Subgroups — #13092380

Order 1472.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#67267	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₂₃)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂³×D₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₃	—	2	2	no	—
9	C₂×C₄×D₂₃	—	2	2	no	—
10	C₂×C₄×D₂₃	C₄	1	2	yes	—
11	C₂×C₄×D₂₃	C₂²	1	2	yes	—
12	M₁₆×C₂₃	—	2	2	no	—
13	C₂×C₈×C₂₃	C₂²	1	2	yes	complemented
14	#67267	—	2	2	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	complemented
16	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₂₃	—	2	2	no	—
18	C₂²×D₂₃	—	2	4	no	—
19	C₂²×D₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	D₄	1	2	yes	—
22	C₂×C₄×C₂₃	D₄	1	2	yes	—
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	2	no	—
25	C₂×Dic₂₃	C₂×C₄	1	2	yes	—
26	C₄×D₂₃	—	2	4	no	—
27	C₄×D₂₃	—	2	2	no	—
28	C₄×D₂₃	—	2	2	no	—
29	C₈×C₂₃	—	2	2	no	—
30	C₂₃⋊C₈	—	2	2	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₂₃	—	2	8	no	—
35	C₂×D₂₃	—	2	2	no	—
36	C₂×D₂₃	—	2	2	no	—
37	C₂×D₂₃	M₁₆	1	4	yes	complemented
38	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
39	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	Dic₂₃	—	2	2	no	—
42	#407	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	2	2	no	—
44	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₂₃	C₂×M₁₆	1	2	yes	—
46	D₂₃	—	2	4	no	—
47	C₂³×C₄	—	23	1	no	—
48	C₂×M₁₆	—	23	1	no	—
49	C₂×M₁₆	D₂₃	1	1	yes	characteristic, complemented
50	C₂²⋊C₈	—	23	4	no	—
51	C₂₃	#407	1	1	yes	characteristic, complemented, Sylow
52	C₂⁴	—	23	1	no	—
53	C₂²×C₄	—	23	1	no	—
54	C₂²×C₄	—	23	2	no	—
55	C₂²×C₄	—	23	2	no	—
56	C₂²×C₄	—	46	2	no	—
57	C₂²×C₄	—	46	2	no	—
58	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	46	2	no	—
61	C₂×C₈	—	23	2	no	—
62	C₂×C₈	C₂×D₂₃	1	2	yes	complemented
63	C₂³	—	23	2	no	—
64	C₂³	—	46	4	no	—
65	C₂³	—	46	2	no	—
66	C₂³	C₄×D₂₃	1	1	yes	characteristic
67	C₂×C₄	—	23	2	no	—
68	C₂×C₄	—	46	2	no	—
69	C₂×C₄	—	46	4	no	—
70	C₂×C₄	—	46	2	no	—
71	C₂×C₄	—	46	2	no	—
72	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
73	C₂×C₄	D₉₂	1	2	yes	—
74	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
75	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	46	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	23	4	no	—
80	C₂²	—	46	8	no	—
81	C₂²	—	46	2	no	—
82	C₂²	—	46	2	no	—
83	C₂²	(C₂×D₂₃)⋊C₄	1	2	yes	—
84	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
85	C₄	—	46	2	no	—
86	C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
87	C₄	C₂×D₉₂	1	1	yes	characteristic
88	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	46	4	no	—
91	C₂	M₁₆×D₂₃	1	2	yes	—
92	C₂	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂×M₁₆×C₂₃	C₂
prime radical	C₂³×D₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₃
hypocenter	C₂₃	#407
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×C₂₃	C₂	semidirect
3. C₂×#67267	C₂	semidirect
4. (C₂×D₂₃)⋊C₈	C₂	semidirect
13. C₂×C₈×C₂₃	C₂²	semidirect
15. C₂×C₂₃⋊C₈	C₂²	semidirect
37. C₂×D₂₃	M₁₆	semidirect
49. C₂×M₁₆	D₂₃	semidirect
51. C₂₃	#407	semidirect
62. C₂×C₈	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₃	2
C₂×M₁₆×C₂₃	2
C₂×#67267	2
(C₂×D₂₃)⋊C₈	2
#407	23