Subgroups — #20292795

Order 6848.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812134	C₂	1	2	yes	maximal, complemented
2	#16812105	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812036	C₂	1	2	yes	maximal, complemented
4	#16812026	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₈	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
8	#13133805	—	2	2	no	—
9	#13133803	—	2	2	no	—
10	#13133802	C₂²	1	2	yes	complemented
11	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic, complemented
12	#41×C₁₀₇	—	2	2	no	—
13	#41×C₁₀₇	C₂²	1	2	yes	—
14	C₂×D₄×C₁₀₇	—	2	2	no	—
15	C₂×D₄×C₁₀₇	—	2	2	no	—
16	C₂×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂³×C₁₀₇	—	2	2	no	—
18	C₂³×C₁₀₇	D₄	1	2	yes	—
19	C₂²×D₁₀₇	—	2	1	no	—
20	Q₈×C₁₀₇	D₄	1	2	yes	complemented
21	D₄×C₁₀₇	—	2	4	no	—
22	D₄×C₁₀₇	—	2	2	no	—
23	D₄×C₁₀₇	—	2	2	no	—
24	D₄×C₁₀₇	D₄	1	2	yes	complemented
25	C₂×C₄×C₁₀₇	—	2	2	no	—
26	C₂×C₄×C₁₀₇	—	2	2	no	—
27	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₂²⋊D₁₀₇	—	4	4	no	—
29	C₂×Dic₁₀₇	—	2	1	no	—
30	D₄₂₈	—	2	2	no	—
31	C₁₀₇⋊C₈	—	2	2	no	—
32	C₂²×C₁₀₇	—	2	2	no	—
33	C₂²×C₁₀₇	—	2	2	no	—
34	C₂²×C₁₀₇	—	4	2	no	—
35	C₂²×C₁₀₇	—	8	2	no	—
36	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₀₇	—	4	2	no	—
38	C₂×D₁₀₇	—	4	1	no	—
39	C₄×C₁₀₇	—	2	2	no	—
40	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
41	Dic₁₀₇	—	2	2	no	—
42	C₂×C₁₀₇	—	2	1	no	—
43	C₂×C₁₀₇	—	4	2	no	—
44	C₂×C₁₀₇	—	4	2	no	—
45	C₂×C₁₀₇	#120	1	1	yes	characteristic
46	D₁₀₇	—	8	1	no	—
47	C₁₀₇	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	107	1	no	maximal, Sylow
49	H(5,𝔽₂)	D₁₀₇	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	—	107	2	no	—
51	#127	—	107	1	no	—
52	#104	—	107	2	no	—
53	#100	—	107	1	no	—
54	#41	—	2	2	no	—
55	#41	C₂×D₁₀₇	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	107	1	no	—
59	C₂×D₄	—	214	2	no	—
60	C₂×D₄	C₂×D₁₀₇	1	1	yes	characteristic
61	SD₁₆	—	214	2	no	—
62	D₈	—	214	2	no	—
63	M₁₆	—	107	2	no	—
64	C₄²	—	107	1	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	214	1	no	—
67	C₂³	C₂²⋊D₁₀₇	1	2	yes	—
68	Q₈	C₂²⋊D₁₀₇	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	428	4	no	—
73	D₄	—	214	2	no	—
74	D₄	C₂²⋊D₁₀₇	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	214	1	no	—
78	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
79	C₈	—	214	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	428	2	no	—
85	C₂²	—	428	1	no	—
86	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	214	2	no	—
89	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	856	1	no	—
94	C₂	#16812078	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16812078
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	H(5,𝔽₂)×C₁₀₇	C₂
socle	C₂×C₁₀₇	#120
hypercenter	H(5,𝔽₂)	D₁₀₇
hypocenter	C₁₀₇	#453
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #16812134	C₂	semidirect
2. #16812105	C₂	semidirect
3. #16812036	C₂	semidirect
4. #16812026	C₂	semidirect
5. H(5,𝔽₂)×C₁₀₇	C₂	semidirect
6. C₂×D₄₂₈	C₂²	semidirect
10. #13133802	C₂²	semidirect
11. C₄×Dic₁₀₇	C₂²	semidirect
20. Q₈×C₁₀₇	D₄	semidirect
24. D₄×C₁₀₇	D₄	semidirect
47. C₁₀₇	#453	semidirect
49. H(5,𝔽₂)	D₁₀₇	semidirect
68. Q₈	C₂²⋊D₁₀₇	semidirect
74. D₄	C₂²⋊D₁₀₇	semidirect