Subgroups — C₂₃×#7842 #19155468

Order 5888.

558 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#322	C₂	1	2	yes	maximal
5	C₂×C₂₃×#322	C₂	1	2	yes	maximal
6	C₂×C₂₃×#97	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
12	C₄³×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	C₄	1	4	yes	—
14	C₂₃×#322	C₄	1	8	yes	complemented
15	C₂₃×#322	C₄	1	8	yes	complemented
16	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#97	—	2	8	no	—
25	C₄×C₈×C₂₃	—	2	8	no	—
26	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
33	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	—	2	4	no	—
37	C₂×C₈×C₂₃	—	2	8	no	—
38	C₂×C₈×C₂₃	—	2	8	no	—
39	C₂×C₈×C₂₃	C₄²	1	4	yes	—
40	C₂×C₈×C₂₃	C₄²	1	8	yes	—
41	C₂×C₈×C₂₃	C₄²	1	8	yes	—
42	C₂×C₈×C₂₃	C₄²	1	4	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	8	yes	—
44	C₄²×C₂₃	—	2	4	no	—
45	C₄²×C₂₃	—	2	4	no	—
46	C₄²×C₂₃	M₁₆	1	8	yes	—
47	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
49	#7842	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	#97	1	8	yes	—
54	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
60	C₂×C₄×C₂₃	#131	1	4	yes	—
61	C₈×C₂₃	—	2	4	no	—
62	C₈×C₂₃	—	2	8	no	—
63	C₈×C₂₃	—	2	8	no	—
64	C₈×C₂₃	#97	1	8	yes	complemented
65	#1744	C₂×C₂₃	1	1	yes	characteristic
66	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
67	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
68	C₂×#322	C₂×C₂₃	1	2	yes	—
69	C₂×#322	C₂×C₂₃	1	2	yes	—
70	C₂²×C₂₃	C₄³	1	1	yes	characteristic
71	C₂²×C₂₃	C₂×#97	1	2	yes	—
72	C₂²×C₂₃	C₄×M₁₆	1	2	yes	—
73	C₂²×C₂₃	#405	1	2	yes	—
74	C₄×C₂₃	—	2	8	no	—
75	C₄×C₂₃	C₂×#97	1	2	yes	—
76	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
77	C₄×C₂₃	C₄×M₁₆	1	4	yes	—
78	C₄×C₂₃	#405	1	4	yes	—
79	C₂×#97	C₄×C₂₃	1	4	yes	—
80	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	—
81	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
85	C₄³	C₂²×C₂₃	1	1	yes	characteristic
86	#336	C₄×C₂₃	1	4	yes	—
87	#322	C₄×C₂₃	1	8	yes	complemented
88	#322	C₄×C₂₃	1	8	yes	complemented
89	C₂×C₂₃	#1451	1	4	yes	—
90	C₂×C₂₃	C₄×#97	1	1	yes	characteristic
91	C₂×C₂₃	C₄×#97	1	1	yes	characteristic
92	C₂×C₂₃	#1726	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
94	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
95	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
96	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
97	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
99	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
100	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
101	#97	—	2	8	no	—
102	C₄×C₈	—	2	8	no	—
103	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
104	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
105	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
106	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
107	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
108	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
109	C₂₃	#7842	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
111	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	—	2	8	no	—
117	C₂×C₈	C₄²×C₂₃	1	4	yes	—
118	C₂×C₈	C₄²×C₂₃	1	8	yes	—
119	C₂×C₈	C₄²×C₂₃	1	8	yes	—
120	C₂×C₈	C₄²×C₂₃	1	4	yes	—
121	C₂×C₈	M₁₆×C₂₃	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	M₁₆×C₂₃	1	8	yes	—
125	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
126	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
127	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₂₃×#97	1	8	yes	—
131	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
134	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
135	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
136	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
137	C₂×C₄	C₂₃×#131	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	2	8	no	—
140	C₈	—	2	8	no	—
141	C₈	C₂₃×#97	1	8	yes	complemented
142	C₂²	C₄³×C₂₃	1	1	yes	characteristic
143	C₂²	C₂×C₂₃×#97	1	2	yes	—
144	C₂²	C₄×M₁₆×C₂₃	1	2	yes	—
145	C₂²	C₂₃×#405	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×C₂₃×#97	1	2	yes	—
148	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
149	C₄	C₄×M₁₆×C₂₃	1	4	yes	—
150	C₄	C₂₃×#405	1	4	yes	—
151	C₂	C₂₃×#1451	1	4	yes	—
152	C₂	C₄×C₂₃×#97	1	1	yes	characteristic
153	C₂	C₄×C₂₃×#97	1	1	yes	characteristic
154	C₂	C₂₃×#1726	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	C₂³
commutator	C₂²	C₄³×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂₃×#322	C₄	semidirect
15. C₂₃×#322	C₄	semidirect
49. #7842	C₂₃	direct (N × H)
64. C₈×C₂₃	#97	semidirect
87. #322	C₄×C₂₃	semidirect
88. #322	C₄×C₂₃	semidirect
109. C₂₃	#7842	direct (N × H)
141. C₈	C₂₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1744	2
C₄×C₂₃×#97	2
C₄²×C₈×C₂₃	2
C₂×C₂₃×#322	2
#7842	23

Subgroups — C23×#7842 #19155468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7842 #19155468