Subgroups — C₂₃×#7159 #19154785

Order 5888.

222 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
5	C₂₃×#431	—	2	1	no	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#346	—	2	1	no	—
9	C₄×C₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#335	—	2	1	no	—
11	C₂₃×#335	—	2	1	no	—
12	C₂₃×#335	C₄	1	2	yes	—
13	C₂₃×#334	C₄	1	2	yes	—
14	C₂²×C₈×C₂₃	—	4	1	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃×M₃₂	—	2	2	no	—
19	C₂₃×M₃₂	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂₃×C₄⋊C₈	—	2	2	no	—
22	C₂₃×C₄⋊C₈	—	2	2	no	—
23	C₂₃×C₄⋊C₈	—	4	1	no	—
24	C₂₃×C₂²⋊C₈	—	4	1	no	—
25	C₄×C₈×C₂₃	—	4	1	no	—
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₂₃	—	2	2	no	—
34	C₂×C₈×C₂₃	—	4	2	no	—
35	C₂×C₈×C₂₃	—	4	2	no	—
36	C₂×C₈×C₂₃	—	8	2	no	—
37	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
43	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₂₃	D₈	1	1	yes	characteristic
46	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
47	C₁₆×C₂₃	—	2	4	no	—
48	#7159	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	—	4	1	no	—
53	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#108	1	2	yes	—
61	C₈×C₂₃	—	2	1	no	—
62	C₈×C₂₃	—	2	1	no	—
63	C₈×C₂₃	—	8	2	no	—
64	C₈×C₂₃	—	8	2	no	—
65	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₂₃	#97	1	1	yes	characteristic
67	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
68	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
69	#1564	C₂×C₂₃	1	1	yes	characteristic
70	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
71	C₂²×C₂₃	—	2	1	no	—
72	C₂²×C₂₃	—	2	1	no	—
73	C₂²×C₂₃	#334	1	2	yes	—
74	C₂²×C₂₃	#340	1	1	yes	characteristic
75	C₄×C₂₃	—	2	1	no	—
76	C₄×C₂₃	—	2	1	no	—
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	#335	1	2	yes	—
79	C₄×C₂₃	#336	1	1	yes	characteristic
80	C₄×C₂₃	#341	1	1	yes	characteristic
81	C₄×C₂₃	#361	1	1	yes	characteristic
82	C₄×C₂₃	#362	1	1	yes	characteristic
83	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
84	#431	—	2	1	no	—
85	C₂×C₄⋊C₈	—	2	1	no	—
86	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
87	#346	—	2	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	#335	—	2	1	no	—
90	#335	—	2	1	no	—
91	#335	C₄×C₂₃	1	2	yes	—
92	#334	C₄×C₂₃	1	2	yes	—
93	C₂×C₂₃	—	2	2	no	—
94	C₂×C₂₃	#1277	1	1	yes	characteristic
95	C₂×C₂₃	#1387	1	1	yes	characteristic
96	C₂×C₂₃	#1389	1	1	yes	characteristic
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	C₈×C₂₃	1	4	yes	complemented
103	C₂×C₁₆	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₂²⋊C₈	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
112	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
113	C₂₃	#7159	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
116	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
126	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
127	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
128	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
129	C₄²	D₈×C₂₃	1	1	yes	characteristic
130	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
131	C₁₆	—	2	4	no	—
132	C₂³	C₂₃×#106	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
137	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
138	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
139	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
140	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
141	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
142	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
143	C₂×C₄	C₂₃×#108	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
149	C₈	C₂₃×#97	1	1	yes	characteristic
150	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	C₂₃×#334	1	2	yes	—
154	C₂²	C₂₃×#340	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	C₂₃×#335	1	2	yes	—
159	C₄	C₂₃×#336	1	1	yes	characteristic
160	C₄	C₂₃×#341	1	1	yes	characteristic
161	C₄	C₂₃×#361	1	1	yes	characteristic
162	C₄	C₂₃×#362	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	C₂₃×#1277	1	1	yes	characteristic
165	C₂	C₂₃×#1387	1	1	yes	characteristic
166	C₂	C₂₃×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#340
prime radical	C₂³×C₂₃	#106
LCS	G ▷ C₈ ▷ 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
19. C₂₃×M₃₂	C₈	semidirect
48. #7159	C₂₃	direct (N × H)
102. M₃₂	C₈×C₂₃	semidirect
113. C₂₃	#7159	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×M₃₂	2
C₂₃×#1564	2
C₂×C₂₃×#335	2
#7159	23

Subgroups — C23×#7159 #19154785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7159 #19154785