Subgroups — C₂₃×#13273 #19160899

Order 5888.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1386	C₂	1	2	yes	maximal
5	C₂₃×#1383	C₂	1	2	yes	maximal
6	C₂×C₂₃×#125	—	2	1	no	—
7	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
8	C₂₃×#499	—	2	2	no	—
9	C₂₃×#499	C₄	1	4	yes	complemented
10	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
11	C₂₃×#427	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#107	—	2	1	no	—
13	C₂×C₂₃×#106	C₂²	1	2	yes	—
14	C₂₃×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#364	—	2	2	no	—
17	C₂₃×#339	—	2	1	no	—
18	C₂×M₁₆×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×#125	—	2	4	no	—
22	C₂₃×#125	—	4	2	no	—
23	C₂₃×#117	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
26	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
27	C₂₃×M₃₂	—	2	2	no	—
28	C₂₃×M₃₂	—	2	2	no	—
29	C₂×C₁₆×C₂₃	—	2	2	no	—
30	C₂₃×#108	—	2	1	no	—
31	C₂₃×#108	Q₈	1	2	yes	—
32	C₂₃×#107	—	2	4	no	—
33	C₂₃×#107	Q₈	1	2	yes	—
34	C₂₃×#106	—	2	2	no	—
35	C₂₃×#106	—	2	1	no	—
36	C₂₃×#106	C₂×C₄	1	4	yes	—
37	C₂₃×#97	—	4	1	no	—
38	C₄×C₈×C₂₃	—	4	1	no	—
39	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₂₃	D₄	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	4	2	no	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₂₃	—	2	2	no	—
45	M₁₆×C₂₃	—	4	1	no	—
46	C₂×C₈×C₂₃	—	4	1	no	—
47	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	4	8	no	—
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₂²⋊C₄×C₂₃	—	4	1	no	—
62	C₄²×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	4	1	no	—
64	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₁₆×C₂₃	—	4	2	no	—
66	#13273	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	4	2	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	4	2	no	—
72	C₂×C₄×C₂₃	—	8	4	no	—
73	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₈×C₂₃	—	4	2	no	—
80	C₈×C₂₃	#103	1	2	yes	—
81	C₈×C₂₃	#103	1	2	yes	—
82	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
84	C₈×C₂₃	#124	1	1	yes	characteristic
85	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
87	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
88	#1772	C₂×C₂₃	1	1	yes	characteristic
89	#1386	C₂×C₂₃	1	2	yes	—
90	#1383	C₂×C₂₃	1	2	yes	—
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	#418	1	1	yes	characteristic
93	C₂²×C₂₃	#476	1	1	yes	characteristic
94	C₂²×C₂₃	#477	1	1	yes	characteristic
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	8	4	no	—
97	C₄×C₂₃	—	8	2	no	—
98	C₄×C₂₃	#391	1	1	yes	characteristic
99	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
100	C₄×C₂₃	#476	1	1	yes	characteristic
101	C₄×C₂₃	#477	1	1	yes	characteristic
102	C₂×#125	—	2	1	no	—
103	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
104	#499	—	2	2	no	—
105	#499	C₄×C₂₃	1	4	yes	complemented
106	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
107	#427	C₂²×C₂₃	1	1	yes	characteristic
108	C₂×#107	—	2	1	no	—
109	C₂×#106	C₂²×C₂₃	1	2	yes	—
110	#405	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
112	#364	—	2	2	no	—
113	#339	—	2	1	no	—
114	C₂×C₂₃	—	2	1	no	—
115	C₂×C₂₃	—	2	1	no	—
116	C₂×C₂₃	#1987	1	1	yes	characteristic
117	C₂×C₂₃	#2231	1	2	yes	—
118	C₂×M₁₆	—	2	1	no	—
119	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
121	#125	—	2	4	no	—
122	#125	—	4	2	no	—
123	#117	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	#108	—	2	1	no	—
131	#108	Q₈×C₂₃	1	2	yes	—
132	#107	—	2	4	no	—
133	#107	Q₈×C₂₃	1	2	yes	—
134	#106	—	2	2	no	—
135	#106	—	2	1	no	—
136	#106	C₂×C₄×C₂₃	1	4	yes	—
137	#97	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
140	C₄×C₈	D₄×C₂₃	1	2	yes	—
141	C₂₃	#13273	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
149	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
152	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
153	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
154	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
155	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
156	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
157	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	4	no	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
177	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	C₂₃×#103	1	2	yes	—
181	C₈	C₂₃×#103	1	2	yes	—
182	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
183	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
184	C₈	C₂₃×#124	1	1	yes	characteristic
185	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	C₂₃×#418	1	1	yes	characteristic
188	C₂²	C₂₃×#476	1	1	yes	characteristic
189	C₂²	C₂₃×#477	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	4	no	—
192	C₄	—	8	2	no	—
193	C₄	C₂₃×#391	1	1	yes	characteristic
194	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
195	C₄	C₂₃×#476	1	1	yes	characteristic
196	C₄	C₂₃×#477	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₂₃×#1987	1	1	yes	characteristic
200	C₂	C₂₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#476
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂₃×#499	C₄	semidirect
66. #13273	C₂₃	direct (N × H)
105. #499	C₄×C₂₃	semidirect
141. C₂₃	#13273	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×#499	2
C₂×C₂₃×#364	2
C₂₃×#1772	2
C₂₃×#1386	2
C₂₃×#1383	2
#13273	23

Subgroups — C23×#13273 #19160899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13273 #19160899