Subgroups — #11764276

Order 1104.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587789	C₂	1	2	yes	maximal, complemented
3	C₂×Dic₆₉	C₂	1	2	yes	maximal, complemented
4	C₄×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₃₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#67290	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₂₃	C₂²	1	2	yes	—
8	C₂×D₆₉	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	Dic₆₉	C₂²	1	2	yes	—
11	Dic₆₉	C₂²	1	1	yes	characteristic
12	D₄×C₂₃	D₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₂₃	—	3	2	no	—
14	C₂×Dic₂₃	—	3	2	no	—
15	C₄×D₂₃	—	3	1	no	—
16	Dic₄₆	—	3	1	no	—
17	C₂×C₃×C₂₃	—	2	2	no	—
18	C₂×C₃×C₂₃	C₂³	1	1	yes	characteristic
19	D₆₉	—	2	1	no	—
20	C₂²×C₂₃	C₂×D₃	1	2	yes	—
21	C₂×D₂₃	—	3	1	no	—
22	C₄×C₂₃	C₂×D₃	1	1	yes	characteristic
23	Dic₂₃	—	3	2	no	—
24	Dic₂₃	—	3	1	no	—
25	C₃×C₂₃	#41	1	1	yes	characteristic, complemented, Hall
26	#237	—	23	1	no	maximal, Hall
27	C₂×C₂₃	—	2	2	no	—
28	C₂×C₂₃	C₂²×D₃	1	1	yes	characteristic
29	D₂₃	—	6	1	no	—
30	C₃×D₄	D₂₃	1	1	yes	characteristic, complemented
31	#67	—	23	2	no	—
32	C₂×Dic₃	—	23	2	no	—
33	C₄×D₃	—	23	1	no	—
34	Dic₆	—	23	1	no	—
35	C₂₃	#237	1	1	yes	characteristic, complemented, Sylow
36	#41	—	69	1	no	Sylow
37	C₂²×C₃	C₂×D₂₃	1	2	yes	—
38	C₂×D₃	—	23	1	no	—
39	C₃×C₄	C₂×D₂₃	1	1	yes	characteristic
40	Dic₃	—	23	2	no	—
41	Dic₃	—	23	1	no	—
42	Q₈	—	69	1	no	—
43	D₄	—	69	2	no	—
44	D₄	D₆₉	1	1	yes	characteristic, complemented
45	C₂×C₄	—	69	2	no	—
46	C₂×C₄	—	69	1	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	C₂²×D₂₃	1	1	yes	characteristic
49	D₃	—	46	1	no	—
50	C₂²	—	69	1	no	—
51	C₂²	C₂×D₆₉	1	2	yes	—
52	C₄	—	69	1	no	—
53	C₄	—	69	2	no	—
54	C₄	C₂×D₆₉	1	1	yes	characteristic
55	C₃	#67290	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	2	no	—
57	C₂	—	138	1	no	—
58	C₂	C₂²×D₆₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₆₉
commutator, socle, cosocle	C₂×C₃×C₂₃	C₂³
Fitting	C₃×D₄×C₂₃	C₂
hypercenter	D₄	D₆₉
hypocenter	C₃×C₂₃	#41
LCS	G ▷ C₂×C₃×C₂₃ ▷ C₃×C₂₃
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₃×D₄×C₂₃	C₂	semidirect
2. #10587789	C₂	semidirect
3. C₂×Dic₆₉	C₂	semidirect
4. C₄×D₆₉	C₂	semidirect
5. Dic₁₃₈	C₂	semidirect
12. D₄×C₂₃	D₃	semidirect
25. C₃×C₂₃	#41	semidirect
30. C₃×D₄	D₂₃	semidirect
35. C₂₃	#237	semidirect
44. D₄	D₆₉	semidirect
55. C₃	#67290	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₄×C₂₃	2
#10587789	2
C₂×Dic₆₉	2
C₄×D₆₉	2
Dic₁₃₈	2
#67290	3
#237	23