Subgroups — C₂₃×#12905 #19160531

Order 5888.

374 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#366	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1845	C₂	1	2	yes	maximal
4	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
5	C₂×C₂₃×#107	—	2	2	no	—
6	C₂×C₂₃×#107	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#384	—	2	2	no	—
9	C₂₃×#366	—	2	8	no	—
10	C₄×C₁₆×C₂₃	C₄	1	4	yes	complemented
11	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
15	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
18	C₂×C₁₆×C₂₃	Q₈	1	4	yes	—
19	C₂₃×#107	—	2	8	no	—
20	C₂₃×#107	—	4	4	no	—
21	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₂₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₃	—	4	4	no	—
24	C₂²×C₄×C₂₃	D₈	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₂₃	D₈	1	2	yes	—
28	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
29	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₂₃	—	4	4	no	—
34	C₄⋊C₄×C₂₃	—	4	4	no	—
35	C₄⋊C₄×C₂₃	—	8	8	no	—
36	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₄²×C₂₃	Q₁₆	1	2	yes	—
38	C₁₆×C₂₃	C₄⋊C₄	1	8	yes	complemented
39	#12905	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	8	4	no	—
42	C₂×C₄×C₂₃	—	8	8	no	—
43	C₂×C₄×C₂₃	#107	1	4	yes	—
44	C₂×C₄×C₂₃	D₁₆	1	4	yes	—
45	C₂×C₄×C₂₃	Q₃₂	1	4	yes	—
46	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₈×C₂₃	#107	1	4	yes	—
53	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
54	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
55	C₂×#366	C₂×C₂₃	1	4	yes	—
56	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
57	#1845	C₂×C₂₃	1	2	yes	—
58	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
59	C₂²×C₂₃	#493	1	1	yes	characteristic
60	C₂²×C₂₃	#500	1	1	yes	characteristic
61	C₂²×C₂₃	C₂×D₁₆	1	2	yes	—
62	C₂²×C₂₃	C₂×Q₃₂	1	2	yes	—
63	C₄×C₂₃	—	16	4	no	—
64	C₄×C₂₃	#366	1	8	yes	complemented
65	C₄×C₂₃	#384	1	1	yes	characteristic
66	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
67	C₄×C₂₃	#494	1	1	yes	characteristic
68	C₄×C₂₃	#500	1	1	yes	characteristic
69	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
70	C₂×#107	—	2	2	no	—
71	C₂×#107	C₂²×C₂₃	1	4	yes	—
72	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
73	#384	—	2	2	no	—
74	#366	—	2	8	no	—
75	C₄×C₁₆	C₄×C₂₃	1	4	yes	complemented
76	C₂×C₂₃	#1845	1	1	yes	characteristic
77	C₂×C₂₃	C₂×#366	1	2	yes	—
78	C₂×C₂₃	#2246	1	1	yes	characteristic
79	C₂×C₂₃	#2247	1	1	yes	characteristic
80	C₂×C₂₃	#2252	1	2	yes	—
81	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
82	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	—	4	4	no	—
85	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
87	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
88	C₂×C₁₆	Q₈×C₂₃	1	4	yes	—
89	#107	—	2	8	no	—
90	#107	—	4	4	no	—
91	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
92	C₄×C₈	Q₈×C₂₃	1	2	yes	—
93	C₂₃	#12905	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	—	4	4	no	—
95	C₂²×C₄	D₈×C₂₃	1	2	yes	—
96	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
98	C₂×C₈	D₈×C₂₃	1	2	yes	—
99	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
100	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
102	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
103	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	8	8	no	—
107	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
108	C₄²	Q₁₆×C₂₃	1	2	yes	—
109	C₁₆	C₄⋊C₄×C₂₃	1	8	yes	complemented
110	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
111	C₂×C₄	—	8	4	no	—
112	C₂×C₄	—	8	8	no	—
113	C₂×C₄	C₂₃×#107	1	4	yes	—
114	C₂×C₄	C₂₃×D₁₆	1	4	yes	—
115	C₂×C₄	C₂₃×Q₃₂	1	4	yes	—
116	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
118	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
122	C₈	C₂₃×#107	1	4	yes	—
123	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
124	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
125	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
126	C₂²	C₂₃×#493	1	1	yes	characteristic
127	C₂²	C₂₃×#500	1	1	yes	characteristic
128	C₂²	C₂×C₂₃×D₁₆	1	2	yes	—
129	C₂²	C₂×C₂₃×Q₃₂	1	2	yes	—
130	C₄	—	16	4	no	—
131	C₄	C₂₃×#366	1	8	yes	complemented
132	C₄	C₂₃×#384	1	1	yes	characteristic
133	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
134	C₄	C₂₃×#494	1	1	yes	characteristic
135	C₄	C₂₃×#500	1	1	yes	characteristic
136	C₂	C₂₃×#1845	1	1	yes	characteristic
137	C₂	C₂×C₂₃×#366	1	2	yes	—
138	C₂	C₂₃×#2246	1	1	yes	characteristic
139	C₂	C₂₃×#2247	1	1	yes	characteristic
140	C₂	C₂₃×#2252	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₂₃	C₄	semidirect
38. C₁₆×C₂₃	C₄⋊C₄	semidirect
39. #12905	C₂₃	direct (N × H)
64. C₄×C₂₃	#366	semidirect
75. C₄×C₁₆	C₄×C₂₃	semidirect
93. C₂₃	#12905	direct (N × H)
109. C₁₆	C₄⋊C₄×C₂₃	semidirect
131. C₄	C₂₃×#366	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#366	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1845	2
#12905	23

Subgroups — C23×#12905 #19160531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12905 #19160531