Subgroups — #15671701

Order 2944.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67262	C₂	1	2	yes	maximal
2	C₄×#67262	C₂	1	1	yes	maximal, characteristic
3	#13092139	C₂	1	2	yes	maximal, complemented
4	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67262	C₄	1	2	yes	—
7	C₂×#67262	C₄	1	4	yes	—
8	C₄²×D₂₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	#10624997	C₄	1	2	yes	—
12	#10624997	C₄	1	4	yes	—
13	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₄²×C₂₃	C₂³	1	1	yes	characteristic
20	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
24	#67262	—	2	4	no	—
25	#67262	—	2	8	no	—
26	C₂²×D₂₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
29	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
30	C₂×Dic₂₃	C₄²	1	2	yes	—
31	C₄×D₂₃	—	2	2	no	—
32	C₄×D₂₃	—	2	4	no	—
33	C₈×C₂₃	—	2	4	no	—
34	C₈×C₂₃	M₁₆	1	4	yes	complemented
35	C₂₃⋊C₈	—	2	2	no	—
36	C₂₃⋊C₈	M₁₆	1	8	yes	complemented
37	#1455	—	23	1	no	maximal, Sylow
38	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×D₂₃	—	2	1	no	—
40	C₂×D₂₃	—	2	2	no	—
41	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
42	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
43	Dic₂₃	—	2	1	no	—
44	Dic₂₃	—	2	2	no	—
45	C₄×M₁₆	—	23	2	no	—
46	C₄×M₁₆	—	23	1	no	—
47	#322	—	23	2	no	—
48	#322	D₂₃	1	1	yes	characteristic, complemented
49	C₈²	—	23	1	no	—
50	C₂×C₂₃	C₄×M₁₆	1	1	yes	characteristic
51	C₂×C₂₃	C₄×M₁₆	1	2	yes	—
52	D₂₃	—	4	1	no	—
53	C₂×M₁₆	—	23	4	no	—
54	C₂×M₁₆	—	23	2	no	—
55	C₂×C₄²	—	23	1	no	—
56	#97	—	23	4	no	—
57	#97	—	23	2	no	—
58	C₄×C₈	—	23	2	no	—
59	C₄×C₈	—	23	1	no	—
60	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
61	C₄×C₈	C₂×D₂₃	1	2	yes	—
62	C₂₃	#1455	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₄	—	23	1	no	—
64	C₂²×C₄	—	23	2	no	—
65	M₁₆	—	46	8	no	—
66	M₁₆	—	46	4	no	—
67	C₂×C₈	—	23	4	no	—
68	C₂×C₈	—	23	2	no	—
69	C₂×C₈	C₄×D₂₃	1	4	yes	—
70	C₂×C₈	C₄×D₂₃	1	2	yes	—
71	C₄²	—	23	1	no	—
72	C₄²	—	23	2	no	—
73	C₄²	C₂²×D₂₃	1	1	yes	characteristic
74	C₂³	—	23	1	no	—
75	C₂×C₄	—	23	1	no	—
76	C₂×C₄	—	23	2	no	—
77	C₂×C₄	—	46	2	no	—
78	C₂×C₄	—	46	4	no	—
79	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	—	23	8	no	—
83	C₈	—	46	2	no	—
84	C₈	#67262	1	4	yes	complemented
85	C₂²	—	46	1	no	—
86	C₂²	—	46	2	no	—
87	C₂²	C₄²×D₂₃	1	1	yes	characteristic
88	C₄	—	46	1	no	—
89	C₄	—	46	2	no	—
90	C₄	C₂×#67262	1	2	yes	—
91	C₄	M₁₆×D₂₃	1	4	yes	—
92	C₂	—	92	1	no	—
93	C₂	C₄×#67262	1	1	yes	characteristic
94	C₂	#13092369	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂×C₄²
Fitting	C₂₃×#322	C₂
prime radical	C₂²×D₂₃	C₄²
cosocle	C₄²×C₂₃	C₂³
hypercenter	#322	D₂₃
hypocenter	C₂₃	#1455
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #13092139	C₂	semidirect
4. C₈×C₂₃⋊C₈	C₂	semidirect
5. C₂₃×#322	C₂	semidirect
34. C₈×C₂₃	M₁₆	semidirect
36. C₂₃⋊C₈	M₁₆	semidirect
48. #322	D₂₃	semidirect
62. C₂₃	#1455	semidirect
84. C₈	#67262	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67262	2
#13092139	2
C₈×C₂₃⋊C₈	2
C₂₃×#322	2
#1455	23