Subgroups — C₂₃×#12596 #19160222

Order 5888.

310 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#347	C₂	1	2	yes	maximal
2	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1377	C₂	1	2	yes	maximal
4	C₂₃×#1376	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#376	C₄	1	1	yes	characteristic
13	C₂₃×#347	C₄	1	8	yes	complemented
14	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₃×M₃₂	—	2	4	no	—
22	C₂₃×M₃₂	C₂×C₄	1	8	yes	—
23	C₂×C₁₆×C₂₃	—	2	4	no	—
24	C₂₃×#97	—	2	2	no	—
25	C₂₃×#97	—	2	2	no	—
26	C₂₃×#97	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₂₃	—	2	2	no	—
28	C₄×C₈×C₂₃	—	2	2	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₂₃×#95	—	2	2	no	—
31	C₂²×C₄×C₂₃	—	2	1	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	—	2	1	no	—
36	C₂×C₈×C₂₃	—	2	1	no	—
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	C₄²	1	2	yes	—
42	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₂₃	#41	1	2	yes	—
46	C₂×C₈×C₂₃	#41	1	2	yes	—
47	C₄²×C₂₃	—	2	1	no	—
48	C₄²×C₂₃	—	2	1	no	—
49	C₄²×C₂₃	—	2	1	no	—
50	C₄²×C₂₃	—	2	1	no	—
51	C₄²×C₂₃	C₄²	1	2	yes	—
52	C₄²×C₂₃	M₁₆	1	2	yes	—
53	C₁₆×C₂₃	—	2	8	no	—
54	#12596	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	4	2	no	—
60	C₂×C₄×C₂₃	—	4	2	no	—
61	C₂×C₄×C₂₃	—	4	2	no	—
62	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₂₃	—	2	2	no	—
69	C₈×C₂₃	—	2	2	no	—
70	C₈×C₂₃	—	2	2	no	—
71	C₈×C₂₃	#117	1	4	yes	—
72	C₂×#347	C₂×C₂₃	1	2	yes	—
73	#1726	C₂×C₂₃	1	1	yes	characteristic
74	#1377	C₂×C₂₃	1	2	yes	—
75	#1376	C₂×C₂₃	1	2	yes	—
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
78	C₂²×C₂₃	#432	1	1	yes	characteristic
79	C₂²×C₂₃	#432	1	1	yes	characteristic
80	C₄×C₂₃	—	4	2	no	—
81	C₄×C₂₃	—	4	2	no	—
82	C₄×C₂₃	—	4	2	no	—
83	C₄×C₂₃	#376	1	1	yes	characteristic
84	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₂₃	#432	1	1	yes	characteristic
86	C₄×C₂₃	#432	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
88	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
89	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
90	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
92	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	#376	C₄×C₂₃	1	1	yes	characteristic
95	#347	C₄×C₂₃	1	8	yes	complemented
96	C₂×C₂₃	—	2	1	no	—
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	#1745	1	1	yes	characteristic
99	C₂×C₂₃	#2166	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
104	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
107	M₃₂	—	2	4	no	—
108	M₃₂	C₂×C₄×C₂₃	1	8	yes	—
109	C₂×C₁₆	—	2	4	no	—
110	#97	—	2	2	no	—
111	#97	—	2	2	no	—
112	#97	C₂×C₄×C₂₃	1	4	yes	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	2	2	no	—
117	C₂₃	#12596	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
121	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	C₄²×C₂₃	1	2	yes	—
129	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
130	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
131	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
132	C₂×C₈	#41×C₂₃	1	2	yes	—
133	C₂×C₈	#41×C₂₃	1	2	yes	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₄²×C₂₃	1	2	yes	—
139	C₄²	M₁₆×C₂₃	1	2	yes	—
140	C₁₆	—	2	8	no	—
141	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
149	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
150	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
151	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
153	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	2	2	no	—
157	C₈	C₂₃×#117	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
160	C₂²	C₂₃×#432	1	1	yes	characteristic
161	C₂²	C₂₃×#432	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	C₂₃×#376	1	1	yes	characteristic
166	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
167	C₄	C₂₃×#432	1	1	yes	characteristic
168	C₄	C₂₃×#432	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₂₃×#1745	1	1	yes	characteristic
172	C₂	C₂₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#432
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#347	C₄	semidirect
54. #12596	C₂₃	direct (N × H)
95. #347	C₄×C₂₃	semidirect
117. C₂₃	#12596	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#347	2
C₂₃×#1726	2
C₂₃×#1377	2
C₂₃×#1376	2
#12596	23

Subgroups — C23×#12596 #19160222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12596 #19160222