Subgroups — #15673528

Order 2944.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092283	C₂	1	2	yes	maximal
2	#13092180	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
7	C₄×C₈×C₂₃	C₄	1	4	yes	complemented
8	C₂₃⋊#95	—	2	4	no	—
9	C₂²×C₄×C₂₃	D₄	1	2	yes	—
10	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
11	C₂²×Dic₂₃	—	2	4	no	—
12	C₂×C₈×C₂₃	—	2	4	no	—
13	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
14	C₄²×C₂₃	C₂×C₄	1	2	yes	—
15	C₄²×C₂₃	Q₈	1	2	yes	—
16	C₄⋊Dic₂₃	—	2	8	no	—
17	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	—	2	4	no	—
19	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₂₃	#41	1	4	yes	—
23	C₂×Dic₂₃	—	4	4	no	—
24	C₂×Dic₂₃	—	4	8	no	—
25	C₈×C₂₃	—	2	4	no	—
26	#1835	—	23	1	no	maximal, Sylow
27	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₃	#124	1	2	yes	—
29	C₂²×C₂₃	#135	1	4	yes	—
30	C₄×C₂₃	—	2	4	no	—
31	C₄×C₂₃	#117	1	2	yes	—
32	C₄×C₂₃	#125	1	2	yes	—
33	Dic₂₃	—	8	4	no	—
34	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
35	#382	—	23	2	no	—
36	#340	—	23	4	no	—
37	C₂×C₂₃	#382	1	1	yes	characteristic
38	C₂×C₂₃	#427	1	2	yes	—
39	C₂×C₂₃	#488	1	4	yes	—
40	C₂²×C₈	C₂×D₂₃	1	2	yes	—
41	C₂×C₄⋊C₄	—	23	4	no	—
42	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
43	C₄×C₈	Dic₂₃	1	4	yes	complemented
44	#95	—	46	4	no	—
45	C₂₃	#1835	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₄	—	46	4	no	—
47	C₂²×C₄	D₉₂	1	2	yes	—
48	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
51	C₄⋊C₄	—	46	8	no	—
52	C₄²	Dic₄₆	1	2	yes	—
53	C₄²	C₂×Dic₂₃	1	2	yes	—
54	C₂³	C₂×D₉₂	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	92	4	no	—
57	C₂×C₄	—	92	8	no	—
58	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
59	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
60	C₂×C₄	#67288	1	4	yes	—
61	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₂²	#10625046	1	2	yes	—
64	C₂²	#10625075	1	4	yes	—
65	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	184	4	no	—
68	C₄	#10625041	1	2	yes	—
69	C₄	#10625097	1	2	yes	—
70	C₂	#13092283	1	1	yes	characteristic
71	C₂	#13092455	1	2	yes	—
72	C₂	#13092772	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1835
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. C₄×C₈×C₂₃	C₄	semidirect
43. C₄×C₈	Dic₂₃	semidirect
45. C₂₃	#1835	semidirect

Maximal subgroups

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

subgroup	index
#13092283	2
#13092180	2
C₂×C₄×C₈×C₂₃	2
#1835	23