Subgroups — #20295065

Order 6976.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182012	C₂	1	1	yes	maximal, characteristic
2	#16812657	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812633	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812630	C₂	1	1	yes	maximal, characteristic, complemented
7	#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₀₉	—	2	1	no	—
9	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	#13182012	C₄	1	4	yes	complemented
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#13182008	—	2	1	no	—
15	#13182006	—	2	1	no	—
16	C₄×Dic₁₀₉	—	2	1	no	—
17	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₀₉	—	2	1	no	—
25	C₂×Dic₁₀₉	—	2	1	no	—
26	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×D₁₀₉	—	4	2	no	—
28	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
29	C₄×D₁₀₉	Q₈	1	2	yes	—
30	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
31	#11724784	C₂×C₄	1	2	yes	complemented
32	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×D₁₀₉	—	2	1	no	—
34	C₂×D₁₀₉	—	2	1	no	—
35	C₂×D₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₄×C₁₀₉	—	4	1	no	—
37	C₄×C₁₀₉	—	4	1	no	—
38	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	Dic₁₀₉	—	4	1	no	—
41	Dic₁₀₉	—	4	1	no	—
42	Dic₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
45	C₂×C₁₀₉	#137	1	1	yes	characteristic
46	D₁₀₉	—	2	2	no	—
47	C₁₀₉	#428	1	1	yes	characteristic, complemented, Sylow
48	#428	—	109	1	no	maximal, Sylow
49	C₂×M₁₆	—	109	1	no	—
50	#117	—	109	1	no	—
51	C₂×C₄⋊C₄	—	109	1	no	—
52	#107	—	109	1	no	—
53	#107	—	109	1	no	—
54	#106	—	109	1	no	—
55	#106	D₁₀₉	1	1	yes	characteristic, complemented
56	C₂²×C₄	—	109	1	no	—
57	C₂²×C₄	—	218	1	no	—
58	M₁₆	—	109	4	no	—
59	C₂×C₈	—	109	1	no	—
60	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	C₄⋊C₄	—	109	1	no	—
62	C₄⋊C₄	—	109	1	no	—
63	C₄⋊C₄	—	218	1	no	—
64	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
66	C₂²⋊C₄	—	218	1	no	—
67	C₄²	—	218	1	no	—
68	C₂³	—	109	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	109	1	no	—
72	C₂×C₄	—	109	2	no	—
73	C₂×C₄	—	109	2	no	—
74	C₂×C₄	—	436	2	no	—
75	C₂×C₄	—	218	1	no	—
76	C₂×C₄	—	218	1	no	—
77	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
78	C₈	—	109	2	no	—
79	C₈	C₄×D₁₀₉	1	2	yes	complemented
80	C₂²	—	109	2	no	—
81	C₂²	—	218	1	no	—
82	C₂²	—	218	1	no	—
83	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	109	2	no	—
87	C₄	—	436	1	no	—
88	C₄	—	436	1	no	—
89	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
90	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
91	C₂	—	218	2	no	—
92	C₂	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
93	C₂	#16812726	1	1	yes	characteristic
94	C₂	#16812728	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#106×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#106	D₁₀₉
hypocenter	C₁₀₉	#428
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
4. #16812633	C₂	semidirect
5. #16812632	C₂	semidirect
6. #16812630	C₂	semidirect
7. #106×C₁₀₉	C₂	semidirect
10. #13182012	C₄	semidirect
30. C₈×C₁₀₉	C₂×C₄	semidirect
31. #11724784	C₂×C₄	semidirect
47. C₁₀₉	#428	semidirect
55. #106	D₁₀₉	semidirect
79. C₈	C₄×D₁₀₉	semidirect