Subgroups — #31819117

Order 17792.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581672	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581312	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581311	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581249	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581166	C₂	1	1	yes	maximal, characteristic
10	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581082	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581053	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
15	#463×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₂₇₈	—	2	1	no	—
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	#17984842	C₂²	1	1	yes	characteristic
19	#17984841	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	#17984835	—	2	1	no	—
24	#17984829	C₂²	1	1	yes	characteristic
25	#17984821	—	2	1	no	—
26	#17984819	—	2	1	no	—
27	#17984819	—	2	1	no	—
28	#17984819	—	2	1	no	—
29	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
35	#17984798	—	2	1	no	—
36	#17984776	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
41	#17984767	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
46	#17984763	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	#17984753	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#124×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×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	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	—	4	4	no	—
63	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
64	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	#14555773	—	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	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
83	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
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	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₉	—	2	1	no	—
97	SD₁₆×C₁₃₉	—	2	2	no	—
98	D₈×C₁₃₉	—	2	2	no	—
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
106	Dic₂₇₈	—	2	2	no	—
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	4	2	no	—
110	Dic₂₇₈	—	4	2	no	—
111	Dic₂₇₈	—	4	2	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	4	2	no	—
117	C₂×Dic₁₃₉	—	4	2	no	—
118	C₂×Dic₁₃₉	—	4	2	no	—
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
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	C₁₃₉⋊C₈	—	2	1	no	—
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₂³×C₁₃₉	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	Q₈×C₁₃₉	—	2	1	no	—
128	Q₈×C₁₃₉	—	2	1	no	—
129	D₄×C₁₃₉	—	2	1	no	—
130	D₄×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	#41	1	2	yes	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	4	2	no	—
136	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₃₉	—	2	1	no	—
140	C₈×C₁₃₉	—	2	1	no	—
141	C₂²×C₁₃₉	—	2	2	no	—
142	C₂²×C₁₃₉	—	4	1	no	—
143	C₂²×C₁₃₉	—	4	1	no	—
144	C₂²×C₁₃₉	—	4	1	no	—
145	C₂²×C₁₃₉	—	4	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₃₉	—	2	1	no	—
155	Dic₁₃₉	—	2	1	no	—
156	Dic₁₃₉	—	2	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₁₃₉	—	4	2	no	—
163	C₂×C₁₃₉	—	8	1	no	—
164	C₂×C₁₃₉	#548	1	1	yes	characteristic
165	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₁₃₉	#578	1	1	yes	characteristic
167	C₁₃₉	#3367	1	1	yes	characteristic, complemented, Sylow
168	#3367	—	139	1	no	maximal, Sylow
169	#556	—	139	1	no	—
170	D₄×Q₈	—	139	1	no	—
171	#489	—	139	1	no	—
172	#489	—	139	1	no	—
173	#487	—	139	1	no	—
174	#477	—	139	1	no	—
175	#477	—	139	1	no	—
176	#474	—	139	1	no	—
177	#463	—	139	1	no	—
178	#463	D₁₃₉	1	1	yes	characteristic
179	#461	—	139	1	no	—
180	#446	—	139	1	no	—
181	#440	—	139	1	no	—
182	#440	—	139	1	no	—
183	C₄×D₈	—	139	1	no	—
184	C₂²×Q₈	—	278	1	no	—
185	C₂×SD₁₆	—	139	1	no	—
186	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
187	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	139	1	no	—
189	C₄⋊Q₈	—	139	1	no	—
190	C₄⋊Q₈	—	139	1	no	—
191	C₄⋊Q₈	—	139	1	no	—
192	#126	—	278	1	no	—
193	#126	—	278	1	no	—
194	#124	—	139	1	no	—
195	#124	C₂×D₁₃₉	1	1	yes	characteristic
196	C₂²⋊Q₈	—	278	1	no	—
197	C₂²⋊Q₈	—	278	1	no	—
198	C₂²⋊Q₈	—	278	1	no	—
199	C₂²⋊Q₈	—	278	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	—	139	1	no	—
202	C₄×Q₈	—	139	1	no	—
203	C₄×D₄	—	139	1	no	—
204	C₄×D₄	—	139	1	no	—
205	C₄×D₄	—	139	1	no	—
206	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
207	#117	—	278	1	no	—
208	#107	—	139	1	no	—
209	#107	—	139	1	no	—
210	#107	—	139	1	no	—
211	C₄⋊C₈	—	139	1	no	—
212	C₄⋊C₈	—	139	1	no	—
213	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	C₂×D₁₃₉	1	1	yes	characteristic
220	#102	—	139	1	no	—
221	#102	—	139	1	no	—
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	C₂×Q₈	—	278	1	no	—
228	C₂×Q₈	—	278	1	no	—
229	C₂×Q₈	—	278	1	no	—
230	C₂×Q₈	—	556	4	no	—
231	C₂×Q₈	—	139	1	no	—
232	C₂×Q₈	—	139	1	no	—
233	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	278	1	no	—
238	C₂²×C₄	—	278	1	no	—
239	C₂²×C₄	—	278	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	278	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	139	1	no	—
244	C₂×C₈	—	139	1	no	—
245	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
247	C₄⋊C₄	—	278	1	no	—
248	C₄⋊C₄	—	278	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	—	139	1	no	—
255	C₄⋊C₄	—	139	1	no	—
256	C₄⋊C₄	—	139	1	no	—
257	C₄⋊C₄	—	139	1	no	—
258	C₄⋊C₄	—	139	1	no	—
259	C₄⋊C₄	—	139	1	no	—
260	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	278	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	C₄²	—	278	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	139	1	no	—
272	C₄²	—	139	1	no	—
273	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	278	2	no	—
279	Q₈	—	278	1	no	—
280	Q₈	—	278	1	no	—
281	Q₈	—	556	2	no	—
282	Q₈	—	556	2	no	—
283	Q₈	—	556	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#14555784	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	278	1	no	—
292	C₂×C₄	—	278	1	no	—
293	C₂×C₄	—	278	1	no	—
294	C₂×C₄	—	278	1	no	—
295	C₂×C₄	—	556	2	no	—
296	C₂×C₄	—	556	2	no	—
297	C₂×C₄	—	556	2	no	—
298	C₂×C₄	—	139	1	no	—
299	C₂×C₄	—	139	1	no	—
300	C₂×C₄	—	139	1	no	—
301	C₂×C₄	—	139	1	no	—
302	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
303	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	278	1	no	—
308	C₈	—	278	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	278	1	no	—
321	C₄	—	278	1	no	—
322	C₄	—	278	1	no	—
323	C₄	—	556	1	no	—
324	C₄	—	556	1	no	—
325	C₄	—	556	1	no	—
326	C₄	—	556	1	no	—
327	C₄	—	139	2	no	—
328	C₄	C₂×#14555784	1	1	yes	characteristic
329	C₄	#17984889	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#22581609	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
334	C₂	#22581767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#463×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₈×C₁₃₉	C₂²
hypercenter	#463	D₁₃₉
hypocenter	C₁₃₉	#3367
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #22581672	C₂	semidirect
2. D₄×Dic₂₇₈	C₂	semidirect
3. #22581312	C₂	semidirect
4. #22581311	C₂	semidirect
5. #22581302	C₂	semidirect
6. #22581249	C₂	semidirect
7. #22581246	C₂	semidirect
8. #22581231	C₂	semidirect
10. #22581156	C₂	semidirect
11. #22581082	C₂	semidirect
12. #22581054	C₂	semidirect
13. #22581053	C₂	semidirect
167. C₁₃₉	#3367	semidirect

Maximal subgroups

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

subgroup	index
#22581672	2
D₄×Dic₂₇₈	2
#22581312	2
#22581311	2
#22581302	2
#22581249	2
#22581246	2
#22581231	2
#22581166	2
#22581156	2
#22581082	2
#22581054	2
#22581053	2
D₈×Dic₁₃₉	2
#463×C₁₃₉	2
#3367	139