Subgroups — C₂₃×#13556 #19161182

Order 5888.

290 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1400	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
5	C₂₃×#504	—	2	2	no	—
6	C₂₃×#504	—	2	2	no	—
7	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₂₃	C₄	1	2	yes	—
10	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×M₆₄	—	2	2	no	—
12	C₂₃×M₆₄	C₂²	1	4	yes	—
13	C₂×C₂₃×C₃₂	—	2	2	no	—
14	C₂×#41×C₂₃	C₈	1	1	yes	characteristic
15	C₂₃×#131	—	2	2	no	—
16	C₂₃×#131	—	2	2	no	—
17	C₂×M₁₆×C₂₃	C₈	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	C₈	1	2	yes	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₃×M₃₂	—	2	2	no	—
22	C₂₃×M₃₂	—	2	4	no	—
23	C₂×C₁₆×C₂₃	—	2	2	no	—
24	C₂×C₁₆×C₂₃	—	2	2	no	—
25	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
28	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
29	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×C₃₂	—	2	4	no	—
31	#41×C₂₃	—	2	2	no	—
32	#41×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	C₁₆	1	1	yes	characteristic
34	C₂×D₄×C₂₃	C₁₆	1	2	yes	—
35	C₂×D₄×C₂₃	C₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₂₃	—	2	2	no	—
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×C₈	1	2	yes	—
44	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₂₃	—	2	2	no	—
48	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
51	#13556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₂×C₁₆	1	2	yes	—
53	C₂³×C₂₃	C₂×C₁₆	1	1	yes	characteristic
54	Q₈×C₂₃	—	2	2	no	—
55	D₄×C₂₃	—	2	4	no	—
56	D₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
62	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
64	C₈×C₂₃	—	2	2	no	—
65	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
66	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
69	C₂×M₆₄	C₂×C₂₃	1	2	yes	complemented
70	#1400	C₂×C₂₃	1	4	yes	complemented
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	4	1	no	—
73	C₂²×C₂₃	—	4	2	no	—
74	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
75	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
78	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
80	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
81	#504	—	2	2	no	—
82	#504	—	2	2	no	—
83	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
84	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
85	C₂²×C₁₆	C₄×C₂₃	1	2	yes	—
86	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
87	M₆₄	—	2	2	no	—
88	M₆₄	C₂²×C₂₃	1	4	yes	—
89	C₂×C₃₂	—	2	2	no	—
90	C₂×C₂₃	—	2	1	no	—
91	C₂×C₂₃	—	2	2	no	—
92	C₂×C₂₃	—	4	2	no	—
93	C₂×C₂₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
94	C₂×#41	C₈×C₂₃	1	1	yes	characteristic
95	#131	—	2	2	no	—
96	#131	—	2	2	no	—
97	C₂×M₁₆	C₈×C₂₃	1	1	yes	characteristic
98	C₂×M₁₆	C₈×C₂₃	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
106	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
107	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
108	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
109	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
110	C₃₂	—	2	4	no	—
111	C₂₃	#13556	1	1	yes	characteristic, direct product factor, Sylow
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	C₂×Q₈	C₁₆×C₂₃	1	1	yes	characteristic
115	C₂×D₄	C₁₆×C₂₃	1	2	yes	—
116	C₂×D₄	C₁₆×C₂₃	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
118	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
124	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
125	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
126	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
127	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
130	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
131	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
132	C₂³	C₂×C₁₆×C₂₃	1	2	yes	—
133	C₂³	C₂×C₁₆×C₂₃	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
141	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
142	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
143	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
146	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
147	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
152	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
155	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
156	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₃	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂×C₂₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₂₃	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₃ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×M₆₄	C₂	semidirect
3. C₂₃×#1400	C₂	semidirect
51. #13556	C₂₃	direct (N × H)
69. C₂×M₆₄	C₂×C₂₃	semidirect
70. #1400	C₂×C₂₃	semidirect
111. C₂₃	#13556	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×#504	2
C₂×C₂₃×M₆₄	2
C₂₃×#1400	2
#13556	23

Subgroups — C23×#13556 #19161182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13556 #19161182