Subgroups — C₂₃×#12901 #19160527

Order 5888.

342 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	4	yes	maximal
2	C₂₃×#2150	C₂	1	2	yes	maximal
3	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
5	C₂×C₂₃×#97	C₄	1	2	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#376	C₄	1	1	yes	characteristic
10	C₂₃×#364	—	2	8	no	—
11	C₂₃×C₄⋊C₁₆	—	2	4	no	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂₃×M₃₂	—	2	8	no	—
18	C₂×C₁₆×C₂₃	—	2	4	no	—
19	C₂₃×#97	—	2	4	no	—
20	C₂₃×#97	C₈	1	4	yes	—
21	C₄×C₈×C₂₃	D₄	1	4	yes	—
22	C₄×C₈×C₂₃	D₄	1	2	yes	—
23	C₄×C₈×C₂₃	Q₈	1	4	yes	—
24	C₄×C₈×C₂₃	Q₈	1	2	yes	—
25	C₂₃×#95	—	2	2	no	—
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
34	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
38	C₄²×C₂₃	—	2	2	no	—
39	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₄²×C₂₃	C₂×C₈	1	2	yes	—
42	C₁₆×C₂₃	—	4	4	no	—
43	#12901	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	—	4	4	no	—
48	C₂×C₄×C₂₃	C₄⋊C₈	1	4	yes	—
49	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
53	C₈×C₂₃	—	2	2	no	—
54	C₈×C₂₃	C₄⋊C₈	1	8	yes	—
55	C₈×C₂₃	#127	1	1	yes	characteristic
56	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
57	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×#364	C₂×C₂₃	1	4	yes	—
59	#2150	C₂×C₂₃	1	2	yes	—
60	#1726	C₂×C₂₃	1	1	yes	characteristic
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₂₃	#423	1	2	yes	—
64	C₄×C₂₃	—	2	4	no	—
65	C₄×C₂₃	—	4	4	no	—
66	C₄×C₂₃	#384	1	1	yes	characteristic
67	C₄×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
68	C₄×C₂₃	#423	1	1	yes	characteristic
69	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
70	C₂×#97	C₄×C₂₃	1	2	yes	—
71	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
74	#376	C₄×C₂₃	1	1	yes	characteristic
75	#364	—	2	8	no	—
76	C₄⋊C₁₆	—	2	4	no	—
77	C₂×C₂₃	—	2	2	no	—
78	C₂×C₂₃	#1842	1	1	yes	characteristic
79	C₂×C₂₃	#2153	1	2	yes	—
80	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
82	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
84	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
85	M₃₂	—	2	8	no	—
86	C₂×C₁₆	—	2	4	no	—
87	#97	—	2	4	no	—
88	#97	C₈×C₂₃	1	4	yes	—
89	C₄×C₈	D₄×C₂₃	1	2	yes	—
90	C₄×C₈	D₄×C₂₃	1	4	yes	—
91	C₄×C₈	Q₈×C₂₃	1	4	yes	—
92	C₄×C₈	Q₈×C₂₃	1	2	yes	—
93	#95	—	2	2	no	—
94	C₂₃	#12901	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
97	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
100	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
101	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
102	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
103	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
105	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
106	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
109	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
110	C₄²	C₂×C₈×C₂₃	1	2	yes	—
111	C₁₆	—	4	4	no	—
112	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂₃×C₄⋊C₈	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₂₃×C₄⋊C₈	1	8	yes	—
123	C₈	C₂₃×#127	1	1	yes	characteristic
124	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
125	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
128	C₂²	C₂₃×#423	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂₃×#384	1	1	yes	characteristic
132	C₄	C₂×C₂₃×C₄⋊C₈	1	2	yes	—
133	C₄	C₂₃×#423	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂₃×#1842	1	1	yes	characteristic
136	C₂	C₂₃×#2153	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. #12901	C₂₃	direct (N × H)
94. C₂₃	#12901	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×#364	2
C₂₃×#2150	2
C₂₃×#1726	2
#12901	23

Subgroups — C23×#12901 #19160527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12901 #19160527