Subgroups — C₂×C₂₃×#1368 #19159624

Order 5888.

254 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×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#334	C₂	1	2	yes	maximal
3	C₂₃×#1368	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#346	C₄	1	2	yes	—
9	C₂₃×#346	C₂²	1	2	yes	—
10	C₂₃×#334	—	2	2	no	—
11	C₂₃×#334	C₂²	1	4	yes	—
12	C₂²×C₈×C₂₃	—	4	2	no	—
13	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₂₃	—	2	2	no	—
17	C₂×C₁₆×C₂₃	C₈	1	4	yes	complemented
18	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂₃×C₄⋊C₈	—	2	4	no	—
21	C₂₃×C₄⋊C₈	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	4	4	no	—
30	C₂×C₈×C₂₃	—	8	4	no	—
31	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
35	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₂₃	SD₁₆	1	2	yes	—
39	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₂₃	—	2	2	no	—
41	C₁₆×C₂₃	C₂×C₈	1	4	yes	complemented
42	C₂×#1368	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₂₃	#106	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
46	C₂×C₄×C₂₃	#106	1	2	yes	—
47	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#108	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
51	C₈×C₂₃	—	2	1	no	—
52	C₈×C₂₃	—	2	1	no	—
53	C₈×C₂₃	—	8	4	no	—
54	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
55	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
58	C₂×#334	C₂×C₂₃	1	2	yes	—
59	#1368	C₂×C₂₃	1	4	yes	direct product factor
60	C₂²×C₂₃	#334	1	2	yes	—
61	C₂²×C₂₃	#365	1	4	yes	—
62	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	—	2	2	no	—
65	C₄×C₂₃	#334	1	2	yes	—
66	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
68	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
69	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
72	#346	C₄×C₂₃	1	2	yes	—
73	#346	C₂²×C₂₃	1	2	yes	—
74	#334	—	2	2	no	—
75	#334	C₂²×C₂₃	1	4	yes	—
76	C₂×C₂₃	#1368	1	4	yes	direct product factor
77	C₂×C₂₃	C₂×#334	1	1	yes	characteristic
78	C₂×C₂₃	C₂×#365	1	2	yes	—
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
82	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	C₈×C₂₃	1	4	yes	complemented
85	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	4	2	no	—
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	C₂×#1368	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
95	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	8	4	no	—
99	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
100	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
102	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
103	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
104	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
105	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
106	C₄²	SD₁₆×C₂₃	1	2	yes	—
107	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
108	C₁₆	—	2	2	no	—
109	C₁₆	C₂×C₈×C₂₃	1	4	yes	complemented
110	C₂³	C₂₃×#106	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
113	C₂×C₄	C₂₃×#106	1	2	yes	—
114	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
115	C₂×C₄	C₂₃×#108	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	8	4	no	—
121	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
122	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
123	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
124	C₂²	C₂₃×#334	1	2	yes	—
125	C₂²	C₂₃×#365	1	4	yes	—
126	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	C₂₃×#334	1	2	yes	—
130	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
131	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
132	C₂	C₂₃×#1368	1	4	yes	direct product factor
133	C₂	C₂×C₂₃×#334	1	1	yes	characteristic
134	C₂	C₂×C₂₃×#365	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#106
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ 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
3. C₂₃×#1368	C₂	direct (N × H)
17. C₂×C₁₆×C₂₃	C₈	semidirect
41. C₁₆×C₂₃	C₂×C₈	semidirect
42. C₂×#1368	C₂₃	direct (N × H)
59. #1368	C₂×C₂₃	direct (N × H)
76. C₂×C₂₃	#1368	direct (N × H)
84. C₂×C₁₆	C₈×C₂₃	semidirect
93. C₂₃	C₂×#1368	direct (N × H)
109. C₁₆	C₂×C₈×C₂₃	semidirect
132. C₂	C₂₃×#1368	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×#346	2
C₂×C₂₃×#334	2
C₂₃×#1368	2
C₂×#1368	23

Subgroups — C2×C23×#1368 #19159624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1368 #19159624