Subgroups — C₁₀₉×#2658 #28359912

Order 13952.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	3	yes	maximal, complemented
2	C₂×#124×C₁₀₉	C₂	1	3	yes	maximal
3	#399×C₁₀₉	C₂	1	8	yes	maximal, complemented
4	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₁₀₉	C₂²	1	3	yes	—
6	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₀₉	—	2	12	no	—
8	#124×C₁₀₉	—	2	12	no	—
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	—
10	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
12	#95×C₁₀₉	C₂²	1	4	yes	—
13	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
14	C₂×Q₈×C₁₀₉	—	2	6	no	—
15	C₂×Q₈×C₁₀₉	—	2	12	no	—
16	C₂×D₄×C₁₀₉	—	2	6	no	—
17	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	—	2	24	no	—
21	C₂²⋊C₄×C₁₀₉	—	2	24	no	—
22	C₄²×C₁₀₉	D₄	1	12	yes	—
23	C₂³×C₁₀₉	—	2	6	no	—
24	C₂³×C₁₀₉	—	2	8	no	—
25	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
26	Q₈×C₁₀₉	—	4	12	no	—
27	D₄×C₁₀₉	—	4	4	no	—
28	C₂×C₄×C₁₀₉	—	2	12	no	—
29	C₂×C₄×C₁₀₉	—	2	6	no	—
30	C₂×C₄×C₁₀₉	—	2	12	no	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
33	C₂²×C₁₀₉	—	4	2	no	—
34	C₂²×C₁₀₉	—	4	12	no	—
35	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
36	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
37	C₄×C₁₀₉	—	2	12	no	—
38	C₄×C₁₀₉	—	4	6	no	—
39	C₄×C₁₀₉	#127	1	4	yes	—
40	C₂×C₁₀₉	—	8	2	no	—
41	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
42	C₂×C₁₀₉	#536	1	6	yes	—
43	#2658	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₁₀₉	#2658	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	3	yes	complemented
46	C₂×#124	C₂×C₁₀₉	1	3	yes	—
47	#399	C₂×C₁₀₉	1	8	yes	complemented
48	#376	C₂×C₁₀₉	1	1	yes	characteristic, complemented
49	C₂²×Q₈	C₂²×C₁₀₉	1	3	yes	—
50	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	12	no	—
52	#124	—	2	12	no	—
53	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	—
54	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	12	yes	—
55	C₂×C₄²	C₂²×C₁₀₉	1	3	yes	—
56	#95	C₂²×C₁₀₉	1	4	yes	—
57	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
58	C₂×Q₈	—	2	6	no	—
59	C₂×Q₈	—	2	12	no	—
60	C₂×D₄	—	2	6	no	—
61	C₂²×C₄	C₂³×C₁₀₉	1	6	yes	—
62	C₂²×C₄	C₂³×C₁₀₉	1	6	yes	—
63	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
64	C₄⋊C₄	—	2	24	no	—
65	C₂²⋊C₄	—	2	24	no	—
66	C₄²	D₄×C₁₀₉	1	12	yes	—
67	C₂³	—	2	6	no	—
68	C₂³	—	2	8	no	—
69	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
70	Q₈	—	4	12	no	—
71	D₄	—	4	4	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	2	6	no	—
74	C₂×C₄	—	2	12	no	—
75	C₂×C₄	C₂×D₄×C₁₀₉	1	12	yes	—
76	C₂×C₄	C₂×D₄×C₁₀₉	1	6	yes	—
77	C₂²	—	4	2	no	—
78	C₂²	—	4	12	no	—
79	C₂²	C₂²×D₄×C₁₀₉	1	3	yes	—
80	C₂²	2₋¹⁺⁴×C₁₀₉	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	4	6	no	—
83	C₄	#127×C₁₀₉	1	4	yes	—
84	C₂	—	8	2	no	—
85	C₂	C₂×#127×C₁₀₉	1	1	yes	characteristic
86	C₂	#536×C₁₀₉	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
3. #399×C₁₀₉	C₂	semidirect
4. #376×C₁₀₉	C₂	semidirect
43. #2658	C₁₀₉	direct (N × H)
44. C₁₀₉	#2658	direct (N × H)
45. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
47. #399	C₂×C₁₀₉	semidirect
48. #376	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#124×C₁₀₉	2
#399×C₁₀₉	2
#376×C₁₀₉	2
#2658	109

Subgroups — C109×#2658 #28359912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2658 #28359912