Subgroups — #20295361

Order 6976.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812678	C₂	1	1	yes	maximal, characteristic
2	#16812632	C₂	1	2	yes	maximal, complemented
3	#16812622	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
7	C₂×#11724784	C₂²	1	2	yes	—
8	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#13182008	—	2	1	no	—
10	C₂²⋊Dic₁₀₉	—	2	1	no	—
11	C₂²×C₄×C₁₀₉	—	2	1	no	—
12	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	—	2	1	no	—
14	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₁₀₉	—	4	2	no	—
19	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	C₂×Dic₁₀₉	—	2	1	no	—
23	Dic₂₁₈	—	2	2	no	—
24	#11724784	—	2	2	no	—
25	C₂²×C₁₀₉	—	2	1	no	—
26	C₂²×C₁₀₉	—	2	1	no	—
27	C₂²×C₁₀₉	Q₁₆	1	2	yes	—
28	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₀₉	—	2	1	no	—
30	C₄×C₁₀₉	—	4	2	no	—
31	C₄×C₁₀₉	#41	1	2	yes	—
32	Dic₁₀₉	—	4	1	no	—
33	Dic₁₀₉	—	4	1	no	—
34	C₂×C₁₀₉	—	2	2	no	—
35	C₂×C₁₀₉	#123	1	1	yes	characteristic
36	C₂×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₁₀₉	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	109	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	109	1	no	—
41	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
42	#107	—	109	2	no	—
43	#103	—	109	2	no	—
44	C₂²⋊C₈	—	109	1	no	—
45	C₂×Q₈	—	109	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
48	C₂×C₈	—	109	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	109	1	no	—
51	C₄⋊C₄	—	218	1	no	—
52	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
53	C₂²⋊C₄	—	218	1	no	—
54	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
55	Q₈	—	218	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	218	1	no	—
60	C₂×C₄	—	218	1	no	—
61	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
63	C₈	—	218	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#13182019	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	436	1	no	—
71	C₄	—	436	1	no	—
72	C₄	#13182029	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#16812682	1	1	yes	characteristic
75	C₂	C₂×#13182019	1	1	yes	characteristic
76	C₂	#16812725	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#484
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #16812632	C₂	semidirect
3. #16812622	C₂	semidirect
38. C₁₀₉	#484	semidirect