Subgroups — #20295267

Order 6976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812678	C₂	1	2	yes	maximal
3	#16812633	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812624	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₈	C₂²	1	2	yes	—
7	Dic₄₃₆	—	2	2	no	—
8	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	#13182008	—	2	2	no	—
10	C₂²⋊Dic₁₀₉	—	2	2	no	—
11	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₀₉	—	2	2	no	—
13	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀₉	—	2	2	no	—
21	C₂×Dic₁₀₉	—	2	2	no	—
22	Dic₂₁₈	—	2	2	no	—
23	Dic₂₁₈	—	2	2	no	—
24	C₈×C₁₀₉	—	2	1	no	—
25	C₈×C₁₀₉	D₄	1	2	yes	—
26	C₂²×C₁₀₉	—	2	1	no	—
27	C₂²×C₁₀₉	—	2	1	no	—
28	C₂²×C₁₀₉	Q₁₆	1	2	yes	complemented
29	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₀₉	—	2	1	no	—
31	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₀₉	#41	1	1	yes	characteristic
33	Dic₁₀₉	—	4	2	no	—
34	Dic₁₀₉	—	4	2	no	—
35	C₂×C₁₀₉	—	2	2	no	—
36	C₂×C₁₀₉	#121	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₁₀₉	#135	1	1	yes	characteristic
39	C₁₀₉	#467	1	1	yes	characteristic, complemented, Sylow
40	#467	—	109	1	no	maximal, Sylow
41	C₂×Q₁₆	—	109	1	no	—
42	C₂²×C₈	D₁₀₉	1	1	yes	characteristic
43	C₂²⋊Q₈	—	109	2	no	—
44	#107	—	109	1	no	—
45	#103	—	109	2	no	—
46	C₂×Q₈	—	109	2	no	—
47	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
48	Q₁₆	—	218	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
52	C₄⋊C₄	—	109	2	no	—
53	C₄⋊C₄	—	218	2	no	—
54	C₂²⋊C₄	—	218	2	no	—
55	C₂³	D₄₃₆	1	1	yes	characteristic
56	Q₈	—	218	2	no	—
57	Q₈	—	218	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	218	2	no	—
61	C₂×C₄	—	218	2	no	—
62	C₂×C₄	D₄₃₆	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	C₂²⋊D₁₀₉	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Dic₄₃₆	1	2	yes	complemented
69	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	436	2	no	—
72	C₄	—	436	2	no	—
73	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
74	C₄	#13182029	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	#16812673	1	1	yes	characteristic
77	C₂	C₂×Dic₄₃₆	1	1	yes	characteristic
78	C₂	#16812721	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	#467
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₄₃₆	C₂	semidirect
3. #16812633	C₂	semidirect
4. #16812624	C₂	semidirect
28. C₂²×C₁₀₉	Q₁₆	semidirect
39. C₁₀₉	#467	semidirect
68. C₂²	Dic₄₃₆	semidirect