Subgroups — #20295334

Order 6976.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812691	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
3	#16812633	C₂	1	1	yes	maximal, characteristic
4	#16812631	C₂	1	1	yes	maximal, characteristic
5	#16812624	C₂	1	1	yes	maximal, characteristic
6	#16812624	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
12	#13182008	—	2	1	no	—
13	#13182008	—	2	1	no	—
14	#13182008	—	2	1	no	—
15	C₄×Dic₁₀₉	—	2	1	no	—
16	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₀₉	—	2	1	no	—
23	C₂×Dic₁₀₉	—	2	1	no	—
24	C₂×Dic₁₀₉	—	2	1	no	—
25	C₂×Dic₁₀₉	—	2	1	no	—
26	Dic₂₁₈	—	2	1	no	—
27	Dic₂₁₈	Q₈	1	2	yes	—
28	C₈×C₁₀₉	—	2	1	no	—
29	C₈×C₁₀₉	—	2	1	no	—
30	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₀₉	—	2	1	no	—
32	C₄×C₁₀₉	—	2	1	no	—
33	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₀₉	#41	1	1	yes	characteristic
35	Dic₁₀₉	—	2	2	no	—
36	Dic₁₀₉	—	4	1	no	—
37	Dic₁₀₉	—	4	1	no	—
38	Dic₁₀₉	—	4	1	no	—
39	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₁₀₉	#135	1	1	yes	characteristic
41	C₂×C₁₀₉	#137	1	1	yes	characteristic
42	C₁₀₉	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	109	1	no	maximal, Sylow
44	#125	—	109	1	no	—
45	C₄×Q₈	—	109	1	no	—
46	#107	—	109	1	no	—
47	#106	—	109	1	no	—
48	C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
49	#103	—	109	1	no	—
50	#103	—	109	1	no	—
51	C₂×Q₈	—	109	1	no	—
52	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
53	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
54	C₄⋊C₄	—	109	1	no	—
55	C₄⋊C₄	—	109	1	no	—
56	C₄⋊C₄	—	109	1	no	—
57	C₄⋊C₄	—	218	1	no	—
58	C₄⋊C₄	—	218	1	no	—
59	C₄⋊C₄	—	218	1	no	—
60	C₄²	—	218	1	no	—
61	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
62	Q₈	—	109	2	no	—
63	Q₈	—	218	1	no	—
64	C₂×C₄	—	218	1	no	—
65	C₂×C₄	—	218	1	no	—
66	C₂×C₄	—	218	1	no	—
67	C₂×C₄	—	218	1	no	—
68	C₂×C₄	D₄₃₆	1	1	yes	characteristic
69	C₂×C₄	D₄₃₆	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	436	1	no	—
77	C₄	—	436	1	no	—
78	C₄	—	436	1	no	—
79	C₄	—	218	2	no	—
80	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
81	C₄	#13182028	1	1	yes	characteristic
82	C₂	#16812679	1	1	yes	characteristic
83	C₂	#16812721	1	1	yes	characteristic
84	C₂	#16812732	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₄⋊C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#479
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
42. C₁₀₉	#479	semidirect