Subgroups — C₂₃×#13638 #19161264

Order 5888.

182 conjugacy classes of subgroups.

The subgroup lattice has 146 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	C₂×C₂₃×#368	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2158	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1426	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×#108	—	2	1	no	—
7	C₂₃×#427	—	2	1	no	—
8	C₂₃×M₆₄	C₄	1	2	yes	complemented
9	C₂₃×M₆₄	C₂²	1	2	yes	—
10	C₂×C₂₃×C₃₂	C₄	1	2	yes	complemented
11	C₂₃×#368	—	2	1	no	—
12	C₂₃×#368	C₂²	1	2	yes	—
13	C₂₃×#367	C₂²	1	2	yes	—
14	C₂₃×#366	—	2	1	no	—
15	C₂×M₁₆×C₂₃	—	4	1	no	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂₃×#117	—	4	1	no	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₂₃	Q₈	1	1	yes	characteristic
22	C₂×C₁₆×C₂₃	Q₈	1	1	yes	characteristic
23	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#108	—	2	2	no	—
25	C₂₃×#108	—	4	1	no	—
26	C₂₃×#107	—	2	2	no	—
27	C₂₃×#106	—	4	1	no	—
28	C₂₃×C₃₂	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
30	M₁₆×C₂₃	—	4	2	no	—
31	M₁₆×C₂₃	—	8	1	no	—
32	C₂×C₈×C₂₃	—	8	1	no	—
33	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
37	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₂₃	—	4	2	no	—
40	C₂²⋊C₄×C₂₃	—	8	1	no	—
41	C₄²×C₂₃	—	8	1	no	—
42	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	#13638	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	SD₃₂	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	8	2	no	—
49	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
51	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
52	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
55	C₈×C₂₃	—	8	2	no	—
56	C₈×C₂₃	#107	1	1	yes	characteristic
57	C₈×C₂₃	#107	1	1	yes	characteristic
58	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×M₆₄	C₂×C₂₃	1	1	yes	characteristic
61	C₂×#368	C₂×C₂₃	1	1	yes	characteristic
62	#2158	C₂×C₂₃	1	1	yes	characteristic
63	#1426	C₂×C₂₃	1	4	yes	complemented
64	C₂²×C₂₃	—	4	1	no	—
65	C₂²×C₂₃	#367	1	1	yes	characteristic
66	C₂²×C₂₃	#367	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
68	C₄×C₂₃	—	16	2	no	—
69	C₄×C₂₃	#367	1	1	yes	characteristic
70	C₄×C₂₃	#367	1	1	yes	characteristic
71	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
72	C₄×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
73	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
74	C₂×#108	—	2	1	no	—
75	#427	—	2	1	no	—
76	M₆₄	C₄×C₂₃	1	2	yes	complemented
77	M₆₄	C₂²×C₂₃	1	2	yes	—
78	C₂×C₃₂	C₄×C₂₃	1	2	yes	complemented
79	#368	—	2	1	no	—
80	#368	C₂²×C₂₃	1	2	yes	—
81	#367	C₂²×C₂₃	1	2	yes	—
82	#366	—	2	1	no	—
83	C₂×C₂₃	—	2	1	no	—
84	C₂×C₂₃	—	2	1	no	—
85	C₂×C₂₃	—	2	1	no	—
86	C₂×C₂₃	C₂×#367	1	1	yes	characteristic
87	C₂×M₁₆	—	4	1	no	—
88	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
89	#117	—	4	1	no	—
90	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
92	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
93	C₂×C₁₆	Q₈×C₂₃	1	1	yes	characteristic
94	C₂×C₁₆	Q₈×C₂₃	1	1	yes	characteristic
95	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
96	#108	—	2	2	no	—
97	#108	—	4	1	no	—
98	#107	—	2	2	no	—
99	#106	—	4	1	no	—
100	C₃₂	C₂×C₄×C₂₃	1	4	yes	—
101	C₂₃	#13638	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
103	M₁₆	—	4	2	no	—
104	M₁₆	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
109	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
110	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
112	C₄⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	8	1	no	—
114	C₄²	—	8	1	no	—
115	C₁₆	C₄⋊C₄×C₂₃	1	1	yes	characteristic
116	C₁₆	C₄⋊C₄×C₂₃	1	1	yes	characteristic
117	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
118	C₁₆	C₂×Q₈×C₂₃	1	1	yes	characteristic
119	C₂³	C₂₃×SD₃₂	1	1	yes	characteristic
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
122	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
123	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
124	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
125	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
127	C₈	—	8	2	no	—
128	C₈	C₂₃×#107	1	1	yes	characteristic
129	C₈	C₂₃×#107	1	1	yes	characteristic
130	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
131	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
132	C₂²	—	4	1	no	—
133	C₂²	C₂₃×#367	1	1	yes	characteristic
134	C₂²	C₂₃×#367	1	1	yes	characteristic
135	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
136	C₄	—	16	2	no	—
137	C₄	C₂₃×#367	1	1	yes	characteristic
138	C₄	C₂₃×#367	1	1	yes	characteristic
139	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
140	C₄	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	C₂×C₂₃×#367	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#367
commutator	C₁₆	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂×C₂₃	C₂×#367
prime radical	C₂³×C₂₃	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1426	C₂	semidirect
8. C₂₃×M₆₄	C₄	semidirect
10. C₂×C₂₃×C₃₂	C₄	semidirect
46. #13638	C₂₃	direct (N × H)
63. #1426	C₂×C₂₃	semidirect
76. M₆₄	C₄×C₂₃	semidirect
78. C₂×C₃₂	C₄×C₂₃	semidirect
101. C₂₃	#13638	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
C₂×C₂₃×#368	2
C₂₃×#2158	2
C₂₃×#1426	2
#13638	23

Subgroups — C23×#13638 #19161264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13638 #19161264