Subgroups — #20294932

Order 6976.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812606	C₂	1	4	yes	maximal
4	C₂×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₈	—	2	2	no	—
6	C₂×Dic₂₁₈	—	2	2	no	—
7	C₂×Dic₂₁₈	C₄	1	4	yes	complemented
8	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
10	C₄⋊Dic₁₀₉	—	2	2	no	—
11	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₀₉	—	2	2	no	—
14	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	—	2	2	no	—
16	C₂×C₄×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
19	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
20	C₂×C₄×C₁₀₉	Q₈	1	4	yes	complemented
21	C₂×Dic₁₀₉	—	2	2	no	—
22	C₂×Dic₁₀₉	—	2	2	no	—
23	C₂×Dic₁₀₉	—	2	2	no	—
24	C₂×Dic₁₀₉	—	2	4	no	—
25	C₂×Dic₁₀₉	C₂×C₄	1	4	yes	—
26	Dic₂₁₈	—	2	8	no	—
27	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₁₀₉	C₂×Q₈	1	2	yes	—
31	C₂²×C₁₀₉	#41	1	2	yes	—
32	C₄×C₁₀₉	—	2	4	no	—
33	C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
34	Dic₁₀₉	—	2	4	no	—
35	Dic₁₀₉	—	4	2	no	—
36	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₁₀₉	C₄×Q₈	1	2	yes	—
38	C₂×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₁₀₉	#124	1	1	yes	characteristic
40	C₂×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
41	C₁₀₉	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	109	1	no	maximal, Sylow
43	C₂²×Q₈	—	109	1	no	—
44	C₂×C₄⋊C₄	—	109	1	no	—
45	C₂×C₄²	D₁₀₉	1	1	yes	characteristic
46	#95	—	109	4	no	—
47	C₂×Q₈	—	109	4	no	—
48	C₂×Q₈	—	218	2	no	—
49	C₂×Q₈	—	218	2	no	—
50	C₂²×C₄	—	109	2	no	—
51	C₂²×C₄	—	109	2	no	—
52	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
53	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
54	C₄⋊C₄	—	218	2	no	—
55	C₄²	—	2	2	no	—
56	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
57	Q₈	—	218	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	109	4	no	—
61	C₂×C₄	—	218	2	no	—
62	C₂×C₄	—	218	4	no	—
63	C₂×C₄	—	218	2	no	—
64	C₂×C₄	—	218	2	no	—
65	C₂×C₄	Dic₂₁₈	1	4	yes	complemented
66	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
67	C₂×C₄	D₄₃₆	1	2	yes	—
68	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
69	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
71	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
72	C₂²	C₂×Dic₂₁₈	1	2	yes	—
73	C₂²	#13182029	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	436	2	no	—
76	C₄	—	218	4	no	—
77	C₄	#13182006	1	4	yes	—
78	C₂	C₂×#13182006	1	1	yes	characteristic
79	C₂	C₄×Dic₂₁₈	1	2	yes	—
80	C₂	#16812678	1	2	yes	—
81	C₂	#16812687	1	1	yes	characteristic
82	C₂	#16812699	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×C₄²×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄²	D₁₀₉
hypocenter	C₁₀₉	#391
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×Dic₂₁₈	C₄	semidirect
20. C₂×C₄×C₁₀₉	Q₈	semidirect
41. C₁₀₉	#391	semidirect
65. C₂×C₄	Dic₂₁₈	semidirect