Subgroups — #518×D₁₀₉ #28376215

Order 13952.

1920 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295501	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295500	C₂	1	2	yes	maximal, complemented
4	#20295499	C₂	1	4	yes	maximal, complemented
5	#20295498	C₂	1	8	yes	maximal, complemented
6	C₄×D₄×D₁₀₉	C₂	1	8	yes	maximal, complemented
7	#117×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₁₀₉	C₂	1	4	yes	maximal, complemented
9	#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₉	C₂²	1	2	yes	—
11	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₉	C₄	1	16	yes	complemented
14	C₂×D₄×D₁₀₉	C₂²	1	4	yes	—
15	C₂×D₄×D₁₀₉	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	C₂²	1	8	yes	complemented
18	C₄×D₄₃₆	C₂²	1	8	yes	complemented
19	D₄×Dic₁₀₉	C₂²	1	8	yes	complemented
20	#16812661	C₂²	1	8	yes	complemented
21	C₄×C₂²⋊D₁₀₉	C₂²	1	16	yes	complemented
22	#16812659	C₂²	1	16	yes	complemented
23	#16812658	C₂²	1	2	yes	complemented
24	#16812657	C₂²	1	4	yes	complemented
25	#16812656	C₂²	1	4	yes	complemented
26	#16812655	C₂²	1	4	yes	complemented
27	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	complemented
29	C₂×#13182006	C₂²	1	8	yes	complemented
30	C₂²⋊C₄×D₁₀₉	C₂²	1	4	yes	—
31	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
32	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
33	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
34	C₄²×D₁₀₉	C₂²	1	4	yes	complemented
35	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	C₂²	1	8	yes	complemented
37	#117×C₁₀₉	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
39	C₂³×D₁₀₉	—	4	8	no	—
40	C₂³×D₁₀₉	C₂×C₄	1	16	yes	complemented
41	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
42	C₂³×D₁₀₉	C₂³	1	4	yes	—
43	C₂³×D₁₀₉	C₂³	1	2	yes	—
44	C₂³×D₁₀₉	C₂³	1	4	yes	—
45	C₂×D₄₃₆	C₂×C₄	1	8	yes	complemented
46	C₂×D₄₃₆	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	16	yes	complemented
48	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	—
49	D₄×D₁₀₉	C₂×C₄	1	32	yes	complemented
50	D₄×D₁₀₉	C₂×C₄	1	32	yes	complemented
51	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₉	C₂³	1	8	yes	complemented
53	C₂×C₄×D₁₀₉	—	4	16	no	—
54	C₂×C₄×D₁₀₉	C₂×C₄	1	8	yes	complemented
55	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
56	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
57	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
58	C₂×C₄×D₁₀₉	C₂³	1	16	yes	complemented
59	C₄⋊Dic₁₀₉	C₂³	1	4	yes	complemented
60	#13182008	C₂³	1	8	yes	complemented
61	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
62	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	complemented
63	#13182006	C₂³	1	8	yes	—
64	#13182006	C₂³	1	16	yes	complemented
65	C₄×Dic₁₀₉	C₂³	1	8	yes	complemented
66	C₄×Dic₁₀₉	C₂³	1	4	yes	complemented
67	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
68	C₂×D₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
69	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
70	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	complemented
72	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
73	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	complemented
75	C₄²×C₁₀₉	C₂³	1	4	yes	complemented
76	C₂³×C₁₀₉	—	4	4	no	—
77	C₂³×C₁₀₉	C₂²×C₄	1	8	yes	complemented
78	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
79	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
80	C₂²×D₁₀₉	—	2	16	no	—
81	C₂²×D₁₀₉	—	2	4	no	—
82	C₂²×D₁₀₉	—	4	32	no	—
83	C₂²×D₁₀₉	—	4	4	no	—
84	C₂²×D₁₀₉	—	4	8	no	—
85	C₂²×D₁₀₉	C₂²×C₄	1	4	yes	—
86	C₂²×D₁₀₉	C₂²×C₄	1	16	yes	—
87	C₂²×D₁₀₉	C₂²×C₄	1	16	yes	complemented
88	C₂²×D₁₀₉	C₂²×C₄	1	8	yes	complemented
89	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
91	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
92	D₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
93	C₂×C₄×C₁₀₉	—	4	8	no	—
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	complemented
95	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
97	C₂²⋊D₁₀₉	C₂²×C₄	1	32	yes	complemented
98	C₂×Dic₁₀₉	—	4	8	no	—
99	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	complemented
100	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
101	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
102	D₄₃₆	C₂²×C₄	1	16	yes	complemented
103	C₄×D₁₀₉	—	2	16	no	—
104	C₄×D₁₀₉	—	4	16	no	—
105	C₄×D₁₀₉	C₂²×C₄	1	8	yes	—
106	C₄×D₁₀₉	C₂²×C₄	1	8	yes	complemented
107	C₂²×C₁₀₉	—	2	8	no	—
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	—	4	8	no	—
110	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
111	C₂²×C₁₀₉	C₂³×C₄	1	8	yes	—
112	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
113	C₂×D₁₀₉	—	2	8	no	—
114	C₂×D₁₀₉	—	2	2	no	—
115	C₂×D₁₀₉	—	2	8	no	—
116	C₂×D₁₀₉	—	2	32	no	—
117	C₂×D₁₀₉	—	4	16	no	—
118	C₂×D₁₀₉	—	4	8	no	—
119	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
120	C₂×D₁₀₉	C₂³×C₄	1	8	yes	—
121	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
122	C₂×D₁₀₉	H(5,𝔽₂)	1	4	yes	—
123	C₄×C₁₀₉	—	2	8	no	—
124	C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
125	Dic₁₀₉	—	2	8	no	—
126	Dic₁₀₉	C₂³×C₄	1	4	yes	—
127	C₂×C₁₀₉	—	2	2	no	—
128	C₂×C₁₀₉	—	2	8	no	—
129	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
130	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
131	D₁₀₉	—	2	2	no	—
132	D₁₀₉	—	2	8	no	—
133	D₁₀₉	#518	1	4	yes	direct product factor
134	C₂×#518	—	109	1	no	maximal, Sylow
135	C₁₀₉	C₂×#518	1	1	yes	characteristic, complemented, Sylow
136	C₂³×D₄	—	109	1	no	—
137	#518	—	109	1	no	—
138	#518	—	109	8	no	—
139	#518	—	109	2	no	—
140	#518	—	109	4	no	—
141	#518	D₁₀₉	1	1	yes	characteristic, direct product factor
142	C₂×C₄×D₄	—	109	8	no	—
143	C₂×#117	—	109	2	no	—
144	C₂²×C₂²⋊C₄	—	109	4	no	—
145	C₂⁵	—	109	2	no	—
146	C₂²×D₄	—	109	2	no	—
147	C₂²×D₄	—	109	16	no	—
148	C₂²×D₄	—	109	4	no	—
149	C₂²×D₄	—	109	4	no	—
150	C₂²×D₄	—	109	1	no	—
151	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
152	C₂³×C₄	—	109	1	no	—
153	C₂³×C₄	—	109	8	no	—
154	C₄×D₄	—	109	8	no	—
155	C₄×D₄	—	109	16	no	—
156	C₄×D₄	—	109	8	no	—
157	C₄×D₄	—	109	16	no	—
158	C₄×D₄	—	109	8	no	—
159	C₄×D₄	C₂×D₁₀₉	1	8	yes	complemented
160	#117	—	109	2	no	—
161	#117	—	109	4	no	—
162	#117	—	109	4	no	—
163	#117	—	109	4	no	—
164	#117	C₂×D₁₀₉	1	2	yes	complemented
165	C₂×C₄⋊C₄	—	109	4	no	—
166	C₂×C₂²⋊C₄	—	109	4	no	—
167	C₂×C₂²⋊C₄	—	109	8	no	—
168	C₂×C₂²⋊C₄	—	109	4	no	—
169	C₂×C₂²⋊C₄	—	109	8	no	—
170	C₂×C₂²⋊C₄	—	109	8	no	—
171	C₂×C₂²⋊C₄	—	109	8	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	4	yes	complemented
173	C₂×C₄²	—	109	4	no	—
174	C₂⁴	—	109	16	no	—
175	C₂⁴	—	109	4	no	—
176	C₂⁴	—	109	2	no	—
177	C₂⁴	—	109	1	no	—
178	C₂⁴	—	109	4	no	—
179	C₂⁴	—	436	8	no	—
180	C₂⁴	C₂²×D₁₀₉	1	2	yes	—
181	C₂×D₄	—	109	16	no	—
182	C₂×D₄	—	109	8	no	—
183	C₂×D₄	—	109	32	no	—
184	C₂×D₄	—	109	32	no	—
185	C₂×D₄	—	109	8	no	—
186	C₂×D₄	—	109	4	no	—
187	C₂×D₄	C₄×D₁₀₉	1	8	yes	complemented
188	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
189	C₂²×C₄	—	109	1	no	—
190	C₂²×C₄	—	109	8	no	—
191	C₂²×C₄	—	109	2	no	—
192	C₂²×C₄	—	109	2	no	—
193	C₂²×C₄	—	109	8	no	—
194	C₂²×C₄	—	109	16	no	—
195	C₂²×C₄	—	109	8	no	—
196	C₂²×C₄	—	436	16	no	—
197	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₀₉	1	8	yes	complemented
199	C₄⋊C₄	—	109	8	no	—
200	C₄⋊C₄	—	109	4	no	—
201	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
202	C₂²⋊C₄	—	109	4	no	—
203	C₂²⋊C₄	—	109	8	no	—
204	C₂²⋊C₄	—	109	8	no	—
205	C₂²⋊C₄	—	109	16	no	—
206	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
207	C₂²⋊C₄	C₂²×D₁₀₉	1	8	yes	complemented
208	C₄²	—	109	8	no	—
209	C₄²	—	109	4	no	—
210	C₄²	C₂²×D₁₀₉	1	4	yes	complemented
211	C₂³	—	4	4	no	—
212	C₂³	—	109	1	no	—
213	C₂³	—	109	16	no	—
214	C₂³	—	109	8	no	—
215	C₂³	—	109	4	no	—
216	C₂³	—	109	16	no	—
217	C₂³	—	109	1	no	—
218	C₂³	—	109	4	no	—
219	C₂³	—	436	8	no	—
220	C₂³	—	436	4	no	—
221	C₂³	—	436	32	no	—
222	C₂³	—	218	16	no	—
223	C₂³	—	218	4	no	—
224	C₂³	C₂×C₄×D₁₀₉	1	8	yes	complemented
225	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
226	C₂³	C₂³×D₁₀₉	1	4	yes	—
227	D₄	—	109	32	no	—
228	D₄	—	109	16	no	—
229	D₄	C₂×C₄×D₁₀₉	1	16	yes	complemented
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	109	4	no	—
232	C₂×C₄	—	109	2	no	—
233	C₂×C₄	—	109	8	no	—
234	C₂×C₄	—	109	8	no	—
235	C₂×C₄	—	109	8	no	—
236	C₂×C₄	—	436	16	no	—
237	C₂×C₄	—	436	8	no	—
238	C₂×C₄	—	218	16	no	—
239	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	complemented
240	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
241	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	109	2	no	—
246	C₂²	—	109	4	no	—
247	C₂²	—	109	2	no	—
248	C₂²	—	109	8	no	—
249	C₂²	—	436	8	no	—
250	C₂²	—	436	16	no	—
251	C₂²	—	218	8	no	—
252	C₂²	—	218	32	no	—
253	C₂²	—	218	8	no	—
254	C₂²	—	218	2	no	—
255	C₂²	C₂²×C₄×D₁₀₉	1	2	yes	—
256	C₂²	C₂²×C₄×D₁₀₉	1	8	yes	—
257	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
258	C₄	—	2	8	no	—
259	C₄	—	109	4	no	—
260	C₄	—	218	8	no	—
261	C₄	C₂²×C₄×D₁₀₉	1	4	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	8	no	—
264	C₂	—	109	4	no	—
265	C₂	—	218	2	no	—
266	C₂	—	218	8	no	—
267	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
268	C₂	H(5,𝔽₂)×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Fitting	#518×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
prime radical	C₂²×D₄×D₁₀₉	C₂
hypercenter	#518	D₁₀₉
hypocenter	C₁₀₉	C₂×#518
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
2. #20295501	C₂	semidirect
3. #20295500	C₂	semidirect
4. #20295499	C₂	semidirect
5. #20295498	C₂	semidirect
6. C₄×D₄×D₁₀₉	C₂	semidirect
7. #117×D₁₀₉	C₂	semidirect
8. C₂×C₂²⋊C₄×D₁₀₉	C₂	semidirect
9. #518×C₁₀₉	C₂	semidirect
13. C₂×D₄×D₁₀₉	C₄	semidirect
17. C₂²×C₄×D₁₀₉	C₂²	semidirect
18. C₄×D₄₃₆	C₂²	semidirect
19. D₄×Dic₁₀₉	C₂²	semidirect
20. #16812661	C₂²	semidirect
21. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
22. #16812659	C₂²	semidirect
23. #16812658	C₂²	semidirect
24. #16812657	C₂²	semidirect
25. #16812656	C₂²	semidirect
26. #16812655	C₂²	semidirect
27. C₄⋊C₄×D₁₀₉	C₂²	semidirect
28. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
29. C₂×#13182006	C₂²	semidirect
31. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
32. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
33. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
34. C₄²×D₁₀₉	C₂²	semidirect
36. C₄×D₄×C₁₀₉	C₂²	semidirect
37. #117×C₁₀₉	C₂²	semidirect
38. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
40. C₂³×D₁₀₉	C₂×C₄	semidirect
45. C₂×D₄₃₆	C₂×C₄	semidirect
47. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
49. D₄×D₁₀₉	C₂×C₄	semidirect
50. D₄×D₁₀₉	C₂×C₄	semidirect
52. C₂²×Dic₁₀₉	C₂³	semidirect
54. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
58. C₂×C₄×D₁₀₉	C₂³	semidirect
59. C₄⋊Dic₁₀₉	C₂³	semidirect
60. #13182008	C₂³	semidirect
62. C₂²⋊Dic₁₀₉	C₂³	semidirect
64. #13182006	C₂³	semidirect
65. C₄×Dic₁₀₉	C₂³	semidirect
66. C₄×Dic₁₀₉	C₂³	semidirect
68. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
71. C₂²×C₄×C₁₀₉	C₂³	semidirect
72. C₄⋊C₄×C₁₀₉	C₂³	semidirect
74. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
75. C₄²×C₁₀₉	C₂³	semidirect
77. C₂³×C₁₀₉	C₂²×C₄	semidirect
87. C₂²×D₁₀₉	C₂²×C₄	semidirect
88. C₂²×D₁₀₉	C₂²×C₄	semidirect
92. D₄×C₁₀₉	C₂²×C₄	semidirect
94. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
97. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
99. C₂×Dic₁₀₉	C₂²×C₄	semidirect
102. D₄₃₆	C₂²×C₄	semidirect
106. C₄×D₁₀₉	C₂²×C₄	semidirect
133. D₁₀₉	#518	direct (N × H)
135. C₁₀₉	C₂×#518	semidirect
141. #518	D₁₀₉	direct (N × H)
159. C₄×D₄	C₂×D₁₀₉	semidirect
164. #117	C₂×D₁₀₉	semidirect
172. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
187. C₂×D₄	C₄×D₁₀₉	semidirect
198. C₂²×C₄	C₂²×D₁₀₉	semidirect
201. C₄⋊C₄	C₂²×D₁₀₉	semidirect
207. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
210. C₄²	C₂²×D₁₀₉	semidirect
224. C₂³	C₂×C₄×D₁₀₉	semidirect
229. D₄	C₂×C₄×D₁₀₉	semidirect
239. C₂×C₄	C₂×C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₁₀₉	2
#20295501	2
#20295500	2
#20295499	2
#20295498	2
C₄×D₄×D₁₀₉	2
#117×D₁₀₉	2
C₂×C₂²⋊C₄×D₁₀₉	2
#518×C₁₀₉	2
C₂×#518	109

Subgroups — #518×D109 #28376215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #518×D₁₀₉ #28376215