Subgroups — #461×D₁₀₉ #28372446

Order 13952.

544 conjugacy classes of subgroups.

The subgroup lattice has 312 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
2	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20295232	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295230	C₂	1	2	yes	maximal, complemented
7	#20295229	C₂	1	2	yes	maximal, complemented
8	#20295228	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₉	—	2	1	no	—
15	C₂×#13182018	C₂²	1	2	yes	—
16	C₂×#13182017	C₂²	1	2	yes	—
17	C₂×#13182016	C₂²	1	2	yes	—
18	SD₁₆×D₁₀₉	—	2	8	no	—
19	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
20	#16812699	C₂²	1	1	yes	characteristic, complemented
21	#16812698	—	2	1	no	—
22	#16812698	—	2	1	no	—
23	#16812697	C₂²	1	2	yes	complemented
24	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
25	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	#16812661	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
28	#16812659	—	2	1	no	—
29	C₄⋊C₄×D₁₀₉	—	2	1	no	—
30	C₄⋊C₄×D₁₀₉	—	2	1	no	—
31	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×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	#16812624	C₂²	1	2	yes	complemented
37	#16812623	C₂²	1	2	yes	complemented
38	#16812622	C₂²	1	2	yes	complemented
39	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
40	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#103×C₁₀₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₉	—	2	1	no	—
45	C₂×Dic₂₁₈	—	2	2	no	—
46	C₂×Dic₂₁₈	C₂³	1	2	yes	—
47	Q₈×D₁₀₉	—	2	4	no	—
48	Q₈×D₁₀₉	—	2	4	no	—
49	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
51	D₄×D₁₀₉	—	2	2	no	—
52	D₄×D₁₀₉	D₄	1	4	yes	—
53	C₂²×Dic₁₀₉	—	2	1	no	—
54	C₂×C₄×D₁₀₉	—	2	1	no	—
55	C₂×C₄×D₁₀₉	—	2	2	no	—
56	C₂×C₄×D₁₀₉	—	2	1	no	—
57	C₂×C₄×D₁₀₉	—	2	1	no	—
58	C₂×C₄×D₁₀₉	—	4	4	no	—
59	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
60	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
62	#13182018	—	2	4	no	—
63	#13182017	—	2	4	no	—
64	#13182016	—	2	4	no	—
65	C₂×#11724784	C₂³	1	2	yes	—
66	C₈×D₁₀₉	—	2	4	no	—
67	C₄⋊Dic₁₀₉	—	2	1	no	—
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
70	#13182008	—	2	2	no	—
71	#13182008	—	2	2	no	—
72	#13182008	—	2	1	no	—
73	C₂²⋊Dic₁₀₉	—	2	1	no	—
74	#13182006	—	2	1	no	—
75	#13182006	—	2	1	no	—
76	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
77	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
78	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
80	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	—	2	1	no	—
82	SD₁₆×C₁₀₉	—	2	4	no	—
83	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₀₉	—	2	1	no	—
85	C₄⋊C₄×C₁₀₉	—	2	1	no	—
86	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
88	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂³×C₁₀₉	—	2	1	no	—
90	C₂²×D₁₀₉	—	2	1	no	—
91	C₂²×D₁₀₉	—	2	4	no	—
92	C₂²×D₁₀₉	—	4	2	no	—
93	C₂²×D₁₀₉	—	4	2	no	—
94	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₁₀₉	—	2	2	no	—
96	Q₈×C₁₀₉	—	2	2	no	—
97	D₄×C₁₀₉	—	2	1	no	—
98	D₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	2	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₀₉	—	2	4	no	—
106	C₂×Dic₁₀₉	—	2	1	no	—
107	C₂×Dic₁₀₉	—	2	2	no	—
108	C₂×Dic₁₀₉	—	4	2	no	—
109	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
111	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
112	D₄₃₆	—	2	1	no	—
113	D₄₃₆	C₂×D₄	1	2	yes	—
114	C₄×D₁₀₉	—	2	2	no	—
115	C₄×D₁₀₉	—	4	4	no	—
116	C₄×D₁₀₉	—	4	2	no	—
117	C₄×D₁₀₉	—	4	2	no	—
118	C₄×D₁₀₉	—	4	2	no	—
119	C₄×D₁₀₉	SD₁₆	1	4	yes	complemented
120	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
121	C₄×D₁₀₉	#41	1	2	yes	—
122	Dic₂₁₈	—	2	2	no	—
123	Dic₂₁₈	—	2	2	no	—
124	Dic₂₁₈	—	4	4	no	—
125	C₈×C₁₀₉	—	2	2	no	—
126	#11724784	—	2	2	no	—
127	C₂²×C₁₀₉	—	2	2	no	—
128	C₂²×C₁₀₉	—	4	1	no	—
129	C₂²×C₁₀₉	—	4	1	no	—
130	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₁₀₉	—	2	2	no	—
132	C₂×D₁₀₉	—	4	1	no	—
133	C₂×D₁₀₉	—	4	1	no	—
134	C₂×D₁₀₉	—	4	8	no	—
135	C₂×D₁₀₉	#121	1	2	yes	—
136	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
137	C₂×D₁₀₉	#137	1	2	yes	—
138	C₄×C₁₀₉	—	2	1	no	—
139	C₄×C₁₀₉	—	4	2	no	—
140	C₄×C₁₀₉	—	4	1	no	—
141	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	complemented
142	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
144	Dic₁₀₉	—	2	1	no	—
145	Dic₁₀₉	—	4	2	no	—
146	Dic₁₀₉	—	4	1	no	—
147	Dic₁₀₉	#121	1	2	yes	—
148	Dic₁₀₉	C₂×SD₁₆	1	2	yes	complemented
149	C₂×C₁₀₉	—	4	2	no	—
150	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
151	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
153	D₁₀₉	—	4	2	no	—
154	D₁₀₉	#461	1	4	yes	direct product factor
155	C₂×#461	—	109	1	no	maximal, Sylow
156	C₁₀₉	C₂×#461	1	1	yes	characteristic, complemented, Sylow
157	C₂²×SD₁₆	—	109	2	no	—
158	C₂×C₄⋊Q₈	—	109	1	no	—
159	C₂×C₄×D₄	—	109	1	no	—
160	#461	—	109	2	no	—
161	#461	—	109	1	no	—
162	#461	—	109	1	no	—
163	#461	—	109	1	no	—
164	#461	—	109	2	no	—
165	#461	D₁₀₉	1	1	yes	characteristic, direct product factor
166	C₂×C₄⋊C₈	—	109	1	no	—
167	C₂×#103	—	109	2	no	—
168	C₂²×Q₈	—	109	2	no	—
169	C₂²×D₄	—	109	1	no	—
170	C₂³×C₄	—	218	1	no	—
171	C₂×SD₁₆	—	109	2	no	—
172	C₂×SD₁₆	—	109	2	no	—
173	C₂×SD₁₆	—	109	2	no	—
174	C₂×SD₁₆	—	218	8	no	—
175	C₂×SD₁₆	C₂×D₁₀₉	1	2	yes	—
176	C₂²×C₈	—	109	2	no	—
177	C₄⋊Q₈	—	109	2	no	—
178	C₄⋊Q₈	—	109	1	no	—
179	C₄⋊Q₈	—	218	1	no	—
180	C₄⋊Q₈	—	218	1	no	—
181	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
182	C₄×D₄	—	109	1	no	—
183	C₄×D₄	—	109	1	no	—
184	C₄×D₄	—	109	1	no	—
185	C₄×D₄	—	218	1	no	—
186	C₄×D₄	—	218	1	no	—
187	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	—	109	1	no	—
189	C₂×C₄⋊C₄	—	218	1	no	—
190	C₂×C₄⋊C₄	—	218	1	no	—
191	C₂×C₂²⋊C₄	—	218	1	no	—
192	C₂×C₄²	—	109	1	no	—
193	C₄⋊C₈	—	109	2	no	—
194	C₄⋊C₈	—	109	1	no	—
195	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
196	#103	—	109	2	no	—
197	#103	—	109	2	no	—
198	#103	—	109	2	no	—
199	#103	C₂×D₁₀₉	1	2	yes	complemented
200	C₂⁴	—	218	1	no	—
201	C₂×Q₈	—	109	2	no	—
202	C₂×Q₈	—	218	2	no	—
203	C₂×Q₈	—	218	4	no	—
204	C₂×Q₈	—	218	4	no	—
205	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
206	C₂×D₄	—	109	4	no	—
207	C₂×D₄	—	109	1	no	—
208	C₂×D₄	—	218	2	no	—
209	C₂×D₄	—	218	1	no	—
210	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	109	1	no	—
213	C₂²×C₄	—	109	1	no	—
214	C₂²×C₄	—	109	1	no	—
215	C₂²×C₄	—	436	4	no	—
216	C₂²×C₄	—	218	1	no	—
217	C₂²×C₄	—	218	1	no	—
218	C₂²×C₄	—	218	2	no	—
219	C₂²×C₄	—	218	1	no	—
220	C₂²×C₄	—	218	1	no	—
221	SD₁₆	—	2	4	no	—
222	SD₁₆	—	218	4	no	—
223	SD₁₆	—	218	4	no	—
224	SD₁₆	—	218	4	no	—
225	C₂×C₈	—	109	2	no	—
226	C₂×C₈	—	218	4	no	—
227	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	109	1	no	—
231	C₄⋊C₄	—	218	1	no	—
232	C₄⋊C₄	—	218	2	no	—
233	C₄⋊C₄	—	218	2	no	—
234	C₄⋊C₄	—	218	1	no	—
235	C₄⋊C₄	—	218	1	no	—
236	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	218	1	no	—
239	C₂²⋊C₄	—	218	1	no	—
240	C₂²⋊C₄	—	218	1	no	—
241	C₄²	—	109	1	no	—
242	C₄²	—	109	1	no	—
243	C₄²	—	109	1	no	—
244	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂³	—	2	1	no	—
246	C₂³	—	109	1	no	—
247	C₂³	—	436	2	no	—
248	C₂³	—	436	2	no	—
249	C₂³	—	218	4	no	—
250	C₂³	—	218	1	no	—
251	Q₈	—	2	2	no	—
252	Q₈	—	2	2	no	—
253	Q₈	—	436	4	no	—
254	Q₈	—	218	2	no	—
255	Q₈	—	218	2	no	—
256	D₄	—	2	1	no	—
257	D₄	—	109	2	no	—
258	D₄	—	218	4	no	—
259	D₄	—	218	1	no	—
260	D₄	D₄×D₁₀₉	1	2	yes	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	109	1	no	—
265	C₂×C₄	—	109	1	no	—
266	C₂×C₄	—	109	4	no	—
267	C₂×C₄	—	109	2	no	—
268	C₂×C₄	—	109	2	no	—
269	C₂×C₄	—	109	1	no	—
270	C₂×C₄	—	436	2	no	—
271	C₂×C₄	—	436	2	no	—
272	C₂×C₄	—	436	4	no	—
273	C₂×C₄	—	436	2	no	—
274	C₂×C₄	—	436	2	no	—
275	C₂×C₄	—	218	2	no	—
276	C₂×C₄	—	218	1	no	—
277	C₂×C₄	—	218	2	no	—
278	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
279	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	218	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	109	2	no	—
287	C₂²	—	109	2	no	—
288	C₂²	—	109	2	no	—
289	C₂²	—	436	8	no	—
290	C₂²	—	436	1	no	—
291	C₂²	—	436	1	no	—
292	C₂²	—	218	2	no	—
293	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	1	no	—
297	C₄	—	109	2	no	—
298	C₄	—	109	2	no	—
299	C₄	—	436	2	no	—
300	C₄	—	436	1	no	—
301	C₄	—	218	1	no	—
302	C₄	SD₁₆×D₁₀₉	1	2	yes	complemented
303	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
304	C₄	#41×D₁₀₉	1	1	yes	characteristic
305	C₂	—	4	2	no	—
306	C₂	—	109	4	no	—
307	C₂	—	436	2	no	—
308	C₂	#121×D₁₀₉	1	1	yes	characteristic
309	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
310	C₂	#137×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	#461×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#461	D₁₀₉
hypocenter	C₁₀₉	C₂×#461
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
2. C₄⋊Q₈×D₁₀₉	C₂	semidirect
4. #20295232	C₂	semidirect
5. #20295231	C₂	semidirect
6. #20295230	C₂	semidirect
7. #20295229	C₂	semidirect
8. #20295228	C₂	semidirect
9. C₄⋊C₈×D₁₀₉	C₂	semidirect
10. #103×D₁₀₉	C₂	semidirect
11. #461×C₁₀₉	C₂	semidirect
20. #16812699	C₂²	semidirect
23. #16812697	C₂²	semidirect
34. C₄⋊#11724784	C₂²	semidirect
35. #16812628	C₂²	semidirect
36. #16812624	C₂²	semidirect
37. #16812623	C₂²	semidirect
38. #16812622	C₂²	semidirect
40. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
42. C₄⋊C₈×C₁₀₉	C₂²	semidirect
43. #103×C₁₀₉	C₂²	semidirect
119. C₄×D₁₀₉	SD₁₆	semidirect
141. C₄×C₁₀₉	C₂×SD₁₆	semidirect
148. Dic₁₀₉	C₂×SD₁₆	semidirect
154. D₁₀₉	#461	direct (N × H)
156. C₁₀₉	C₂×#461	semidirect
165. #461	D₁₀₉	direct (N × H)
181. C₄⋊Q₈	C₂×D₁₀₉	semidirect
195. C₄⋊C₈	C₂×D₁₀₉	semidirect
199. #103	C₂×D₁₀₉	semidirect
302. 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
C₄⋊Q₈×D₁₀₉	2
C₄×D₄×D₁₀₉	2
#20295232	2
#20295231	2
#20295230	2
#20295229	2
#20295228	2
C₄⋊C₈×D₁₀₉	2
#103×D₁₀₉	2
#461×C₁₀₉	2
C₂×#461	109

Subgroups — #461×D109 #28372446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #461×D₁₀₉ #28372446