Subgroups — C₄×#108×C₁₀₉ #28359031

Order 13952.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×C₁₀₉	C₂	1	2	yes	maximal
2	C₄×M₁₆×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#341×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×M₁₆×C₁₀₉	C₂²	1	4	yes	—
6	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
9	#108×C₁₀₉	C₄	1	8	yes	direct product factor
10	#97×C₁₀₉	—	2	2	no	—
11	C₄×C₈×C₁₀₉	—	2	2	no	—
12	C₄×C₈×C₁₀₉	C₄	1	4	yes	—
13	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
14	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
15	M₁₆×C₁₀₉	—	2	4	no	—
16	M₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
17	C₂×C₈×C₁₀₉	—	2	4	no	—
18	C₂×C₈×C₁₀₉	—	2	2	no	—
19	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
24	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	—	2	4	no	—
27	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	#41	1	2	yes	—
33	C₈×C₁₀₉	—	2	2	no	—
34	C₈×C₁₀₉	—	2	8	no	—
35	C₈×C₁₀₉	C₄²	1	4	yes	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
39	C₄×C₁₀₉	—	2	2	no	—
40	C₄×C₁₀₉	#108	1	4	yes	direct product factor
41	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₁₀₉	#117	1	1	yes	characteristic
43	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
44	C₂×C₁₀₉	—	2	2	no	—
45	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂×#108	1	2	yes	—
47	C₄×#108	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₁₀₉	C₄×#108	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×#108	C₂×C₁₀₉	1	2	yes	—
50	C₄×M₁₆	C₂×C₁₀₉	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
52	#341	C₂×C₁₀₉	1	2	yes	—
53	C₂×M₁₆	C₂²×C₁₀₉	1	4	yes	—
54	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
56	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
57	#108	C₄×C₁₀₉	1	8	yes	direct product factor
58	#97	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₄×C₁₀₉	1	4	yes	—
61	C₂²×C₄	Q₈×C₁₀₉	1	2	yes	—
62	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
68	C₂×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
69	C₂×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
70	C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
71	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
72	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
73	C₂³	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
76	C₂×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
77	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
79	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
80	C₂×C₄	#41×C₁₀₉	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	2	8	no	—
83	C₈	C₄²×C₁₀₉	1	4	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
86	C₂²	C₄×Q₈×C₁₀₉	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#108×C₁₀₉	1	4	yes	direct product factor
89	C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
90	C₄	#117×C₁₀₉	1	1	yes	characteristic
91	C₄	C₄×D₄×C₁₀₉	1	2	yes	—
92	C₂	—	2	2	no	—
93	C₂	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
94	C₂	C₂×#108×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₀₉	D₄
commutator	C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
9. #108×C₁₀₉	C₄	direct (N × H)
40. C₄×C₁₀₉	#108	direct (N × H)
47. C₄×#108	C₁₀₉	direct (N × H)
48. C₁₀₉	C₄×#108	direct (N × H)
57. #108	C₄×C₁₀₉	direct (N × H)
88. C₄	#108×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#108×C₁₀₉	2
C₄×M₁₆×C₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
#341×C₁₀₉	2
C₄×#108	109

Subgroups — C4×#108×C109 #28359031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#108×C₁₀₉ #28359031