Subgroups — C₂₃×#12032 #19159658

Order 5888.

222 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1371	C₂	1	4	yes	maximal, complemented
5	C₂₃×#423	—	2	1	no	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂₃×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#335	—	2	1	no	—
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#335	C₂²	1	2	yes	—
14	C₂₃×#335	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₃	—	4	1	no	—
16	C₂²×C₈×C₂₃	—	4	1	no	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
21	C₂₃×C₄⋊C₈	—	2	2	no	—
22	C₂₃×C₄⋊C₈	—	2	2	no	—
23	C₂₃×C₄⋊C₈	—	4	1	no	—
24	C₂₃×C₄⋊C₈	—	4	1	no	—
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₂₃	—	8	2	no	—
33	C₂×C₈×C₂₃	—	2	2	no	—
34	C₂×C₈×C₂₃	—	4	1	no	—
35	C₂×C₈×C₂₃	—	4	2	no	—
36	C₂×C₈×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	8	2	no	—
38	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₂₃	D₈	1	1	yes	characteristic
44	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
45	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₂₃	—	2	4	no	—
47	#12032	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₂₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₂₃	—	2	1	no	—
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	—	4	1	no	—
52	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
55	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
56	C₂×C₄×C₂₃	Q₃₂	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₂₃	—	2	1	no	—
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	#131	1	2	yes	—
66	#2162	C₂×C₂₃	1	1	yes	characteristic
67	#1567	C₂×C₂₃	1	1	yes	characteristic
68	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
69	#1371	C₂×C₂₃	1	4	yes	complemented
70	C₂²×C₂₃	—	2	1	no	—
71	C₂²×C₂₃	—	2	1	no	—
72	C₂²×C₂₃	#366	1	2	yes	—
73	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
74	C₄×C₂₃	—	2	1	no	—
75	C₄×C₂₃	—	2	1	no	—
76	C₄×C₂₃	—	2	1	no	—
77	C₄×C₂₃	#366	1	2	yes	—
78	C₄×C₂₃	#424	1	1	yes	characteristic
79	C₄×C₂₃	#430	1	1	yes	characteristic
80	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
81	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
82	#423	—	2	1	no	—
83	C₂×C₄⋊C₈	—	2	1	no	—
84	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
85	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
86	C₂²⋊C₁₆	C₄×C₂₃	1	2	yes	—
87	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
88	#335	—	2	1	no	—
89	#335	—	2	1	no	—
90	#335	C₂²×C₂₃	1	2	yes	—
91	#335	C₂²×C₂₃	1	2	yes	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	#1571	1	1	yes	characteristic
94	C₂×C₂₃	C₂×#366	1	1	yes	characteristic
95	C₂×C₂₃	#2161	1	1	yes	characteristic
96	C₂×M₁₆	—	4	1	no	—
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	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
107	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
108	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
109	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
110	C₂₃	#12032	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
114	M₁₆	—	8	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
124	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
125	C₄²	D₈×C₂₃	1	1	yes	characteristic
126	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
127	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₂³	C₂₃×#107	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
134	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
135	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
136	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
137	C₂×C₄	C₂₃×Q₃₂	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	2	no	—
146	C₈	C₂₃×#131	1	2	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂₃×#366	1	2	yes	—
150	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	C₂₃×#366	1	2	yes	—
155	C₄	C₂₃×#424	1	1	yes	characteristic
156	C₄	C₂₃×#430	1	1	yes	characteristic
157	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
158	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	C₂₃×#1571	1	1	yes	characteristic
161	C₂	C₂×C₂₃×#366	1	1	yes	characteristic
162	C₂	C₂₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂³×C₂₃	#107
LCS	G ▷ C₂×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
4. C₂₃×#1371	C₂	semidirect
47. #12032	C₂₃	direct (N × H)
69. #1371	C₂×C₂₃	semidirect
110. C₂₃	#12032	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2162	2
C₂₃×#1567	2
C₂×C₂₃×#335	2
C₂₃×#1371	2
#12032	23

Subgroups — C23×#12032 #19159658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12032 #19159658