×#364	C₂²	1	4	yes	—
10	C₂₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
15	C₂₃×M₃₂	—	2	2	no	—
16	C₂₃×M₃₂	—	2	4	no	—
17	C₂₃×M₃₂	—	4	2	no	—
18	C₂₃×M₃₂	C₈	1	4	yes	—
19	C₂×C₁₆×C₂₃	—	4	2	no	—
20	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
32	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
33	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
34	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₂₃	—	2	1	no	—
37	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₂₃	—	2	2	no	—
40	C₁₆×C₂₃	—	4	4	no	—
41	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
42	#12002	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₂₃	#108	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	1	no	—
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₂₃	#106	1	2	yes	—
49	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
52	C₈×C₂₃	#106	1	1	yes	characteristic
53	C₈×C₂₃	#106	1	1	yes	characteristic
54	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
57	C₂×#364	C₂×C₂₃	1	2	yes	—
58	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
59	#1369	C₂×C₂₃	1	4	yes	complemented
60	C₂²×C₂₃	—	4	1	no	—
61	C₂²×C₂₃	#334	1	2	yes	—
62	C₂²×C₂₃	C₂×#108	1	1	yes	characteristic
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	—	4	1	no	—
65	C₄×C₂₃	#334	1	2	yes	—
66	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
68	C₂×M₃₂	—	2	2	no	—
69	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
70	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
72	#364	—	2	2	no	—
73	#364	C₂²×C₂₃	1	4	yes	—
74	#346	C₂²×C₂₃	1	2	yes	—
75	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	2	2	no	—
78	C₂×C₂₃	C₂×#334	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
80	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
81	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
82	M₃₂	—	2	2	no	—
83	M₃₂	—	2	4	no	—
84	M₃₂	—	4	2	no	—
85	M₃₂	C₈×C₂₃	1	4	yes	—
86	C₂×C₁₆	—	4	2	no	—
87	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
89	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
90	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
91	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
92	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
93	C₂₃	#12002	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
99	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
100	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
101	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
106	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	4	4	no	—
109	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
110	C₂³	C₂₃×#108	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
115	C₂×C₄	C₂₃×#106	1	2	yes	—
116	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
118	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
119	C₈	C₂₃×#106	1	1	yes	characteristic
120	C₈	C₂₃×#106	1	1	yes	characteristic
121	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
122	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
123	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
124	C₂²	—	4	1	no	—
125	C₂²	C₂₃×#334	1	2	yes	—
126	C₂²	C₂×C₂₃×#108	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	C₂₃×#334	1	2	yes	—
130	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
131	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×C₂₃×#334	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#106
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂×C₂₃	C₂×#334
prime radical	C₂³×C₂₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1369	C₂	semidirect
42. #12002	C₂₃	direct (N × H)
59. #1369	C₂×C₂₃	semidirect
93. C₂₃	#12002	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×#364	2
C₄×C₂₃×M₃₂	2
C₂₃×#1369	2
#12002	23

Subgroups — C23×#12002 #19159628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12002 #19159628