Subgroups — C₂₃×#33414 #19181040

Order 5888.

622 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2916	C₂	1	2	yes	maximal
3	C₂₃×#2161	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#368	C₂	1	6	yes	maximal, complemented
5	C₂₃×#568	—	2	2	no	—
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂₃×#504	C₄	1	8	yes	—
8	C₂×C₂₃×M₃₂	C₂²	1	3	yes	—
9	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	—
10	C₂₃×#430	—	2	6	no	—
11	C₂×C₂₃×#108	C₂²	1	12	yes	—
12	C₂₃×#368	C₂²	1	4	yes	—
13	C₂₃×#368	C₂²	1	12	yes	—
14	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
15	C₂₃×#131	—	4	8	no	—
16	C₂₃×#131	C₂×C₄	1	4	yes	—
17	C₂₃×#131	Q₈	1	4	yes	—
18	C₂×M₁₆×C₂₃	—	2	12	no	—
19	C₂×M₁₆×C₂₃	D₄	1	3	yes	—
20	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
21	C₂₃×M₃₂	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₆×C₂₃	C₂³	1	3	yes	—
25	C₂₃×#108	—	2	12	no	—
26	C₂₃×#108	C₂³	1	8	yes	—
27	#41×C₂₃	C₄⋊C₄	1	4	yes	—
28	#41×C₂₃	D₈	1	4	yes	—
29	C₂×Q₈×C₂₃	Q₁₆	1	1	yes	characteristic
30	C₂×D₄×C₂₃	Q₁₆	1	3	yes	—
31	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
32	M₁₆×C₂₃	—	2	8	no	—
33	M₁₆×C₂₃	—	4	12	no	—
34	M₁₆×C₂₃	C₄⋊C₄	1	12	yes	—
35	C₂×C₈×C₂₃	—	4	12	no	—
36	C₂×C₈×C₂₃	C₂²×C₄	1	6	yes	—
37	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
38	C₂×C₈×C₂₃	C₂×Q₈	1	6	yes	—
39	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
40	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
41	C₁₆×C₂₃	C₂²×C₄	1	6	yes	—
42	#33414	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₂₃	C₂×Q₁₆	1	3	yes	—
44	Q₈×C₂₃	#107	1	4	yes	—
45	D₄×C₂₃	#107	1	12	yes	—
46	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
47	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
48	C₂×C₄×C₂₃	C₂×Q₁₆	1	3	yes	—
49	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
50	C₈×C₂₃	—	4	8	no	—
51	C₈×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
52	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
53	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
55	#2916	C₂×C₂₃	1	2	yes	—
56	#2161	C₂×C₂₃	1	6	yes	complemented
57	C₂×#368	C₂×C₂₃	1	6	yes	complemented
58	C₂²×C₂₃	—	4	3	no	—
59	C₂²×C₂₃	C₂×#107	1	6	yes	—
60	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₃	C₂×#107	1	6	yes	—
62	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
64	#568	—	2	2	no	—
65	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
66	#504	C₄×C₂₃	1	8	yes	—
67	C₂×M₃₂	C₂²×C₂₃	1	3	yes	—
68	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	—
69	#430	—	2	6	no	—
70	C₂×#108	C₂²×C₂₃	1	12	yes	—
71	#368	C₂²×C₂₃	1	4	yes	—
72	#368	C₂²×C₂₃	1	12	yes	—
73	C₂×C₂₃	—	2	1	no	—
74	C₂×C₂₃	—	2	6	no	—
75	C₂×C₂₃	C₂²×#107	1	1	yes	characteristic
76	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
77	#131	—	4	8	no	—
78	#131	C₂×C₄×C₂₃	1	4	yes	—
79	#131	Q₈×C₂₃	1	4	yes	—
80	C₂×M₁₆	—	2	12	no	—
81	C₂×M₁₆	D₄×C₂₃	1	3	yes	—
82	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
83	M₃₂	C₂×C₄×C₂₃	1	12	yes	—
84	C₂×C₁₆	C₂×C₄×C₂₃	1	12	yes	—
85	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂³×C₂₃	1	3	yes	—
87	#108	—	2	12	no	—
88	#108	C₂³×C₂₃	1	8	yes	—
89	C₂₃	#33414	1	1	yes	characteristic, direct product factor, Sylow
90	#41	C₄⋊C₄×C₂₃	1	4	yes	—
91	#41	D₈×C₂₃	1	4	yes	—
92	C₂×Q₈	Q₁₆×C₂₃	1	1	yes	characteristic
93	C₂×D₄	Q₁₆×C₂₃	1	3	yes	—
94	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
95	M₁₆	—	2	8	no	—
96	M₁₆	—	4	12	no	—
97	M₁₆	C₄⋊C₄×C₂₃	1	12	yes	—
98	C₂×C₈	—	4	12	no	—
99	C₂×C₈	C₂²×C₄×C₂₃	1	6	yes	—
100	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
101	C₂×C₈	C₂×Q₈×C₂₃	1	6	yes	—
102	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
103	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
104	C₁₆	C₂²×C₄×C₂₃	1	6	yes	—
105	C₂³	C₂×Q₁₆×C₂₃	1	3	yes	—
106	Q₈	C₂₃×#107	1	4	yes	—
107	D₄	C₂₃×#107	1	12	yes	—
108	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
109	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
110	C₂×C₄	C₂×Q₁₆×C₂₃	1	3	yes	—
111	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
112	C₈	—	4	8	no	—
113	C₈	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
114	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
115	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
116	C₂²	—	4	3	no	—
117	C₂²	C₂×C₂₃×#107	1	6	yes	—
118	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
119	C₄	C₂×C₂₃×#107	1	6	yes	—
120	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
121	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	6	no	—
124	C₂	C₂²×C₂₃×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂²×#107
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×#41×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2161	C₂	semidirect
4. C₂×C₂₃×#368	C₂	semidirect
42. #33414	C₂₃	direct (N × H)
56. #2161	C₂×C₂₃	semidirect
57. C₂×#368	C₂×C₂₃	semidirect
89. C₂₃	#33414	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×#504	2
C₂₃×#2916	2
C₂₃×#2161	2
C₂×C₂₃×#368	2
#33414	23

Subgroups — C23×#33414 #19181040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33414 #19181040