Subgroups — C₂₃×#33726 #19181352

Order 5888.

490 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	C₂₃×#3160	C₂	1	2	yes	maximal
2	C₂₃×#2254	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2253	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2252	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2158	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#533	—	2	2	no	—
8	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#500	C₂²	1	4	yes	—
10	C₂₃×#499	—	2	2	no	—
11	C₂₃×#499	C₂²	1	4	yes	—
12	C₂₃×#498	—	2	2	no	—
13	C₂₃×#427	—	2	2	no	—
14	C₂₃×#427	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#367	C₂²	1	8	yes	—
17	C₂₃×#366	C₂²	1	8	yes	—
18	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
20	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	—	2	4	no	—
24	C₂₃×#125	—	2	4	no	—
25	C₂₃×C₂²⋊Q₈	—	4	4	no	—
26	C₄×Q₈×C₂₃	—	4	4	no	—
27	C₂₃×#117	—	2	4	no	—
28	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₂₃	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
31	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
32	C₂₃×#107	—	2	4	no	—
33	C₂₃×#107	C₂³	1	8	yes	—
34	C₂₃×#106	—	2	4	no	—
35	C₂₃×#106	—	2	4	no	—
36	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₂₃	—	4	4	no	—
41	C₂²×C₄×C₂₃	D₈	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
44	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	8	no	—
50	C₄⋊C₄×C₂₃	—	4	8	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₂²⋊C₄×C₂₃	—	4	4	no	—
53	C₄²×C₂₃	—	4	4	no	—
54	C₄²×C₂₃	D₈	1	2	yes	—
55	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₂₃	C₂×Q₈	1	4	yes	—
58	C₁₆×C₂₃	C₂×Q₈	1	4	yes	—
59	#33726	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
61	Q₈×C₂₃	—	8	8	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	4	8	no	—
64	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
70	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
71	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
72	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
73	C₈×C₂₃	C₂²×Q₈	1	2	yes	—
74	#3160	C₂×C₂₃	1	2	yes	—
75	#2254	C₂×C₂₃	1	2	yes	complemented
76	#2253	C₂×C₂₃	1	4	yes	complemented
77	#2252	C₂×C₂₃	1	2	yes	complemented
78	#2158	C₂×C₂₃	1	4	yes	—
79	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	#500	1	2	yes	—
82	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
83	C₄×C₂₃	—	2	2	no	—
84	C₄×C₂₃	—	8	8	no	—
85	C₄×C₂₃	#500	1	2	yes	—
86	C₄×C₂₃	#508	1	4	yes	—
87	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
89	#533	—	2	2	no	—
90	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
91	#500	C₂²×C₂₃	1	4	yes	—
92	#499	—	2	2	no	—
93	#499	C₂²×C₂₃	1	4	yes	—
94	#498	—	2	2	no	—
95	#427	—	2	2	no	—
96	#427	C₂²×C₂₃	1	4	yes	—
97	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	#367	C₂²×C₂₃	1	8	yes	—
99	#366	C₂²×C₂₃	1	8	yes	—
100	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
101	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
102	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
103	C₂×C₂₃	—	2	2	no	—
104	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
105	C₂×C₂₃	C₂×#508	1	2	yes	—
106	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	4	no	—
109	#125	—	2	4	no	—
110	C₂²⋊Q₈	—	4	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	#117	—	2	4	no	—
113	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
114	C₂×C₁₆	Q₈×C₂₃	1	8	yes	complemented
115	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
116	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
117	#107	—	2	4	no	—
118	#107	C₂³×C₂₃	1	8	yes	—
119	#106	—	2	4	no	—
120	#106	—	2	4	no	—
121	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
122	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
125	C₂₃	#33726	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	D₈×C₂₃	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
130	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
133	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
134	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	D₈×C₂₃	1	2	yes	—
141	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
143	C₁₆	C₂×Q₈×C₂₃	1	4	yes	—
144	C₁₆	C₂×Q₈×C₂₃	1	4	yes	—
145	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
146	Q₈	—	8	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
150	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
151	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
153	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
154	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
155	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
156	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
157	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
158	C₈	C₂²×Q₈×C₂₃	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂₃×#500	1	2	yes	—
161	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	8	8	no	—
164	C₄	C₂₃×#500	1	2	yes	—
165	C₄	C₂₃×#508	1	4	yes	—
166	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
167	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
170	C₂	C₂×C₂₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₈
prime radical	C₂³×C₂₃	C₂×D₈
LCS	G ▷ C₂×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
2. C₂₃×#2254	C₂	semidirect
3. C₂₃×#2253	C₂	semidirect
4. C₂₃×#2252	C₂	semidirect
29. C₂×C₁₆×C₂₃	Q₈	semidirect
59. #33726	C₂₃	direct (N × H)
75. #2254	C₂×C₂₃	semidirect
76. #2253	C₂×C₂₃	semidirect
77. #2252	C₂×C₂₃	semidirect
114. C₂×C₁₆	Q₈×C₂₃	semidirect
125. C₂₃	#33726	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3160	2
C₂₃×#2254	2
C₂₃×#2253	2
C₂₃×#2252	2
C₂₃×#2158	2
C₂×C₄×C₁₆×C₂₃	2
#33726	23

Subgroups — C23×#33726 #19181352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33726 #19181352