Subgroups — #28371430

Order 13952.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812699	C₂	1	6	yes	maximal
2	#20294896	C₂	1	8	yes	maximal
3	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
5	#16812699	—	2	24	no	—
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	24	yes	—
7	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
8	C₂×C₄²×C₁₀₉	C₂²	1	4	yes	—
9	C₂×Dic₂₁₈	—	2	24	no	—
10	C₂²×Dic₁₀₉	C₂³	1	8	yes	—
11	C₄⋊Dic₁₀₉	—	2	48	no	—
12	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
14	C₄²×C₁₀₉	D₄	1	12	yes	—
15	C₄²×C₁₀₉	Q₈	1	16	yes	complemented
16	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
18	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
19	C₂×C₄×C₁₀₉	C₂×Q₈	1	24	yes	—
20	C₂×Dic₁₀₉	—	2	24	no	—
21	Dic₂₁₈	—	4	16	no	—
22	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
23	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	—
24	C₄×C₁₀₉	#127	1	4	yes	—
25	C₄×C₁₀₉	C₄⋊Q₈	1	24	yes	complemented
26	Dic₁₀₉	—	4	8	no	—
27	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
28	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	6	yes	—
29	#2868	—	109	1	no	maximal, Sylow
30	C₁₀₉	#2868	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₄⋊Q₈	—	109	6	no	—
32	#384	—	109	8	no	—
33	C₄³	D₁₀₉	1	1	yes	characteristic
34	C₂²×Q₈	—	109	4	no	—
35	C₄⋊Q₈	—	218	24	no	—
36	C₂×C₄⋊C₄	—	109	24	no	—
37	C₂×C₄²	C₂×D₁₀₉	1	4	yes	—
38	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
39	C₂×Q₈	—	218	24	no	—
40	C₂²×C₄	—	109	8	no	—
41	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
42	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
43	C₄⋊C₄	—	218	48	no	—
44	C₄²	Dic₂₁₈	1	16	yes	complemented
45	C₄²	D₄₃₆	1	12	yes	—
46	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
47	Q₈	—	436	16	no	—
48	C₂×C₄	—	218	24	no	—
49	C₂×C₄	C₂×D₄₃₆	1	6	yes	—
50	C₂×C₄	C₂×D₄₃₆	1	12	yes	—
51	C₂×C₄	C₂×Dic₂₁₈	1	24	yes	—
52	C₂²	C₂²×D₄₃₆	1	3	yes	—
53	C₂²	C₂²×Dic₂₁₈	1	4	yes	—
54	C₄	—	436	8	no	—
55	C₄	#16812696	1	4	yes	—
56	C₄	#16812699	1	24	yes	complemented
57	C₂	C₂×#16812696	1	1	yes	characteristic
58	C₂	C₂×#16812699	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₄³×C₁₀₉	C₂
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#2868
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
15. C₄²×C₁₀₉	Q₈	semidirect
25. C₄×C₁₀₉	C₄⋊Q₈	semidirect
30. C₁₀₉	#2868	semidirect
44. C₄²	Dic₂₁₈	semidirect
56. C₄	#16812699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812699	2
#20294896	2
C₄³×C₁₀₉	2
#2868	109