Subgroups — #20296031

Order 6976.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182039	C₂	1	2	yes	maximal
2	#16812766	C₂	1	4	yes	maximal, complemented
3	C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#13182039	—	2	4	no	—
5	C₂×#11724786	C₂²	1	4	yes	—
6	C₂×C₄×D₁₀₉	C₄	1	2	yes	—
7	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₄×Dic₁₀₉	C₂²	1	2	yes	—
10	C₄²×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
15	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
16	C₄×D₁₀₉	—	2	4	no	—
17	C₄×D₁₀₉	D₄	1	2	yes	—
18	C₄×D₁₀₉	Q₈	1	2	yes	—
19	#11724786	—	2	4	no	—
20	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₀₉	—	2	2	no	—
22	C₂×D₁₀₉	C₄⋊C₄	1	2	yes	—
23	C₄×C₁₀₉	—	2	2	no	—
24	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
25	Dic₁₀₉	M₁₆	1	4	yes	complemented
26	Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
27	Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₀₉	C₂×M₁₆	1	2	yes	—
30	D₁₀₉	—	2	2	no	—
31	C₁₀₉	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	109	1	no	maximal, Sylow
33	C₂×M₁₆	—	109	2	no	—
34	C₂×C₄²	—	109	1	no	—
35	C₄⋊C₈	—	109	4	no	—
36	C₂²×C₄	—	109	1	no	—
37	C₂²×C₄	—	109	2	no	—
38	M₁₆	—	218	4	no	—
39	C₂×C₈	—	109	4	no	—
40	C₄²	—	109	1	no	—
41	C₄²	—	109	2	no	—
42	C₄²	#89137	1	1	yes	characteristic
43	C₂³	—	109	1	no	—
44	C₂×C₄	—	109	1	no	—
45	C₂×C₄	—	109	2	no	—
46	C₂×C₄	—	109	2	no	—
47	C₂×C₄	—	109	2	no	—
48	C₂×C₄	—	218	4	no	—
49	C₂×C₄	C₂×#89137	1	2	yes	—
50	C₂×C₄	C₂×#89137	1	1	yes	characteristic
51	C₈	—	218	4	no	—
52	C₂²	—	109	2	no	—
53	C₂²	—	218	2	no	—
54	C₂²	C₂²×#89137	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	109	1	no	—
57	C₄	—	109	1	no	—
58	C₄	—	109	4	no	—
59	C₄	C₄⋊#89137	1	2	yes	—
60	C₂	—	218	2	no	—
61	C₂	C₂×C₄⋊#89137	1	1	yes	characteristic
62	C₂	D₁₀₉⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₄²×C₁₀₉	C₄
hypercenter	C₄²	#89137
hypocenter	C₁₀₉	#423
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #16812766	C₂	semidirect
25. Dic₁₀₉	M₁₆	semidirect
31. C₁₀₉	#423	semidirect