Subgroups — #405×D₁₀₉ #28371442

Order 13952.

592 conjugacy classes of subgroups.

The subgroup lattice has 248 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	1	yes	maximal, characteristic
2	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294984	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294983	C₂	1	2	yes	maximal, complemented
6	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
7	#20294981	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294980	C₂	1	2	yes	maximal, complemented
9	#97×D₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#13182013	C₂²	1	1	yes	characteristic
14	C₂×#13182012	C₂²	1	2	yes	—
15	M₁₆×D₁₀₉	C₄	1	8	yes	complemented
16	C₂²×#11724784	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₄	1	8	yes	complemented
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
23	#16812658	C₂²	1	1	yes	characteristic
24	#16812657	C₄	1	2	yes	—
25	#16812656	C₂²	1	2	yes	—
26	#16812655	C₄	1	2	yes	—
27	C₄⋊C₄×D₁₀₉	C₄	1	2	yes	—
28	C₂²⋊C₄×D₁₀₉	C₄	1	2	yes	—
29	C₄²×D₁₀₉	C₂²	1	2	yes	—
30	#16812610	C₂²	1	2	yes	complemented
31	#16812609	C₂²	1	2	yes	complemented
32	#16812609	C₂²	1	2	yes	complemented
33	C₄×#11724784	C₂²	1	2	yes	complemented
34	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
35	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	#117×C₁₀₉	C₂²	1	1	yes	characteristic
39	#97×C₁₀₉	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
41	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
50	#13182013	C₂×C₄	1	4	yes	complemented
51	#13182012	C₂×C₄	1	8	yes	complemented
52	C₂×#11724784	C₂×C₄	1	4	yes	complemented
53	C₂×#11724784	C₂³	1	1	yes	characteristic
54	C₂×#11724784	C₂³	1	2	yes	—
55	C₂×#11724784	C₂³	1	1	yes	characteristic
56	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
57	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
58	C₈×D₁₀₉	C₂×C₄	1	8	yes	—
59	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
60	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
61	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
62	#13182008	C₂×C₄	1	4	yes	—
63	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
64	#13182006	C₂×C₄	1	4	yes	—
65	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
66	C₄×Dic₁₀₉	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
71	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
74	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
75	C₄²×C₁₀₉	C₂³	1	2	yes	—
76	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₀₉	—	2	4	no	—
78	C₂²×D₁₀₉	C₄²	1	4	yes	—
79	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₄²	1	4	yes	—
87	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
88	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₁₀₉	—	2	8	no	—
91	C₄×D₁₀₉	C₄²	1	4	yes	—
92	C₄×D₁₀₉	C₄²	1	4	yes	—
93	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
94	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
95	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
96	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
97	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
98	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
99	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
100	#11724784	C₂²×C₄	1	2	yes	—
101	#11724784	C₂²×C₄	1	2	yes	—
102	#11724784	C₂²×C₄	1	4	yes	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
105	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₀₉	—	2	2	no	—
107	C₂×D₁₀₉	—	2	8	no	—
108	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
109	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
110	C₂×D₁₀₉	#131	1	4	yes	—
111	C₄×C₁₀₉	—	2	4	no	—
112	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
113	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
115	Dic₁₀₉	—	2	4	no	—
116	Dic₁₀₉	C₂×C₄²	1	2	yes	—
117	Dic₁₀₉	C₂×C₄²	1	2	yes	—
118	C₂×C₁₀₉	—	2	2	no	—
119	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
120	C₂×C₁₀₉	C₂×#131	1	2	yes	—
121	D₁₀₉	—	2	2	no	—
122	D₁₀₉	#405	1	4	yes	direct product factor
123	C₂×#405	—	109	1	no	maximal, Sylow
124	C₁₀₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	109	1	no	—
126	C₂³×C₈	—	109	1	no	—
127	C₂×#117	—	109	1	no	—
128	#405	—	109	1	no	—
129	#405	—	109	1	no	—
130	#405	—	109	1	no	—
131	#405	—	109	2	no	—
132	#405	—	109	2	no	—
133	#405	D₁₀₉	1	1	yes	characteristic, direct product factor
134	C₂×#97	—	109	2	no	—
135	C₂×C₄×C₈	—	109	2	no	—
136	C₂³×C₄	—	109	1	no	—
137	C₂×M₁₆	—	109	1	no	—
138	C₂×M₁₆	—	109	8	no	—
139	C₂×M₁₆	—	109	2	no	—
140	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	—	109	1	no	—
142	C₂²×C₈	—	109	8	no	—
143	C₂²×C₈	—	109	1	no	—
144	C₂²×C₈	—	109	1	no	—
145	C₂²×C₈	—	109	2	no	—
146	C₂²×C₈	—	109	1	no	—
147	C₂²×C₈	—	109	1	no	—
148	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
149	#117	—	109	1	no	—
150	#117	—	109	2	no	—
151	#117	—	109	2	no	—
152	#117	—	109	2	no	—
153	#117	C₂×D₁₀₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	109	2	no	—
155	C₂×C₂²⋊C₄	—	109	2	no	—
156	C₂×C₄²	—	109	2	no	—
157	#97	—	109	2	no	—
158	#97	—	109	2	no	—
159	#97	—	109	2	no	—
160	#97	C₂×D₁₀₉	1	2	yes	complemented
161	C₄×C₈	—	109	2	no	—
162	C₄×C₈	—	109	2	no	—
163	C₄×C₈	—	109	2	no	—
164	C₄×C₈	C₂×D₁₀₉	1	2	yes	complemented
165	C₂⁴	—	109	1	no	—
166	C₂²×C₄	—	109	1	no	—
167	C₂²×C₄	—	109	1	no	—
168	C₂²×C₄	—	109	1	no	—
169	C₂²×C₄	—	109	4	no	—
170	C₂²×C₄	—	109	4	no	—
171	C₂²×C₄	—	109	1	no	—
172	C₂²×C₄	—	109	1	no	—
173	C₂²×C₄	—	109	4	no	—
174	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
175	M₁₆	—	109	4	no	—
176	M₁₆	—	109	8	no	—
177	M₁₆	C₄×D₁₀₉	1	4	yes	complemented
178	C₂×C₈	—	109	4	no	—
179	C₂×C₈	—	109	1	no	—
180	C₂×C₈	—	109	2	no	—
181	C₂×C₈	—	109	1	no	—
182	C₂×C₈	—	109	4	no	—
183	C₂×C₈	—	109	4	no	—
184	C₂×C₈	—	109	8	no	—
185	C₂×C₈	—	109	4	no	—
186	C₂×C₈	—	109	4	no	—
187	C₂×C₈	C₄×D₁₀₉	1	4	yes	complemented
188	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
189	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
190	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₄⋊C₄	—	109	4	no	—
192	C₄⋊C₄	—	109	2	no	—
193	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
194	C₂²⋊C₄	—	109	2	no	—
195	C₂²⋊C₄	—	109	4	no	—
196	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
197	C₄²	—	109	4	no	—
198	C₄²	—	109	2	no	—
199	C₄²	C₂²×D₁₀₉	1	2	yes	—
200	C₂³	—	109	1	no	—
201	C₂³	—	109	4	no	—
202	C₂³	—	109	1	no	—
203	C₂³	—	218	4	no	—
204	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
205	C₂×C₄	—	109	4	no	—
206	C₂×C₄	—	109	1	no	—
207	C₂×C₄	—	109	1	no	—
208	C₂×C₄	—	109	4	no	—
209	C₂×C₄	—	109	4	no	—
210	C₂×C₄	—	109	2	no	—
211	C₂×C₄	—	109	2	no	—
212	C₂×C₄	—	109	2	no	—
213	C₂×C₄	—	109	2	no	—
214	C₂×C₄	—	109	4	no	—
215	C₂×C₄	—	218	8	no	—
216	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
220	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
221	C₈	—	109	2	no	—
222	C₈	—	109	2	no	—
223	C₈	—	109	4	no	—
224	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
225	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
226	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	109	2	no	—
229	C₂²	—	109	4	no	—
230	C₂²	—	109	2	no	—
231	C₂²	—	218	8	no	—
232	C₂²	—	218	2	no	—
233	C₂²	C₄²×D₁₀₉	1	2	yes	—
234	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	109	2	no	—
237	C₄	—	109	2	no	—
238	C₄	—	218	4	no	—
239	C₄	C₄²×D₁₀₉	1	2	yes	—
240	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	109	4	no	—
244	C₂	—	218	2	no	—
245	C₂	C₂×C₄²×D₁₀₉	1	1	yes	characteristic
246	C₂	#131×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#405×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂³×D₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	C₂×#405
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #20294984	C₂	semidirect
5. #20294983	C₂	semidirect
6. #20294982	C₂	semidirect
7. #20294981	C₂	semidirect
8. #20294980	C₂	semidirect
9. #97×D₁₀₉	C₂	semidirect
10. C₄×C₈×D₁₀₉	C₂	semidirect
11. #405×C₁₀₉	C₂	semidirect
15. M₁₆×D₁₀₉	C₄	semidirect
17. C₂×C₈×D₁₀₉	C₄	semidirect
30. #16812610	C₂²	semidirect
31. #16812609	C₂²	semidirect
32. #16812609	C₂²	semidirect
33. C₄×#11724784	C₂²	semidirect
34. C₈×Dic₁₀₉	C₂²	semidirect
35. C₈×Dic₁₀₉	C₂²	semidirect
39. #97×C₁₀₉	C₂²	semidirect
40. C₄×C₈×C₁₀₉	C₂²	semidirect
50. #13182013	C₂×C₄	semidirect
51. #13182012	C₂×C₄	semidirect
52. C₂×#11724784	C₂×C₄	semidirect
59. C₈×D₁₀₉	C₂×C₄	semidirect
60. C₈×D₁₀₉	C₂×C₄	semidirect
68. M₁₆×C₁₀₉	C₂×C₄	semidirect
69. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
122. D₁₀₉	#405	direct (N × H)
124. C₁₀₉	C₂×#405	semidirect
133. #405	D₁₀₉	direct (N × H)
160. #97	C₂×D₁₀₉	semidirect
164. C₄×C₈	C₂×D₁₀₉	semidirect
177. M₁₆	C₄×D₁₀₉	semidirect
187. C₂×C₈	C₄×D₁₀₉	semidirect

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
C₂²×C₈×D₁₀₉	2
#117×D₁₀₉	2
#20294984	2
#20294983	2
#20294982	2
#20294981	2
#20294980	2
#97×D₁₀₉	2
C₄×C₈×D₁₀₉	2
#405×C₁₀₉	2
C₂×#405	109

Subgroups — #405×D109 #28371442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #405×D₁₀₉ #28371442