Subgroups — #428×D₁₀₉ #28371613

Order 13952.

536 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295067	C₂	1	2	yes	maximal, complemented
7	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295065	C₂	1	2	yes	maximal, complemented
9	#107×D₁₀₉	C₂	1	2	yes	maximal, complemented
10	#106×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₉	—	2	1	no	—
13	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#13182013	C₂²	1	1	yes	characteristic
15	C₂×#13182012	C₂²	1	2	yes	—
16	M₁₆×D₁₀₉	C₄	1	8	yes	complemented
17	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
18	#16812658	C₂²	1	1	yes	characteristic
19	#16812657	C₂²	1	1	yes	characteristic
20	#16812657	C₂²	1	1	yes	characteristic
21	#16812656	—	2	1	no	—
22	#16812655	—	2	1	no	—
23	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×#13182008	—	2	1	no	—
25	C₄⋊C₄×D₁₀₉	—	2	1	no	—
26	C₄⋊C₄×D₁₀₉	—	2	1	no	—
27	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
29	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
32	C₄²×D₁₀₉	—	2	1	no	—
33	#16812633	C₂²	1	2	yes	complemented
34	#16812632	C₂²	1	2	yes	complemented
35	#16812632	C₂²	1	2	yes	complemented
36	#16812631	C₂²	1	2	yes	complemented
37	#16812630	C₂²	1	2	yes	complemented
38	#16812630	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	#107×C₁₀₉	C₂²	1	2	yes	complemented
43	#106×C₁₀₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
45	C₂²×Dic₁₀₉	—	2	1	no	—
46	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	C₂×C₄×D₁₀₉	—	2	2	no	—
51	C₂×C₄×D₁₀₉	—	4	4	no	—
52	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
53	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	Q₈	1	4	yes	—
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
58	#13182013	C₂×C₄	1	4	yes	complemented
59	#13182012	C₂×C₄	1	8	yes	complemented
60	C₂×#11724784	C₂³	1	2	yes	—
61	C₈×D₁₀₉	C₂×C₄	1	8	yes	complemented
62	C₄⋊Dic₁₀₉	—	2	1	no	—
63	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
65	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
66	#13182008	—	2	2	no	—
67	#13182008	—	2	2	no	—
68	#13182008	—	2	1	no	—
69	#13182008	—	2	1	no	—
70	C₂²⋊Dic₁₀₉	—	2	1	no	—
71	#13182006	—	2	1	no	—
72	#13182006	—	2	1	no	—
73	C₄×Dic₁₀₉	—	2	1	no	—
74	C₄×Dic₁₀₉	—	2	1	no	—
75	C₄×Dic₁₀₉	—	2	1	no	—
76	C₂²×C₄×C₁₀₉	—	2	1	no	—
77	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₀₉	—	2	1	no	—
81	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
85	C₄²×C₁₀₉	—	2	1	no	—
86	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×D₁₀₉	—	2	2	no	—
88	C₂²×D₁₀₉	—	2	2	no	—
89	C₂²×D₁₀₉	C₄⋊C₄	1	4	yes	—
90	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₁₀₉	—	2	1	no	—
101	C₂×Dic₁₀₉	—	2	1	no	—
102	C₂×Dic₁₀₉	—	2	2	no	—
103	C₂×Dic₁₀₉	—	4	2	no	—
104	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
105	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₄×D₁₀₉	—	4	4	no	—
108	C₄×D₁₀₉	—	4	2	no	—
109	C₄×D₁₀₉	—	4	4	no	—
110	C₄×D₁₀₉	—	4	2	no	—
111	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
112	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
113	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
114	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
115	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
116	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
117	C₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
118	#11724784	C₂²×C₄	1	4	yes	complemented
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₀₉	—	2	1	no	—
124	C₂×D₁₀₉	—	2	1	no	—
125	C₂×D₁₀₉	—	2	8	no	—
126	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×D₁₀₉	#137	1	2	yes	—
130	C₄×C₁₀₉	—	4	2	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
134	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
136	Dic₁₀₉	—	4	2	no	—
137	Dic₁₀₉	—	4	1	no	—
138	Dic₁₀₉	—	4	1	no	—
139	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
140	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₁₀₉	—	2	2	no	—
142	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
145	D₁₀₉	—	2	2	no	—
146	D₁₀₉	#428	1	4	yes	direct product factor
147	C₂×#428	—	109	1	no	maximal, Sylow
148	C₁₀₉	C₂×#428	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	109	1	no	—
150	C₂×#117	—	109	1	no	—
151	C₂²×C₄⋊C₄	—	109	1	no	—
152	#428	—	109	1	no	—
153	#428	—	109	1	no	—
154	#428	—	109	1	no	—
155	#428	—	109	2	no	—
156	#428	—	109	2	no	—
157	#428	D₁₀₉	1	1	yes	characteristic, direct product factor
158	C₂×#107	—	109	2	no	—
159	C₂×#106	—	109	2	no	—
160	C₂³×C₄	—	109	1	no	—
161	C₂³×C₄	—	218	1	no	—
162	C₂×M₁₆	—	109	1	no	—
163	C₂×M₁₆	—	109	8	no	—
164	C₂×M₁₆	—	109	2	no	—
165	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
166	C₂²×C₈	—	109	2	no	—
167	#117	—	109	1	no	—
168	#117	—	109	1	no	—
169	#117	—	109	1	no	—
170	#117	—	218	1	no	—
171	#117	—	218	1	no	—
172	#117	C₂×D₁₀₉	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	109	1	no	—
174	C₂×C₄⋊C₄	—	109	2	no	—
175	C₂×C₄⋊C₄	—	109	2	no	—
176	C₂×C₄⋊C₄	—	109	1	no	—
177	C₂×C₄⋊C₄	—	109	1	no	—
178	C₂×C₄⋊C₄	—	218	1	no	—
179	C₂×C₄⋊C₄	—	218	1	no	—
180	C₂×C₄⋊C₄	—	218	1	no	—
181	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	218	1	no	—
183	C₂×C₄²	—	218	1	no	—
184	#107	—	109	2	no	—
185	#107	—	109	2	no	—
186	#107	—	109	2	no	—
187	#107	C₂×D₁₀₉	1	2	yes	complemented
188	#106	—	109	2	no	—
189	#106	—	109	2	no	—
190	#106	—	109	2	no	—
191	#106	C₂×D₁₀₉	1	2	yes	complemented
192	C₂⁴	—	109	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	109	1	no	—
195	C₂²×C₄	—	109	4	no	—
196	C₂²×C₄	—	109	1	no	—
197	C₂²×C₄	—	109	1	no	—
198	C₂²×C₄	—	109	4	no	—
199	C₂²×C₄	—	109	1	no	—
200	C₂²×C₄	—	109	1	no	—
201	C₂²×C₄	—	436	4	no	—
202	C₂²×C₄	—	218	1	no	—
203	C₂²×C₄	—	218	1	no	—
204	C₂²×C₄	—	218	2	no	—
205	C₂²×C₄	—	218	2	no	—
206	C₂²×C₄	—	218	1	no	—
207	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
208	M₁₆	—	109	4	no	—
209	M₁₆	—	109	8	no	—
210	M₁₆	C₄×D₁₀₉	1	4	yes	complemented
211	C₂×C₈	—	109	2	no	—
212	C₂×C₈	—	109	8	no	—
213	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₄⋊C₄	—	109	2	no	—
217	C₄⋊C₄	—	109	1	no	—
218	C₄⋊C₄	—	218	2	no	—
219	C₄⋊C₄	—	218	2	no	—
220	C₄⋊C₄	—	218	1	no	—
221	C₄⋊C₄	—	218	1	no	—
222	C₄⋊C₄	—	218	1	no	—
223	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
225	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	218	1	no	—
228	C₂²⋊C₄	—	218	1	no	—
229	C₂²⋊C₄	—	218	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	218	1	no	—
232	C₄²	—	218	1	no	—
233	C₄²	—	218	1	no	—
234	C₂³	—	109	1	no	—
235	C₂³	—	109	4	no	—
236	C₂³	—	109	1	no	—
237	C₂³	—	218	2	no	—
238	C₂³	—	218	2	no	—
239	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	109	4	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	4	no	—
248	C₂×C₄	—	109	4	no	—
249	C₂×C₄	—	109	2	no	—
250	C₂×C₄	—	109	2	no	—
251	C₂×C₄	—	109	2	no	—
252	C₂×C₄	—	109	2	no	—
253	C₂×C₄	—	436	4	no	—
254	C₂×C₄	—	436	2	no	—
255	C₂×C₄	—	436	2	no	—
256	C₂×C₄	—	436	4	no	—
257	C₂×C₄	—	436	2	no	—
258	C₂×C₄	—	218	1	no	—
259	C₂×C₄	—	218	1	no	—
260	C₂×C₄	—	218	2	no	—
261	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
262	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
263	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
264	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
265	C₈	—	109	4	no	—
266	C₈	C₂×C₄×D₁₀₉	1	4	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	109	2	no	—
270	C₂²	—	109	2	no	—
271	C₂²	—	109	2	no	—
272	C₂²	—	109	2	no	—
273	C₂²	—	218	8	no	—
274	C₂²	—	218	1	no	—
275	C₂²	—	218	1	no	—
276	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
277	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	109	2	no	—
282	C₄	—	109	2	no	—
283	C₄	—	436	2	no	—
284	C₄	—	436	1	no	—
285	C₄	—	436	1	no	—
286	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
287	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
288	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	109	4	no	—
291	C₂	—	218	2	no	—
292	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
293	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
294	C₂	#137×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#428×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	C₂×#428
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #20295069	C₂	semidirect
5. #20295068	C₂	semidirect
6. #20295067	C₂	semidirect
7. #20295066	C₂	semidirect
8. #20295065	C₂	semidirect
9. #107×D₁₀₉	C₂	semidirect
10. #106×D₁₀₉	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
16. M₁₆×D₁₀₉	C₄	semidirect
33. #16812633	C₂²	semidirect
34. #16812632	C₂²	semidirect
35. #16812632	C₂²	semidirect
36. #16812631	C₂²	semidirect
37. #16812630	C₂²	semidirect
38. #16812630	C₂²	semidirect
42. #107×C₁₀₉	C₂²	semidirect
43. #106×C₁₀₉	C₂²	semidirect
58. #13182013	C₂×C₄	semidirect
59. #13182012	C₂×C₄	semidirect
61. C₈×D₁₀₉	C₂×C₄	semidirect
78. M₁₆×C₁₀₉	C₂×C₄	semidirect
117. C₈×C₁₀₉	C₂²×C₄	semidirect
118. #11724784	C₂²×C₄	semidirect
146. D₁₀₉	#428	direct (N × H)
148. C₁₀₉	C₂×#428	semidirect
157. #428	D₁₀₉	direct (N × H)
187. #107	C₂×D₁₀₉	semidirect
191. #106	C₂×D₁₀₉	semidirect
210. M₁₆	C₄×D₁₀₉	semidirect
266. C₈	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
C₂×M₁₆×D₁₀₉	2
#117×D₁₀₉	2
C₂×C₄⋊C₄×D₁₀₉	2
#20295069	2
#20295068	2
#20295067	2
#20295066	2
#20295065	2
#107×D₁₀₉	2
#106×D₁₀₉	2
#428×C₁₀₉	2
C₂×#428	109

Subgroups — #428×D109 #28371613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #428×D₁₀₉ #28371613