Subgroups — C₂₃×#12000 #19159626

Order 5888.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1368	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂₃×#431	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#334	—	2	1	no	—
14	C₂₃×#334	C₂²	1	2	yes	—
15	C₂₃×#334	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₃	—	4	1	no	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂₃×M₃₂	—	2	2	no	—
21	C₂₃×M₃₂	C₈	1	4	yes	complemented
22	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	4	1	no	—
27	C₂₃×C₂²⋊C₈	—	4	1	no	—
28	C₄×C₈×C₂₃	—	4	1	no	—
29	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₂×C₈×C₂₃	—	8	2	no	—
40	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
48	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₂₃	—	2	2	no	—
51	C₁₆×C₂₃	C₂×C₈	1	4	yes	complemented
52	#12000	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	4	1	no	—
56	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#108	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	1	no	—
66	C₈×C₂₃	—	8	2	no	—
67	C₈×C₂₃	—	8	2	no	—
68	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
69	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
71	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
72	#1564	C₂×C₂₃	1	1	yes	characteristic
73	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
74	#1368	C₂×C₂₃	1	4	yes	complemented
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	—	2	1	no	—
77	C₂²×C₂₃	#334	1	2	yes	—
78	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	—	2	1	no	—
82	C₄×C₂₃	#334	1	2	yes	—
83	C₄×C₂₃	#365	1	2	yes	—
84	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
86	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
87	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
88	#431	—	2	1	no	—
89	C₂×C₄⋊C₈	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
91	#346	C₂²×C₂₃	1	2	yes	—
92	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
93	#335	—	2	1	no	—
94	#334	—	2	1	no	—
95	#334	C₂²×C₂₃	1	2	yes	—
96	#334	C₂²×C₂₃	1	2	yes	—
97	C₂×C₂₃	—	2	2	no	—
98	C₂×C₂₃	C₂×#334	1	1	yes	characteristic
99	C₂×C₂₃	C₂×#365	1	1	yes	characteristic
100	C₂×C₂₃	#2155	1	1	yes	characteristic
101	C₂²×C₈	—	4	1	no	—
102	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
104	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
105	M₃₂	—	2	2	no	—
106	M₃₂	C₈×C₂₃	1	4	yes	complemented
107	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
108	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
115	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
117	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
118	C₂₃	#12000	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	2	no	—
126	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
129	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
130	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
132	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
133	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
134	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
135	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×C₈×C₂₃	1	4	yes	complemented
138	C₂³	C₂₃×#106	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
142	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
143	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
144	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#108	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	2	no	—
153	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
154	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
155	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₂₃×#334	1	2	yes	—
159	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	C₂₃×#334	1	2	yes	—
164	C₄	C₂₃×#365	1	2	yes	—
165	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
166	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×C₂₃×#334	1	1	yes	characteristic
169	C₂	C₂×C₂₃×#365	1	1	yes	characteristic
170	C₂	C₂₃×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#106
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#106
prime radical	C₂³×C₂₃	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
4. C₂₃×#1368	C₂	semidirect
21. C₂₃×M₃₂	C₈	semidirect
51. C₁₆×C₂₃	C₂×C₈	semidirect
52. #12000	C₂₃	direct (N × H)
74. #1368	C₂×C₂₃	semidirect
106. M₃₂	C₈×C₂₃	semidirect
118. C₂₃	#12000	direct (N × H)
137. C₁₆	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₂₃×#1564	2
C₂×C₂₃×#334	2
C₂₃×#1368	2
#12000	23

Subgroups — C23×#12000 #19159626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12000 #19159626