Subgroups — #460×D₁₀₉ #28372460

Order 13952.

584 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×D₁₀₉	C₂	1	2	yes	maximal, complemented
2	#127×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295227	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295226	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295225	C₂	1	2	yes	maximal, complemented
7	#20295224	C₂	1	2	yes	maximal, complemented
8	#20295223	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#460×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₁₀₉	—	2	1	no	—
14	C₂×D₄×D₁₀₉	—	2	1	no	—
15	C₂×D₄×D₁₀₉	C₂²	1	2	yes	—
16	C₂×#13182018	C₂²	1	2	yes	complemented
17	C₂×#13182017	C₂²	1	2	yes	complemented
18	C₂×#13182016	C₂²	1	2	yes	complemented
19	SD₁₆×D₁₀₉	—	2	8	no	—
20	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
21	#16812696	C₂²	1	1	yes	characteristic
22	#16812695	—	2	1	no	—
23	#16812695	—	2	1	no	—
24	#16812695	C₂²	1	2	yes	—
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	C₄×Dic₂₁₈	—	2	1	no	—
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
28	#16812664	—	2	1	no	—
29	#16812664	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×D₁₀₉	—	2	1	no	—
31	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
32	C₄²×D₁₀₉	—	2	1	no	—
33	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
35	#16812628	C₂²	1	2	yes	complemented
36	#16812621	C₂²	1	2	yes	complemented
37	#16812620	C₂²	1	2	yes	complemented
38	#16812619	C₂²	1	2	yes	complemented
39	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
40	#127×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#102×C₁₀₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₉	—	2	2	no	—
45	C₂×Dic₂₁₈	—	2	1	no	—
46	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
47	Q₈×D₁₀₉	—	2	2	no	—
48	Q₈×D₁₀₉	D₄	1	4	yes	complemented
49	C₂×D₄₃₆	—	2	1	no	—
50	C₂×D₄₃₆	—	2	1	no	—
51	C₂×D₄₃₆	C₂³	1	2	yes	—
52	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
53	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
54	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
55	D₄×D₁₀₉	—	2	4	no	—
56	D₄×D₁₀₉	—	2	4	no	—
57	D₄×D₁₀₉	—	4	4	no	—
58	D₄×D₁₀₉	—	4	4	no	—
59	C₂×C₄×D₁₀₉	—	2	1	no	—
60	C₂×C₄×D₁₀₉	—	2	1	no	—
61	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
64	#13182018	—	2	4	no	—
65	#13182017	—	2	4	no	—
66	#13182016	—	2	4	no	—
67	C₂×#11724784	C₂³	1	2	yes	—
68	C₈×D₁₀₉	—	2	4	no	—
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
71	#13182008	—	2	1	no	—
72	#13182008	—	2	1	no	—
73	#13182008	—	2	1	no	—
74	C₄×Dic₁₀₉	—	2	1	no	—
75	C₄×Dic₁₀₉	—	2	1	no	—
76	C₄×Dic₁₀₉	—	2	1	no	—
77	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
78	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
79	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₁₀₉	—	2	1	no	—
82	C₂×D₄×C₁₀₉	—	2	1	no	—
83	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
84	SD₁₆×C₁₀₉	—	2	4	no	—
85	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
86	C₄⋊C₄×C₁₀₉	—	2	1	no	—
87	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₄²×C₁₀₉	—	2	1	no	—
89	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂³×C₁₀₉	—	2	2	no	—
91	C₂²×D₁₀₉	—	2	2	no	—
92	C₂²×D₁₀₉	—	4	4	no	—
93	C₂²×D₁₀₉	—	4	4	no	—
94	C₂²×D₁₀₉	—	4	4	no	—
95	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₁₀₉	—	2	1	no	—
97	Q₈×C₁₀₉	C₂×D₄	1	2	yes	complemented
98	D₄×C₁₀₉	—	2	2	no	—
99	D₄×C₁₀₉	—	2	2	no	—
100	D₄×C₁₀₉	—	4	2	no	—
101	D₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₁₀₉	—	4	4	no	—
108	C₂²⋊D₁₀₉	—	4	4	no	—
109	C₂²⋊D₁₀₉	—	4	4	no	—
110	C₂×Dic₁₀₉	—	2	1	no	—
111	C₂×Dic₁₀₉	—	2	1	no	—
112	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
115	D₄₃₆	—	2	2	no	—
116	D₄₃₆	—	2	2	no	—
117	D₄₃₆	—	4	2	no	—
118	D₄₃₆	—	4	2	no	—
119	C₄×D₁₀₉	—	2	4	no	—
120	C₄×D₁₀₉	—	2	2	no	—
121	C₄×D₁₀₉	—	4	2	no	—
122	C₄×D₁₀₉	SD₁₆	1	4	yes	complemented
123	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
124	C₄×D₁₀₉	#41	1	2	yes	—
125	Dic₂₁₈	—	2	1	no	—
126	Dic₂₁₈	—	2	4	no	—
127	Dic₂₁₈	C₂×D₄	1	2	yes	complemented
128	C₈×C₁₀₉	—	2	2	no	—
129	#11724784	—	2	2	no	—
130	C₂²×C₁₀₉	—	4	2	no	—
131	C₂²×C₁₀₉	—	4	2	no	—
132	C₂²×C₁₀₉	—	4	2	no	—
133	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₁₀₉	—	4	2	no	—
135	C₂×D₁₀₉	—	4	2	no	—
136	C₂×D₁₀₉	—	4	2	no	—
137	C₂×D₁₀₉	—	8	8	no	—
138	C₂×D₁₀₉	#121	1	2	yes	—
139	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
140	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
141	C₄×C₁₀₉	—	2	2	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	complemented
145	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
147	Dic₁₀₉	—	2	2	no	—
148	Dic₁₀₉	—	2	1	no	—
149	Dic₁₀₉	—	4	1	no	—
150	Dic₁₀₉	#121	1	2	yes	—
151	Dic₁₀₉	C₂×SD₁₆	1	2	yes	complemented
152	C₂×C₁₀₉	—	8	2	no	—
153	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
154	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
155	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	D₁₀₉	—	8	2	no	—
157	D₁₀₉	#460	1	4	yes	direct product factor
158	C₂×#460	—	109	1	no	maximal, Sylow
159	C₁₀₉	C₂×#460	1	1	yes	characteristic, complemented, Sylow
160	C₂²×SD₁₆	—	109	2	no	—
161	C₂×#127	—	109	1	no	—
162	C₂×C₄×Q₈	—	109	1	no	—
163	#460	—	109	2	no	—
164	#460	—	109	1	no	—
165	#460	—	109	1	no	—
166	#460	—	109	1	no	—
167	#460	—	109	2	no	—
168	#460	D₁₀₉	1	1	yes	characteristic, direct product factor
169	C₂×C₄⋊C₈	—	109	1	no	—
170	C₂×#102	—	109	2	no	—
171	C₂²×Q₈	—	109	1	no	—
172	C₂²×D₄	—	109	2	no	—
173	C₂²×D₄	—	218	1	no	—
174	C₂²×D₄	—	218	1	no	—
175	C₂×SD₁₆	—	109	2	no	—
176	C₂×SD₁₆	—	109	2	no	—
177	C₂×SD₁₆	—	109	2	no	—
178	C₂×SD₁₆	—	218	8	no	—
179	C₂×SD₁₆	C₂×D₁₀₉	1	2	yes	complemented
180	C₂²×C₈	—	109	2	no	—
181	#127	—	109	2	no	—
182	#127	—	109	1	no	—
183	#127	—	218	1	no	—
184	#127	—	218	1	no	—
185	#127	C₂×D₁₀₉	1	1	yes	characteristic
186	C₄×Q₈	—	109	1	no	—
187	C₄×Q₈	—	109	1	no	—
188	C₄×Q₈	—	109	1	no	—
189	C₄×Q₈	—	218	1	no	—
190	C₄×Q₈	—	218	1	no	—
191	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
192	C₂×C₄⋊C₄	—	109	1	no	—
193	C₂×C₄⋊C₄	—	218	1	no	—
194	C₂×C₄²	—	109	1	no	—
195	C₂×C₄²	—	218	1	no	—
196	C₄⋊C₈	—	109	2	no	—
197	C₄⋊C₈	—	109	1	no	—
198	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
199	#102	—	109	2	no	—
200	#102	—	109	2	no	—
201	#102	—	109	2	no	—
202	#102	C₂×D₁₀₉	1	2	yes	complemented
203	C₂⁴	—	218	2	no	—
204	C₂×Q₈	—	109	4	no	—
205	C₂×Q₈	—	109	1	no	—
206	C₂×Q₈	—	218	1	no	—
207	C₂×Q₈	—	218	2	no	—
208	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	109	2	no	—
212	C₂×D₄	—	436	4	no	—
213	C₂×D₄	—	436	4	no	—
214	C₂×D₄	—	218	4	no	—
215	C₂×D₄	—	218	4	no	—
216	C₂×D₄	—	218	2	no	—
217	C₂×D₄	—	218	2	no	—
218	C₂×D₄	—	218	2	no	—
219	C₂×D₄	—	218	1	no	—
220	C₂×D₄	—	218	1	no	—
221	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
222	C₂²×C₄	—	109	1	no	—
223	C₂²×C₄	—	109	1	no	—
224	C₂²×C₄	—	109	1	no	—
225	C₂²×C₄	—	218	1	no	—
226	C₂²×C₄	—	218	1	no	—
227	SD₁₆	—	2	4	no	—
228	SD₁₆	—	218	4	no	—
229	SD₁₆	—	218	4	no	—
230	SD₁₆	—	218	4	no	—
231	C₂×C₈	—	109	2	no	—
232	C₂×C₈	—	218	4	no	—
233	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	109	1	no	—
236	C₄⋊C₄	—	218	1	no	—
237	C₄⋊C₄	—	218	1	no	—
238	C₄⋊C₄	—	218	1	no	—
239	C₄⋊C₄	—	218	1	no	—
240	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	—	109	1	no	—
243	C₄²	—	109	1	no	—
244	C₄²	—	109	1	no	—
245	C₄²	—	218	1	no	—
246	C₄²	—	218	1	no	—
247	C₄²	—	218	1	no	—
248	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	109	1	no	—
251	C₂³	—	436	4	no	—
252	C₂³	—	436	4	no	—
253	C₂³	—	436	4	no	—
254	C₂³	—	218	2	no	—
255	Q₈	—	2	1	no	—
256	Q₈	—	109	2	no	—
257	Q₈	—	218	4	no	—
258	Q₈	—	218	1	no	—
259	Q₈	D₄×D₁₀₉	1	2	yes	complemented
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	436	2	no	—
265	D₄	—	436	4	no	—
266	D₄	—	436	4	no	—
267	D₄	—	436	4	no	—
268	D₄	—	436	2	no	—
269	D₄	—	218	2	no	—
270	D₄	—	218	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	109	1	no	—
274	C₂×C₄	—	109	1	no	—
275	C₂×C₄	—	109	4	no	—
276	C₂×C₄	—	109	2	no	—
277	C₂×C₄	—	109	2	no	—
278	C₂×C₄	—	109	1	no	—
279	C₂×C₄	—	436	2	no	—
280	C₂×C₄	—	218	2	no	—
281	C₂×C₄	—	218	4	no	—
282	C₂×C₄	—	218	1	no	—
283	C₂×C₄	—	218	1	no	—
284	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
285	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
286	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	218	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	872	8	no	—
293	C₂²	—	109	2	no	—
294	C₂²	—	109	2	no	—
295	C₂²	—	109	2	no	—
296	C₂²	—	436	2	no	—
297	C₂²	—	436	2	no	—
298	C₂²	—	436	2	no	—
299	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	109	2	no	—
304	C₄	—	109	2	no	—
305	C₄	—	436	1	no	—
306	C₄	—	218	2	no	—
307	C₄	—	218	1	no	—
308	C₄	SD₁₆×D₁₀₉	1	2	yes	complemented
309	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
310	C₄	#41×D₁₀₉	1	1	yes	characteristic
311	C₂	—	8	2	no	—
312	C₂	—	872	2	no	—
313	C₂	—	109	4	no	—
314	C₂	#121×D₁₀₉	1	1	yes	characteristic
315	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
316	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#460×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#127×D₁₀₉	C₂
hypercenter	#460	D₁₀₉
hypocenter	C₁₀₉	C₂×#460
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×D₁₀₉	C₂	semidirect
3. C₄×Q₈×D₁₀₉	C₂	semidirect
4. #20295227	C₂	semidirect
5. #20295226	C₂	semidirect
6. #20295225	C₂	semidirect
7. #20295224	C₂	semidirect
8. #20295223	C₂	semidirect
9. C₄⋊C₈×D₁₀₉	C₂	semidirect
10. #102×D₁₀₉	C₂	semidirect
11. #460×C₁₀₉	C₂	semidirect
16. C₂×#13182018	C₂²	semidirect
17. C₂×#13182017	C₂²	semidirect
18. C₂×#13182016	C₂²	semidirect
25. Q₈×Dic₁₀₉	C₂²	semidirect
27. C₄×Dic₂₁₈	C₂²	semidirect
29. #16812664	C₂²	semidirect
34. C₄⋊#11724784	C₂²	semidirect
35. #16812628	C₂²	semidirect
36. #16812621	C₂²	semidirect
37. #16812620	C₂²	semidirect
38. #16812619	C₂²	semidirect
39. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
41. C₄×Q₈×C₁₀₉	C₂²	semidirect
42. C₄⋊C₈×C₁₀₉	C₂²	semidirect
43. #102×C₁₀₉	C₂²	semidirect
48. Q₈×D₁₀₉	D₄	semidirect
97. Q₈×C₁₀₉	C₂×D₄	semidirect
122. C₄×D₁₀₉	SD₁₆	semidirect
127. Dic₂₁₈	C₂×D₄	semidirect
144. C₄×C₁₀₉	C₂×SD₁₆	semidirect
151. Dic₁₀₉	C₂×SD₁₆	semidirect
157. D₁₀₉	#460	direct (N × H)
159. C₁₀₉	C₂×#460	semidirect
168. #460	D₁₀₉	direct (N × H)
179. C₂×SD₁₆	C₂×D₁₀₉	semidirect
191. C₄×Q₈	C₂×D₁₀₉	semidirect
198. C₄⋊C₈	C₂×D₁₀₉	semidirect
202. #102	C₂×D₁₀₉	semidirect
259. Q₈	D₄×D₁₀₉	semidirect
308. C₄	SD₁₆×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×D₁₀₉	2
#127×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
#20295227	2
#20295226	2
#20295225	2
#20295224	2
#20295223	2
C₄⋊C₈×D₁₀₉	2
#102×D₁₀₉	2
#460×C₁₀₉	2
C₂×#460	109

Subgroups — #460×D109 #28372460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #460×D₁₀₉ #28372460