Subgroups — #15681814

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₃	C₂	1	2	yes	maximal
2	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
3	#13092374	C₂	1	4	yes	maximal, complemented
4	#13092371	C₂	1	4	yes	maximal, complemented
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×Dic₂₃	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×(C₂×D₂₃)⋊C₄	C₄	1	2	yes	—
12	C₂×Dic₂₃⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	2	yes	—
15	M₁₆×D₂₃	C₄	1	16	yes	complemented
16	C₂×#67262	C₂²	1	4	yes	—
17	C₂×C₈×D₂₃	C₂²	1	4	yes	—
18	#10625043	C₄	1	4	yes	—
19	#10625043	C₂²	1	2	yes	—
20	#10625043	C₂²	1	2	yes	—
21	C₂₃×#97	C₂²	1	2	yes	complemented
22	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
23	#10624997	C₂²	1	4	yes	complemented
24	C₈×Dic₂₃	C₂²	1	4	yes	complemented
25	#10624985	C₂²	1	2	yes	complemented
26	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
27	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₂₃	C₂×C₄	1	8	yes	—
34	C₂×C₄×D₂₃	C₂³	1	2	yes	—
35	M₁₆×C₂₃	C₂×C₄	1	8	yes	—
36	C₂×C₈×C₂₃	C₂³	1	4	yes	—
37	C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
41	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
42	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
43	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
44	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
45	C₄×Dic₂₃	C₂³	1	2	yes	—
46	#67267	C₂×C₄	1	8	yes	—
47	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
48	#67262	C₂×C₄	1	16	yes	complemented
49	C₈×D₂₃	C₂×C₄	1	16	yes	complemented
50	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×D₂₃	—	4	2	no	—
52	C₂²×D₂₃	C₄²	1	4	yes	—
53	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₃	—	4	2	no	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₂₃	—	4	2	no	—
61	C₂×Dic₂₃	C₄²	1	4	yes	—
62	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
63	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
64	C₄×D₂₃	C₄²	1	8	yes	—
65	C₄×D₂₃	C₂²×C₄	1	4	yes	—
66	C₄×D₂₃	C₂²×C₄	1	4	yes	—
67	C₈×C₂₃	C₂²×C₄	1	8	yes	—
68	C₂₃⋊C₈	C₂²×C₄	1	8	yes	—
69	#2873	—	23	1	no	maximal, Sylow
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	C₂×C₄²	1	2	yes	—
72	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₂₃	—	2	4	no	—
74	C₂×D₂₃	—	4	4	no	—
75	C₂×D₂₃	C₂×C₄²	1	4	yes	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	2	2	no	—
78	C₄×C₂₃	C₂×C₄²	1	2	yes	—
79	C₄×C₂₃	C₂³×C₄	1	2	yes	—
80	Dic₂₃	—	2	4	no	—
81	Dic₂₃	C₂×C₄²	1	4	yes	—
82	C₂²×M₁₆	—	23	2	no	—
83	C₂×#117	—	23	1	no	—
84	#405	—	23	4	no	—
85	#405	—	23	4	no	—
86	C₄×M₁₆	—	23	2	no	—
87	C₄×M₁₆	—	23	1	no	—
88	C₄×M₁₆	D₂₃	1	1	yes	characteristic, complemented
89	C₂×C₂₃	—	2	2	no	—
90	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
91	C₂×C₂₃	#568	1	2	yes	—
92	D₂₃	—	2	4	no	—
93	C₂³×C₄	—	23	1	no	—
94	C₂×M₁₆	—	23	2	no	—
95	C₂×M₁₆	—	23	16	no	—
96	C₂×M₁₆	—	23	4	no	—
97	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
98	C₂²×C₈	—	23	4	no	—
99	#117	—	23	4	no	—
100	#117	—	23	2	no	—
101	#117	—	23	2	no	—
102	C₂×C₄⋊C₄	—	23	2	no	—
103	C₂×C₂²⋊C₄	—	23	2	no	—
104	C₂×C₄²	—	23	1	no	—
105	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
106	#97	—	23	4	no	—
107	#97	—	23	2	no	—
108	#97	C₂×D₂₃	1	2	yes	complemented
109	C₄×C₈	—	23	4	no	—
110	C₄×C₈	—	23	2	no	—
111	C₄×C₈	C₂×D₂₃	1	2	yes	complemented
112	C₂₃	#2873	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴	—	23	1	no	—
114	C₂²×C₄	—	23	1	no	—
115	C₂²×C₄	—	23	2	no	—
116	C₂²×C₄	—	23	8	no	—
117	C₂²×C₄	—	23	2	no	—
118	C₂²×C₄	—	23	2	no	—
119	C₂²×C₄	C₄×D₂₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	M₁₆	—	23	8	no	—
122	M₁₆	—	23	16	no	—
123	M₁₆	C₄×D₂₃	1	8	yes	—
124	C₂×C₈	—	23	4	no	—
125	C₂×C₈	—	23	16	no	—
126	C₂×C₈	C₂²×D₂₃	1	4	yes	—
127	C₄⋊C₄	—	23	4	no	—
128	C₄⋊C₄	—	23	4	no	—
129	C₂²⋊C₄	—	23	4	no	—
130	C₂²⋊C₄	—	23	4	no	—
131	C₄²	—	23	2	no	—
132	C₄²	—	23	2	no	—
133	C₄²	C₄×D₂₃	1	2	yes	—
134	C₄²	C₂²×D₂₃	1	1	yes	characteristic
135	C₄²	C₂²×D₂₃	1	1	yes	characteristic
136	C₂³	—	23	2	no	—
137	C₂³	—	23	4	no	—
138	C₂³	—	92	2	no	—
139	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	23	4	no	—
142	C₂×C₄	—	23	2	no	—
143	C₂×C₄	—	23	8	no	—
144	C₂×C₄	—	23	4	no	—
145	C₂×C₄	—	23	4	no	—
146	C₂×C₄	—	23	4	no	—
147	C₂×C₄	—	92	2	no	—
148	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
149	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
150	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
152	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
153	C₈	—	23	8	no	—
154	C₈	C₂×C₄×D₂₃	1	8	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	23	4	no	—
157	C₂²	—	46	4	no	—
158	C₂²	—	92	4	no	—
159	C₂²	C₄²×D₂₃	1	2	yes	—
160	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	23	4	no	—
164	C₄	—	46	4	no	—
165	C₄	C₄²×D₂₃	1	2	yes	—
166	C₄	C₂²×C₄×D₂₃	1	2	yes	—
167	C₂	—	2	2	no	—
168	C₂	—	46	4	no	—
169	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
170	C₂	#13093183	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Fitting	C₄×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#2873
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13092374	C₂	semidirect
4. #13092371	C₂	semidirect
5. C₄×#67267	C₂	semidirect
6. M₁₆×Dic₂₃	C₂	semidirect
7. C₄×M₁₆×C₂₃	C₂	semidirect
15. M₁₆×D₂₃	C₄	semidirect
21. C₂₃×#97	C₂²	semidirect
22. C₄×C₈×C₂₃	C₂²	semidirect
23. #10624997	C₂²	semidirect
24. C₈×Dic₂₃	C₂²	semidirect
25. #10624985	C₂²	semidirect
26. C₄×C₂₃⋊C₈	C₂²	semidirect
48. #67262	C₂×C₄	semidirect
49. C₈×D₂₃	C₂×C₄	semidirect
88. C₄×M₁₆	D₂₃	semidirect
108. #97	C₂×D₂₃	semidirect
111. C₄×C₈	C₂×D₂₃	semidirect
112. C₂₃	#2873	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₃	2
C₂×#10625043	2
#13092374	2
#13092371	2
C₄×#67267	2
M₁₆×Dic₂₃	2
C₄×M₁₆×C₂₃	2
#2873	23