Subgroups — C₂₃×#33312 #19180938

Order 5888.

1006 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	4	yes	maximal
2	C₂₃×#2868	C₂	1	2	yes	maximal
3	C₂₃×#1845	C₂	1	8	yes	maximal
4	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×C₄⋊Q₈	—	2	4	no	—
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
7	C₂₃×#500	—	2	16	no	—
8	C₂×C₂₃×#107	C₂²	1	16	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#384	—	2	4	no	—
12	C₂₃×#384	C₂²	1	8	yes	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	—	2	4	no	—
15	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
16	C₂₃×C₄⋊Q₈	—	2	16	no	—
17	C₂₃×C₄⋊Q₈	—	4	16	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
20	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
21	C₂×C₄²×C₂₃	D₄	1	2	yes	—
22	C₂×C₄²×C₂₃	D₄	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
25	C₂₃×#107	—	2	32	no	—
26	C₄×C₈×C₂₃	D₄	1	8	yes	—
27	C₄×C₈×C₂₃	Q₈	1	16	yes	complemented
28	C₂×Q₈×C₂₃	—	4	8	no	—
29	C₂×Q₈×C₂₃	—	4	16	no	—
30	C₂²×C₄×C₂₃	—	2	8	no	—
31	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
34	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
36	C₂×C₈×C₂₃	C₂×Q₈	1	16	yes	—
37	C₄⋊C₄×C₂₃	—	2	16	no	—
38	C₄⋊C₄×C₂₃	—	4	8	no	—
39	C₄⋊C₄×C₂₃	—	4	32	no	—
40	C₄²×C₂₃	D₈	1	8	yes	—
41	C₄²×C₂₃	Q₁₆	1	8	yes	—
42	C₄²×C₂₃	C₂×D₄	1	4	yes	—
43	C₄²×C₂₃	C₂×Q₈	1	8	yes	—
44	#33312	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
46	Q₈×C₂₃	—	8	16	no	—
47	C₂×C₄×C₂₃	—	4	8	no	—
48	C₂×C₄×C₂₃	—	4	16	no	—
49	C₂×C₄×C₂₃	#127	1	2	yes	—
50	C₂×C₄×C₂₃	#127	1	2	yes	—
51	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
52	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₂₃	C₂×D₈	1	8	yes	—
54	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
55	C₂×C₄×C₂₃	C₂×Q₁₆	1	8	yes	—
56	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
58	C₈×C₂₃	C₄⋊Q₈	1	16	yes	complemented
59	C₂×#500	C₂×C₂₃	1	4	yes	—
60	#2868	C₂×C₂₃	1	2	yes	—
61	#1845	C₂×C₂₃	1	8	yes	—
62	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
63	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
65	C₂²×C₂₃	C₂²×D₈	1	2	yes	—
66	C₂²×C₂₃	C₂²×Q₁₆	1	2	yes	—
67	C₄×C₂₃	—	8	8	no	—
68	C₄×C₂₃	#493	1	4	yes	—
69	C₄×C₂₃	#494	1	4	yes	—
70	C₄×C₂₃	#500	1	16	yes	complemented
71	C₄×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
72	C₂×C₄⋊Q₈	—	2	4	no	—
73	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
74	#500	—	2	16	no	—
75	C₂×#107	C₂²×C₂₃	1	16	yes	—
76	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
77	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
78	#384	—	2	4	no	—
79	#384	C₂²×C₂₃	1	8	yes	—
80	C₄³	C₂²×C₂₃	1	1	yes	characteristic
81	C₂×C₂₃	#2868	1	1	yes	characteristic
82	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
83	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
84	C₂×C₂₃	C₂×#500	1	4	yes	—
85	C₂²×Q₈	—	2	4	no	—
86	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
87	C₄⋊Q₈	—	2	16	no	—
88	C₄⋊Q₈	—	4	16	no	—
89	C₂×C₄⋊C₄	—	2	16	no	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
92	C₂×C₄²	D₄×C₂₃	1	2	yes	—
93	C₂×C₄²	D₄×C₂₃	1	2	yes	—
94	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
95	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
96	#107	—	2	32	no	—
97	C₄×C₈	D₄×C₂₃	1	8	yes	—
98	C₄×C₈	Q₈×C₂₃	1	16	yes	complemented
99	C₂₃	#33312	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×Q₈	—	4	8	no	—
101	C₂×Q₈	—	4	16	no	—
102	C₂²×C₄	—	2	8	no	—
103	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
106	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
108	C₂×C₈	C₂×Q₈×C₂₃	1	16	yes	—
109	C₄⋊C₄	—	2	16	no	—
110	C₄⋊C₄	—	4	8	no	—
111	C₄⋊C₄	—	4	32	no	—
112	C₄²	D₈×C₂₃	1	8	yes	—
113	C₄²	Q₁₆×C₂₃	1	8	yes	—
114	C₄²	C₂×D₄×C₂₃	1	4	yes	—
115	C₄²	C₂×Q₈×C₂₃	1	8	yes	—
116	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
117	Q₈	—	8	16	no	—
118	C₂×C₄	—	4	16	no	—
119	C₂×C₄	—	4	8	no	—
120	C₂×C₄	C₂₃×#127	1	2	yes	—
121	C₂×C₄	C₂₃×#127	1	2	yes	—
122	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
123	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
124	C₂×C₄	C₂×D₈×C₂₃	1	8	yes	—
125	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
126	C₂×C₄	C₂×Q₁₆×C₂₃	1	8	yes	—
127	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
128	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
129	C₈	C₂₃×C₄⋊Q₈	1	16	yes	complemented
130	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
131	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
132	C₂²	C₂²×D₈×C₂₃	1	2	yes	—
133	C₂²	C₂²×Q₁₆×C₂₃	1	2	yes	—
134	C₄	—	8	8	no	—
135	C₄	C₂₃×#493	1	4	yes	—
136	C₄	C₂₃×#494	1	4	yes	—
137	C₄	C₂₃×#500	1	16	yes	complemented
138	C₄	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
139	C₂	C₂₃×#2868	1	1	yes	characteristic
140	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
141	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
142	C₂	C₂×C₂₃×#500	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×C₈×C₂₃	Q₈	semidirect
44. #33312	C₂₃	direct (N × H)
58. C₈×C₂₃	C₄⋊Q₈	semidirect
70. C₄×C₂₃	#500	semidirect
98. C₄×C₈	Q₈×C₂₃	semidirect
99. C₂₃	#33312	direct (N × H)
129. C₈	C₂₃×C₄⋊Q₈	semidirect
137. C₄	C₂₃×#500	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#500	2
C₂₃×#2868	2
C₂₃×#1845	2
C₄²×C₈×C₂₃	2
#33312	23

Subgroups — C23×#33312 #19180938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33312 #19180938