Subgroups — C₂₃×#8814 #19156440

Order 5888.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1845	C₂	1	2	yes	maximal
2	C₂₃×#1844	C₂	1	2	yes	maximal
3	C₂₃×#1834	C₂	1	2	yes	maximal
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#107	—	2	4	no	—
6	C₂×C₂₃×#106	—	2	4	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#384	C₂²	1	4	yes	—
11	C₂₃×#340	—	2	4	no	—
12	C₈²×C₂₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₂₃	D₄	1	2	yes	—
14	C₂²×C₈×C₂₃	D₄	1	2	yes	—
15	C₂²×C₈×C₂₃	D₄	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
19	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃×#107	—	4	8	no	—
21	C₂₃×#106	—	4	8	no	—
22	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₂₃	Q₈	1	2	yes	—
26	C₄×C₈×C₂₃	Q₈	1	2	yes	—
27	C₄×C₈×C₂₃	Q₈	1	2	yes	—
28	C₂²×C₄×C₂₃	—	4	4	no	—
29	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₂₃	D₈	1	4	yes	—
37	C₂×C₈×C₂₃	SD₁₆	1	8	yes	—
38	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
39	C₄⋊C₄×C₂₃	—	4	8	no	—
40	C₄⋊C₄×C₂₃	—	4	8	no	—
41	C₄⋊C₄×C₂₃	—	4	8	no	—
42	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	#8814	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	8	4	no	—
49	C₂×C₄×C₂₃	—	8	4	no	—
50	C₂×C₄×C₂₃	—	8	4	no	—
51	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
59	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
60	C₂×C₄×C₂₃	#135	1	4	yes	—
61	C₈×C₂₃	—	2	4	no	—
62	C₈×C₂₃	#106	1	8	yes	complemented
63	C₈×C₂₃	#107	1	8	yes	complemented
64	#1845	C₂×C₂₃	1	2	yes	—
65	#1844	C₂×C₂₃	1	2	yes	—
66	#1834	C₂×C₂₃	1	2	yes	—
67	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
68	C₂²×C₂₃	#384	1	1	yes	characteristic
69	C₂²×C₂₃	#492	1	2	yes	—
70	C₂²×C₂₃	#493	1	1	yes	characteristic
71	C₂²×C₂₃	#494	1	1	yes	characteristic
72	C₂²×C₂₃	#495	1	2	yes	—
73	C₄×C₂₃	—	16	4	no	—
74	C₄×C₂₃	C₂×#106	1	2	yes	—
75	C₄×C₂₃	C₂×#107	1	2	yes	—
76	C₄×C₂₃	#427	1	2	yes	—
77	C₄×C₂₃	#498	1	2	yes	—
78	C₄×C₂₃	#499	1	2	yes	—
79	C₄×C₂₃	#500	1	2	yes	—
80	C₂×#107	—	2	4	no	—
81	C₂×#106	—	2	4	no	—
82	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
85	#384	C₂²×C₂₃	1	4	yes	—
86	#340	—	2	4	no	—
87	C₈²	C₄×C₂₃	1	4	yes	complemented
88	C₂×C₂₃	#1706	1	2	yes	—
89	C₂×C₂₃	#1707	1	2	yes	—
90	C₂×C₂₃	#1844	1	1	yes	characteristic
91	C₂×C₂₃	#1845	1	1	yes	characteristic
92	C₂×C₂₃	#1846	1	1	yes	characteristic
93	C₂²×C₈	D₄×C₂₃	1	2	yes	—
94	C₂²×C₈	D₄×C₂₃	1	2	yes	—
95	C₂²×C₈	D₄×C₂₃	1	2	yes	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	2	4	no	—
99	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
100	#107	—	4	8	no	—
101	#106	—	4	8	no	—
102	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
103	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
104	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
105	C₄×C₈	Q₈×C₂₃	1	2	yes	—
106	C₄×C₈	Q₈×C₂₃	1	2	yes	—
107	C₄×C₈	Q₈×C₂₃	1	2	yes	—
108	C₂₃	#8814	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
115	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
116	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
117	C₂×C₈	D₈×C₂₃	1	4	yes	—
118	C₂×C₈	SD₁₆×C₂₃	1	8	yes	—
119	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
120	C₄⋊C₄	—	4	8	no	—
121	C₄⋊C₄	—	4	8	no	—
122	C₄⋊C₄	—	4	8	no	—
123	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
124	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
125	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
126	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
127	C₂³	C₂₃×#127	1	1	yes	characteristic
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
138	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
139	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
140	C₂×C₄	C₂₃×#135	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	C₂₃×#106	1	8	yes	complemented
143	C₈	C₂₃×#107	1	8	yes	complemented
144	C₂²	C₂₃×#384	1	1	yes	characteristic
145	C₂²	C₂₃×#492	1	2	yes	—
146	C₂²	C₂₃×#493	1	1	yes	characteristic
147	C₂²	C₂₃×#494	1	1	yes	characteristic
148	C₂²	C₂₃×#495	1	2	yes	—
149	C₄	—	16	4	no	—
150	C₄	C₂×C₂₃×#106	1	2	yes	—
151	C₄	C₂×C₂₃×#107	1	2	yes	—
152	C₄	C₂₃×#427	1	2	yes	—
153	C₄	C₂₃×#498	1	2	yes	—
154	C₄	C₂₃×#499	1	2	yes	—
155	C₄	C₂₃×#500	1	2	yes	—
156	C₂	C₂₃×#1706	1	2	yes	—
157	C₂	C₂₃×#1707	1	2	yes	—
158	C₂	C₂₃×#1844	1	1	yes	characteristic
159	C₂	C₂₃×#1845	1	1	yes	characteristic
160	C₂	C₂₃×#1846	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₈²×C₂₃	C₄	semidirect
46. #8814	C₂₃	direct (N × H)
62. C₈×C₂₃	#106	semidirect
63. C₈×C₂₃	#107	semidirect
87. C₈²	C₄×C₂₃	semidirect
108. C₂₃	#8814	direct (N × H)
142. C₈	C₂₃×#106	semidirect
143. C₈	C₂₃×#107	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1845	2
C₂₃×#1844	2
C₂₃×#1834	2
C₂×C₈²×C₂₃	2
#8814	23

Subgroups — C23×#8814 #19156440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8814 #19156440