Subgroups — #20295301

Order 6976.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812729	C₂	1	2	yes	maximal
2	C₂×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812635	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812617	C₂	1	2	yes	maximal, complemented
5	#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₈	C₂²	1	2	yes	—
7	Dic₄₃₆	—	2	1	no	—
8	Dic₄₃₆	—	2	1	no	—
9	#13182019	—	2	2	no	—
10	#13182017	—	2	2	no	—
11	#13182013	C₂²	1	2	yes	—
12	#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	M₁₆×C₁₀₉	—	2	1	no	—
14	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₁₀₉	—	2	1	no	—
16	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
18	D₄×C₁₀₉	—	2	1	no	—
19	D₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₀₉	—	2	2	no	—
23	Dic₂₁₈	—	2	2	no	—
24	Dic₂₁₈	—	2	2	no	—
25	C₈×C₁₀₉	—	2	1	no	—
26	C₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₈×C₁₀₉	D₄	1	1	yes	characteristic
28	#11724784	—	2	2	no	—
29	C₂²×C₁₀₉	—	2	1	no	—
30	C₂²×C₁₀₉	#41	1	1	yes	characteristic
31	C₄×C₁₀₉	—	2	1	no	—
32	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	Dic₁₀₉	—	4	2	no	—
35	C₂×C₁₀₉	—	2	1	no	—
36	C₂×C₁₀₉	—	4	1	no	—
37	C₂×C₁₀₉	#121	1	1	yes	characteristic
38	C₁₀₉	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	109	1	no	maximal, Sylow
40	#137	—	109	2	no	—
41	C₂×Q₁₆	—	109	1	no	—
42	#131	D₁₀₉	1	1	yes	characteristic
43	#108	—	109	1	no	—
44	#101	—	109	2	no	—
45	#41	C₂×D₁₀₉	1	1	yes	characteristic
46	C₂×Q₈	—	109	2	no	—
47	Q₁₆	—	218	2	no	—
48	Q₁₆	—	218	1	no	—
49	Q₁₆	—	218	1	no	—
50	SD₁₆	—	218	2	no	—
51	M₁₆	—	2	1	no	—
52	M₁₆	—	109	2	no	—
53	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
56	Q₈	—	218	2	no	—
57	Q₈	—	218	2	no	—
58	Q₈	D₄₃₆	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	D₄₃₆	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	218	2	no	—
63	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	218	2	no	—
66	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
67	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	#13182029	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	436	2	no	—
72	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
73	C₄	C₂×D₄₃₆	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	#16812673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16812673
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#131×C₁₀₉	C₂
socle	C₂×C₁₀₉	#121
hypercenter	#131	D₁₀₉
hypocenter	C₁₀₉	#473
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₃₆	C₂	semidirect
3. #16812635	C₂	semidirect
4. #16812617	C₂	semidirect
38. C₁₀₉	#473	semidirect