Subgroups — #20292914

Order 6848.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₂₈	C₂	1	1	yes	maximal, characteristic
2	#16812040	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812029	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₀₇	—	2	1	no	—
8	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
10	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₀₇	—	2	1	no	—
12	C₄⋊C₄×C₁₀₇	—	2	1	no	—
13	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
14	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₀₇	—	2	1	no	—
16	Q₈×C₁₀₇	—	4	2	no	—
17	C₂×C₄×C₁₀₇	—	2	2	no	—
18	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₇	—	2	1	no	—
22	D₄₂₈	—	2	1	no	—
23	D₄₂₈	Q₈	1	2	yes	complemented
24	C₄×D₁₀₇	—	4	2	no	—
25	C₁₀₇⋊C₈	—	2	2	no	—
26	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₀₇	—	2	2	no	—
28	C₂×D₁₀₇	—	4	1	no	—
29	C₂×D₁₀₇	—	4	1	no	—
30	C₄×C₁₀₇	—	2	1	no	—
31	C₄×C₁₀₇	—	4	2	no	—
32	C₄×C₁₀₇	SD₁₆	1	2	yes	—
33	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₀₇	#41	1	1	yes	characteristic
35	Dic₁₀₇	—	4	1	no	—
36	C₂×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₁₀₇	—	4	2	no	—
40	C₁₀₇	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	107	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic, complemented
43	C₄×D₄	—	107	1	no	—
44	#106	—	107	2	no	—
45	C₄⋊C₈	—	107	1	no	—
46	#102	—	107	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	107	1	no	—
49	C₂²×C₄	—	214	1	no	—
50	C₂×C₈	—	107	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	107	1	no	—
53	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
54	C₂²⋊C₄	—	214	1	no	—
55	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
56	C₂³	—	214	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	107	2	no	—
59	D₄	—	214	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	428	2	no	—
62	C₂×C₄	—	214	1	no	—
63	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
66	C₈	—	214	2	no	—
67	C₂²	—	428	1	no	—
68	C₂²	—	428	1	no	—
69	C₂²	—	214	2	no	—
70	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	428	1	no	—
74	C₄	#13133805	1	2	yes	—
75	C₄	Q₈×D₁₀₇	1	1	yes	characteristic
76	C₄	#13133819	1	1	yes	characteristic
77	C₂	—	428	2	no	—
78	C₂	#16812085	1	1	yes	characteristic
79	C₂	C₂×#13133805	1	1	yes	characteristic
80	C₂	#16812135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₄⋊Q₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#476
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16812040	C₂	semidirect
3. C₄⋊(C₁₀₇⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₁₀₇	C₂	semidirect
23. D₄₂₈	Q₈	semidirect
40. C₁₀₇	#476	semidirect
42. C₄⋊Q₈	D₁₀₇	semidirect