Subgroups — C₂×C₄×C₂²⋊Dic₁₀₉ #28365362

Order 13952.

1360 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Dic₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₁₀₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812606	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₀₉	C₄	1	16	yes	direct product factor
8	C₂×C₂²⋊Dic₁₀₉	C₂²	1	8	yes	—
9	C₂×C₄×Dic₁₀₉	—	2	8	no	—
10	C₂×C₄×Dic₁₀₉	C₂²	1	8	yes	complemented
11	#16812606	C₂²	1	8	yes	complemented
12	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₀₉	—	2	8	no	—
14	C₂³×C₄×C₁₀₉	C₄	1	8	yes	complemented
15	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₉	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	—	2	8	no	—
19	C₂²×Dic₁₀₉	—	2	8	no	—
20	C₂²×Dic₁₀₉	C₂×C₄	1	16	yes	complemented
21	C₂²×Dic₁₀₉	C₂³	1	8	yes	—
22	C₂²⋊Dic₁₀₉	C₂×C₄	1	32	yes	direct product factor
23	C₄×Dic₁₀₉	—	2	16	no	—
24	C₂⁴×C₁₀₉	—	2	4	no	—
25	C₂⁴×C₁₀₉	—	2	4	no	—
26	C₂⁴×C₁₀₉	C₂×C₄	1	8	yes	—
27	C₂⁴×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
29	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	—	2	4	no	—
31	C₂²×C₄×C₁₀₉	—	2	32	no	—
32	C₂²×C₄×C₁₀₉	—	2	16	no	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
36	C₂²×C₄×C₁₀₉	D₄	1	8	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂³×C₁₀₉	—	2	8	no	—
40	C₂³×C₁₀₉	—	2	8	no	—
41	C₂³×C₁₀₉	—	2	32	no	—
42	C₂³×C₁₀₉	—	2	4	no	—
43	C₂³×C₁₀₉	—	2	4	no	—
44	C₂³×C₁₀₉	C₄²	1	16	yes	complemented
45	C₂³×C₁₀₉	C₂²×C₄	1	8	yes	—
46	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
47	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
50	C₂³×C₁₀₉	#41	1	4	yes	—
51	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	—	2	8	no	—
53	C₂×C₄×C₁₀₉	—	2	4	no	—
54	C₂×C₄×C₁₀₉	—	2	32	no	—
55	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	16	yes	complemented
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₀₉	C₂×D₄	1	8	yes	—
59	C₂×Dic₁₀₉	—	2	32	no	—
60	C₂×Dic₁₀₉	—	2	8	no	—
61	C₂×Dic₁₀₉	—	2	8	no	—
62	C₂×Dic₁₀₉	C₂²×C₄	1	16	yes	complemented
63	C₂²×C₁₀₉	—	2	4	no	—
64	C₂²×C₁₀₉	—	2	4	no	—
65	C₂²×C₁₀₉	—	2	16	no	—
66	C₂²×C₁₀₉	—	2	32	no	—
67	C₂²×C₁₀₉	C₂×C₄²	1	4	yes	—
68	C₂²×C₁₀₉	C₂×C₄²	1	8	yes	complemented
69	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
70	C₂²×C₁₀₉	#117	1	4	yes	—
71	C₂²×C₁₀₉	C₄×D₄	1	16	yes	—
72	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
75	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
76	C₄×C₁₀₉	—	2	4	no	—
77	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	8	yes	complemented
78	Dic₁₀₉	—	2	16	no	—
79	C₂×C₁₀₉	—	2	8	no	—
80	C₂×C₁₀₉	C₄×C₂²⋊C₄	1	8	yes	complemented
81	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
82	C₂×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂×C₄×D₄	1	4	yes	—
85	C₂×C₄×C₂²⋊C₄	—	109	1	no	maximal, Sylow
86	C₁₀₉	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic
88	C₂²×C₂²⋊C₄	—	109	2	no	—
89	C₂²×C₄²	—	109	2	no	—
90	C₄×C₂²⋊C₄	—	109	8	no	—
91	C₂×#95	—	109	2	no	—
92	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
93	C₂³×C₄	—	2	8	no	—
94	C₂³×C₄	—	109	4	no	—
95	C₂³×C₄	Dic₁₀₉	1	8	yes	complemented
96	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
97	C₂³×C₄	C₂×D₁₀₉	1	4	yes	—
98	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	—	109	16	no	—
100	C₂×C₂²⋊C₄	—	109	8	no	—
101	C₂×C₄²	—	109	8	no	—
102	C₂×C₄²	—	218	8	no	—
103	#95	—	109	8	no	—
104	C₂⁴	—	2	4	no	—
105	C₂⁴	—	2	4	no	—
106	C₂⁴	C₄×D₁₀₉	1	8	yes	—
107	C₂⁴	C₂×Dic₁₀₉	1	4	yes	—
108	C₂⁴	C₂²×D₁₀₉	1	2	yes	—
109	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	32	no	—
112	C₂²×C₄	—	2	16	no	—
113	C₂²×C₄	—	109	16	no	—
114	C₂²×C₄	—	109	8	no	—
115	C₂²×C₄	—	218	8	no	—
116	C₂²×C₄	—	218	8	no	—
117	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
118	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
119	C₂²×C₄	C₂×Dic₁₀₉	1	16	yes	complemented
120	C₂²×C₄	C₂²⋊D₁₀₉	1	8	yes	—
121	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
123	C₂²⋊C₄	—	109	32	no	—
124	C₄²	—	218	16	no	—
125	C₂³	—	2	8	no	—
126	C₂³	—	2	8	no	—
127	C₂³	—	2	32	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	C₄×Dic₁₀₉	1	16	yes	complemented
131	C₂³	C₂×C₄×D₁₀₉	1	8	yes	—
132	C₂³	C₂×C₄×D₁₀₉	1	4	yes	—
133	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
134	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
135	C₂³	C₂×C₂²⋊D₁₀₉	1	4	yes	—
136	C₂³	#13182029	1	4	yes	—
137	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	109	16	no	—
142	C₂×C₄	—	218	32	no	—
143	C₂×C₄	—	218	8	no	—
144	C₂×C₄	—	218	8	no	—
145	C₂×C₄	C₂²⋊Dic₁₀₉	1	16	yes	direct product factor
146	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
147	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	8	yes	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	16	no	—
152	C₂²	—	2	32	no	—
153	C₂²	C₂×C₄×Dic₁₀₉	1	4	yes	—
154	C₂²	C₂×C₄×Dic₁₀₉	1	8	yes	complemented
155	C₂²	C₂×C₂²⋊Dic₁₀₉	1	4	yes	—
156	C₂²	#16812658	1	4	yes	—
157	C₂²	C₄×C₂²⋊D₁₀₉	1	16	yes	—
158	C₂²	C₂²×C₄×D₁₀₉	1	2	yes	—
159	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
160	C₂²	C₂²×C₂²⋊D₁₀₉	1	2	yes	—
161	C₂²	C₂×#13182029	1	2	yes	—
162	C₄	—	2	4	no	—
163	C₄	—	218	16	no	—
164	C₄	C₂×C₂²⋊Dic₁₀₉	1	8	yes	direct product factor
165	C₂	—	2	8	no	—
166	C₂	C₄×C₂²⋊Dic₁₀₉	1	8	yes	direct product factor
167	C₂	C₂²×C₄×Dic₁₀₉	1	1	yes	characteristic
168	C₂	C₂²×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
169	C₂	C₂×#16812658	1	1	yes	characteristic
170	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂⁴×C₄×C₁₀₉	C₂
socle	C₂⁴×C₁₀₉	C₂³
prime radical	C₂⁵×C₁₀₉	C₂²
cosocle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂⁴×C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₁₀₉	C₂	semidirect
3. C₄×C₂²⋊Dic₁₀₉	C₂	direct (N × H)
4. C₂×#16812606	C₂	semidirect
7. C₂×C₂²⋊Dic₁₀₉	C₄	direct (N × H)
10. C₂×C₄×Dic₁₀₉	C₂²	semidirect
11. #16812606	C₂²	semidirect
14. C₂³×C₄×C₁₀₉	C₄	semidirect
20. C₂²×Dic₁₀₉	C₂×C₄	semidirect
22. C₂²⋊Dic₁₀₉	C₂×C₄	direct (N × H)
35. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
44. C₂³×C₁₀₉	C₄²	semidirect
55. C₂×C₄×C₁₀₉	C₂²⋊C₄	semidirect
62. C₂×Dic₁₀₉	C₂²×C₄	semidirect
68. C₂²×C₁₀₉	C₂×C₄²	semidirect
77. C₄×C₁₀₉	C₂×C₂²⋊C₄	semidirect
80. C₂×C₁₀₉	C₄×C₂²⋊C₄	semidirect
86. C₁₀₉	C₂×C₄×C₂²⋊C₄	semidirect
95. C₂³×C₄	Dic₁₀₉	semidirect
119. C₂²×C₄	C₂×Dic₁₀₉	semidirect
130. C₂³	C₄×Dic₁₀₉	semidirect
145. C₂×C₄	C₂²⋊Dic₁₀₉	direct (N × H)
154. C₂²	C₂×C₄×Dic₁₀₉	semidirect
164. C₄	C₂×C₂²⋊Dic₁₀₉	direct (N × H)
166. C₂	C₄×C₂²⋊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×C₂²⋊Dic₁₀₉	2
C₂²×C₄×Dic₁₀₉	2
C₄×C₂²⋊Dic₁₀₉	2
C₂×#16812606	2
C₂⁴×C₄×C₁₀₉	2
C₂×C₄×C₂²⋊C₄	109

Subgroups — C2×C4×C22⋊Dic109 #28365362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂²⋊Dic₁₀₉ #28365362