Subgroups — #31818986

Order 17792.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581671	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581603	C₂	1	1	yes	maximal, characteristic
3	#22581357	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581289	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581275	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581198	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581197	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581028	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580996	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580992	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	—	2	1	no	—
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	C₂×#14555769	C₂²	1	1	yes	characteristic
19	C₂×#14555768	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984836	—	2	1	no	—
23	#17984835	—	2	1	no	—
24	#17984828	—	2	1	no	—
25	#17984824	—	2	1	no	—
26	#17984823	—	2	1	no	—
27	#17984822	C₂²	1	1	yes	characteristic
28	#17984821	C₂²	1	1	yes	characteristic
29	#17984819	—	2	1	no	—
30	#17984813	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₇₈	—	2	1	no	—
32	#17984807	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
35	#17984800	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984776	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984775	C₂²	1	1	yes	characteristic
42	#17984775	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	#17984773	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984767	C₂²	1	1	yes	characteristic
48	#17984765	C₂²	1	1	yes	characteristic
49	#17984763	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	#121×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555784	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
69	#14555773	—	2	2	no	—
70	#14555769	—	2	2	no	—
71	#14555768	—	2	2	no	—
72	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	—	2	1	no	—
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	Dic₁₃₉⋊C₄	—	2	1	no	—
84	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	—	2	1	no	—
96	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	—	2	1	no	—
106	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	4	2	no	—
110	Dic₂₇₈	—	4	2	no	—
111	C₂²⋊D₁₃₉	—	2	2	no	—
112	C₂²⋊D₁₃₉	—	4	2	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	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	C₄×D₁₃₉	—	2	1	no	—
123	C₄×D₁₃₉	—	2	1	no	—
124	C₄×D₁₃₉	—	4	2	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₁₃₉⋊C₈	#41	1	2	yes	complemented
127	C₂³×C₁₃₉	—	2	1	no	—
128	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
129	D₄×C₁₃₉	—	2	1	no	—
130	D₄×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	4	2	no	—
132	C₂×C₄×C₁₃₉	—	2	1	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	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	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	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	C₂×D₁₃₉	—	2	1	no	—
149	C₂×D₁₃₉	—	2	1	no	—
150	C₂×D₁₃₉	—	2	1	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₁₃₉	—	2	1	no	—
158	Dic₁₃₉	—	2	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₁₃₉	—	4	1	no	—
165	C₂×C₁₃₉	—	8	1	no	—
166	C₂×C₁₃₉	#547	1	1	yes	characteristic
167	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₁₃₉	#578	1	1	yes	characteristic
169	D₁₃₉	—	4	1	no	—
170	C₁₃₉	#3353	1	1	yes	characteristic, complemented, Sylow
171	#3353	—	139	1	no	maximal, Sylow
172	#556	—	139	1	no	—
173	#547	—	139	1	no	—
174	#500	—	139	1	no	—
175	#484	—	139	1	no	—
176	#484	—	139	1	no	—
177	#482	—	139	1	no	—
178	#482	D₁₃₉	1	1	yes	characteristic, complemented
179	#475	—	139	1	no	—
180	#474	—	139	1	no	—
181	#468	—	139	1	no	—
182	#468	—	139	1	no	—
183	C₄×SD₁₆	—	139	1	no	—
184	#436	—	139	1	no	—
185	#428	—	139	1	no	—
186	#428	—	139	1	no	—
187	C₂×#41	—	278	1	no	—
188	C₂×SD₁₆	—	139	1	no	—
189	C₂×M₁₆	—	139	1	no	—
190	C₂×M₁₆	—	139	1	no	—
191	C₄⋊Q₈	—	139	1	no	—
192	C₄⋊Q₈	—	139	1	no	—
193	#126	—	278	1	no	—
194	#126	—	278	1	no	—
195	#124	—	278	1	no	—
196	#123	—	278	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	#121	—	139	1	no	—
202	#121	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄×Q₈	—	278	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×D₄	—	139	1	no	—
206	C₄×D₄	—	139	1	no	—
207	#117	—	139	1	no	—
208	#117	C₂×D₁₃₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	139	1	no	—
210	C₂×C₄⋊C₄	—	139	1	no	—
211	#107	—	139	1	no	—
212	#107	—	139	1	no	—
213	#107	—	139	1	no	—
214	#107	—	139	1	no	—
215	#107	—	139	1	no	—
216	#107	C₂×D₁₃₉	1	1	yes	characteristic
217	#106	—	139	1	no	—
218	#106	—	139	1	no	—
219	#106	C₂×D₁₃₉	1	1	yes	characteristic
220	C₄⋊C₈	—	139	1	no	—
221	#103	—	139	1	no	—
222	#103	—	139	1	no	—
223	#103	—	139	1	no	—
224	#102	—	139	1	no	—
225	#102	C₂×D₁₃₉	1	1	yes	characteristic
226	#102	C₂×D₁₃₉	1	1	yes	characteristic
227	C₂²⋊C₈	—	139	1	no	—
228	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
229	C₄×C₈	—	139	1	no	—
230	#41	—	556	4	no	—
231	C₂×Q₈	—	278	1	no	—
232	C₂×Q₈	—	278	1	no	—
233	C₂×Q₈	—	139	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	139	1	no	—
237	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂²×C₄	—	278	1	no	—
239	C₂²×C₄	—	278	1	no	—
240	C₂²×C₄	—	278	1	no	—
241	C₂²×C₄	—	139	1	no	—
242	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	SD₁₆	—	278	2	no	—
244	M₁₆	—	278	2	no	—
245	M₁₆	—	278	2	no	—
246	C₂×C₈	—	139	1	no	—
247	C₂×C₈	—	139	1	no	—
248	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
258	C₄⋊C₄	—	139	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	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	278	1	no	—
271	C₂²⋊C₄	—	139	1	no	—
272	C₂²⋊C₄	—	139	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	278	1	no	—
275	C₄²	—	278	1	no	—
276	C₄²	—	139	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	139	1	no	—
279	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
280	Q₈	—	278	1	no	—
281	Q₈	—	278	1	no	—
282	Q₈	—	556	2	no	—
283	Q₈	—	556	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	278	2	no	—
288	D₄	—	556	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	1	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	—	278	1	no	—
300	C₂×C₄	—	556	2	no	—
301	C₂×C₄	—	556	2	no	—
302	C₂×C₄	—	556	2	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	—	139	1	no	—
306	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	278	1	no	—
311	C₈	—	139	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	278	1	no	—
319	C₂²	—	278	1	no	—
320	C₂²	—	278	1	no	—
321	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	278	1	no	—
327	C₄	—	278	1	no	—
328	C₄	—	556	1	no	—
329	C₄	—	556	1	no	—
330	C₄	—	556	1	no	—
331	C₄	—	556	1	no	—
332	C₄	—	139	2	no	—
333	C₄	#41×D₁₃₉	1	1	yes	characteristic
334	C₄	#17984889	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	556	1	no	—
338	C₂	#22581601	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
340	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	#482×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581603	C₂
hypercenter	#482	D₁₃₉
hypocenter	C₁₃₉	#3353
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #22581671	C₂	semidirect
3. #22581357	C₂	semidirect
4. #22581290	C₂	semidirect
5. #22581289	C₂	semidirect
6. #22581275	C₂	semidirect
7. #22581237	C₂	semidirect
8. #22581231	C₂	semidirect
9. #22581198	C₂	semidirect
10. #22581197	C₂	semidirect
11. #22581044	C₂	semidirect
12. #22581028	C₂	semidirect
13. #22580996	C₂	semidirect
14. #22580992	C₂	semidirect
15. #482×C₁₃₉	C₂	semidirect
126. C₁₃₉⋊C₈	#41	semidirect
170. C₁₃₉	#3353	semidirect
178. #482	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581671	2
#22581603	2
#22581357	2
#22581290	2
#22581289	2
#22581275	2
#22581237	2
#22581231	2
#22581198	2
#22581197	2
#22581044	2
#22581028	2
#22580996	2
#22580992	2
#482×C₁₃₉	2
#3353	139