Subgroups — #28373636

Order 13952.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295580	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295563	C₂	1	1	yes	maximal, characteristic
3	#20295434	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295419	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295393	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295387	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295283	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295281	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295275	C₂	1	1	yes	maximal, characteristic
10	#20295273	C₂	1	1	yes	maximal, characteristic
11	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182016	C₂²	1	1	yes	characteristic
19	C₂×#13182012	C₂²	1	1	yes	characteristic
20	C₂×#13182012	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic
22	#16812697	C₂²	1	1	yes	characteristic
23	#16812694	—	2	1	no	—
24	#16812690	C₂²	1	1	yes	characteristic
25	#16812689	—	2	1	no	—
26	#16812688	C₂²	1	1	yes	characteristic
27	#16812684	—	2	1	no	—
28	#16812679	C₂²	1	1	yes	characteristic
29	#16812677	—	2	1	no	—
30	#16812675	C₂²	1	1	yes	characteristic
31	#16812675	C₂²	1	1	yes	characteristic
32	#16812674	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	—	2	1	no	—
34	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
35	C₄×D₄₃₆	—	2	1	no	—
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	#16812656	—	2	1	no	—
38	#16812656	—	2	1	no	—
39	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₄×#11724784	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	#13182019	—	2	2	no	—
68	#13182018	—	2	2	no	—
69	#13182016	—	2	2	no	—
70	#13182012	—	2	2	no	—
71	#13182012	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	2	1	no	—
129	D₄₃₆	—	4	2	no	—
130	C₄×D₁₀₉	—	2	2	no	—
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	2	1	no	—
133	C₄×D₁₀₉	—	4	1	no	—
134	C₄×D₁₀₉	—	4	2	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	4	2	no	—
138	C₈×C₁₀₉	—	2	1	no	—
139	C₈×C₁₀₉	#41	1	2	yes	—
140	#11724784	—	2	1	no	—
141	#11724784	#41	1	2	yes	—
142	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₀₉	—	2	1	no	—
144	C₂×D₁₀₉	—	2	1	no	—
145	C₂×D₁₀₉	—	2	1	no	—
146	C₂×D₁₀₉	—	4	1	no	—
147	C₂×D₁₀₉	—	4	1	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	#137	1	2	yes	—
155	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	2	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	C₂×C₁₀₉	#532	1	1	yes	characteristic
164	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
165	C₂×C₁₀₉	#575	1	1	yes	characteristic
166	D₁₀₉	—	4	1	no	—
167	D₁₀₉	—	8	1	no	—
168	#3172	—	109	1	no	maximal, Sylow
169	C₁₀₉	#3172	1	1	yes	characteristic, complemented, Sylow
170	#533	—	109	1	no	—
171	#529	—	109	1	no	—
172	#501	—	109	1	no	—
173	#497	—	109	1	no	—
174	#491	—	109	1	no	—
175	#489	—	109	1	no	—
176	#470	—	109	1	no	—
177	#470	—	109	1	no	—
178	#468	—	109	1	no	—
179	#468	—	109	1	no	—
180	C₄×Q₁₆	—	109	1	no	—
181	C₄×Q₁₆	D₁₀₉	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	109	1	no	—
183	C₄×SD₁₆	—	109	1	no	—
184	C₄×M₁₆	—	109	1	no	—
185	C₂×Q₁₆	—	109	1	no	—
186	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
187	C₂×SD₁₆	—	109	1	no	—
188	C₂×SD₁₆	—	109	1	no	—
189	C₂×M₁₆	—	109	1	no	—
190	C₂×M₁₆	—	109	1	no	—
191	C₄⋊Q₈	—	109	1	no	—
192	C₄⋊Q₈	—	109	1	no	—
193	#126	—	218	1	no	—
194	#125	—	109	1	no	—
195	#125	—	218	1	no	—
196	#124	—	109	1	no	—
197	#123	—	218	1	no	—
198	C₂²⋊Q₈	—	109	1	no	—
199	C₂²⋊Q₈	—	109	1	no	—
200	C₂²⋊Q₈	—	109	1	no	—
201	C₂²⋊Q₈	—	218	1	no	—
202	#121	—	109	1	no	—
203	C₄×Q₈	—	109	1	no	—
204	C₄×Q₈	—	218	1	no	—
205	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
206	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	218	1	no	—
209	#117	—	218	1	no	—
210	#117	—	218	1	no	—
211	C₂×C₄²	—	109	1	no	—
212	#107	—	109	1	no	—
213	#107	C₂×D₁₀₉	1	1	yes	characteristic
214	#106	—	109	1	no	—
215	#106	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	C₂×D₁₀₉	1	1	yes	characteristic
221	#103	C₂×D₁₀₉	1	1	yes	characteristic
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	#97	—	109	1	no	—
225	#97	—	109	1	no	—
226	C₄×C₈	—	109	1	no	—
227	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	C₂×Q₈	—	109	1	no	—
229	C₂×Q₈	—	218	1	no	—
230	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂×D₄	—	109	1	no	—
233	C₂×D₄	—	218	1	no	—
234	C₂²×C₄	—	109	1	no	—
235	C₂²×C₄	—	109	1	no	—
236	C₂²×C₄	—	109	1	no	—
237	C₂²×C₄	—	218	1	no	—
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	218	2	no	—
240	SD₁₆	—	218	2	no	—
241	SD₁₆	—	218	2	no	—
242	M₁₆	—	218	2	no	—
243	M₁₆	—	218	2	no	—
244	C₂×C₈	—	109	1	no	—
245	C₂×C₈	—	109	1	no	—
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₂²⋊C₄	—	218	1	no	—
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	109	1	no	—
273	C₄²	—	109	1	no	—
274	C₄²	—	109	1	no	—
275	C₄²	—	218	1	no	—
276	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂³	—	109	1	no	—
278	C₂³	—	218	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	436	2	no	—
284	Q₈	—	218	1	no	—
285	Q₈	—	218	1	no	—
286	D₄	—	436	2	no	—
287	D₄	—	218	1	no	—
288	D₄	—	218	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	109	1	no	—
294	C₂×C₄	—	109	1	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	436	1	no	—
297	C₂×C₄	—	436	2	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	—	218	2	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
305	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	109	2	no	—
309	C₈	—	218	1	no	—
310	C₈	#13182029	1	2	yes	—
311	C₂²	—	436	1	no	—
312	C₂²	—	436	1	no	—
313	C₂²	—	436	1	no	—
314	C₂²	—	218	1	no	—
315	C₂²	—	218	1	no	—
316	C₂²	—	218	1	no	—
317	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	—	218	1	no	—
327	C₄	—	218	1	no	—
328	C₄	—	218	1	no	—
329	C₄	#16812731	1	2	yes	—
330	C₄	#41×D₁₀₉	1	1	yes	characteristic
331	C₄	C₂×#13182029	1	1	yes	characteristic
332	C₂	—	872	1	no	—
333	C₂	—	436	1	no	—
334	C₂	#20295573	1	1	yes	characteristic
335	C₂	C₂×#16812731	1	1	yes	characteristic
336	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×Q₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812674	C₂²
hypercenter	C₄×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#3172
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #20295580	C₂	semidirect
3. #20295434	C₂	semidirect
4. #20295419	C₂	semidirect
5. #20295393	C₂	semidirect
6. #20295387	C₂	semidirect
7. #20295283	C₂	semidirect
8. #20295281	C₂	semidirect
11. C₄×#13182019	C₂	semidirect
12. C₄×#13182018	C₂	semidirect
13. C₄×#13182016	C₂	semidirect
14. C₄×#13182012	C₂	semidirect
15. C₄×Q₁₆×C₁₀₉	C₂	semidirect
169. C₁₀₉	#3172	semidirect
181. C₄×Q₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295580	2
#20295563	2
#20295434	2
#20295419	2
#20295393	2
#20295387	2
#20295283	2
#20295281	2
#20295275	2
#20295273	2
C₄×#13182019	2
C₄×#13182018	2
C₄×#13182016	2
C₄×#13182012	2
C₄×Q₁₆×C₁₀₉	2
#3172	109