Subgroups — #550×C₁₀₉ #20294708

Order 6976.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#127×C₁₀₉	C₂	1	3	yes	maximal, complemented
3	#121×C₁₀₉	C₂	1	6	yes	maximal, complemented
4	C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×C₁₀₉	C₂	1	3	yes	maximal, complemented
6	#41×C₁₀₉	—	2	8	no	—
7	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₁₀₉	C₂²	1	3	yes	—
9	C₂×D₄×C₁₀₉	C₂²	1	6	yes	—
10	C₂×D₄×C₁₀₉	C₂²	1	6	yes	—
11	C₂²×C₄×C₁₀₉	C₂²	1	6	yes	—
12	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
14	C₂²⋊C₄×C₁₀₉	C₂²	1	6	yes	—
15	C₄²×C₁₀₉	C₂²	1	3	yes	—
16	C₂³×C₁₀₉	C₂³	1	6	yes	—
17	Q₈×C₁₀₉	D₄	1	4	yes	complemented
18	D₄×C₁₀₉	—	2	12	no	—
19	D₄×C₁₀₉	—	2	6	no	—
20	D₄×C₁₀₉	—	4	6	no	—
21	C₂×C₄×C₁₀₉	—	2	12	no	—
22	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
23	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
24	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
25	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₀₉	—	2	6	no	—
27	C₂²×C₁₀₉	—	2	6	no	—
28	C₂²×C₁₀₉	—	2	6	no	—
29	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
30	C₄×C₁₀₉	—	2	2	no	—
31	C₄×C₁₀₉	—	2	3	no	—
32	C₄×C₁₀₉	C₂×D₄	1	6	yes	—
33	C₄×C₁₀₉	#41	1	2	yes	complemented
34	C₂×C₁₀₉	—	4	6	no	—
35	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
36	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
37	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
38	C₁₀₉	#550	1	1	yes	characteristic, direct product factor, Sylow
39	#550	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×#41	C₂×C₁₀₉	1	2	yes	complemented
41	#127	C₂×C₁₀₉	1	3	yes	complemented
42	#121	C₂×C₁₀₉	1	6	yes	complemented
43	C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
44	C₄×D₄	C₂×C₁₀₉	1	3	yes	complemented
45	#41	—	2	8	no	—
46	C₂×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
47	C₂×D₄	C₂²×C₁₀₉	1	3	yes	—
48	C₂×D₄	C₂²×C₁₀₉	1	6	yes	—
49	C₂×D₄	C₂²×C₁₀₉	1	6	yes	—
50	C₂²×C₄	C₂²×C₁₀₉	1	6	yes	—
51	C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
52	C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
53	C₂²⋊C₄	C₂²×C₁₀₉	1	6	yes	—
54	C₄²	C₂²×C₁₀₉	1	3	yes	—
55	C₂³	C₂³×C₁₀₉	1	6	yes	—
56	Q₈	D₄×C₁₀₉	1	4	yes	complemented
57	D₄	—	2	12	no	—
58	D₄	—	2	6	no	—
59	D₄	—	4	6	no	—
60	C₂×C₄	—	2	12	no	—
61	C₂×C₄	C₂³×C₁₀₉	1	3	yes	—
62	C₂×C₄	C₂³×C₁₀₉	1	2	yes	—
63	C₂×C₄	C₂³×C₁₀₉	1	3	yes	—
64	C₂×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
65	C₂²	—	2	6	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	6	no	—
68	C₂²	C₂⁴×C₁₀₉	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	3	no	—
71	C₄	C₂×D₄×C₁₀₉	1	6	yes	—
72	C₄	#41×C₁₀₉	1	2	yes	complemented
73	C₂	—	4	6	no	—
74	C₂	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
75	C₂	C₂×#41×C₁₀₉	1	1	yes	characteristic
76	C₂	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#41×C₁₀₉	C₂	semidirect
2. #127×C₁₀₉	C₂	semidirect
3. #121×C₁₀₉	C₂	semidirect
4. C₄×Q₈×C₁₀₉	C₂	semidirect
5. C₄×D₄×C₁₀₉	C₂	semidirect
17. Q₈×C₁₀₉	D₄	semidirect
33. C₄×C₁₀₉	#41	semidirect
38. C₁₀₉	#550	direct (N × H)
39. #550	C₁₀₉	direct (N × H)
40. C₂×#41	C₂×C₁₀₉	semidirect
41. #127	C₂×C₁₀₉	semidirect
42. #121	C₂×C₁₀₉	semidirect
43. C₄×Q₈	C₂×C₁₀₉	semidirect
44. C₄×D₄	C₂×C₁₀₉	semidirect
56. Q₈	D₄×C₁₀₉	semidirect
72. C₄	#41×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#41×C₁₀₉	2
#127×C₁₀₉	2
#121×C₁₀₉	2
C₄×Q₈×C₁₀₉	2
C₄×D₄×C₁₀₉	2
#550	109

Subgroups — #550×C109 #20294708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #550×C₁₀₉ #20294708