Subgroups — C₂₃×#7163 #19154789

Order 5888.

238 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×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂₃×#431	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
10	C₄×C₁₆×C₂₃	—	2	1	no	—
11	C₄×C₁₆×C₂₃	—	2	1	no	—
12	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
13	C₂₃×#322	—	2	1	no	—
14	C₈²×C₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	C₈	1	2	yes	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	—	2	4	no	—
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂₃×C₄⋊C₈	C₈	1	2	yes	—
23	C₂₃×C₂²⋊C₈	—	2	1	no	—
24	C₂₃×#97	—	2	1	no	—
25	C₄×C₈×C₂₃	—	2	1	no	—
26	C₄×C₈×C₂₃	—	2	1	no	—
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	2	no	—
35	M₁₆×C₂₃	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	2	no	—
38	C₂×C₈×C₂₃	—	2	2	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₄²	1	1	yes	characteristic
45	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
48	C₁₆×C₂₃	—	2	4	no	—
49	C₁₆×C₂₃	—	4	2	no	—
50	#7163	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	1	no	—
53	C₂×C₄×C₂₃	—	2	1	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
63	C₈×C₂₃	—	2	1	no	—
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	4	no	—
66	C₈×C₂₃	—	4	2	no	—
67	C₈×C₂₃	#104	1	2	yes	—
68	C₈×C₂₃	#108	1	2	yes	—
69	#2162	C₂×C₂₃	1	1	yes	characteristic
70	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
71	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	—	2	1	no	—
73	C₂²×C₂₃	—	2	1	no	—
74	C₂²×C₂₃	#336	1	1	yes	characteristic
75	C₂²×C₂₃	C₄⋊C₁₆	1	2	yes	—
76	C₄×C₂₃	—	2	1	no	—
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	#339	1	1	yes	characteristic
79	C₄×C₂₃	#341	1	1	yes	characteristic
80	C₄×C₂₃	C₄×C₁₆	1	1	yes	characteristic
81	C₄×C₂₃	#346	1	1	yes	characteristic
82	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
83	C₄×C₂₃	#350	1	2	yes	—
84	C₂²×C₁₆	—	2	1	no	—
85	#431	C₄×C₂₃	1	1	yes	characteristic
86	C₄×M₁₆	C₄×C₂₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
88	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
93	#322	—	2	1	no	—
94	C₈²	—	2	1	no	—
95	C₂×C₂₃	—	2	2	no	—
96	C₂×C₂₃	#1278	1	1	yes	characteristic
97	C₂×C₂₃	#1374	1	1	yes	characteristic
98	C₂×C₂₃	#1379	1	1	yes	characteristic
99	C₂×M₁₆	C₈×C₂₃	1	2	yes	—
100	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
101	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₄⋊C₈	C₈×C₂₃	1	2	yes	—
107	C₂²⋊C₈	—	2	1	no	—
108	#97	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
114	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂₃	#7163	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
118	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	C₁₆×C₂₃	1	4	yes	complemented
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
129	C₄²	C₄²×C₂₃	1	1	yes	characteristic
130	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
132	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₁₆	—	4	2	no	—
135	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
140	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
141	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
142	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
143	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
144	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
146	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	2	no	—
151	C₈	C₂₃×#104	1	2	yes	—
152	C₈	C₂₃×#108	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	C₂₃×#336	1	1	yes	characteristic
156	C₂²	C₂₃×C₄⋊C₁₆	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	C₂₃×#339	1	1	yes	characteristic
160	C₄	C₂₃×#341	1	1	yes	characteristic
161	C₄	C₄×C₁₆×C₂₃	1	1	yes	characteristic
162	C₄	C₂₃×#346	1	1	yes	characteristic
163	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
164	C₄	C₂₃×#350	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂₃×#1278	1	1	yes	characteristic
167	C₂	C₂₃×#1374	1	1	yes	characteristic
168	C₂	C₂₃×#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₂₃	D₄
commutator	C₄	C₄×C₁₆×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. M₁₆×C₂₃	C₁₆	semidirect
50. #7163	C₂₃	direct (N × H)
115. C₂₃	#7163	direct (N × H)
120. M₁₆	C₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2162	2
C₂×C₄×C₁₆×C₂₃	2
C₈×M₁₆×C₂₃	2
#7163	23

Subgroups — C23×#7163 #19154789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7163 #19154789