Subgroups — C₂₃×#19604 #19167230

Order 5888.

706 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	C₂₃×#3170	C₂	1	4	yes	maximal
2	C₂₃×#3160	C₂	1	2	yes	maximal
3	C₂₃×#1713	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1710	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1707	C₂	1	4	yes	maximal, complemented
6	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#533	C₂²	1	4	yes	—
8	C₂₃×#500	C₂²	1	4	yes	—
9	C₂₃×#500	C₂²	1	4	yes	—
10	C₂₃×#499	—	2	4	no	—
11	C₂₃×#498	C₂²	1	4	yes	—
12	C₂₃×#494	C₂²	1	4	yes	—
13	C₂₃×#487	C₂²	1	8	yes	—
14	C₂₃×#467	—	2	8	no	—
15	C₄×Q₁₆×C₂₃	—	2	8	no	—
16	C₂₃×#427	—	2	4	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₈²×C₂₃	C₂²	1	2	yes	—
20	C₈²×C₂₃	C₂²	1	1	yes	characteristic
21	C₈²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₂₃	—	2	8	no	—
23	C₂²×C₈×C₂₃	D₄	1	4	yes	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
26	C₂₃×#125	—	2	4	no	—
27	C₂₃×C₂²⋊Q₈	—	2	8	no	—
28	C₄×Q₈×C₂₃	—	2	8	no	—
29	C₂₃×#117	—	2	4	no	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×#107	—	2	8	no	—
32	C₂₃×#107	—	2	8	no	—
33	C₂₃×#106	—	2	8	no	—
34	C₂₃×#103	—	2	16	no	—
35	C₄×C₈×C₂₃	D₄	1	4	yes	—
36	C₄×C₈×C₂₃	D₄	1	2	yes	—
37	C₄×C₈×C₂₃	C₂³	1	2	yes	—
38	C₄×C₈×C₂₃	C₂³	1	2	yes	—
39	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	—	2	8	no	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	Q₁₆×C₂₃	—	4	16	no	—
45	C₂×C₈×C₂₃	—	2	8	no	—
46	C₂×C₈×C₂₃	Q₁₆	1	8	yes	complemented
47	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
48	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
51	C₄⋊C₄×C₂₃	—	2	8	no	—
52	C₄⋊C₄×C₂₃	—	2	8	no	—
53	C₄⋊C₄×C₂₃	—	4	8	no	—
54	C₂²⋊C₄×C₂₃	—	4	4	no	—
55	C₄²×C₂₃	—	4	4	no	—
56	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₂₃	C₂×D₄	1	2	yes	—
58	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
59	#19604	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	#127	1	1	yes	characteristic
61	Q₈×C₂₃	—	4	16	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	4	8	no	—
64	C₂×C₄×C₂₃	#127	1	2	yes	—
65	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
67	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
69	C₈×C₂₃	—	2	4	no	—
70	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
71	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
72	C₈×C₂₃	#135	1	8	yes	—
73	#3170	C₂×C₂₃	1	4	yes	—
74	#3160	C₂×C₂₃	1	2	yes	—
75	#1713	C₂×C₂₃	1	2	yes	complemented
76	#1710	C₂×C₂₃	1	2	yes	complemented
77	#1707	C₂×C₂₃	1	4	yes	complemented
78	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
79	C₂²×C₂₃	—	2	2	no	—
80	C₂²×C₂₃	#494	1	2	yes	—
81	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
82	C₄×C₂₃	—	2	2	no	—
83	C₄×C₂₃	—	8	8	no	—
84	C₄×C₂₃	#494	1	2	yes	—
85	C₄×C₂₃	C₂²×Q₁₆	1	2	yes	—
86	C₄×C₂₃	C₂×#135	1	4	yes	—
87	#533	C₂²×C₂₃	1	4	yes	—
88	#500	C₂²×C₂₃	1	4	yes	—
89	#500	C₂²×C₂₃	1	4	yes	—
90	#499	—	2	4	no	—
91	#498	C₂²×C₂₃	1	4	yes	—
92	#494	C₂²×C₂₃	1	4	yes	—
93	#487	C₂²×C₂₃	1	8	yes	—
94	#467	—	2	8	no	—
95	C₄×Q₁₆	—	2	8	no	—
96	#427	—	2	4	no	—
97	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
99	C₈²	C₂²×C₂₃	1	2	yes	—
100	C₈²	C₂²×C₂₃	1	1	yes	characteristic
101	C₈²	C₂²×C₂₃	1	1	yes	characteristic
102	C₂×C₂₃	—	2	2	no	—
103	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
104	C₂×C₂₃	#3147	1	2	yes	—
105	C₂×Q₁₆	—	2	8	no	—
106	C₂²×C₈	D₄×C₂₃	1	4	yes	—
107	C₂²×C₈	D₄×C₂₃	1	2	yes	—
108	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
109	#125	—	2	4	no	—
110	C₂²⋊Q₈	—	2	8	no	—
111	C₄×Q₈	—	2	8	no	—
112	#117	—	2	4	no	—
113	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
114	#107	—	2	8	no	—
115	#107	—	2	8	no	—
116	#106	—	2	8	no	—
117	#103	—	2	16	no	—
118	C₄×C₈	D₄×C₂₃	1	4	yes	—
119	C₄×C₈	D₄×C₂₃	1	2	yes	—
120	C₄×C₈	C₂³×C₂₃	1	2	yes	—
121	C₄×C₈	C₂³×C₂₃	1	2	yes	—
122	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	C₂₃	#19604	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×Q₈	—	2	8	no	—
126	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
128	Q₁₆	—	4	16	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	Q₁₆×C₂₃	1	8	yes	complemented
131	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
132	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
133	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
134	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	8	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₂×D₄×C₂₃	1	2	yes	—
141	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
143	C₂³	C₂₃×#127	1	1	yes	characteristic
144	Q₈	—	4	16	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	C₂₃×#127	1	2	yes	—
148	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
149	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
150	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
154	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
155	C₈	C₂₃×#135	1	8	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂₃×#494	1	2	yes	—
158	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	8	8	no	—
161	C₄	C₂₃×#494	1	2	yes	—
162	C₄	C₂²×Q₁₆×C₂₃	1	2	yes	—
163	C₄	C₂×C₂₃×#135	1	4	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
166	C₂	C₂₃×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#127
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1713	C₂	semidirect
4. C₂₃×#1710	C₂	semidirect
5. C₂₃×#1707	C₂	semidirect
46. C₂×C₈×C₂₃	Q₁₆	semidirect
59. #19604	C₂₃	direct (N × H)
75. #1713	C₂×C₂₃	semidirect
76. #1710	C₂×C₂₃	semidirect
77. #1707	C₂×C₂₃	semidirect
124. C₂₃	#19604	direct (N × H)
130. C₂×C₈	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3170	2
C₂₃×#3160	2
C₂₃×#1713	2
C₂₃×#1710	2
C₂₃×#1707	2
C₂×C₈²×C₂₃	2
#19604	23

Subgroups — C23×#19604 #19167230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19604 #19167230