Subgroups — C₄×M₁₆×D₁₀₉ #28371437

Order 13952.

656 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₀₉	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294978	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₄×#13182012	C₂	1	2	yes	maximal, complemented
7	#97×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×C₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₀₉	C₄	1	2	yes	—
11	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#13182013	C₂²	1	2	yes	—
13	C₂×#13182012	C₂²	1	4	yes	—
14	M₁₆×D₁₀₉	C₄	1	16	yes	direct product factor
15	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₄²×D₁₀₉	C₄	1	2	yes	—
19	C₄²×D₁₀₉	C₄	1	2	yes	—
20	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
22	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
23	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
24	#16812610	C₂²	1	2	yes	complemented
25	#16812609	C₂²	1	4	yes	complemented
26	C₄×#11724784	C₂²	1	2	yes	complemented
27	C₈×Dic₁₀₉	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
30	#97×C₁₀₉	C₂²	1	2	yes	complemented
31	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
32	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₀₉	—	2	8	no	—
36	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₀₉	C₂×C₄	1	8	yes	—
42	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
45	#13182013	C₂×C₄	1	8	yes	complemented
46	#13182012	C₂×C₄	1	16	yes	complemented
47	C₂×#11724784	C₂³	1	4	yes	—
48	C₈×D₁₀₉	C₂×C₄	1	16	yes	complemented
49	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
55	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
59	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
60	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
61	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×D₁₀₉	—	2	4	no	—
65	C₂²×D₁₀₉	C₄²	1	4	yes	—
66	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₁₀₉	—	2	4	no	—
75	C₂×Dic₁₀₉	C₄²	1	4	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
80	C₄×D₁₀₉	—	2	4	no	—
81	C₄×D₁₀₉	—	2	4	no	—
82	C₄×D₁₀₉	—	2	4	no	—
83	C₄×D₁₀₉	C₄²	1	4	yes	—
84	C₄×D₁₀₉	C₄²	1	4	yes	—
85	C₄×D₁₀₉	M₁₆	1	8	yes	direct product factor
86	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
87	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
88	C₈×C₁₀₉	C₂²×C₄	1	8	yes	complemented
89	#11724784	C₂²×C₄	1	8	yes	complemented
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
92	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₀₉	—	2	2	no	—
94	C₂×D₁₀₉	—	2	8	no	—
95	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
96	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
97	C₂×D₁₀₉	C₂×M₁₆	1	4	yes	—
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
100	C₄×C₁₀₉	C₂×M₁₆	1	4	yes	complemented
101	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
102	Dic₁₀₉	—	2	2	no	—
103	Dic₁₀₉	C₂×C₄²	1	2	yes	—
104	Dic₁₀₉	C₂×C₄²	1	2	yes	—
105	Dic₁₀₉	C₂×M₁₆	1	4	yes	complemented
106	C₂×C₁₀₉	—	2	2	no	—
107	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
108	C₂×C₁₀₉	C₂²×M₁₆	1	2	yes	—
109	D₁₀₉	—	2	2	no	—
110	D₁₀₉	C₄×M₁₆	1	4	yes	direct product factor
111	C₂×C₄×M₁₆	—	109	1	no	maximal, Sylow
112	C₁₀₉	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×M₁₆	—	109	2	no	—
114	C₂²×C₄²	—	109	1	no	—
115	C₄×M₁₆	—	109	2	no	—
116	C₄×M₁₆	—	109	1	no	—
117	C₄×M₁₆	—	109	2	no	—
118	C₄×M₁₆	—	109	2	no	—
119	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic, direct product factor
120	C₂×#97	—	109	2	no	—
121	C₂×C₄×C₈	—	109	2	no	—
122	C₂³×C₄	—	109	1	no	—
123	C₂³×C₄	—	109	2	no	—
124	C₂×M₁₆	—	109	2	no	—
125	C₂×M₁₆	—	109	16	no	—
126	C₂×M₁₆	—	109	4	no	—
127	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
128	C₂²×C₈	—	109	4	no	—
129	C₂×C₄²	—	109	2	no	—
130	C₂×C₄²	—	109	1	no	—
131	C₂×C₄²	—	109	2	no	—
132	C₂×C₄²	—	109	2	no	—
133	C₂×C₄²	—	109	1	no	—
134	C₂×C₄²	—	109	1	no	—
135	C₂×C₄²	—	109	1	no	—
136	C₂×C₄²	—	109	1	no	—
137	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
138	#97	—	109	4	no	—
139	#97	—	109	2	no	—
140	#97	C₂×D₁₀₉	1	2	yes	complemented
141	C₄×C₈	—	109	4	no	—
142	C₄×C₈	—	109	2	no	—
143	C₄×C₈	C₂×D₁₀₉	1	2	yes	complemented
144	C₂⁴	—	109	1	no	—
145	C₂²×C₄	—	109	1	no	—
146	C₂²×C₄	—	109	1	no	—
147	C₂²×C₄	—	109	1	no	—
148	C₂²×C₄	—	109	2	no	—
149	C₂²×C₄	—	109	8	no	—
150	C₂²×C₄	—	109	1	no	—
151	C₂²×C₄	—	109	1	no	—
152	C₂²×C₄	—	109	2	no	—
153	C₂²×C₄	—	109	2	no	—
154	C₂²×C₄	—	109	2	no	—
155	C₂²×C₄	—	109	2	no	—
156	C₂²×C₄	—	218	8	no	—
157	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
159	M₁₆	—	109	8	no	—
160	M₁₆	—	109	16	no	—
161	M₁₆	C₄×D₁₀₉	1	8	yes	direct product factor
162	C₂×C₈	—	109	4	no	—
163	C₂×C₈	—	109	16	no	—
164	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
165	C₄²	—	109	2	no	—
166	C₄²	—	109	2	no	—
167	C₄²	—	109	2	no	—
168	C₄²	—	109	2	no	—
169	C₄²	—	109	2	no	—
170	C₄²	—	109	1	no	—
171	C₄²	—	109	1	no	—
172	C₄²	C₄×D₁₀₉	1	2	yes	—
173	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
174	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
175	C₂³	—	109	1	no	—
176	C₂³	—	109	4	no	—
177	C₂³	—	109	1	no	—
178	C₂³	—	218	4	no	—
179	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	109	4	no	—
182	C₂×C₄	—	109	1	no	—
183	C₂×C₄	—	109	1	no	—
184	C₂×C₄	—	109	2	no	—
185	C₂×C₄	—	109	4	no	—
186	C₂×C₄	—	109	4	no	—
187	C₂×C₄	—	109	8	no	—
188	C₂×C₄	—	109	4	no	—
189	C₂×C₄	—	109	4	no	—
190	C₂×C₄	—	109	2	no	—
191	C₂×C₄	—	218	4	no	—
192	C₂×C₄	—	218	4	no	—
193	C₂×C₄	—	218	4	no	—
194	C₂×C₄	—	218	4	no	—
195	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
196	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
199	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
200	C₈	—	109	8	no	—
201	C₈	C₂×C₄×D₁₀₉	1	8	yes	complemented
202	C₂²	—	2	2	no	—
203	C₂²	—	109	2	no	—
204	C₂²	—	109	4	no	—
205	C₂²	—	109	2	no	—
206	C₂²	—	218	8	no	—
207	C₂²	—	218	2	no	—
208	C₂²	C₄²×D₁₀₉	1	2	yes	—
209	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	109	2	no	—
212	C₄	—	109	2	no	—
213	C₄	—	109	4	no	—
214	C₄	—	218	2	no	—
215	C₄	C₄²×D₁₀₉	1	2	yes	—
216	C₄	M₁₆×D₁₀₉	1	4	yes	direct product factor
217	C₄	C₂²×C₄×D₁₀₉	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	109	4	no	—
220	C₂	—	218	2	no	—
221	C₂	C₂×C₄²×D₁₀₉	1	1	yes	characteristic
222	C₂	C₂×M₁₆×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂³×D₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#13182013	C₂	semidirect
4. #20294978	C₂	semidirect
5. M₁₆×Dic₁₀₉	C₂	semidirect
6. C₄×#13182012	C₂	semidirect
7. #97×D₁₀₉	C₂	semidirect
8. C₄×C₈×D₁₀₉	C₂	semidirect
9. C₄×M₁₆×C₁₀₉	C₂	semidirect
14. M₁₆×D₁₀₉	C₄	direct (N × H)
24. #16812610	C₂²	semidirect
25. #16812609	C₂²	semidirect
26. C₄×#11724784	C₂²	semidirect
27. C₈×Dic₁₀₉	C₂²	semidirect
30. #97×C₁₀₉	C₂²	semidirect
31. C₄×C₈×C₁₀₉	C₂²	semidirect
45. #13182013	C₂×C₄	semidirect
46. #13182012	C₂×C₄	semidirect
48. C₈×D₁₀₉	C₂×C₄	semidirect
58. M₁₆×C₁₀₉	C₂×C₄	semidirect
85. C₄×D₁₀₉	M₁₆	direct (N × H)
88. C₈×C₁₀₉	C₂²×C₄	semidirect
89. #11724784	C₂²×C₄	semidirect
100. C₄×C₁₀₉	C₂×M₁₆	semidirect
105. Dic₁₀₉	C₂×M₁₆	semidirect
110. D₁₀₉	C₄×M₁₆	direct (N × H)
112. C₁₀₉	C₂×C₄×M₁₆	semidirect
119. C₄×M₁₆	D₁₀₉	direct (N × H)
140. #97	C₂×D₁₀₉	semidirect
143. C₄×C₈	C₂×D₁₀₉	semidirect
161. M₁₆	C₄×D₁₀₉	direct (N × H)
201. C₈	C₂×C₄×D₁₀₉	semidirect
216. C₄	M₁₆×D₁₀₉	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×M₁₆×D₁₀₉	2
C₂×C₄²×D₁₀₉	2
C₄×#13182013	2
#20294978	2
M₁₆×Dic₁₀₉	2
C₄×#13182012	2
#97×D₁₀₉	2
C₄×C₈×D₁₀₉	2
C₄×M₁₆×C₁₀₉	2
C₂×C₄×M₁₆	109

Subgroups — C4×M16×D109 #28371437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×D₁₀₉ #28371437