Subgroups — C₂₃×#7276 #19154902

Order 5888.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂₃×#431	—	2	1	no	—
6	C₂×C₂₃×#108	—	2	1	no	—
7	C₂×C₂₃×#108	C₄	1	2	yes	—
8	C₂₃×#423	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#334	C₄	1	2	yes	—
14	C₂×M₁₆×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₂₃	—	2	2	no	—
19	C₂×C₁₆×C₂₃	—	4	1	no	—
20	C₂×C₁₆×C₂₃	—	4	1	no	—
21	C₂₃×#108	—	4	2	no	—
22	C₂₃×#108	—	4	1	no	—
23	C₂₃×#108	—	4	1	no	—
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	4	1	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	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₂₃	—	4	2	no	—
37	M₁₆×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	4	2	no	—
41	C₂×C₈×C₂₃	—	4	2	no	—
42	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
50	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₂₃	D₈	1	1	yes	characteristic
53	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
54	C₁₆×C₂₃	—	4	2	no	—
55	#7276	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	4	1	no	—
59	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#101	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
69	C₈×C₂₃	—	2	1	no	—
70	C₈×C₂₃	—	2	1	no	—
71	C₈×C₂₃	—	8	2	no	—
72	C₈×C₂₃	—	8	2	no	—
73	C₈×C₂₃	#102	1	1	yes	characteristic
74	C₈×C₂₃	#103	1	1	yes	characteristic
75	C₈×C₂₃	#106	1	1	yes	characteristic
76	C₈×C₂₃	#107	1	1	yes	characteristic
77	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
78	#1848	C₂×C₂₃	1	1	yes	characteristic
79	#1564	C₂×C₂₃	1	1	yes	characteristic
80	C₂²×C₂₃	—	2	1	no	—
81	C₂²×C₂₃	—	2	1	no	—
82	C₂²×C₂₃	#332	1	1	yes	characteristic
83	C₂²×C₂₃	#333	1	1	yes	characteristic
84	C₂²×C₂₃	#340	1	1	yes	characteristic
85	C₄×C₂₃	—	2	1	no	—
86	C₄×C₂₃	—	2	1	no	—
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	#331	1	1	yes	characteristic
89	C₄×C₂₃	#332	1	1	yes	characteristic
90	C₄×C₂₃	#340	1	1	yes	characteristic
91	C₄×C₂₃	#343	1	1	yes	characteristic
92	C₄×C₂₃	#368	1	2	yes	—
93	C₂²×C₁₆	—	2	1	no	—
94	#431	—	2	1	no	—
95	C₂×#108	—	2	1	no	—
96	C₂×#108	C₄×C₂₃	1	2	yes	—
97	#423	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	1	no	—
100	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
101	#335	—	2	1	no	—
102	#334	C₄×C₂₃	1	2	yes	—
103	C₂×C₂₃	—	2	1	no	—
104	C₂×C₂₃	—	2	1	no	—
105	C₂×C₂₃	#1296	1	1	yes	characteristic
106	C₂×C₂₃	#1381	1	1	yes	characteristic
107	C₂×C₂₃	#1389	1	1	yes	characteristic
108	C₂×M₁₆	—	2	2	no	—
109	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	4	1	no	—
114	C₂×C₁₆	—	4	1	no	—
115	#108	—	4	2	no	—
116	#108	—	4	1	no	—
117	#108	—	4	1	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₂²⋊C₈	—	4	1	no	—
122	C₄×C₈	—	4	1	no	—
123	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
126	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
127	C₂₃	#7276	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
130	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
131	M₁₆	—	4	2	no	—
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
142	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
143	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
144	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
145	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
146	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₄²	D₈×C₂₃	1	1	yes	characteristic
148	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	C₂³	C₂₃×#103	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
154	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#101	1	1	yes	characteristic
156	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
157	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
158	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
159	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
160	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
161	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
162	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	8	2	no	—
167	C₈	C₂₃×#102	1	1	yes	characteristic
168	C₈	C₂₃×#103	1	1	yes	characteristic
169	C₈	C₂₃×#106	1	1	yes	characteristic
170	C₈	C₂₃×#107	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	C₂₃×#332	1	1	yes	characteristic
174	C₂²	C₂₃×#333	1	1	yes	characteristic
175	C₂²	C₂₃×#340	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₂₃×#331	1	1	yes	characteristic
180	C₄	C₂₃×#332	1	1	yes	characteristic
181	C₄	C₂₃×#340	1	1	yes	characteristic
182	C₄	C₂₃×#343	1	1	yes	characteristic
183	C₄	C₂₃×#368	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	C₂₃×#1296	1	1	yes	characteristic
187	C₂	C₂₃×#1381	1	1	yes	characteristic
188	C₂	C₂₃×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#340
prime radical	C₂³×C₂₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×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
55. #7276	C₂₃	direct (N × H)
127. C₂₃	#7276	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×C₄⋊C₁₆	2
C₂₃×#1848	2
C₂₃×#1564	2
#7276	23

Subgroups — C23×#7276 #19154902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7276 #19154902