Subgroups — C₂₃×#7154 #19154780

Order 5888.

242 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×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#335	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₂₃	C₄	1	2	yes	—
4	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₂₃	—	2	2	no	—
7	C₂₃×#335	—	2	2	no	—
8	C₂₃×#335	C₄	1	4	yes	—
9	C₂²×C₈×C₂₃	—	4	2	no	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₂₃	—	2	2	no	—
14	C₂×C₁₆×C₂₃	—	2	2	no	—
15	C₂×C₁₆×C₂₃	C₈	1	4	yes	complemented
16	C₂₃×C₄⋊C₈	—	2	4	no	—
17	C₂₃×C₄⋊C₈	—	4	2	no	—
18	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	4	4	no	—
25	C₂×C₈×C₂₃	—	8	2	no	—
26	C₂×C₈×C₂₃	—	8	2	no	—
27	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
28	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
32	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₄²×C₂₃	SD₁₆	1	2	yes	—
34	C₁₆×C₂₃	—	2	4	no	—
35	#7154	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₂₃	#107	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₂₃	#102	1	2	yes	—
40	C₂×C₄×C₂₃	#103	1	2	yes	—
41	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
42	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#108	1	2	yes	—
44	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
45	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
46	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
47	C₈×C₂₃	—	2	2	no	—
48	C₈×C₂₃	—	8	4	no	—
49	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
50	C₈×C₂₃	#97	1	1	yes	characteristic
51	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
52	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
53	C₂×#335	C₂×C₂₃	1	2	yes	—
54	C₂²×C₂₃	#335	1	2	yes	—
55	C₂²×C₂₃	#340	1	1	yes	characteristic
56	C₂²×C₂₃	#366	1	1	yes	characteristic
57	C₂²×C₂₃	#367	1	1	yes	characteristic
58	C₂²×C₂₃	#368	1	2	yes	—
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	#334	1	2	yes	—
61	C₄×C₂₃	#336	1	1	yes	characteristic
62	C₄×C₂₃	#341	1	1	yes	characteristic
63	C₄×C₂₃	#357	1	2	yes	—
64	C₄×C₂₃	#358	1	2	yes	—
65	C₂²×C₁₆	C₄×C₂₃	1	2	yes	—
66	C₂×C₄⋊C₈	—	2	2	no	—
67	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
68	C₄×C₁₆	—	2	2	no	—
69	#335	—	2	2	no	—
70	#335	C₄×C₂₃	1	4	yes	—
71	C₂×C₂₃	#1277	1	1	yes	characteristic
72	C₂×C₂₃	#1371	1	2	yes	—
73	C₂×C₂₃	#1372	1	2	yes	—
74	C₂×C₂₃	#1380	1	1	yes	characteristic
75	C₂×C₂₃	#1381	1	1	yes	characteristic
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
78	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
79	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₈×C₂₃	1	4	yes	complemented
83	C₄⋊C₈	—	2	4	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
86	C₄×C₈	D₄×C₂₃	1	2	yes	—
87	C₂₃	#7154	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
89	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
90	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
98	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
99	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
100	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
101	C₄²	SD₁₆×C₂₃	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	C₂₃×#107	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
106	C₂×C₄	C₂₃×#102	1	2	yes	—
107	C₂×C₄	C₂₃×#103	1	2	yes	—
108	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
109	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
110	C₂×C₄	C₂₃×#108	1	2	yes	—
111	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
112	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
113	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	8	4	no	—
116	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
117	C₈	C₂₃×#97	1	1	yes	characteristic
118	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
119	C₂²	C₂₃×#335	1	2	yes	—
120	C₂²	C₂₃×#340	1	1	yes	characteristic
121	C₂²	C₂₃×#366	1	1	yes	characteristic
122	C₂²	C₂₃×#367	1	1	yes	characteristic
123	C₂²	C₂₃×#368	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	C₂₃×#334	1	2	yes	—
126	C₄	C₂₃×#336	1	1	yes	characteristic
127	C₄	C₂₃×#341	1	1	yes	characteristic
128	C₄	C₂₃×#357	1	2	yes	—
129	C₄	C₂₃×#358	1	2	yes	—
130	C₂	C₂₃×#1277	1	1	yes	characteristic
131	C₂	C₂₃×#1371	1	2	yes	—
132	C₂	C₂₃×#1372	1	2	yes	—
133	C₂	C₂₃×#1380	1	1	yes	characteristic
134	C₂	C₂₃×#1381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	D₈
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle, prime radical	C₂³×C₂₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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
15. C₂×C₁₆×C₂₃	C₈	semidirect
35. #7154	C₂₃	direct (N × H)
82. C₂×C₁₆	C₈×C₂₃	semidirect
87. C₂₃	#7154	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×C₄×C₁₆×C₂₃	2
C₂×C₂₃×#335	2
#7154	23

Subgroups — C23×#7154 #19154780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7154 #19154780