Subgroups — #15673468

Order 2944.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
2	#13092172	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#117	—	2	4	no	—
6	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
7	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
10	C₂²⋊(C₂₃⋊C₈)	—	2	4	no	—
11	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	—	2	4	no	—
13	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	4	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃	—	2	4	no	—
18	C₂²⋊C₄×C₂₃	—	2	4	no	—
19	C₄²×C₂₃	—	2	4	no	—
20	C₂×C₂₃⋊C₈	—	2	4	no	—
21	C₂×C₂₃⋊C₈	—	2	8	no	—
22	C₂³×C₂₃	—	2	2	no	—
23	C₂³×C₂₃	—	2	4	no	—
24	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	2	8	no	—
28	C₂×C₄×C₂₃	—	2	4	no	—
29	C₂×C₄×C₂₃	—	2	8	no	—
30	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₂₃	#41	1	4	yes	—
33	C₂₃⋊C₈	—	4	4	no	—
34	#1818	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	—	2	2	no	—
36	C₂²×C₂₃	—	2	4	no	—
37	C₂²×C₂₃	—	4	4	no	—
38	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₃	#117	1	2	yes	—
40	C₂²×C₂₃	#131	1	4	yes	—
41	C₄×C₂₃	—	2	2	no	—
42	C₄×C₂₃	—	4	4	no	—
43	C₄×C₂₃	#123	1	4	yes	—
44	C₂×#117	D₂₃	1	1	yes	characteristic
45	C₂×C₂²⋊C₈	—	23	2	no	—
46	#336	—	23	4	no	—
47	C₂×C₂₃	—	4	2	no	—
48	C₂×C₂₃	#381	1	1	yes	characteristic
49	C₂×C₂₃	#408	1	2	yes	—
50	C₂×C₂₃	#433	1	4	yes	—
51	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
52	C₂²×C₈	—	23	4	no	—
53	#117	—	2	4	no	—
54	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
55	C₂×C₂²⋊C₄	Dic₂₃	1	2	yes	—
56	C₂×C₄²	C₂×D₂₃	1	2	yes	—
57	C₂²⋊C₈	—	46	4	no	—
58	C₂₃	#1818	1	1	yes	characteristic, complemented, Sylow
59	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
62	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
64	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
65	C₂×C₈	—	46	8	no	—
66	C₂×C₈	—	46	4	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	4	no	—
72	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
73	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
79	C₂×C₄	#67288	1	4	yes	—
80	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
81	C₈	—	92	4	no	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	4	no	—
85	C₂²	#10625097	1	2	yes	—
86	C₂²	#10625119	1	4	yes	—
87	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#10625100	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#13092280	1	1	yes	characteristic
93	C₂	#13092384	1	2	yes	—
94	C₂	#13092478	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×#117	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#1818
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13092172	C₂	semidirect
58. C₂₃	#1818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092172	2
C₂×C₂₃×#117	2
#1818	23