Subgroups — #20295062

Order 6976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11724784	C₂	1	1	yes	maximal, characteristic
2	#16812658	C₂	1	1	yes	maximal, characteristic
3	#16812632	C₂	1	2	yes	maximal, complemented
4	#16812630	C₂	1	2	yes	maximal, complemented
5	#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11724784	C₄	1	4	yes	complemented
7	C₂×#11724784	C₂²	1	1	yes	characteristic
8	C₂×#11724784	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂²⋊Dic₁₀₉	—	2	1	no	—
11	C₄×Dic₁₀₉	—	2	1	no	—
12	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
15	C₄²×C₁₀₉	—	2	1	no	—
16	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
21	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₀₉	—	2	2	no	—
23	#11724784	C₂×C₄	1	2	yes	—
24	#11724784	C₂×C₄	1	2	yes	—
25	C₂²×C₁₀₉	—	2	2	no	—
26	C₂²×C₁₀₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₀₉	—	4	2	no	—
29	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
30	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
32	Dic₁₀₉	—	4	2	no	—
33	C₂×C₁₀₉	—	2	2	no	—
34	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₀₉	#135	1	2	yes	—
36	C₁₀₉	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	109	1	no	maximal, Sylow
38	C₂²×C₈	—	109	1	no	—
39	#117	—	109	1	no	—
40	#117	D₁₀₉	1	1	yes	characteristic
41	#107	—	109	2	no	—
42	#106	—	109	2	no	—
43	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
44	C₂×C₈	—	109	4	no	—
45	C₂×C₈	—	109	1	no	—
46	C₂×C₈	—	109	1	no	—
47	C₄⋊C₄	—	109	2	no	—
48	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	218	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	218	1	no	—
53	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	218	2	no	—
56	C₂×C₄	Dic₂₁₈	1	2	yes	—
57	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
58	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
60	C₈	—	109	2	no	—
61	C₈	—	109	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	#13182008	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	436	2	no	—
67	C₄	#13182008	1	2	yes	—
68	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
69	C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×#13182008	1	1	yes	characteristic
72	C₂	#16812719	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#117×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#117	D₁₀₉
hypocenter	C₁₀₉	#427
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #16812632	C₂	semidirect
4. #16812630	C₂	semidirect
6. C₂×#11724784	C₄	semidirect
36. C₁₀₉	#427	semidirect