Subgroups — #20293375

Order 6848.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812097	C₂	1	2	yes	maximal, complemented
2	#16812094	C₂	1	4	yes	maximal, complemented
3	#16812087	C₂	1	4	yes	maximal, complemented
4	#16812084	C₂	1	4	yes	maximal, complemented
5	#125×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₄	C₂²	1	2	yes	—
7	C₂×D₄₂₈	C₂²	1	4	yes	—
8	C₂×D₄₂₈	C₂²	1	2	yes	—
9	C₂×C₄×D₁₀₇	C₂²	1	4	yes	—
10	Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₀₇	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
16	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂²×D₁₀₇	C₂³	1	4	yes	—
18	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
20	C₂×C₄×C₁₀₇	C₂³	1	4	yes	—
21	C₂×Dic₁₀₇	C₂³	1	4	yes	—
22	D₄₂₈	—	4	2	no	—
23	D₄₂₈	—	4	4	no	—
24	C₄×D₁₀₇	—	4	4	no	—
25	Dic₂₁₄	—	4	2	no	—
26	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₀₇	—	2	4	no	—
28	C₂×D₁₀₇	—	2	4	no	—
29	C₂×D₁₀₇	—	2	4	no	—
30	C₄×C₁₀₇	—	2	2	no	—
31	C₄×C₁₀₇	—	2	4	no	—
32	C₄×C₁₀₇	—	2	1	no	—
33	Dic₁₀₇	—	2	4	no	—
34	C₂×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₀₇	—	4	4	no	—
37	C₁₀₇	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	107	1	no	maximal, Sylow
39	#125	D₁₀₇	1	1	yes	characteristic, complemented
40	#124	—	107	2	no	—
41	#123	—	107	4	no	—
42	C₂²⋊Q₈	—	107	4	no	—
43	#121	—	107	4	no	—
44	C₂×Q₈	—	107	2	no	—
45	C₂×D₄	—	107	4	no	—
46	C₂×D₄	—	107	2	no	—
47	C₂²×C₄	—	107	4	no	—
48	C₄⋊C₄	—	107	4	no	—
49	C₄⋊C₄	C₂×D₁₀₇	1	4	yes	—
50	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
51	C₂²⋊C₄	—	107	4	no	—
52	C₂²⋊C₄	—	107	4	no	—
53	C₂²⋊C₄	—	107	4	no	—
54	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
55	C₂³	—	107	4	no	—
56	Q₈	—	428	2	no	—
57	D₄	—	428	4	no	—
58	D₄	—	428	2	no	—
59	C₂×C₄	—	107	4	no	—
60	C₂×C₄	—	428	4	no	—
61	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
63	C₂×C₄	C₂²×D₁₀₇	1	4	yes	—
64	C₂²	—	214	4	no	—
65	C₂²	—	214	4	no	—
66	C₂²	—	214	4	no	—
67	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	214	4	no	—
72	C₂	—	428	4	no	—
73	C₂	#16812155	1	2	yes	—
74	C₂	#16812157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#125×C₁₀₇	C₂
hypercenter	#125	D₁₀₇
hypocenter	C₁₀₇	#562
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #16812097	C₂	semidirect
2. #16812094	C₂	semidirect
3. #16812087	C₂	semidirect
4. #16812084	C₂	semidirect
5. #125×C₁₀₇	C₂	semidirect
37. C₁₀₇	#562	semidirect
39. #125	D₁₀₇	semidirect