Subgroups — #28363192

Order 13952.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812633	C₂	1	4	yes	maximal
2	#20294896	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₀₉	—	2	4	no	—
5	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
6	#16812633	—	2	8	no	—
7	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₀₉	C₄	1	4	yes	complemented
10	C₂²×Dic₁₀₉	—	2	4	no	—
11	C₄⋊Dic₁₀₉	—	2	8	no	—
12	C₄⋊Dic₁₀₉	—	4	8	no	—
13	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
14	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
17	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
18	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₄²×C₁₀₉	Q₈	1	2	yes	—
20	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₁₀₉	D₈	1	4	yes	—
23	C₂×C₄×C₁₀₉	Q₁₆	1	4	yes	—
24	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
26	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
27	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	4	4	no	—
29	C₂×Dic₁₀₉	—	4	8	no	—
30	C₈×C₁₀₉	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₁₀₉	#127	1	1	yes	characteristic
33	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₁₀₉	C₂×D₈	1	2	yes	—
35	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
36	C₄×C₁₀₉	#107	1	8	yes	complemented
37	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
39	Dic₁₀₉	—	8	4	no	—
40	C₂×C₁₀₉	#384	1	1	yes	characteristic
41	C₂×C₁₀₉	C₂×#107	1	2	yes	—
42	C₂×C₁₀₉	#493	1	1	yes	characteristic
43	C₂×C₁₀₉	#494	1	1	yes	characteristic
44	C₂×C₁₀₉	#500	1	2	yes	—
45	#1845	—	109	1	no	maximal, Sylow
46	C₁₀₉	#1845	1	1	yes	characteristic, complemented, Sylow
47	C₂×#107	—	109	4	no	—
48	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
49	#384	—	109	2	no	—
50	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
51	C₂×C₄⋊C₄	—	109	4	no	—
52	C₂×C₄⋊C₄	—	218	4	no	—
53	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
54	#107	—	218	8	no	—
55	C₄×C₈	Dic₁₀₉	1	4	yes	complemented
56	C₂²×C₄	—	218	4	no	—
57	C₂²×C₄	D₄₃₆	1	2	yes	—
58	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
59	C₂×C₈	Dic₂₁₈	1	4	yes	—
60	C₂×C₈	D₄₃₆	1	4	yes	—
61	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
62	C₄⋊C₄	—	436	8	no	—
63	C₄⋊C₄	—	218	8	no	—
64	C₄²	Dic₂₁₈	1	2	yes	—
65	C₄²	C₂×Dic₁₀₉	1	2	yes	—
66	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
67	C₂×C₄	—	436	4	no	—
68	C₂×C₄	—	436	8	no	—
69	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
70	C₂×C₄	D₈₇₂	1	4	yes	—
71	C₂×C₄	Dic₄₃₆	1	4	yes	—
72	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
74	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
75	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
76	C₈	C₄⋊Dic₁₀₉	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
78	C₂²	#16812696	1	1	yes	characteristic
79	C₂²	#16812699	1	1	yes	characteristic
80	C₂²	C₂×D₈₇₂	1	2	yes	—
81	C₂²	C₂×Dic₄₃₆	1	2	yes	—
82	C₄	—	872	4	no	—
83	C₄	#16812633	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₁₀₉	1	2	yes	—
85	C₄	#16812699	1	2	yes	—
86	C₂	#20294896	1	1	yes	characteristic
87	C₂	C₂×#16812633	1	2	yes	—
88	C₂	#20295403	1	1	yes	characteristic
89	C₂	#20295406	1	1	yes	characteristic
90	C₂	#20295431	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#1845
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₁₀₉	C₄	semidirect
30. C₈×C₁₀₉	C₄⋊C₄	semidirect
36. C₄×C₁₀₉	#107	semidirect
46. C₁₀₉	#1845	semidirect
55. C₄×C₈	Dic₁₀₉	semidirect
76. C₈	C₄⋊Dic₁₀₉	semidirect
83. C₄	#16812633	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812633	2
#20294896	2
C₂×C₄×C₈×C₁₀₉	2
#1845	109