Subgroups — #20292973

Order 6848.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812109	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812030	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₄	—	2	1	no	—
6	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₀₇	—	2	1	no	—
8	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	C₂×D₄×C₁₀₇	—	2	1	no	—
10	C₂×D₄×C₁₀₇	—	2	1	no	—
11	C₂×D₄×C₁₀₇	C₂²	1	2	yes	—
12	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂³×C₁₀₇	—	2	2	no	—
14	D₄×C₁₀₇	—	2	4	no	—
15	D₄×C₁₀₇	—	4	2	no	—
16	D₄×C₁₀₇	—	4	2	no	—
17	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀₇	—	2	2	no	—
21	Dic₂₁₄	—	4	2	no	—
22	C₁₀₇⋊C₈	—	2	2	no	—
23	C₂²×C₁₀₇	—	4	2	no	—
24	C₂²×C₁₀₇	—	4	2	no	—
25	C₂²×C₁₀₇	—	4	2	no	—
26	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₀₇	D₈	1	2	yes	—
28	C₄×C₁₀₇	SD₁₆	1	2	yes	—
29	C₄×C₁₀₇	#41	1	2	yes	—
30	Dic₁₀₇	—	4	2	no	—
31	C₂×C₁₀₇	—	8	2	no	—
32	C₂×C₁₀₇	#124	1	1	yes	characteristic
33	C₂×C₁₀₇	C₂×D₈	1	1	yes	characteristic
34	C₂×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
35	C₁₀₇	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	107	1	no	maximal, Sylow
37	C₄⋊Q₈	—	107	1	no	—
38	#127	D₁₀₇	1	1	yes	characteristic
39	#102	—	107	4	no	—
40	C₄×C₈	—	107	1	no	—
41	C₂×Q₈	—	214	1	no	—
42	C₂×D₄	—	2	1	no	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂×D₁₀₇	1	2	yes	—
45	C₂×C₈	—	107	2	no	—
46	C₄⋊C₄	—	107	2	no	—
47	C₄⋊C₄	—	214	1	no	—
48	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	Q₈	—	428	2	no	—
51	D₄	—	2	4	no	—
52	D₄	—	4	2	no	—
53	D₄	—	4	2	no	—
54	C₂×C₄	—	214	2	no	—
55	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
58	C₈	—	214	2	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
63	C₄	—	428	2	no	—
64	C₄	#13133803	1	2	yes	—
65	C₄	#13133806	1	2	yes	—
66	C₄	#13133817	1	2	yes	—
67	C₂	—	8	2	no	—
68	C₂	#16812096	1	1	yes	characteristic
69	C₂	C₂×#13133803	1	1	yes	characteristic
70	C₂	C₂×#13133806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	#127×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	#127	D₁₀₇
hypocenter	C₁₀₇	#486
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #16812109	C₂	semidirect
2. #16812030	C₂	semidirect
3. C₄×C₁₀₇⋊C₈	C₂	semidirect
35. C₁₀₇	#486	semidirect