Subgroups — C₂₃×#9623 #19157249

Order 5888.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1950	C₂	1	2	yes	maximal
2	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1301	C₂	1	2	yes	maximal
6	C₂₃×#431	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#108	—	2	2	no	—
8	C₂₃×#423	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#341	—	2	2	no	—
13	C₂₃×#334	—	2	1	no	—
14	C₂₃×#334	—	2	1	no	—
15	C₂₃×#334	C₄	1	2	yes	—
16	C₂₃×#334	C₄	1	2	yes	—
17	C₂₃×#322	—	2	1	no	—
18	C₈²×C₂₃	—	2	1	no	—
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₂₃×#108	—	4	4	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	2	1	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₂₃×C₂²⋊C₈	Q₈	1	2	yes	—
32	C₂₃×#97	—	2	1	no	—
33	C₄×C₈×C₂₃	—	2	1	no	—
34	C₄×C₈×C₂₃	—	2	1	no	—
35	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₂₃	—	2	2	no	—
43	M₁₆×C₂₃	—	4	2	no	—
44	M₁₆×C₂₃	—	4	2	no	—
45	M₁₆×C₂₃	SD₁₆	1	4	yes	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	4	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	4	1	no	—
51	C₂×C₈×C₂₃	—	4	2	no	—
52	C₂×C₈×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	4	2	no	—
54	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₂₃	#41	1	1	yes	characteristic
61	C₄²×C₂₃	#41	1	1	yes	characteristic
62	#9623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	4	1	no	—
67	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₂₃	#135	1	2	yes	—
75	C₈×C₂₃	—	2	1	no	—
76	C₈×C₂₃	—	2	1	no	—
77	C₈×C₂₃	—	2	1	no	—
78	C₈×C₂₃	—	2	4	no	—
79	C₈×C₂₃	—	4	2	no	—
80	C₈×C₂₃	—	4	4	no	—
81	C₈×C₂₃	—	8	2	no	—
82	C₈×C₂₃	#108	1	2	yes	complemented
83	#1950	C₂×C₂₃	1	2	yes	—
84	#1566	C₂×C₂₃	1	1	yes	characteristic
85	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
86	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
87	#1301	C₂×C₂₃	1	2	yes	—
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	#389	1	1	yes	characteristic
91	C₂²×C₂₃	#498	1	1	yes	characteristic
92	C₂²×C₂₃	#499	1	1	yes	characteristic
93	C₄×C₂₃	—	2	1	no	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
97	C₄×C₂₃	#430	1	1	yes	characteristic
98	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
99	C₄×C₂₃	#465	1	2	yes	—
100	#431	C₂²×C₂₃	1	1	yes	characteristic
101	C₂×#108	—	2	2	no	—
102	#423	C₂²×C₂₃	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
104	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	#341	—	2	2	no	—
107	#334	—	2	1	no	—
108	#334	—	2	1	no	—
109	#334	C₄×C₂₃	1	2	yes	—
110	#334	C₄×C₂₃	1	2	yes	—
111	#322	—	2	1	no	—
112	C₈²	—	2	1	no	—
113	C₂×C₂₃	—	2	1	no	—
114	C₂×C₂₃	—	2	1	no	—
115	C₂×C₂₃	#1941	1	1	yes	characteristic
116	C₂×C₂₃	#1950	1	1	yes	characteristic
117	C₂×C₂₃	#1953	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
122	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
130	C₂²⋊C₈	Q₈×C₂₃	1	2	yes	—
131	#97	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
137	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
138	C₂₃	#9623	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	SD₁₆×C₂₃	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
159	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₄²	#41×C₂₃	1	1	yes	characteristic
161	C₄²	#41×C₂₃	1	1	yes	characteristic
162	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
170	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
173	C₂×C₄	C₂₃×#135	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	4	no	—
178	C₈	—	4	2	no	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₈	C₂₃×#108	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	C₂₃×#389	1	1	yes	characteristic
185	C₂²	C₂₃×#498	1	1	yes	characteristic
186	C₂²	C₂₃×#499	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
191	C₄	C₂₃×#430	1	1	yes	characteristic
192	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
193	C₄	C₂₃×#465	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₂₃×#1941	1	1	yes	characteristic
197	C₂	C₂₃×#1950	1	1	yes	characteristic
198	C₂	C₂₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#389
prime radical	C₂³×C₂₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
62. #9623	C₂₃	direct (N × H)
82. C₈×C₂₃	#108	semidirect
138. C₂₃	#9623	direct (N × H)
181. C₈	C₂₃×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1950	2
C₂₃×#1566	2
C₂×C₂₃×#334	2
C₈×M₁₆×C₂₃	2
C₂₃×#1301	2
#9623	23

Subgroups — C23×#9623 #19157249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9623 #19157249