Subgroups — C₂×C₄×C₄⋊Dic₁₀₉ #28365367

Order 13952.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₀₉	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₁₀₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812606	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₉	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	8	yes	—
9	C₂×C₄×Dic₁₀₉	—	2	8	no	—
10	C₂×C₄×Dic₁₀₉	C₂²	1	8	yes	—
11	#16812606	C₂²	1	8	yes	—
12	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₉	C₄	1	8	yes	complemented
16	C₂×C₄²×C₁₀₉	C₂²	1	4	yes	—
17	C₂²×Dic₁₀₉	—	2	8	no	—
18	C₂²×Dic₁₀₉	—	2	8	no	—
19	C₂²×Dic₁₀₉	C₂×C₄	1	16	yes	—
20	C₂²×Dic₁₀₉	C₂³	1	8	yes	—
21	C₄⋊Dic₁₀₉	C₂×C₄	1	32	yes	direct product factor
22	C₄×Dic₁₀₉	—	2	16	no	—
23	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₉	—	2	4	no	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
30	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	16	yes	complemented
35	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
36	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
37	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
38	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
39	C₂³×C₁₀₉	#41	1	4	yes	—
40	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	8	no	—
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	C₄²	1	16	yes	complemented
44	C₂×C₄×C₁₀₉	C₄⋊C₄	1	16	yes	complemented
45	C₂×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
51	C₂×Dic₁₀₉	—	2	32	no	—
52	C₂×Dic₁₀₉	—	2	8	no	—
53	C₂×Dic₁₀₉	—	2	8	no	—
54	C₂×Dic₁₀₉	C₂²×C₄	1	16	yes	—
55	C₂²×C₁₀₉	C₂×C₄²	1	4	yes	—
56	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₁₀₉	#117	1	4	yes	—
58	C₂²×C₁₀₉	C₄×D₄	1	8	yes	—
59	C₂²×C₁₀₉	C₄×Q₈	1	8	yes	—
60	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
61	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
64	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
65	C₄×C₁₀₉	—	2	4	no	—
66	C₄×C₁₀₉	C₂×C₄²	1	8	yes	complemented
67	C₄×C₁₀₉	C₂×C₄⋊C₄	1	8	yes	complemented
68	Dic₁₀₉	—	2	16	no	—
69	C₂×C₁₀₉	C₄×C₄⋊C₄	1	8	yes	complemented
70	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
73	C₂×C₁₀₉	C₂×C₄×D₄	1	2	yes	—
74	C₂×C₁₀₉	C₂×C₄×Q₈	1	2	yes	—
75	C₂×C₄×C₄⋊C₄	—	109	1	no	maximal, Sylow
76	C₁₀₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄⋊C₄	—	109	2	no	—
78	C₂²×C₄²	—	109	2	no	—
79	C₂²×C₄²	D₁₀₉	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	109	8	no	—
81	C₂×#95	—	109	2	no	—
82	C₂³×C₄	—	109	4	no	—
83	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
85	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	109	16	no	—
87	C₂×C₄⋊C₄	—	109	8	no	—
88	C₂×C₄²	—	109	8	no	—
89	C₂×C₄²	—	218	8	no	—
90	C₂×C₄²	Dic₁₀₉	1	8	yes	complemented
91	C₂×C₄²	C₂×D₁₀₉	1	4	yes	—
92	#95	—	109	8	no	—
93	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	109	16	no	—
96	C₂²×C₄	—	109	8	no	—
97	C₂²×C₄	—	218	8	no	—
98	C₂²×C₄	—	218	8	no	—
99	C₂²×C₄	Dic₂₁₈	1	4	yes	—
100	C₂²×C₄	C₄×D₁₀₉	1	8	yes	—
101	C₂²×C₄	D₄₃₆	1	4	yes	—
102	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
103	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
104	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
105	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
108	C₄⋊C₄	—	109	32	no	—
109	C₄²	—	218	16	no	—
110	C₄²	C₂×Dic₁₀₉	1	16	yes	complemented
111	C₂³	C₂×C₄×D₁₀₉	1	4	yes	—
112	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
113	C₂³	C₂×D₄₃₆	1	2	yes	—
114	C₂³	C₂×Dic₂₁₈	1	2	yes	—
115	C₂³	#13182029	1	4	yes	—
116	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
117	C₂×C₄	—	2	8	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	109	16	no	—
120	C₂×C₄	—	218	32	no	—
121	C₂×C₄	—	218	8	no	—
122	C₂×C₄	—	218	8	no	—
123	C₂×C₄	C₄×Dic₁₀₉	1	16	yes	complemented
124	C₂×C₄	C₄⋊Dic₁₀₉	1	16	yes	direct product factor
125	C₂×C₄	C₂×C₄×D₁₀₉	1	8	yes	—
126	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
127	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
128	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
129	C₂×C₄	C₂×D₄₃₆	1	4	yes	—
130	C₂×C₄	C₂×Dic₂₁₈	1	4	yes	—
131	C₂²	C₂×C₄×Dic₁₀₉	1	4	yes	—
132	C₂²	C₂×C₄⋊Dic₁₀₉	1	4	yes	—
133	C₂²	#16812658	1	4	yes	—
134	C₂²	C₄×D₄₃₆	1	8	yes	—
135	C₂²	C₄×Dic₂₁₈	1	8	yes	—
136	C₂²	C₂²×C₄×D₁₀₉	1	2	yes	—
137	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
138	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
139	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
140	C₂²	C₂×#13182029	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	218	16	no	—
143	C₄	C₂×C₄×Dic₁₀₉	1	8	yes	complemented
144	C₄	C₂×C₄⋊Dic₁₀₉	1	8	yes	direct product factor
145	C₂	C₄×C₄⋊Dic₁₀₉	1	8	yes	direct product factor
146	C₂	C₂²×C₄×Dic₁₀₉	1	1	yes	characteristic
147	C₂	C₂²×C₄⋊Dic₁₀₉	1	1	yes	characteristic
148	C₂	C₂×#16812658	1	1	yes	characteristic
149	C₂	C₂×C₄×D₄₃₆	1	2	yes	—
150	C₂	C₂×C₄×Dic₂₁₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂²×C₄²×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₂³
cosocle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂²×C₄²	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊Dic₁₀₉	C₂	direct (N × H)
7. C₂×C₄⋊Dic₁₀₉	C₄	direct (N × H)
15. C₂×C₄²×C₁₀₉	C₄	semidirect
21. C₄⋊Dic₁₀₉	C₂×C₄	direct (N × H)
34. C₄²×C₁₀₉	C₂×C₄	semidirect
43. C₂×C₄×C₁₀₉	C₄²	semidirect
44. C₂×C₄×C₁₀₉	C₄⋊C₄	semidirect
66. C₄×C₁₀₉	C₂×C₄²	semidirect
67. C₄×C₁₀₉	C₂×C₄⋊C₄	semidirect
69. C₂×C₁₀₉	C₄×C₄⋊C₄	semidirect
76. C₁₀₉	C₂×C₄×C₄⋊C₄	semidirect
90. C₂×C₄²	Dic₁₀₉	semidirect
110. C₄²	C₂×Dic₁₀₉	semidirect
123. C₂×C₄	C₄×Dic₁₀₉	semidirect
124. C₂×C₄	C₄⋊Dic₁₀₉	direct (N × H)
143. C₄	C₂×C₄×Dic₁₀₉	semidirect
144. C₄	C₂×C₄⋊Dic₁₀₉	direct (N × H)
145. C₂	C₄×C₄⋊Dic₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₀₉	2
C₂²×C₄×Dic₁₀₉	2
C₄×C₄⋊Dic₁₀₉	2
C₂×#16812606	2
C₂²×C₄²×C₁₀₉	2
C₂×C₄×C₄⋊C₄	109

Subgroups — C2×C4×C4⋊Dic109 #28365367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊Dic₁₀₉ #28365367