Subgroups — #31817003

Order 17792.

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	#22581505	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581484	C₂	1	1	yes	maximal, characteristic
3	#22581361	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581344	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581320	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581311	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581211	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581209	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581197	C₂	1	1	yes	maximal, characteristic
10	#22581047	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	M₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
17	C₂×#14555775	C₂²	1	1	yes	characteristic
18	C₂×#14555773	C₂²	1	1	yes	characteristic
19	C₂×#14555769	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984840	C₂²	1	1	yes	characteristic
22	#17984835	—	2	1	no	—
23	#17984833	C₂²	1	1	yes	characteristic
24	#17984832	—	2	1	no	—
25	#17984829	C₂²	1	1	yes	characteristic
26	#17984823	—	2	1	no	—
27	#17984821	C₂²	1	1	yes	characteristic
28	#17984821	C₂²	1	1	yes	characteristic
29	#17984819	—	2	1	no	—
30	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	#17984807	—	2	1	no	—
32	#17984807	C₂²	1	1	yes	characteristic
33	#17984807	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₃₉	—	2	1	no	—
35	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
36	#17984798	—	2	1	no	—
37	#17984798	—	2	1	no	—
38	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984752	C₂²	1	1	yes	characteristic
49	#17984752	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#107×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
65	Dic₅₅₆	—	2	2	no	—
66	#14555775	—	2	2	no	—
67	#14555773	—	2	2	no	—
68	#14555769	—	2	2	no	—
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	Dic₁₃₉⋊C₄	—	2	1	no	—
76	Dic₁₃₉⋊C₄	—	2	1	no	—
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	C₄×Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	SD₁₆×C₁₃₉	—	2	2	no	—
100	M₁₆×C₁₃₉	—	2	2	no	—
101	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	—	2	1	no	—
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	2	1	no	—
112	Dic₂₇₈	—	4	2	no	—
113	C₂×Dic₁₃₉	—	2	2	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	4	1	no	—
119	C₂×Dic₁₃₉	—	4	2	no	—
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₁₃₉⋊C₈	#41	1	2	yes	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	—	2	1	no	—
130	D₄×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	—	4	2	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₃₉	—	2	1	no	—
141	C₈×C₁₃₉	#41	1	2	yes	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	4	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	—	4	1	no	—
148	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
155	Dic₁₃₉	—	2	1	no	—
156	Dic₁₃₉	—	2	1	no	—
157	Dic₁₃₉	—	2	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	#137	1	2	yes	—
163	C₂×C₁₃₉	—	4	1	no	—
164	C₂×C₁₃₉	—	8	1	no	—
165	C₂×C₁₃₉	#532	1	1	yes	characteristic
166	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
167	C₂×C₁₃₉	#575	1	1	yes	characteristic
168	C₁₃₉	#3172	1	1	yes	characteristic, complemented, Sylow
169	#3172	—	139	1	no	maximal, Sylow
170	#533	—	139	1	no	—
171	#529	—	139	1	no	—
172	#501	—	139	1	no	—
173	#497	—	139	1	no	—
174	#491	—	139	1	no	—
175	#489	—	139	1	no	—
176	#470	—	139	1	no	—
177	#470	—	139	1	no	—
178	#468	—	139	1	no	—
179	#468	D₁₃₉	1	1	yes	characteristic
180	C₄×Q₁₆	—	139	1	no	—
181	C₄×Q₁₆	—	139	1	no	—
182	C₄×SD₁₆	—	139	1	no	—
183	C₄×SD₁₆	—	139	1	no	—
184	C₄×M₁₆	—	139	1	no	—
185	C₂×Q₁₆	—	139	1	no	—
186	C₂×Q₁₆	—	139	1	no	—
187	C₂×SD₁₆	—	139	1	no	—
188	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
189	C₂×M₁₆	—	139	1	no	—
190	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	139	1	no	—
192	C₄⋊Q₈	—	139	1	no	—
193	#126	—	278	1	no	—
194	#125	—	278	1	no	—
195	#125	—	139	1	no	—
196	#124	—	139	1	no	—
197	#123	—	278	1	no	—
198	C₂²⋊Q₈	—	278	1	no	—
199	C₂²⋊Q₈	—	139	1	no	—
200	C₂²⋊Q₈	—	139	1	no	—
201	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
202	#121	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄×Q₈	—	278	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×Q₈	—	139	1	no	—
206	C₄×Q₈	—	139	1	no	—
207	C₄×D₄	—	278	1	no	—
208	C₄×D₄	—	139	1	no	—
209	#117	—	278	1	no	—
210	#117	—	278	1	no	—
211	C₂×C₄²	—	139	1	no	—
212	#107	—	139	1	no	—
213	#107	C₂×D₁₃₉	1	1	yes	characteristic
214	#106	—	139	1	no	—
215	#106	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	—	139	1	no	—
221	#103	C₂×D₁₃₉	1	1	yes	characteristic
222	#102	—	139	1	no	—
223	#102	C₂×D₁₃₉	1	1	yes	characteristic
224	#97	—	139	1	no	—
225	#97	—	139	1	no	—
226	C₄×C₈	—	139	1	no	—
227	C₄×C₈	—	139	1	no	—
228	C₂×Q₈	—	278	1	no	—
229	C₂×Q₈	—	139	1	no	—
230	C₂×Q₈	—	139	1	no	—
231	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂²×C₄	—	278	1	no	—
235	C₂²×C₄	—	139	1	no	—
236	C₂²×C₄	—	139	1	no	—
237	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	Q₁₆	—	278	2	no	—
239	Q₁₆	—	278	2	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	278	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	278	2	no	—
244	C₂×C₈	—	139	1	no	—
245	C₂×C₈	—	139	1	no	—
246	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	278	1	no	—
251	C₄⋊C₄	—	278	1	no	—
252	C₄⋊C₄	—	278	1	no	—
253	C₄⋊C₄	—	278	1	no	—
254	C₄⋊C₄	—	278	1	no	—
255	C₄⋊C₄	—	278	1	no	—
256	C₄⋊C₄	—	278	1	no	—
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	139	1	no	—
260	C₄⋊C₄	—	139	1	no	—
261	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	278	1	no	—
266	C₂²⋊C₄	—	278	1	no	—
267	C₂²⋊C₄	—	278	1	no	—
268	C₂²⋊C₄	—	278	1	no	—
269	C₂²⋊C₄	—	278	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	—	278	1	no	—
273	C₄²	—	139	1	no	—
274	C₄²	—	139	1	no	—
275	C₄²	—	139	1	no	—
276	C₄²	—	139	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	278	1	no	—
282	Q₈	—	278	1	no	—
283	Q₈	—	278	1	no	—
284	Q₈	—	278	1	no	—
285	Q₈	—	556	2	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	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	—	2	1	no	—
294	C₂×C₄	—	278	2	no	—
295	C₂×C₄	—	278	1	no	—
296	C₂×C₄	—	278	1	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	556	1	no	—
300	C₂×C₄	—	556	2	no	—
301	C₂×C₄	—	139	1	no	—
302	C₂×C₄	—	139	1	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	278	1	no	—
309	C₈	—	139	2	no	—
310	C₈	#14555784	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₁₃₉	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	—	278	1	no	—
323	C₄	—	278	1	no	—
324	C₄	—	278	1	no	—
325	C₄	—	556	1	no	—
326	C₄	—	556	1	no	—
327	C₄	—	556	1	no	—
328	C₄	—	556	1	no	—
329	C₄	—	139	2	no	—
330	C₄	#41×D₁₃₉	1	1	yes	characteristic
331	C₄	C₂×#14555784	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	#22581498	1	1	yes	characteristic
335	C₂	#137×D₁₃₉	1	1	yes	characteristic
336	C₂	#22581749	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#468×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#121×C₁₃₉	C₂²
hypercenter	#468	D₁₃₉
hypocenter	C₁₃₉	#3172
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #22581505	C₂	semidirect
3. #22581361	C₂	semidirect
4. #22581344	C₂	semidirect
5. #22581320	C₂	semidirect
6. #22581311	C₂	semidirect
7. #22581211	C₂	semidirect
8. #22581209	C₂	semidirect
10. #22581047	C₂	semidirect
11. #22581045	C₂	semidirect
12. #22581044	C₂	semidirect
13. SD₁₆×Dic₁₃₉	C₂	semidirect
14. M₁₆×Dic₁₃₉	C₂	semidirect
168. C₁₃₉	#3172	semidirect

Maximal subgroups

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

subgroup	index
#22581505	2
#22581484	2
#22581361	2
#22581344	2
#22581320	2
#22581311	2
#22581211	2
#22581209	2
#22581197	2
#22581047	2
#22581045	2
#22581044	2
SD₁₆×Dic₁₃₉	2
M₁₆×Dic₁₃₉	2
#468×C₁₃₉	2
#3172	139