Subgroups — C₄×#347×C₁₀₉ #42130534

Order 27904.

418 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#347×C₁₀₉	C₂	1	4	yes	maximal
2	C₄×M₃₂×C₁₀₉	C₂	1	2	yes	maximal
3	C₄×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
5	C₂×#97×C₁₀₉	C₄	1	2	yes	—
6	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
10	#347×C₁₀₉	C₄	1	16	yes	direct product factor
11	#346×C₁₀₉	C₄	1	4	yes	complemented
12	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	—	2	2	no	—
17	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₁₀₉	C₂×C₄	1	16	yes	—
20	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
21	#97×C₁₀₉	—	2	4	no	—
22	#97×C₁₀₉	C₂×C₄	1	8	yes	—
23	C₄×C₈×C₁₀₉	—	2	2	no	—
24	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	C₄²	1	4	yes	—
32	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
33	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
34	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
36	C₄²×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	—	4	4	no	—
38	C₄²×C₁₀₉	C₄²	1	4	yes	—
39	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
41	C₄²×C₁₀₉	M₁₆	1	2	yes	—
42	C₁₆×C₁₀₉	C₄²	1	16	yes	complemented
43	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	4	4	no	—
47	C₂×C₄×C₁₀₉	#97	1	2	yes	—
48	C₂×C₄×C₁₀₉	#97	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	C₈×C₁₀₉	—	2	4	no	—
54	C₈×C₁₀₉	#97	1	4	yes	—
55	C₈×C₁₀₉	C₂×C₄²	1	4	yes	—
56	C₂²×C₁₀₉	—	2	2	no	—
57	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₄×M₁₆	1	2	yes	—
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	4	4	no	—
61	C₄×C₁₀₉	#347	1	4	yes	direct product factor
62	C₄×C₁₀₉	C₄³	1	1	yes	characteristic
63	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
64	C₄×C₁₀₉	C₄×M₁₆	1	2	yes	—
65	C₄×#347	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₀₉	—	2	2	no	—
67	C₂×C₁₀₉	C₄×#97	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#347	1	2	yes	—
69	C₂×#347	C₂×C₁₀₉	1	4	yes	—
70	C₄×M₃₂	C₂×C₁₀₉	1	2	yes	—
71	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic
72	C₁₀₉	C₄×#347	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
74	C₂×#97	C₄×C₁₀₉	1	2	yes	—
75	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
76	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
78	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
79	#347	C₄×C₁₀₉	1	16	yes	direct product factor
80	#346	C₄×C₁₀₉	1	4	yes	complemented
81	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
82	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
83	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
84	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
85	C₂×C₄²	—	2	2	no	—
86	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
87	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
88	M₃₂	C₂×C₄×C₁₀₉	1	16	yes	—
89	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
90	#97	—	2	4	no	—
91	#97	C₂×C₄×C₁₀₉	1	8	yes	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
94	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
97	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₄²×C₁₀₉	1	4	yes	—
101	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
102	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
103	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
104	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
105	C₄²	—	2	2	no	—
106	C₄²	—	4	4	no	—
107	C₄²	C₄²×C₁₀₉	1	4	yes	—
108	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
109	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
110	C₄²	M₁₆×C₁₀₉	1	2	yes	—
111	C₁₆	C₄²×C₁₀₉	1	16	yes	complemented
112	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#97×C₁₀₉	1	2	yes	—
117	C₂×C₄	#97×C₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
120	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
121	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	#97×C₁₀₉	1	4	yes	—
124	C₈	C₂×C₄²×C₁₀₉	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
127	C₂²	C₄×M₁₆×C₁₀₉	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	—	4	4	no	—
130	C₄	#347×C₁₀₉	1	4	yes	direct product factor
131	C₄	C₄³×C₁₀₉	1	1	yes	characteristic
132	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
133	C₄	C₄×M₁₆×C₁₀₉	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	C₄×#97×C₁₀₉	1	1	yes	characteristic
136	C₂	C₂×#347×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₀₉	C₄²
commutator	C₄	C₄³×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#97
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. #347×C₁₀₉	C₄	direct (N × H)
11. #346×C₁₀₉	C₄	semidirect
12. C₄×C₁₆×C₁₀₉	C₄	semidirect
42. C₁₆×C₁₀₉	C₄²	semidirect
61. C₄×C₁₀₉	#347	direct (N × H)
65. C₄×#347	C₁₀₉	direct (N × H)
72. C₁₀₉	C₄×#347	direct (N × H)
79. #347	C₄×C₁₀₉	direct (N × H)
80. #346	C₄×C₁₀₉	semidirect
81. C₄×C₁₆	C₄×C₁₀₉	semidirect
111. C₁₆	C₄²×C₁₀₉	semidirect
130. C₄	#347×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×#347×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
C₄×#97×C₁₀₉	2
C₄×#347	109

Subgroups — C4×#347×C109 #42130534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#347×C₁₀₉ #42130534