Subgroups — #20295049

Order 6976.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182013	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812628	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₉	C₄	1	2	yes	—
6	#13182013	—	2	2	no	—
7	C₂×#11724784	C₂²	1	2	yes	—
8	C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₄×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	M₁₆×C₁₀₉	—	2	2	no	—
13	C₂×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
17	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
18	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₀₉	—	2	4	no	—
20	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
22	C₈×C₁₀₉	—	2	2	no	—
23	#11724784	—	2	2	no	—
24	C₂²×C₁₀₉	—	2	2	no	—
25	C₂²×C₁₀₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
28	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
30	Dic₁₀₉	—	2	2	no	—
31	Dic₁₀₉	M₁₆	1	4	yes	complemented
32	C₂×C₁₀₉	—	2	2	no	—
33	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₁₀₉	C₂×M₁₆	1	2	yes	—
35	C₁₀₉	#423	1	1	yes	characteristic, complemented, Sylow
36	#423	—	109	1	no	maximal, Sylow
37	C₂×M₁₆	—	109	1	no	—
38	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic
39	C₂×C₄²	—	109	1	no	—
40	C₄⋊C₈	—	109	4	no	—
41	C₂²×C₄	—	109	2	no	—
42	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	218	2	no	—
45	C₂×C₈	—	109	2	no	—
46	C₂×C₈	C₂×D₁₀₉	1	2	yes	—
47	C₄²	—	109	1	no	—
48	C₄²	—	109	1	no	—
49	C₄²	—	109	2	no	—
50	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
51	C₂×C₄	—	109	2	no	—
52	C₂×C₄	—	109	2	no	—
53	C₂×C₄	—	218	4	no	—
54	C₂×C₄	Dic₂₁₈	1	2	yes	—
55	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
57	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	218	2	no	—
60	C₂²	—	2	2	no	—
61	C₂²	#13182008	1	2	yes	—
62	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
63	C₄	—	109	4	no	—
64	C₄	—	218	2	no	—
65	C₄	#13182008	1	2	yes	—
66	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
67	C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	C₂×#13182008	1	1	yes	characteristic
70	C₂	M₁₆×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×M₁₆×C₁₀₉	C₂
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#423
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16812628	C₂	semidirect
31. Dic₁₀₉	M₁₆	semidirect
35. C₁₀₉	#423	semidirect