Subgroups — C₂₃×#9609 #19157235

Order 5888.

294 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1565	C₂	1	2	yes	maximal
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1283	C₂	1	2	yes	maximal
6	C₂₃×#431	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#108	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#341	C₄	1	1	yes	characteristic
13	C₂₃×#341	C₄	1	1	yes	characteristic
14	C₂₃×#335	—	2	2	no	—
15	C₂₃×#334	—	2	2	no	—
16	C₂₃×#334	—	2	2	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#108	—	2	4	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	1	no	—
28	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
29	C₂₃×C₂²⋊C₈	—	2	1	no	—
30	C₂₃×C₂²⋊C₈	Q₈	1	2	yes	—
31	C₂₃×#97	—	2	2	no	—
32	C₄×C₈×C₂₃	—	2	1	no	—
33	C₄×C₈×C₂₃	—	2	1	no	—
34	C₄×C₈×C₂₃	—	2	2	no	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₄×C₈×C₂₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
40	M₁₆×C₂₃	—	2	2	no	—
41	M₁₆×C₂₃	—	2	2	no	—
42	M₁₆×C₂₃	—	2	2	no	—
43	M₁₆×C₂₃	M₁₆	1	4	yes	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	8	2	no	—
52	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₂₃	#41	1	1	yes	characteristic
58	#9609	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	4	1	no	—
62	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₂₃	#131	1	2	yes	—
70	C₈×C₂₃	—	2	2	no	—
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	—	2	4	no	—
73	C₈×C₂₃	—	4	2	no	—
74	C₈×C₂₃	—	4	2	no	—
75	C₈×C₂₃	—	4	2	no	—
76	C₈×C₂₃	—	8	2	no	—
77	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
78	#1565	C₂×C₂₃	1	2	yes	—
79	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
80	#1452	C₂×C₂₃	1	1	yes	characteristic
81	#1283	C₂×C₂₃	1	2	yes	—
82	C₂²×C₂₃	—	2	1	no	—
83	C₂²×C₂₃	—	2	1	no	—
84	C₂²×C₂₃	#389	1	1	yes	characteristic
85	C₂²×C₂₃	#446	1	2	yes	—
86	C₄×C₂₃	—	2	1	no	—
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	#423	1	1	yes	characteristic
90	C₄×C₂₃	#424	1	1	yes	characteristic
91	C₄×C₂₃	#435	1	2	yes	—
92	C₄×C₂₃	#471	1	2	yes	—
93	#431	C₂²×C₂₃	1	1	yes	characteristic
94	#431	C₂²×C₂₃	1	1	yes	characteristic
95	C₂×#108	C₄×C₂₃	1	2	yes	—
96	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
97	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	#341	C₄×C₂₃	1	1	yes	characteristic
100	#341	C₄×C₂₃	1	1	yes	characteristic
101	#335	—	2	2	no	—
102	#334	—	2	2	no	—
103	#334	—	2	2	no	—
104	#322	—	2	2	no	—
105	C₂×C₂₃	—	2	2	no	—
106	C₂×C₂₃	#1940	1	1	yes	characteristic
107	C₂×C₂₃	#1951	1	1	yes	characteristic
108	C₂×C₂₃	#1953	1	1	yes	characteristic
109	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
110	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
115	#108	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
120	C₂²⋊C₈	—	2	1	no	—
121	C₂²⋊C₈	Q₈×C₂₃	1	2	yes	—
122	#97	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	D₄×C₂₃	1	2	yes	—
127	C₄×C₈	Q₈×C₂₃	1	2	yes	—
128	C₂₃	#9609	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
131	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
132	M₁₆	—	2	2	no	—
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	M₁₆×C₂₃	1	4	yes	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	8	2	no	—
144	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
145	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
146	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
148	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
149	C₄²	#41×C₂₃	1	1	yes	characteristic
150	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
156	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
158	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
160	C₂×C₄	C₂₃×#131	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	4	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	2	no	—
167	C₈	—	8	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₂₃×#389	1	1	yes	characteristic
171	C₂²	C₂₃×#446	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂₃×#423	1	1	yes	characteristic
176	C₄	C₂₃×#424	1	1	yes	characteristic
177	C₄	C₂₃×#435	1	2	yes	—
178	C₄	C₂₃×#471	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₂₃×#1940	1	1	yes	characteristic
181	C₂	C₂₃×#1951	1	1	yes	characteristic
182	C₂	C₂₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#389
prime radical	C₂³×C₂₃	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
58. #9609	C₂₃	direct (N × H)
128. C₂₃	#9609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#108	2
C₂₃×#1565	2
C₂×C₂₃×#334	2
C₂₃×#1452	2
C₂₃×#1283	2
#9609	23

Subgroups — C23×#9609 #19157235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9609 #19157235