Subgroups — #344×D₁₀₉ #28362772

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 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	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294798	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294797	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294796	C₂	1	1	yes	maximal, characteristic, complemented
7	#344×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#13182013	C₂²	1	1	yes	characteristic
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂×#13182012	—	2	1	no	—
12	C₂×#13182012	C₄	1	2	yes	complemented
13	M₁₆×D₁₀₉	—	2	1	no	—
14	M₁₆×D₁₀₉	—	2	1	no	—
15	M₁₆×D₁₀₉	—	2	1	no	—
16	M₁₆×D₁₀₉	—	2	1	no	—
17	M₁₆×D₁₀₉	—	2	1	no	—
18	M₁₆×D₁₀₉	—	2	1	no	—
19	M₁₆×D₁₀₉	C₄	1	2	yes	complemented
20	M₁₆×D₁₀₉	C₄	1	2	yes	complemented
21	C₂×C₈×D₁₀₉	—	2	1	no	—
22	C₂×C₈×D₁₀₉	C₄	1	2	yes	complemented
23	#16812658	C₂²	1	1	yes	characteristic
24	#16812657	—	2	1	no	—
25	#16812656	—	2	1	no	—
26	#16812655	C₄	1	2	yes	—
27	C₄⋊C₄×D₁₀₉	—	2	1	no	—
28	C₂²⋊C₄×D₁₀₉	C₄	1	2	yes	—
29	C₄²×D₁₀₉	—	2	1	no	—
30	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
32	#117×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₀₉	—	2	2	no	—
36	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
41	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
42	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
48	#13182013	—	2	1	no	—
49	#13182013	—	2	1	no	—
50	#13182013	—	2	1	no	—
51	#13182013	C₂×C₄	1	2	yes	complemented
52	#13182012	—	2	2	no	—
53	#13182012	—	2	2	no	—
54	#13182012	—	2	2	no	—
55	#13182012	—	2	2	no	—
56	C₂×#11724784	—	2	1	no	—
57	C₂×#11724784	C₂×C₄	1	2	yes	complemented
58	C₈×D₁₀₉	—	2	2	no	—
59	C₈×D₁₀₉	—	2	2	no	—
60	C₈×D₁₀₉	—	2	2	no	—
61	C₈×D₁₀₉	—	2	2	no	—
62	C₄⋊Dic₁₀₉	—	2	1	no	—
63	#13182008	—	2	2	no	—
64	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
65	#13182006	—	2	2	no	—
66	C₄×Dic₁₀₉	—	2	1	no	—
67	C₄×Dic₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	M₁₆×C₁₀₉	—	2	1	no	—
70	M₁₆×C₁₀₉	—	2	1	no	—
71	M₁₆×C₁₀₉	—	2	1	no	—
72	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
73	C₂×C₈×C₁₀₉	—	2	1	no	—
74	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
75	C₄⋊C₄×C₁₀₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₀₉	—	4	2	no	—
80	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
81	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
82	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
93	C₂×Dic₁₀₉	—	2	2	no	—
94	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
95	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
96	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₁₀₉	—	4	4	no	—
101	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
110	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
111	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
112	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
113	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
114	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
115	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
116	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₁₀₉	—	2	2	no	—
118	C₈×C₁₀₉	—	2	2	no	—
119	#11724784	—	2	2	no	—
120	#11724784	—	2	2	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
123	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×D₁₀₉	—	2	2	no	—
126	C₂×D₁₀₉	—	2	2	no	—
127	C₂×D₁₀₉	—	2	2	no	—
128	C₂×D₁₀₉	—	4	1	no	—
129	C₂×D₁₀₉	—	4	1	no	—
130	C₂×D₁₀₉	—	4	1	no	—
131	C₂×D₁₀₉	#95	1	1	yes	characteristic
132	C₂×D₁₀₉	#95	1	1	yes	characteristic
133	C₂×D₁₀₉	#95	1	1	yes	characteristic
134	C₂×D₁₀₉	#95	1	1	yes	characteristic
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	Dic₁₀₉	—	4	2	no	—
141	Dic₁₀₉	#95	1	1	yes	characteristic
142	Dic₁₀₉	#95	1	1	yes	characteristic
143	Dic₁₀₉	#95	1	1	yes	characteristic
144	Dic₁₀₉	#95	1	1	yes	characteristic
145	C₂×C₁₀₉	—	2	1	no	—
146	C₂×C₁₀₉	—	2	1	no	—
147	C₂×C₁₀₉	—	2	1	no	—
148	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
149	D₁₀₉	—	2	1	no	—
150	D₁₀₉	—	2	1	no	—
151	D₁₀₉	—	2	1	no	—
152	D₁₀₉	#344	1	2	yes	direct product factor
153	C₂×#344	—	109	1	no	maximal, Sylow
154	C₁₀₉	C₂×#344	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	109	1	no	—
156	C₂²×M₁₆	—	109	1	no	—
157	C₂×#117	—	109	1	no	—
158	#344	—	109	1	no	—
159	#344	—	109	1	no	—
160	#344	—	109	1	no	—
161	#344	D₁₀₉	1	1	yes	characteristic, direct product factor
162	C₂³×C₄	—	109	1	no	—
163	C₂×M₁₆	—	109	1	no	—
164	C₂×M₁₆	—	109	2	no	—
165	C₂×M₁₆	—	109	1	no	—
166	C₂×M₁₆	—	109	2	no	—
167	C₂×M₁₆	—	109	2	no	—
168	C₂×M₁₆	—	218	1	no	—
169	C₂×M₁₆	—	218	1	no	—
170	C₂×M₁₆	—	218	1	no	—
171	C₂×M₁₆	—	218	1	no	—
172	C₂×M₁₆	—	218	1	no	—
173	C₂×M₁₆	—	218	1	no	—
174	C₂×M₁₆	—	218	1	no	—
175	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
176	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
177	C₂²×C₈	—	109	2	no	—
178	C₂²×C₈	—	218	1	no	—
179	#117	—	109	1	no	—
180	#117	—	109	2	no	—
181	#117	—	218	1	no	—
182	#117	—	218	1	no	—
183	#117	C₂×D₁₀₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	—	218	1	no	—
185	C₂×C₂²⋊C₄	—	109	2	no	—
186	C₂×C₄²	—	218	1	no	—
187	C₂⁴	—	109	1	no	—
188	C₂²×C₄	—	109	1	no	—
189	C₂²×C₄	—	109	1	no	—
190	C₂²×C₄	—	109	1	no	—
191	C₂²×C₄	—	109	1	no	—
192	C₂²×C₄	—	109	1	no	—
193	C₂²×C₄	—	109	1	no	—
194	C₂²×C₄	—	109	1	no	—
195	C₂²×C₄	—	109	1	no	—
196	C₂²×C₄	—	109	1	no	—
197	C₂²×C₄	—	109	1	no	—
198	C₂²×C₄	—	109	1	no	—
199	C₂²×C₄	—	109	1	no	—
200	C₂²×C₄	—	109	1	no	—
201	C₂²×C₄	—	218	2	no	—
202	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	109	2	no	—
207	M₁₆	—	218	2	no	—
208	M₁₆	—	218	2	no	—
209	M₁₆	—	218	2	no	—
210	M₁₆	—	218	2	no	—
211	M₁₆	—	218	1	no	—
212	M₁₆	—	218	1	no	—
213	M₁₆	—	218	1	no	—
214	M₁₆	C₄×D₁₀₉	1	2	yes	complemented
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	109	2	no	—
217	C₂×C₈	—	218	2	no	—
218	C₂×C₈	—	218	2	no	—
219	C₂×C₈	—	218	2	no	—
220	C₂×C₈	—	218	2	no	—
221	C₂×C₈	—	218	1	no	—
222	C₂×C₈	C₄×D₁₀₉	1	2	yes	complemented
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	218	2	no	—
225	C₄⋊C₄	—	218	1	no	—
226	C₂²⋊C₄	—	109	2	no	—
227	C₂²⋊C₄	—	218	2	no	—
228	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
229	C₄²	—	2	1	no	—
230	C₄²	—	218	2	no	—
231	C₄²	—	218	1	no	—
232	C₂³	—	109	1	no	—
233	C₂³	—	109	1	no	—
234	C₂³	—	109	1	no	—
235	C₂³	—	109	1	no	—
236	C₂³	—	109	1	no	—
237	C₂³	—	109	1	no	—
238	C₂³	—	436	2	no	—
239	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	109	1	no	—
244	C₂×C₄	—	109	1	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	1	no	—
248	C₂×C₄	—	109	1	no	—
249	C₂×C₄	—	109	1	no	—
250	C₂×C₄	—	109	1	no	—
251	C₂×C₄	—	109	1	no	—
252	C₂×C₄	—	109	1	no	—
253	C₂×C₄	—	109	1	no	—
254	C₂×C₄	—	109	1	no	—
255	C₂×C₄	—	109	1	no	—
256	C₂×C₄	—	109	1	no	—
257	C₂×C₄	—	109	1	no	—
258	C₂×C₄	—	109	1	no	—
259	C₂×C₄	—	109	1	no	—
260	C₂×C₄	—	109	1	no	—
261	C₂×C₄	—	109	1	no	—
262	C₂×C₄	—	109	1	no	—
263	C₂×C₄	—	436	4	no	—
264	C₂×C₄	—	218	2	no	—
265	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
267	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
268	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
269	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
270	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	218	2	no	—
274	C₈	—	218	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	109	1	no	—
277	C₂²	—	109	1	no	—
278	C₂²	—	109	1	no	—
279	C₂²	—	109	1	no	—
280	C₂²	—	436	1	no	—
281	C₂²	—	436	1	no	—
282	C₂²	—	436	1	no	—
283	C₂²	—	218	2	no	—
284	C₂²	—	218	2	no	—
285	C₂²	—	218	2	no	—
286	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
287	C₂²	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
288	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	109	1	no	—
291	C₄	—	109	1	no	—
292	C₄	—	109	1	no	—
293	C₄	—	109	1	no	—
294	C₄	—	436	2	no	—
295	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
296	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
297	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
298	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	109	2	no	—
303	C₂	—	218	1	no	—
304	C₂	—	218	1	no	—
305	C₂	—	218	1	no	—
306	C₂	#95×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#344×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#95
prime radical	C₂³×D₁₀₉	C₂×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#344	D₁₀₉
hypocenter	C₁₀₉	C₂×#344
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. #20294798	C₂	semidirect
5. #20294797	C₂	semidirect
6. #20294796	C₂	semidirect
7. #344×C₁₀₉	C₂	semidirect
12. C₂×#13182012	C₄	semidirect
19. M₁₆×D₁₀₉	C₄	semidirect
20. M₁₆×D₁₀₉	C₄	semidirect
22. C₂×C₈×D₁₀₉	C₄	semidirect
51. #13182013	C₂×C₄	semidirect
57. C₂×#11724784	C₂×C₄	semidirect
72. M₁₆×C₁₀₉	C₂×C₄	semidirect
74. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
152. D₁₀₉	#344	direct (N × H)
154. C₁₀₉	C₂×#344	semidirect
161. #344	D₁₀₉	direct (N × H)
214. M₁₆	C₄×D₁₀₉	semidirect
222. C₂×C₈	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
#20294798	2
#20294797	2
#20294796	2
#344×C₁₀₉	2
C₂×#344	109

Subgroups — #344×D109 #28362772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #344×D₁₀₉ #28362772