Subgroups — C₂×M₃₂×D₂₃ #15686491

Order 2944.

400 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10624980	C₂	1	2	yes	maximal, complemented
4	M₃₂×D₂₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₈×D₂₃	C₄	1	4	yes	—
11	C₂×C₈×D₂₃	C₄	1	1	yes	characteristic
12	C₂×C₈×D₂₃	C₄	1	1	yes	characteristic
13	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₃	C₂²	1	4	yes	—
15	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×M₃₂	C₂²	1	4	yes	complemented
17	C₂×C₁₆×C₂₃	C₂²	1	2	yes	complemented
18	#10624994	C₂²	1	4	yes	complemented
19	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	complemented
20	#10624980	C₂²	1	8	yes	complemented
21	C₁₆×D₂₃	C₂²	1	8	yes	complemented
22	C₂³×D₂₃	C₈	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₂₃	C₈	1	1	yes	characteristic
25	C₂×C₄×D₂₃	C₈	1	1	yes	characteristic
26	C₂×C₄×D₂₃	C₈	1	4	yes	—
27	C₂×C₄×D₂₃	C₈	1	1	yes	characteristic
28	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₈×D₂₃	C₂×C₄	1	2	yes	—
39	C₈×D₂₃	C₂×C₄	1	4	yes	—
40	C₈×D₂₃	C₂×C₄	1	4	yes	—
41	C₈×D₂₃	C₂×C₄	1	2	yes	—
42	C₈×D₂₃	C₂³	1	2	yes	—
43	C₈×D₂₃	C₂³	1	2	yes	—
44	C₁₆×C₂₃	C₂³	1	4	yes	complemented
45	C₂₃⋊C₁₆	C₂³	1	4	yes	complemented
46	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×D₂₃	—	2	4	no	—
48	C₂²×D₂₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₂₃	C₂×C₈	1	4	yes	—
50	C₂²×D₂₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
56	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
57	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
58	C₄×D₂₃	C₂×C₈	1	4	yes	—
59	C₄×D₂₃	C₂×C₈	1	2	yes	—
60	C₄×D₂₃	C₂×C₈	1	2	yes	—
61	C₄×D₂₃	C₂×C₈	1	4	yes	—
62	C₄×D₂₃	C₂²×C₄	1	2	yes	—
63	C₄×D₂₃	C₂²×C₄	1	2	yes	—
64	C₈×C₂₃	C₂²×C₄	1	2	yes	—
65	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
68	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
69	C₂²×M₃₂	—	23	1	no	maximal, Sylow
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
72	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×D₂₃	—	2	2	no	—
74	C₂×D₂₃	—	2	8	no	—
75	C₂×D₂₃	M₃₂	1	4	yes	direct product factor
76	C₂×D₂₃	C₂²×C₈	1	2	yes	—
77	C₂×D₂₃	C₂²×C₈	1	2	yes	—
78	C₄×C₂₃	C₂²×C₈	1	2	yes	—
79	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
80	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	Dic₂₃	C₂²×C₈	1	2	yes	—
82	Dic₂₃	C₂²×C₈	1	2	yes	—
83	C₂³×C₈	—	23	1	no	—
84	C₂×M₃₂	—	23	1	no	—
85	C₂×M₃₂	—	23	8	no	—
86	C₂×M₃₂	—	23	2	no	—
87	C₂×M₃₂	D₂₃	1	1	yes	characteristic, direct product factor
88	C₂²×C₁₆	—	23	2	no	—
89	C₂×C₂₃	—	2	2	no	—
90	C₂×C₂₃	C₂×M₃₂	1	2	yes	complemented
91	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
92	D₂₃	—	2	2	no	—
93	D₂₃	C₂×M₃₂	1	4	yes	direct product factor
94	C₂³×C₄	—	23	1	no	—
95	C₂²×C₈	—	23	1	no	—
96	C₂²×C₈	—	23	4	no	—
97	C₂²×C₈	—	23	1	no	—
98	C₂²×C₈	—	23	1	no	—
99	C₂²×C₈	—	23	4	no	—
100	C₂²×C₈	—	23	1	no	—
101	C₂²×C₈	—	23	1	no	—
102	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
103	M₃₂	—	23	4	no	—
104	M₃₂	—	23	8	no	—
105	M₃₂	C₂×D₂₃	1	4	yes	direct product factor
106	C₂×C₁₆	—	23	2	no	—
107	C₂×C₁₆	—	23	8	no	—
108	C₂×C₁₆	C₂×D₂₃	1	2	yes	complemented
109	C₂₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
110	C₂⁴	—	23	1	no	—
111	C₂²×C₄	—	23	1	no	—
112	C₂²×C₄	—	23	1	no	—
113	C₂²×C₄	—	23	4	no	—
114	C₂²×C₄	—	23	1	no	—
115	C₂²×C₄	—	23	4	no	—
116	C₂²×C₄	—	23	1	no	—
117	C₂²×C₄	—	23	1	no	—
118	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
119	C₂×C₈	—	23	4	no	—
120	C₂×C₈	—	23	1	no	—
121	C₂×C₈	—	23	1	no	—
122	C₂×C₈	—	23	4	no	—
123	C₂×C₈	—	23	4	no	—
124	C₂×C₈	—	23	2	no	—
125	C₂×C₈	—	23	2	no	—
126	C₂×C₈	—	23	2	no	—
127	C₂×C₈	—	23	2	no	—
128	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
129	C₂×C₈	C₄×D₂₃	1	2	yes	—
130	C₂×C₈	C₂²×D₂₃	1	2	yes	—
131	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
132	C₁₆	—	23	4	no	—
133	C₁₆	C₂²×D₂₃	1	4	yes	complemented
134	C₂³	—	23	1	no	—
135	C₂³	—	23	4	no	—
136	C₂³	—	23	1	no	—
137	C₂³	—	46	4	no	—
138	C₂³	C₈×D₂₃	1	1	yes	characteristic
139	C₂×C₄	—	23	4	no	—
140	C₂×C₄	—	23	1	no	—
141	C₂×C₄	—	23	1	no	—
142	C₂×C₄	—	23	4	no	—
143	C₂×C₄	—	23	2	no	—
144	C₂×C₄	—	23	4	no	—
145	C₂×C₄	—	23	2	no	—
146	C₂×C₄	—	23	2	no	—
147	C₂×C₄	—	23	2	no	—
148	C₂×C₄	C₈×D₂₃	1	2	yes	—
149	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
152	C₈	—	23	2	no	—
153	C₈	—	23	2	no	—
154	C₈	C₂×C₄×D₂₃	1	2	yes	—
155	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
156	C₈	C₂³×D₂₃	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	—	23	2	no	—
159	C₂²	—	23	4	no	—
160	C₂²	—	23	2	no	—
161	C₂²	—	46	8	no	—
162	C₂²	—	46	2	no	—
163	C₂²	C₂×C₈×D₂₃	1	2	yes	—
164	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
165	C₄	—	23	2	no	—
166	C₄	—	23	2	no	—
167	C₄	C₂×C₈×D₂₃	1	2	yes	—
168	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
169	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	23	4	no	—
172	C₂	—	46	2	no	—
173	C₂	M₃₂×D₂₃	1	2	yes	direct product factor
174	C₂	C₂²×C₈×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂³×C₈
Frattini	C₈	C₂³×D₂₃
Fitting	C₂×C₂₃×M₃₂	C₂
socle	C₂²×C₂₃	C₂²×C₈
prime radical	C₂³×D₂₃	C₈
cosocle	C₈×C₂₃	C₂⁴
hypercenter	C₂×M₃₂	D₂₃
hypocenter	C₂₃	C₂²×M₃₂
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#10624994	C₂	semidirect
3. C₂×#10624980	C₂	semidirect
4. M₃₂×D₂₃	C₂	direct (N × H)
5. C₂×C₁₆×D₂₃	C₂	semidirect
6. C₂×C₂₃×M₃₂	C₂	semidirect
16. C₂₃×M₃₂	C₂²	semidirect
17. C₂×C₁₆×C₂₃	C₂²	semidirect
18. #10624994	C₂²	semidirect
19. C₂×C₂₃⋊C₁₆	C₂²	semidirect
20. #10624980	C₂²	semidirect
21. C₁₆×D₂₃	C₂²	semidirect
44. C₁₆×C₂₃	C₂³	semidirect
45. C₂₃⋊C₁₆	C₂³	semidirect
75. C₂×D₂₃	M₃₂	direct (N × H)
87. C₂×M₃₂	D₂₃	direct (N × H)
90. C₂×C₂₃	C₂×M₃₂	semidirect
93. D₂₃	C₂×M₃₂	direct (N × H)
105. M₃₂	C₂×D₂₃	direct (N × H)
108. C₂×C₁₆	C₂×D₂₃	semidirect
109. C₂₃	C₂²×M₃₂	semidirect
133. C₁₆	C₂²×D₂₃	semidirect
173. C₂	M₃₂×D₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₂₃	2
C₂×#10624994	2
C₂×#10624980	2
M₃₂×D₂₃	2
C₂×C₁₆×D₂₃	2
C₂×C₂₃×M₃₂	2
C₂²×M₃₂	23

Subgroups — C2×M32×D23 #15686491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₃₂×D₂₃ #15686491