Subgroups — #473×D₁₀₉ #28372741

Order 13952.

448 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#137×D₁₀₉	C₂	1	2	yes	maximal
2	C₂×Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20295302	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295301	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295300	C₂	1	2	yes	maximal, complemented
7	#20295299	C₂	1	2	yes	maximal, complemented
8	#20295298	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#101×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#473×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
14	#16812732	C₂²	1	2	yes	—
15	#16812731	—	2	2	no	—
16	#16812730	C₂²	1	2	yes	—
17	#16812729	C₂²	1	2	yes	—
18	#16812728	—	2	2	no	—
19	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13182019	C₂²	1	2	yes	complemented
21	Q₁₆×D₁₀₉	—	2	2	no	—
22	Q₁₆×D₁₀₉	—	2	1	no	—
23	Q₁₆×D₁₀₉	—	2	1	no	—
24	Q₁₆×D₁₀₉	—	2	1	no	—
25	Q₁₆×D₁₀₉	—	2	1	no	—
26	SD₁₆×D₁₀₉	—	2	2	no	—
27	#16812707	—	2	1	no	—
28	#16812707	C₂²	1	1	yes	characteristic
29	#16812706	C₂²	1	1	yes	characteristic
30	#16812705	—	2	1	no	—
31	#16812705	C₂²	1	1	yes	characteristic
32	M₁₆×D₁₀₉	—	2	1	no	—
33	M₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
34	M₁₆×D₁₀₉	C₂²	1	2	yes	—
35	C₂×C₈×D₁₀₉	—	2	1	no	—
36	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812635	C₂²	1	1	yes	characteristic, complemented
38	#16812634	C₂²	1	2	yes	complemented
39	#16812618	C₂²	1	2	yes	complemented
40	#16812617	C₂²	1	2	yes	complemented
41	#16812617	C₂²	1	2	yes	complemented
42	#137×C₁₀₉	C₂²	1	2	yes	—
43	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
44	#131×C₁₀₉	C₂²	1	1	yes	characteristic
45	#108×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
46	#101×C₁₀₉	C₂²	1	2	yes	complemented
47	#13182030	D₄	1	1	yes	characteristic
48	#13182029	—	2	1	no	—
49	#13182029	C₂³	1	1	yes	characteristic
50	#13182028	—	2	1	no	—
51	#13182028	D₄	1	1	yes	characteristic
52	C₂×Dic₂₁₈	—	2	2	no	—
53	C₂×Dic₂₁₈	C₂³	1	2	yes	—
54	Q₈×D₁₀₉	—	2	2	no	—
55	Q₈×D₁₀₉	—	2	2	no	—
56	Q₈×D₁₀₉	—	2	2	no	—
57	Q₈×D₁₀₉	—	2	2	no	—
58	Q₈×D₁₀₉	D₄	1	1	yes	characteristic
59	D₄×D₁₀₉	—	2	1	no	—
60	D₄×D₁₀₉	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	—	2	1	no	—
62	C₂×C₄×D₁₀₉	—	2	2	no	—
63	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
64	Dic₄₃₆	—	2	2	no	—
65	Dic₄₃₆	—	2	1	no	—
66	Dic₄₃₆	—	2	1	no	—
67	#13182019	—	2	2	no	—
68	#13182019	—	2	2	no	—
69	#13182019	—	2	2	no	—
70	#13182019	—	2	2	no	—
71	#13182018	—	2	2	no	—
72	#13182017	—	2	2	no	—
73	#13182016	—	2	2	no	—
74	#13182013	—	2	1	no	—
75	#13182013	C₂³	1	1	yes	characteristic
76	#13182013	C₂³	1	2	yes	—
77	#13182012	—	2	1	no	—
78	#13182012	—	2	2	no	—
79	#13182012	—	2	1	no	—
80	#13182012	—	2	1	no	—
81	C₂×#11724784	—	2	1	no	—
82	C₂×#11724784	C₂³	1	1	yes	characteristic
83	C₈×D₁₀₉	—	2	1	no	—
84	C₈×D₁₀₉	—	2	2	no	—
85	C₈×D₁₀₉	—	2	1	no	—
86	C₈×D₁₀₉	—	2	1	no	—
87	C₈×D₁₀₉	D₄	1	1	yes	characteristic
88	C₈×D₁₀₉	D₄	1	1	yes	characteristic
89	C₈×D₁₀₉	D₄	1	1	yes	characteristic
90	C₈×D₁₀₉	D₄	1	1	yes	characteristic
91	#41×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
93	Q₁₆×C₁₀₉	—	2	2	no	—
94	Q₁₆×C₁₀₉	—	2	1	no	—
95	Q₁₆×C₁₀₉	—	2	1	no	—
96	SD₁₆×C₁₀₉	—	2	2	no	—
97	M₁₆×C₁₀₉	—	2	1	no	—
98	M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₀₉	C₂³	1	2	yes	—
100	C₂×C₈×C₁₀₉	—	2	1	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²×D₁₀₉	—	2	1	no	—
103	C₂²×D₁₀₉	#41	1	1	yes	characteristic
104	Q₈×C₁₀₉	—	2	2	no	—
105	Q₈×C₁₀₉	—	2	2	no	—
106	Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	D₄×C₁₀₉	—	2	1	no	—
108	D₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₀₉	—	2	1	no	—
113	C₂²⋊D₁₀₉	—	2	1	no	—
114	C₂²⋊D₁₀₉	—	2	1	no	—
115	C₂×Dic₁₀₉	—	2	1	no	—
116	C₂×Dic₁₀₉	—	2	2	no	—
117	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
118	D₄₃₆	—	2	1	no	—
119	D₄₃₆	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₀₉	—	2	1	no	—
121	C₄×D₁₀₉	—	2	1	no	—
122	C₄×D₁₀₉	—	2	1	no	—
123	C₄×D₁₀₉	—	4	2	no	—
124	C₄×D₁₀₉	—	4	2	no	—
125	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
129	Dic₂₁₈	—	2	1	no	—
130	Dic₂₁₈	—	2	2	no	—
131	Dic₂₁₈	—	2	2	no	—
132	Dic₂₁₈	—	4	2	no	—
133	Dic₂₁₈	—	4	2	no	—
134	Dic₂₁₈	C₂×D₄	1	1	yes	characteristic
135	C₈×C₁₀₉	—	2	1	no	—
136	C₈×C₁₀₉	—	2	2	no	—
137	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
139	#11724784	—	2	1	no	—
140	#11724784	—	2	2	no	—
141	#11724784	C₂×D₄	1	1	yes	characteristic
142	#11724784	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
145	C₂×D₁₀₉	—	2	2	no	—
146	C₂×D₁₀₉	—	2	1	no	—
147	C₂×D₁₀₉	—	4	2	no	—
148	C₂×D₁₀₉	#121	1	1	yes	characteristic
149	C₂×D₁₀₉	#121	1	1	yes	characteristic
150	C₄×C₁₀₉	—	2	1	no	—
151	C₄×C₁₀₉	—	4	2	no	—
152	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
154	Dic₁₀₉	—	2	1	no	—
155	Dic₁₀₉	—	4	2	no	—
156	Dic₁₀₉	#121	1	1	yes	characteristic
157	Dic₁₀₉	#121	1	1	yes	characteristic
158	C₂×C₁₀₉	—	2	1	no	—
159	C₂×C₁₀₉	—	4	1	no	—
160	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
161	D₁₀₉	—	2	1	no	—
162	D₁₀₉	—	4	1	no	—
163	D₁₀₉	#473	1	2	yes	direct product factor
164	C₂×#473	—	109	1	no	maximal, Sylow
165	C₁₀₉	C₂×#473	1	1	yes	characteristic, complemented, Sylow
166	C₂×#137	—	109	2	no	—
167	C₂²×Q₁₆	—	109	1	no	—
168	C₂×#131	—	109	1	no	—
169	#473	—	109	2	no	—
170	#473	—	109	1	no	—
171	#473	—	109	1	no	—
172	#473	—	109	1	no	—
173	#473	—	109	2	no	—
174	#473	D₁₀₉	1	1	yes	characteristic, direct product factor
175	C₂×#108	—	109	1	no	—
176	C₂×#101	—	109	2	no	—
177	C₂×#41	—	109	1	no	—
178	C₂²×Q₈	—	109	2	no	—
179	#137	—	109	2	no	—
180	#137	—	109	2	no	—
181	#137	—	109	2	no	—
182	#137	—	218	2	no	—
183	#137	—	218	2	no	—
184	#137	C₂×D₁₀₉	1	2	yes	—
185	C₂×Q₁₆	—	109	2	no	—
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	—	218	1	no	—
188	C₂×Q₁₆	—	218	1	no	—
189	C₂×Q₁₆	—	218	1	no	—
190	C₂×Q₁₆	—	218	1	no	—
191	C₂×Q₁₆	—	218	2	no	—
192	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic, complemented
193	C₂×SD₁₆	—	218	2	no	—
194	#131	—	109	1	no	—
195	#131	—	109	1	no	—
196	#131	—	109	1	no	—
197	#131	—	218	1	no	—
198	#131	—	218	1	no	—
199	#131	C₂×D₁₀₉	1	1	yes	characteristic
200	C₂×M₁₆	—	109	1	no	—
201	C₂×M₁₆	—	109	2	no	—
202	C₂×M₁₆	—	218	1	no	—
203	C₂²×C₈	—	109	1	no	—
204	C₂²×C₈	—	218	1	no	—
205	#108	—	109	2	no	—
206	#108	—	109	1	no	—
207	#108	C₂×D₁₀₉	1	1	yes	characteristic, complemented
208	#101	—	109	2	no	—
209	#101	—	109	2	no	—
210	#101	—	109	2	no	—
211	#101	C₂×D₁₀₉	1	2	yes	complemented
212	#41	—	109	1	no	—
213	#41	—	109	1	no	—
214	#41	—	109	1	no	—
215	#41	—	218	1	no	—
216	#41	—	218	1	no	—
217	#41	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₂×Q₈	—	109	1	no	—
219	C₂×Q₈	—	109	2	no	—
220	C₂×Q₈	—	218	2	no	—
221	C₂×Q₈	—	218	2	no	—
222	C₂×Q₈	—	218	2	no	—
223	C₂×Q₈	—	218	2	no	—
224	C₂×Q₈	—	218	2	no	—
225	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
226	C₂×D₄	—	109	1	no	—
227	C₂×D₄	—	218	1	no	—
228	C₂²×C₄	—	109	1	no	—
229	C₂²×C₄	—	218	1	no	—
230	C₂²×C₄	—	218	2	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	1	no	—
233	Q₁₆	—	2	1	no	—
234	Q₁₆	—	218	2	no	—
235	Q₁₆	—	218	2	no	—
236	Q₁₆	—	218	2	no	—
237	Q₁₆	—	218	2	no	—
238	Q₁₆	—	218	2	no	—
239	Q₁₆	—	218	1	no	—
240	Q₁₆	—	218	1	no	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	218	2	no	—
243	SD₁₆	—	218	2	no	—
244	SD₁₆	—	218	2	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	109	1	no	—
247	M₁₆	—	109	2	no	—
248	M₁₆	—	218	1	no	—
249	M₁₆	—	218	1	no	—
250	M₁₆	—	218	2	no	—
251	M₁₆	—	218	1	no	—
252	M₁₆	—	218	1	no	—
253	M₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
254	M₁₆	C₂²×D₁₀₉	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	109	1	no	—
257	C₂×C₈	—	109	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	109	1	no	—
260	C₂×C₈	—	109	1	no	—
261	C₂×C₈	—	218	1	no	—
262	C₂×C₈	—	218	1	no	—
263	C₂×C₈	—	218	2	no	—
264	C₂×C₈	—	218	1	no	—
265	C₂×C₈	—	218	1	no	—
266	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂³	—	109	1	no	—
268	C₂³	—	218	1	no	—
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	109	1	no	—
272	Q₈	—	436	2	no	—
273	Q₈	—	436	2	no	—
274	Q₈	—	218	1	no	—
275	Q₈	—	218	2	no	—
276	Q₈	—	218	2	no	—
277	Q₈	D₄×D₁₀₉	1	1	yes	characteristic
278	D₄	—	2	1	no	—
279	D₄	—	109	1	no	—
280	D₄	—	218	1	no	—
281	D₄	—	218	1	no	—
282	D₄	—	218	1	no	—
283	D₄	—	218	1	no	—
284	D₄	D₄×D₁₀₉	1	1	yes	characteristic
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	109	1	no	—
288	C₂×C₄	—	109	1	no	—
289	C₂×C₄	—	109	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	109	1	no	—
292	C₂×C₄	—	436	2	no	—
293	C₂×C₄	—	436	2	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	—	218	2	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	2	no	—
302	C₈	—	109	1	no	—
303	C₈	—	109	1	no	—
304	C₈	—	218	1	no	—
305	C₈	—	218	2	no	—
306	C₈	D₄×D₁₀₉	1	1	yes	characteristic
307	C₈	D₄×D₁₀₉	1	1	yes	characteristic
308	C₂²	—	2	1	no	—
309	C₂²	—	109	1	no	—
310	C₂²	—	109	1	no	—
311	C₂²	—	436	2	no	—
312	C₂²	—	218	2	no	—
313	C₂²	—	218	1	no	—
314	C₂²	#41×D₁₀₉	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	109	1	no	—
318	C₄	—	109	1	no	—
319	C₄	—	436	2	no	—
320	C₄	—	218	1	no	—
321	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
322	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	4	1	no	—
325	C₂	—	109	2	no	—
326	C₂	—	436	1	no	—
327	C₂	—	218	1	no	—
328	C₂	#121×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#121×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#473×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#121
prime radical	#41×D₁₀₉	C₂²
hypercenter	#473	D₁₀₉
hypocenter	C₁₀₉	C₂×#473
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #473

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₁₀₉	C₂	semidirect
4. #20295302	C₂	semidirect
5. #20295301	C₂	semidirect
6. #20295300	C₂	semidirect
7. #20295299	C₂	semidirect
8. #20295298	C₂	semidirect
9. #108×D₁₀₉	C₂	semidirect
10. #101×D₁₀₉	C₂	semidirect
11. #473×C₁₀₉	C₂	semidirect
19. C₂×Dic₄₃₆	C₂²	semidirect
20. C₂×#13182019	C₂²	semidirect
37. #16812635	C₂²	semidirect
38. #16812634	C₂²	semidirect
39. #16812618	C₂²	semidirect
40. #16812617	C₂²	semidirect
41. #16812617	C₂²	semidirect
43. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
45. #108×C₁₀₉	C₂²	semidirect
46. #101×C₁₀₉	C₂²	semidirect
163. D₁₀₉	#473	direct (N × H)
165. C₁₀₉	C₂×#473	semidirect
174. #473	D₁₀₉	direct (N × H)
192. C₂×Q₁₆	C₂×D₁₀₉	semidirect
207. #108	C₂×D₁₀₉	semidirect
211. #101	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#137×D₁₀₉	2
C₂×Q₁₆×D₁₀₉	2
#131×D₁₀₉	2
#20295302	2
#20295301	2
#20295300	2
#20295299	2
#20295298	2
#108×D₁₀₉	2
#101×D₁₀₉	2
#473×C₁₀₉	2
C₂×#473	109

Subgroups — #473×D109 #28372741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #473×D₁₀₉ #28372741