Subgroups — C₂₃×#7281 #19154907

Order 5888.

226 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	—	2	1	no	—
5	C₂×C₂₃×#108	—	2	1	no	—
6	C₂×C₂₃×#108	C₄	1	2	yes	—
7	C₂₃×#423	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	—	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×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	—	4	1	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
19	C₂₃×M₃₂	—	4	1	no	—
20	C₂₃×M₃₂	—	4	1	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂₃×#108	—	4	2	no	—
23	C₂₃×#108	—	4	1	no	—
24	C₂₃×#108	—	4	1	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	4	1	no	—
27	C₂₃×C₄⋊C₈	—	4	1	no	—
28	C₂₃×C₄⋊C₈	—	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	—	8	2	no	—
43	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
51	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₂₃	D₈	1	1	yes	characteristic
54	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
55	C₁₆×C₂₃	—	4	2	no	—
56	#7281	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	4	1	no	—
61	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#101	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
71	C₈×C₂₃	—	2	1	no	—
72	C₈×C₂₃	—	2	1	no	—
73	C₈×C₂₃	—	8	2	no	—
74	C₈×C₂₃	—	8	2	no	—
75	C₈×C₂₃	#102	1	1	yes	characteristic
76	C₈×C₂₃	#103	1	1	yes	characteristic
77	C₈×C₂₃	#106	1	1	yes	characteristic
78	C₈×C₂₃	#107	1	1	yes	characteristic
79	#2150	C₂×C₂₃	1	1	yes	characteristic
80	#1848	C₂×C₂₃	1	1	yes	characteristic
81	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
82	C₂²×C₂₃	—	2	1	no	—
83	C₂²×C₂₃	—	2	1	no	—
84	C₂²×C₂₃	#332	1	1	yes	characteristic
85	C₂²×C₂₃	#333	1	1	yes	characteristic
86	C₂²×C₂₃	#340	1	1	yes	characteristic
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	#331	1	1	yes	characteristic
91	C₄×C₂₃	#332	1	1	yes	characteristic
92	C₄×C₂₃	#340	1	1	yes	characteristic
93	C₄×C₂₃	#343	1	1	yes	characteristic
94	C₄×C₂₃	#361	1	1	yes	characteristic
95	C₄×C₂₃	#362	1	1	yes	characteristic
96	C₂×M₃₂	—	2	1	no	—
97	C₂×#108	—	2	1	no	—
98	C₂×#108	C₄×C₂₃	1	2	yes	—
99	#423	—	2	1	no	—
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
102	C₄⋊C₁₆	—	2	1	no	—
103	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
104	#335	—	2	1	no	—
105	#335	C₄×C₂₃	1	2	yes	—
106	C₂×C₂₃	—	2	1	no	—
107	C₂×C₂₃	—	2	1	no	—
108	C₂×C₂₃	#1296	1	1	yes	characteristic
109	C₂×C₂₃	#1385	1	1	yes	characteristic
110	C₂×C₂₃	#1387	1	1	yes	characteristic
111	C₂×M₁₆	—	2	2	no	—
112	C₂²×C₈	—	4	1	no	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
116	M₃₂	—	4	1	no	—
117	M₃₂	—	4	1	no	—
118	C₂×C₁₆	—	2	2	no	—
119	#108	—	4	2	no	—
120	#108	—	4	1	no	—
121	#108	—	4	1	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄⋊C₈	—	4	1	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
129	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
130	C₂₃	#7281	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
134	M₁₆	—	4	2	no	—
135	M₁₆	—	4	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
146	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
148	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
149	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
150	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
151	C₄²	D₈×C₂₃	1	1	yes	characteristic
152	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
153	C₁₆	—	4	2	no	—
154	C₂³	C₂₃×#103	1	1	yes	characteristic
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
159	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
160	C₂×C₄	C₂₃×#101	1	1	yes	characteristic
161	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
162	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
163	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
164	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
165	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
166	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
167	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	8	2	no	—
171	C₈	—	8	2	no	—
172	C₈	C₂₃×#102	1	1	yes	characteristic
173	C₈	C₂₃×#103	1	1	yes	characteristic
174	C₈	C₂₃×#106	1	1	yes	characteristic
175	C₈	C₂₃×#107	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₂₃×#332	1	1	yes	characteristic
179	C₂²	C₂₃×#333	1	1	yes	characteristic
180	C₂²	C₂₃×#340	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂₃×#331	1	1	yes	characteristic
185	C₄	C₂₃×#332	1	1	yes	characteristic
186	C₄	C₂₃×#340	1	1	yes	characteristic
187	C₄	C₂₃×#343	1	1	yes	characteristic
188	C₄	C₂₃×#361	1	1	yes	characteristic
189	C₄	C₂₃×#362	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	C₂₃×#1296	1	1	yes	characteristic
193	C₂	C₂₃×#1385	1	1	yes	characteristic
194	C₂	C₂₃×#1387	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#340
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
prime radical	C₂³×C₂₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
56. #7281	C₂₃	direct (N × H)
130. C₂₃	#7281	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×#2150	2
C₂₃×#1848	2
C₂×C₂₃×#335	2
#7281	23

Subgroups — C23×#7281 #19154907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7281 #19154907