Subgroups — C₂₃×#13267 #19160893

Order 5888.

334 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1379	C₂	1	4	yes	maximal
5	C₂₃×#533	C₄	1	1	yes	characteristic
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂₃×#446	—	2	2	no	—
9	C₈×Q₈×C₂₃	—	2	2	no	—
10	C₂₃×#424	C₄	1	2	yes	—
11	C₂₃×#423	C₄	1	1	yes	characteristic
12	C₂₃×#405	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂₃×#346	—	2	2	no	—
16	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₈	1	2	yes	—
20	C₂₃×#125	C₈	1	2	yes	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₂₃	—	2	2	no	—
23	C₂₃×#117	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×M₃₂	—	2	4	no	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	—	2	2	no	—
28	C₂×C₁₆×C₂₃	—	2	2	no	—
29	C₂×C₁₆×C₂₃	Q₈	1	4	yes	complemented
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×#97	—	2	2	no	—
33	C₄×C₈×C₂₃	—	2	2	no	—
34	C₄×C₈×C₂₃	D₄	1	2	yes	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₂×Q₈×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₂₃	#41	1	2	yes	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	—	2	2	no	—
52	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₂₃	—	2	4	no	—
56	C₁₆×C₂₃	—	4	2	no	—
57	#13267	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
59	Q₈×C₂₃	—	4	4	no	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	4	2	no	—
62	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#131	1	2	yes	—
68	C₈×C₂₃	—	2	2	no	—
69	C₈×C₂₃	—	4	2	no	—
70	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
71	C₈×C₂₃	#124	1	1	yes	characteristic
72	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
73	#2960	C₂×C₂₃	1	1	yes	characteristic
74	#2150	C₂×C₂₃	1	1	yes	characteristic
75	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
76	#1379	C₂×C₂₃	1	4	yes	—
77	C₂²×C₂₃	—	2	2	no	—
78	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂²×C₂₃	#446	1	2	yes	—
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	2	2	no	—
82	C₄×C₂₃	—	4	4	no	—
83	C₄×C₂₃	#391	1	1	yes	characteristic
84	C₄×C₂₃	#408	1	1	yes	characteristic
85	C₄×C₂₃	C₈×Q₈	1	2	yes	—
86	#533	C₄×C₂₃	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
89	#446	—	2	2	no	—
90	C₈×Q₈	—	2	2	no	—
91	#424	C₄×C₂₃	1	2	yes	—
92	#423	C₄×C₂₃	1	1	yes	characteristic
93	#405	C₂²×C₂₃	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	2	no	—
96	#346	—	2	2	no	—
97	C₂×C₂₃	—	2	2	no	—
98	C₂×C₂₃	#1984	1	1	yes	characteristic
99	C₂×C₂₃	#2117	1	2	yes	—
100	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
101	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
103	C₄⋊Q₈	C₈×C₂₃	1	2	yes	—
104	#125	C₈×C₂₃	1	2	yes	—
105	C₂²⋊Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	#117	C₂×C₄×C₂₃	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
109	M₃₂	—	2	4	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	Q₈×C₂₃	1	4	yes	complemented
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	D₄×C₂₃	1	2	yes	—
119	C₄×C₈	D₄×C₂₃	1	2	yes	—
120	C₂₃	#13267	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×Q₈	—	2	2	no	—
122	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
123	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
131	C₂×C₈	#41×C₂₃	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
138	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
139	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
140	C₁₆	—	2	4	no	—
141	C₁₆	—	4	2	no	—
142	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
143	Q₈	—	4	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
147	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
150	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂₃×#131	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
155	C₈	C₂₃×#124	1	1	yes	characteristic
156	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²	C₂₃×#446	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	C₂₃×#391	1	1	yes	characteristic
164	C₄	C₂₃×#408	1	1	yes	characteristic
165	C₄	C₈×Q₈×C₂₃	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	C₂₃×#1984	1	1	yes	characteristic
168	C₂	C₂₃×#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₂×C₁₆×C₂₃	Q₈	semidirect
57. #13267	C₂₃	direct (N × H)
113. C₂×C₁₆	Q₈×C₂₃	semidirect
120. C₂₃	#13267	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×#2960	2
C₂₃×#2150	2
C₂×C₂₃×#346	2
C₂₃×#1379	2
#13267	23

Subgroups — C23×#13267 #19160893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13267 #19160893