Subgroups — #578×D₁₀₉ #28377794

Order 13952.

1408 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×D₁₀₉	C₂	1	2	yes	maximal, complemented
2	#20295843	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295842	C₂	1	6	yes	maximal, complemented
4	#20295841	C₂	1	2	yes	maximal, complemented
5	#20295840	C₂	1	3	yes	maximal, complemented
6	#20295839	C₂	1	3	yes	maximal, complemented
7	#137×D₁₀₉	C₂	1	6	yes	maximal, complemented
8	#135×D₁₀₉	C₂	1	3	yes	maximal, complemented
9	C₂×Q₁₆×D₁₀₉	C₂	1	3	yes	maximal, complemented
10	#131×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#578×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812747	—	2	2	no	—
13	#16812747	C₂²	1	6	yes	complemented
14	#16812746	—	2	6	no	—
15	#16812746	C₂²	1	2	yes	complemented
16	#16812746	C₂²	1	6	yes	complemented
17	#41×D₁₀₉	—	2	6	no	—
18	#41×D₁₀₉	C₂²	1	6	yes	—
19	#41×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₁₀₉	—	2	2	no	—
21	C₂×Q₈×D₁₀₉	C₂²	1	6	yes	—
22	#16812732	C₂²	1	6	yes	complemented
23	#16812731	C₂²	1	12	yes	complemented
24	#16812730	C₂²	1	6	yes	complemented
25	#16812729	C₂²	1	6	yes	complemented
26	#16812728	C₂²	1	12	yes	complemented
27	#16812722	C₂²	1	3	yes	complemented
28	#16812721	C₂²	1	3	yes	complemented
29	#16812720	C₂²	1	6	yes	complemented
30	#16812719	C₂²	1	6	yes	complemented
31	#16812718	C₂²	1	3	yes	complemented
32	C₂×Dic₄₃₆	C₂²	1	3	yes	complemented
33	C₂×#13182019	C₂²	1	6	yes	complemented
34	Q₁₆×D₁₀₉	C₂²	1	3	yes	—
35	Q₁₆×D₁₀₉	C₂²	1	6	yes	—
36	Q₁₆×D₁₀₉	C₂²	1	6	yes	complemented
37	Q₁₆×D₁₀₉	C₂²	1	6	yes	complemented
38	SD₁₆×D₁₀₉	C₂²	1	6	yes	—
39	D₈×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812707	C₂²	1	3	yes	complemented
41	#16812706	C₂²	1	1	yes	characteristic, complemented
42	#16812705	C₂²	1	3	yes	complemented
43	M₁₆×D₁₀₉	C₂²	1	3	yes	—
44	C₂×C₈×D₁₀₉	C₂²	1	3	yes	—
45	2₋¹⁺⁴×C₁₀₉	C₂²	1	2	yes	complemented
46	#137×C₁₀₉	C₂²	1	6	yes	complemented
47	#135×C₁₀₉	C₂²	1	3	yes	complemented
48	C₂×Q₁₆×C₁₀₉	C₂²	1	3	yes	complemented
49	#131×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	#13182030	—	2	6	no	—
51	#13182030	D₄	1	1	yes	characteristic
52	#13182030	C₂³	1	6	yes	—
53	#13182029	—	2	12	no	—
54	#13182029	—	2	6	no	—
55	#13182029	—	2	6	no	—
56	#13182029	C₂³	1	3	yes	—
57	#13182029	C₂³	1	6	yes	—
58	#13182028	—	2	6	no	—
59	#13182028	—	2	12	no	—
60	#13182028	—	2	6	no	—
61	#13182028	D₄	1	3	yes	—
62	#13182028	C₂³	1	6	yes	—
63	C₂×Dic₂₁₈	—	2	6	no	—
64	C₂×Dic₂₁₈	—	2	6	no	—
65	C₂×Dic₂₁₈	C₂³	1	6	yes	—
66	Q₈×D₁₀₉	—	2	2	no	—
67	Q₈×D₁₀₉	—	2	6	no	—
68	Q₈×D₁₀₉	—	2	6	no	—
69	Q₈×D₁₀₉	—	2	6	no	—
70	Q₈×D₁₀₉	D₄	1	1	yes	characteristic
71	Q₈×D₁₀₉	C₂³	1	6	yes	—
72	Q₈×D₁₀₉	C₂³	1	12	yes	—
73	D₄×D₁₀₉	—	2	6	no	—
74	D₄×D₁₀₉	D₄	1	3	yes	—
75	D₄×D₁₀₉	C₂³	1	2	yes	—
76	C₂×C₄×D₁₀₉	—	2	6	no	—
77	C₂×C₄×D₁₀₉	—	2	6	no	—
78	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
79	Dic₄₃₆	C₂³	1	3	yes	—
80	Dic₄₃₆	C₂³	1	6	yes	—
81	#13182019	C₂³	1	6	yes	—
82	#13182019	C₂³	1	12	yes	—
83	#13182018	C₂³	1	6	yes	—
84	#13182017	C₂³	1	6	yes	—
85	#13182016	C₂³	1	6	yes	—
86	D₈₇₂	C₂³	1	1	yes	characteristic
87	#13182014	C₂³	1	2	yes	—
88	#13182013	C₂³	1	3	yes	—
89	#13182012	C₂³	1	6	yes	—
90	C₂×#11724784	C₂³	1	3	yes	—
91	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
92	C₈×D₁₀₉	C₂³	1	3	yes	—
93	C₈×D₁₀₉	C₂³	1	3	yes	—
94	C₈×D₁₀₉	C₂³	1	3	yes	—
95	#41×C₁₀₉	—	2	6	no	—
96	#41×C₁₀₉	C₂³	1	6	yes	—
97	#41×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
100	Q₁₆×C₁₀₉	C₂³	1	3	yes	—
101	Q₁₆×C₁₀₉	C₂³	1	6	yes	—
102	SD₁₆×C₁₀₉	C₂³	1	6	yes	—
103	D₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	M₁₆×C₁₀₉	C₂³	1	3	yes	—
105	C₂×C₈×C₁₀₉	C₂³	1	3	yes	—
106	C₂²×D₁₀₉	—	2	2	no	—
107	C₂²×D₁₀₉	C₂×D₄	1	3	yes	—
108	Q₈×C₁₀₉	—	2	6	no	—
109	Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	C₂⁴	1	6	yes	—
111	D₄×C₁₀₉	—	2	6	no	—
112	D₄×C₁₀₉	C₂×D₄	1	3	yes	—
113	D₄×C₁₀₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₀₉	—	2	6	no	—
115	C₂×C₄×C₁₀₉	—	2	6	no	—
116	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
117	C₂²⋊D₁₀₉	—	2	6	no	—
118	C₂²⋊D₁₀₉	—	2	6	no	—
119	C₂²⋊D₁₀₉	—	2	2	no	—
120	C₂²⋊D₁₀₉	C₂×D₄	1	6	yes	—
121	C₂×Dic₁₀₉	—	2	6	no	—
122	C₂×Dic₁₀₉	—	2	6	no	—
123	C₂×Dic₁₀₉	C₂×D₄	1	3	yes	—
124	D₄₃₆	—	2	6	no	—
125	D₄₃₆	C₂×D₄	1	3	yes	—
126	D₄₃₆	C₂⁴	1	2	yes	—
127	C₄×D₁₀₉	—	2	6	no	—
128	C₄×D₁₀₉	—	2	12	no	—
129	C₄×D₁₀₉	—	2	6	no	—
130	C₄×D₁₀₉	—	2	6	no	—
131	C₄×D₁₀₉	C₂×D₄	1	3	yes	—
132	C₄×D₁₀₉	C₂×D₄	1	3	yes	—
133	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
134	C₄×D₁₀₉	C₂⁴	1	3	yes	—
135	Dic₂₁₈	—	2	6	no	—
136	Dic₂₁₈	—	2	6	no	—
137	Dic₂₁₈	—	2	12	no	—
138	Dic₂₁₈	C₂×D₄	1	3	yes	—
139	Dic₂₁₈	C₂⁴	1	6	yes	—
140	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₀₉	C₂⁴	1	3	yes	—
142	#11724784	C₂⁴	1	1	yes	characteristic
143	#11724784	C₂⁴	1	3	yes	—
144	C₂²×C₁₀₉	—	2	2	no	—
145	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
146	C₂×D₁₀₉	—	2	6	no	—
147	C₂×D₁₀₉	—	2	2	no	—
148	C₂×D₁₀₉	—	4	4	no	—
149	C₂×D₁₀₉	C₂²×D₄	1	3	yes	—
150	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₁₀₉	—	2	6	no	—
152	C₄×C₁₀₉	C₂²×D₄	1	3	yes	—
153	C₄×C₁₀₉	C₂⁵	1	1	yes	characteristic
154	Dic₁₀₉	—	2	6	no	—
155	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	Dic₁₀₉	C₂²×D₄	1	3	yes	—
157	C₂×C₁₀₉	—	2	3	no	—
158	C₂×C₁₀₉	—	4	2	no	—
159	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
160	D₁₀₉	—	2	3	no	—
161	D₁₀₉	—	4	2	no	—
162	D₁₀₉	#578	1	2	yes	direct product factor
163	C₂×#578	—	109	1	no	maximal, Sylow
164	C₁₀₉	C₂×#578	1	1	yes	characteristic, complemented, Sylow
165	C₂×2₋¹⁺⁴	—	109	2	no	—
166	#578	—	109	2	no	—
167	#578	—	109	6	no	—
168	#578	—	109	3	no	—
169	#578	—	109	3	no	—
170	#578	—	109	1	no	—
171	#578	D₁₀₉	1	1	yes	characteristic, direct product factor
172	C₂×#137	—	109	6	no	—
173	C₂×#135	—	109	3	no	—
174	C₂²×Q₁₆	—	109	3	no	—
175	C₂×#131	—	109	1	no	—
176	2₋¹⁺⁴	—	109	6	no	—
177	2₋¹⁺⁴	—	109	2	no	—
178	2₋¹⁺⁴	—	109	6	no	—
179	2₋¹⁺⁴	—	218	2	no	—
180	2₋¹⁺⁴	—	218	6	no	—
181	2₋¹⁺⁴	C₂×D₁₀₉	1	2	yes	complemented
182	C₂×#41	—	109	1	no	—
183	C₂×#41	—	109	6	no	—
184	C₂×#41	—	218	6	no	—
185	C₂²×Q₈	—	109	6	no	—
186	C₂²×Q₈	—	218	2	no	—
187	#137	—	109	6	no	—
188	#137	—	109	6	no	—
189	#137	—	109	12	no	—
190	#137	—	109	12	no	—
191	#137	—	109	6	no	—
192	#137	C₂×D₁₀₉	1	6	yes	complemented
193	#135	—	109	6	no	—
194	#135	—	109	3	no	—
195	#135	—	109	6	no	—
196	#135	—	109	3	no	—
197	#135	—	109	3	no	—
198	#135	C₂×D₁₀₉	1	3	yes	complemented
199	C₂×Q₁₆	—	109	6	no	—
200	C₂×Q₁₆	—	109	3	no	—
201	C₂×Q₁₆	—	109	6	no	—
202	C₂×Q₁₆	—	109	6	no	—
203	C₂×Q₁₆	—	109	6	no	—
204	C₂×Q₁₆	—	109	3	no	—
205	C₂×Q₁₆	C₂×D₁₀₉	1	3	yes	complemented
206	C₂×SD₁₆	—	109	6	no	—
207	C₂×D₈	—	109	1	no	—
208	#131	—	109	1	no	—
209	#131	—	109	3	no	—
210	#131	—	109	3	no	—
211	#131	C₂×D₁₀₉	1	1	yes	characteristic, complemented
212	C₂×M₁₆	—	109	3	no	—
213	C₂²×C₈	—	109	3	no	—
214	#41	—	2	6	no	—
215	#41	—	109	1	no	—
216	#41	—	109	3	no	—
217	#41	—	109	3	no	—
218	#41	—	109	6	no	—
219	#41	—	109	6	no	—
220	#41	—	109	6	no	—
221	#41	—	218	6	no	—
222	#41	—	218	12	no	—
223	#41	—	218	6	no	—
224	#41	—	218	6	no	—
225	#41	—	218	12	no	—
226	#41	—	218	6	no	—
227	#41	—	218	6	no	—
228	#41	C₂²×D₁₀₉	1	6	yes	—
229	#41	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	109	1	no	—
232	C₂×Q₈	—	109	6	no	—
233	C₂×Q₈	—	109	12	no	—
234	C₂×Q₈	—	109	6	no	—
235	C₂×Q₈	—	218	6	no	—
236	C₂×Q₈	—	218	6	no	—
237	C₂×Q₈	—	218	6	no	—
238	C₂×Q₈	—	218	6	no	—
239	C₂×Q₈	—	218	6	no	—
240	C₂×Q₈	—	218	2	no	—
241	C₂×Q₈	C₂²×D₁₀₉	1	6	yes	—
242	C₂×D₄	—	109	3	no	—
243	C₂×D₄	—	109	2	no	—
244	C₂×D₄	—	218	6	no	—
245	C₂²×C₄	—	109	3	no	—
246	C₂²×C₄	—	218	6	no	—
247	C₂²×C₄	—	218	6	no	—
248	Q₁₆	—	109	6	no	—
249	Q₁₆	—	109	12	no	—
250	Q₁₆	—	109	3	no	—
251	Q₁₆	—	109	6	no	—
252	Q₁₆	C₂²×D₁₀₉	1	3	yes	—
253	Q₁₆	C₂²×D₁₀₉	1	6	yes	—
254	SD₁₆	—	109	6	no	—
255	SD₁₆	—	109	6	no	—
256	SD₁₆	—	109	6	no	—
257	SD₁₆	C₂²×D₁₀₉	1	6	yes	—
258	D₈	—	109	2	no	—
259	D₈	—	109	1	no	—
260	D₈	C₂²×D₁₀₉	1	1	yes	characteristic
261	M₁₆	—	109	3	no	—
262	M₁₆	—	109	6	no	—
263	M₁₆	C₂²×D₁₀₉	1	3	yes	—
264	C₂×C₈	—	109	3	no	—
265	C₂×C₈	—	109	1	no	—
266	C₂×C₈	—	109	3	no	—
267	C₂×C₈	—	109	3	no	—
268	C₂×C₈	—	109	3	no	—
269	C₂×C₈	C₂²×D₁₀₉	1	3	yes	—
270	C₂³	—	109	3	no	—
271	C₂³	—	218	2	no	—
272	Q₈	—	2	6	no	—
273	Q₈	—	109	3	no	—
274	Q₈	—	109	6	no	—
275	Q₈	—	218	6	no	—
276	Q₈	—	218	12	no	—
277	Q₈	—	218	6	no	—
278	Q₈	D₄×D₁₀₉	1	1	yes	characteristic
279	Q₈	C₂³×D₁₀₉	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	109	6	no	—
282	D₄	—	109	3	no	—
283	D₄	—	109	2	no	—
284	D₄	—	218	6	no	—
285	D₄	—	218	2	no	—
286	D₄	—	218	6	no	—
287	D₄	—	218	6	no	—
288	D₄	D₄×D₁₀₉	1	3	yes	—
289	D₄	C₂³×D₁₀₉	1	2	yes	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	2	6	no	—
292	C₂×C₄	—	109	3	no	—
293	C₂×C₄	—	109	3	no	—
294	C₂×C₄	—	109	3	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	109	3	no	—
297	C₂×C₄	—	218	6	no	—
298	C₂×C₄	—	218	6	no	—
299	C₂×C₄	—	218	12	no	—
300	C₂×C₄	—	218	6	no	—
301	C₂×C₄	—	218	6	no	—
302	C₂×C₄	—	218	6	no	—
303	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
304	C₈	—	109	1	no	—
305	C₈	—	109	3	no	—
306	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
307	C₈	C₂³×D₁₀₉	1	3	yes	—
308	C₂²	—	2	2	no	—
309	C₂²	—	109	1	no	—
310	C₂²	—	109	3	no	—
311	C₂²	—	436	4	no	—
312	C₂²	—	218	6	no	—
313	C₂²	—	218	2	no	—
314	C₂²	C₂×D₄×D₁₀₉	1	3	yes	—
315	C₄	—	2	6	no	—
316	C₄	—	109	1	no	—
317	C₄	—	109	3	no	—
318	C₄	—	218	6	no	—
319	C₄	C₂×D₄×D₁₀₉	1	3	yes	—
320	C₄	C₂⁴×D₁₀₉	1	1	yes	characteristic
321	C₂	—	2	3	no	—
322	C₂	—	4	2	no	—
323	C₂	—	109	2	no	—
324	C₂	—	436	2	no	—
325	C₂	—	218	3	no	—
326	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₉
commutator, cosocle	C₄×C₁₀₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₉
Fitting	#578×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂³×D₄
hypercenter	#578	D₁₀₉
hypocenter	C₁₀₉	C₂×#578
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×D₁₀₉	C₂	semidirect
2. #20295843	C₂	semidirect
3. #20295842	C₂	semidirect
4. #20295841	C₂	semidirect
5. #20295840	C₂	semidirect
6. #20295839	C₂	semidirect
7. #137×D₁₀₉	C₂	semidirect
8. #135×D₁₀₉	C₂	semidirect
9. C₂×Q₁₆×D₁₀₉	C₂	semidirect
10. #131×D₁₀₉	C₂	semidirect
11. #578×C₁₀₉	C₂	semidirect
13. #16812747	C₂²	semidirect
15. #16812746	C₂²	semidirect
16. #16812746	C₂²	semidirect
22. #16812732	C₂²	semidirect
23. #16812731	C₂²	semidirect
24. #16812730	C₂²	semidirect
25. #16812729	C₂²	semidirect
26. #16812728	C₂²	semidirect
27. #16812722	C₂²	semidirect
28. #16812721	C₂²	semidirect
29. #16812720	C₂²	semidirect
30. #16812719	C₂²	semidirect
31. #16812718	C₂²	semidirect
32. C₂×Dic₄₃₆	C₂²	semidirect
33. C₂×#13182019	C₂²	semidirect
36. Q₁₆×D₁₀₉	C₂²	semidirect
37. Q₁₆×D₁₀₉	C₂²	semidirect
40. #16812707	C₂²	semidirect
41. #16812706	C₂²	semidirect
42. #16812705	C₂²	semidirect
45. 2₋¹⁺⁴×C₁₀₉	C₂²	semidirect
46. #137×C₁₀₉	C₂²	semidirect
47. #135×C₁₀₉	C₂²	semidirect
48. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
49. #131×C₁₀₉	C₂²	semidirect
162. D₁₀₉	#578	direct (N × H)
164. C₁₀₉	C₂×#578	semidirect
171. #578	D₁₀₉	direct (N × H)
181. 2₋¹⁺⁴	C₂×D₁₀₉	semidirect
192. #137	C₂×D₁₀₉	semidirect
198. #135	C₂×D₁₀₉	semidirect
205. C₂×Q₁₆	C₂×D₁₀₉	semidirect
211. #131	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×D₁₀₉	2
#20295843	2
#20295842	2
#20295841	2
#20295840	2
#20295839	2
#137×D₁₀₉	2
#135×D₁₀₉	2
C₂×Q₁₆×D₁₀₉	2
#131×D₁₀₉	2
#578×C₁₀₉	2
C₂×#578	109

Subgroups — #578×D109 #28377794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #578×D₁₀₉ #28377794