Subgroups — #31818976

Order 17792.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581644	C₂	1	1	yes	maximal, characteristic
2	#22581641	C₂	1	1	yes	maximal, characteristic
3	#22581352	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581297	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581292	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581287	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581261	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581210	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581210	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14555775	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581030	C₂	1	1	yes	maximal, characteristic
13	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580992	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	#17984837	—	2	1	no	—
20	#17984836	—	2	1	no	—
21	#17984834	C₂²	1	1	yes	characteristic
22	#17984833	—	2	1	no	—
23	#17984832	—	2	1	no	—
24	#17984827	—	2	1	no	—
25	#17984827	—	2	1	no	—
26	#17984822	C₂²	1	1	yes	characteristic
27	#17984822	C₂²	1	1	yes	characteristic
28	#17984818	C₂²	1	1	yes	characteristic
29	#17984818	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
31	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
32	#17984800	—	2	1	no	—
33	#17984800	C₂²	1	1	yes	characteristic
34	#17984800	C₂²	1	1	yes	characteristic
35	#17984799	—	2	1	no	—
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984775	C₂²	1	1	yes	characteristic
42	#17984774	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	#17984773	C₂²	1	1	yes	characteristic
45	#17984767	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984766	C₂²	1	1	yes	characteristic
48	#17984765	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#103×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₅₅₆	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₃₉	—	2	1	no	—
62	C₂×C₄×D₁₃₉	—	2	1	no	—
63	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
65	#14555775	—	2	2	no	—
66	#14555768	—	2	2	no	—
67	#14555768	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
90	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₃₉	—	2	1	no	—
98	C₄⋊C₄×C₁₃₉	—	2	1	no	—
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
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	C₄²×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	D₅₅₆	—	4	1	no	—
110	C₂×Dic₁₃₉	—	2	1	no	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₄×D₁₃₉	—	2	1	no	—
117	C₄×D₁₃₉	—	2	1	no	—
118	C₄×D₁₃₉	—	2	1	no	—
119	C₄×D₁₃₉	—	2	1	no	—
120	C₄×D₁₃₉	—	4	2	no	—
121	C₄×D₁₃₉	—	4	2	no	—
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₁₃₉⋊C₈	#41	1	2	yes	—
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	—	2	1	no	—
126	C₂×C₄×C₁₃₉	—	2	1	no	—
127	C₂×C₄×C₁₃₉	—	2	1	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₃₉	—	2	1	no	—
134	C₈×C₁₃₉	—	2	1	no	—
135	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₃₉	—	2	1	no	—
137	C₂×D₁₃₉	—	2	1	no	—
138	C₂×D₁₃₉	—	2	1	no	—
139	C₂×D₁₃₉	—	2	1	no	—
140	C₂×D₁₃₉	—	2	1	no	—
141	C₂×D₁₃₉	—	2	1	no	—
142	C₄×C₁₃₉	—	2	1	no	—
143	C₄×C₁₃₉	—	2	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	4	1	no	—
146	C₄×C₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	—	4	1	no	—
148	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₃₉	—	2	1	no	—
151	Dic₁₃₉	—	2	1	no	—
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	—	4	1	no	—
154	Dic₁₃₉	—	4	1	no	—
155	Dic₁₃₉	—	4	1	no	—
156	C₂×C₁₃₉	#547	1	1	yes	characteristic
157	C₂×C₁₃₉	#575	1	1	yes	characteristic
158	C₂×C₁₃₉	#578	1	1	yes	characteristic
159	D₁₃₉	—	4	1	no	—
160	D₁₃₉	—	4	1	no	—
161	C₁₃₉	#3352	1	1	yes	characteristic, complemented, Sylow
162	#3352	—	139	1	no	maximal, Sylow
163	#552	—	139	1	no	—
164	#552	—	139	1	no	—
165	#499	—	139	1	no	—
166	#485	—	139	1	no	—
167	#485	—	139	1	no	—
168	#484	—	139	1	no	—
169	#484	—	139	1	no	—
170	#479	—	139	1	no	—
171	#479	D₁₃₉	1	1	yes	characteristic, complemented
172	#470	—	139	1	no	—
173	#470	—	139	1	no	—
174	C₄×Q₁₆	—	139	1	no	—
175	#436	—	139	1	no	—
176	#428	—	139	1	no	—
177	#428	—	139	1	no	—
178	C₂×Q₁₆	—	139	1	no	—
179	C₂×M₁₆	—	139	1	no	—
180	C₂×M₁₆	—	139	1	no	—
181	#126	—	278	1	no	—
182	#126	—	278	1	no	—
183	#125	—	278	1	no	—
184	#125	—	278	1	no	—
185	#125	—	139	1	no	—
186	#125	C₂×D₁₃₉	1	1	yes	characteristic
187	#123	—	278	1	no	—
188	#123	—	278	1	no	—
189	C₂²⋊Q₈	—	139	1	no	—
190	C₂²⋊Q₈	—	139	1	no	—
191	C₂²⋊Q₈	—	139	1	no	—
192	C₂²⋊Q₈	—	139	1	no	—
193	C₄×Q₈	—	139	1	no	—
194	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
195	C₄×D₄	—	139	1	no	—
196	#117	—	278	1	no	—
197	#117	—	278	1	no	—
198	#117	—	139	1	no	—
199	#117	—	139	1	no	—
200	C₂×C₄⋊C₄	—	139	1	no	—
201	C₂×C₄⋊C₄	—	139	1	no	—
202	#107	—	139	1	no	—
203	#107	—	139	1	no	—
204	#107	—	139	1	no	—
205	#107	—	139	1	no	—
206	#107	C₂×D₁₃₉	1	1	yes	characteristic
207	#106	—	139	1	no	—
208	#106	—	139	1	no	—
209	#106	—	139	1	no	—
210	#106	C₂×D₁₃₉	1	1	yes	characteristic
211	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	C₂×D₁₃₉	1	1	yes	characteristic
217	#103	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	139	1	no	—
219	C₂²⋊C₈	—	139	1	no	—
220	C₄×C₈	—	139	1	no	—
221	C₂×Q₈	—	139	1	no	—
222	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
223	C₂×D₄	—	139	1	no	—
224	C₂²×C₄	—	278	1	no	—
225	C₂²×C₄	—	278	1	no	—
226	C₂²×C₄	—	139	1	no	—
227	C₂²×C₄	—	139	1	no	—
228	Q₁₆	—	278	2	no	—
229	M₁₆	—	278	2	no	—
230	M₁₆	—	278	2	no	—
231	C₂×C₈	—	139	1	no	—
232	C₂×C₈	—	139	1	no	—
233	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	278	1	no	—
239	C₄⋊C₄	—	278	1	no	—
240	C₄⋊C₄	—	278	1	no	—
241	C₄⋊C₄	—	278	1	no	—
242	C₄⋊C₄	—	278	1	no	—
243	C₄⋊C₄	—	278	1	no	—
244	C₄⋊C₄	—	278	1	no	—
245	C₄⋊C₄	—	278	1	no	—
246	C₄⋊C₄	—	278	1	no	—
247	C₄⋊C₄	—	139	1	no	—
248	C₄⋊C₄	—	139	1	no	—
249	C₄⋊C₄	—	139	1	no	—
250	C₄⋊C₄	—	139	1	no	—
251	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	278	1	no	—
255	C₂²⋊C₄	—	278	1	no	—
256	C₂²⋊C₄	—	278	1	no	—
257	C₂²⋊C₄	—	278	1	no	—
258	C₂²⋊C₄	—	278	1	no	—
259	C₂²⋊C₄	—	278	1	no	—
260	C₂²⋊C₄	—	139	1	no	—
261	C₂²⋊C₄	—	139	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	278	1	no	—
264	C₄²	—	278	1	no	—
265	C₄²	—	278	1	no	—
266	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₂³	—	139	1	no	—
268	C₂³	—	139	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	278	1	no	—
272	Q₈	—	278	1	no	—
273	D₄	—	556	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	278	1	no	—
279	C₂×C₄	—	278	1	no	—
280	C₂×C₄	—	278	1	no	—
281	C₂×C₄	—	278	1	no	—
282	C₂×C₄	—	278	1	no	—
283	C₂×C₄	—	278	1	no	—
284	C₂×C₄	—	278	1	no	—
285	C₂×C₄	—	278	1	no	—
286	C₂×C₄	—	556	2	no	—
287	C₂×C₄	—	556	2	no	—
288	C₂×C₄	—	139	1	no	—
289	C₂×C₄	—	139	1	no	—
290	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	278	1	no	—
296	C₈	—	139	2	no	—
297	C₂²	—	278	1	no	—
298	C₂²	—	278	1	no	—
299	C₂²	—	278	1	no	—
300	C₂²	—	278	1	no	—
301	C₂²	—	278	1	no	—
302	C₂²	—	278	1	no	—
303	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	278	1	no	—
311	C₄	—	278	1	no	—
312	C₄	—	556	1	no	—
313	C₄	—	556	1	no	—
314	C₄	—	556	1	no	—
315	C₄	—	556	1	no	—
316	C₄	#41×D₁₃₉	1	1	yes	characteristic
317	C₄	#17984890	1	1	yes	characteristic
318	C₂	—	556	1	no	—
319	C₂	—	556	1	no	—
320	C₂	#22581605	1	1	yes	characteristic
321	C₂	#22581750	1	1	yes	characteristic
322	C₂	#22581769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#479×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₅₅₆	D₄
hypercenter	#479	D₁₃₉
hypocenter	C₁₃₉	#3352
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #22581352	C₂	semidirect
4. #22581297	C₂	semidirect
5. #22581292	C₂	semidirect
6. #22581290	C₂	semidirect
7. #22581287	C₂	semidirect
8. #22581261	C₂	semidirect
9. #22581210	C₂	semidirect
10. #22581210	C₂	semidirect
11. C₄×#14555775	C₂	semidirect
13. #22580994	C₂	semidirect
14. #22580992	C₂	semidirect
15. #479×C₁₃₉	C₂	semidirect
161. C₁₃₉	#3352	semidirect
171. #479	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581644	2
#22581641	2
#22581352	2
#22581297	2
#22581292	2
#22581290	2
#22581287	2
#22581261	2
#22581210	2
C₄×#14555775	2
#22581030	2
#22580994	2
#22580992	2
#479×C₁₃₉	2
#3352	139