Subgroups — #20295392

Order 6976.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812696	C₂	1	1	yes	maximal, characteristic
2	#16812619	C₂	1	4	yes	maximal, complemented
3	#16812610	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₃₆	—	2	2	no	—
6	C₂×D₄₃₆	C₂²	1	2	yes	—
7	C₂×#11724784	C₂²	1	2	yes	—
8	C₄⋊C₄×C₁₀₉	—	2	2	no	—
9	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₁₀₉	—	2	2	no	—
12	C₂×C₄×C₁₀₉	—	2	2	no	—
13	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
14	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
15	D₄₃₆	—	2	4	no	—
16	D₄₃₆	—	4	4	no	—
17	#11724784	—	2	2	no	—
18	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
19	C₂×D₁₀₉	—	4	2	no	—
20	C₂×D₁₀₉	—	4	4	no	—
21	C₄×C₁₀₉	—	2	2	no	—
22	C₄×C₁₀₉	—	4	2	no	—
23	C₄×C₁₀₉	#41	1	2	yes	—
24	C₂×C₁₀₉	#124	1	1	yes	characteristic
25	C₂×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
26	D₁₀₉	—	8	2	no	—
27	C₁₀₉	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	109	1	no	maximal, Sylow
29	#127	—	109	1	no	—
30	#125	D₁₀₉	1	1	yes	characteristic, complemented
31	#102	—	109	4	no	—
32	#97	—	109	1	no	—
33	C₂×D₄	—	109	2	no	—
34	C₂×D₄	—	218	2	no	—
35	C₂×C₈	—	109	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
38	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
39	C₂³	—	218	2	no	—
40	D₄	—	436	4	no	—
41	D₄	—	218	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
44	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
45	C₈	—	218	2	no	—
46	C₂²	—	436	2	no	—
47	C₂²	—	436	4	no	—
48	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#13182030	1	2	yes	—
52	C₂	—	872	2	no	—
53	C₂	#16812688	1	1	yes	characteristic
54	C₂	#16812724	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#125×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#125	D₁₀₉
hypocenter	C₁₀₉	#490
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #16812619	C₂	semidirect
3. #16812610	C₂	semidirect
4. #125×C₁₀₉	C₂	semidirect
27. C₁₀₉	#490	semidirect
30. #125	D₁₀₉	semidirect