Subgroups — #31816239

Order 17792.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581580	C₂	1	1	yes	maximal, characteristic
2	#22581429	C₂	1	1	yes	maximal, characteristic
3	#22581263	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581261	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581260	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581258	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581245	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580992	C₂	1	1	yes	maximal, characteristic
11	#22580988	C₂	1	1	yes	maximal, characteristic
12	#22580978	C₂	1	1	yes	maximal, characteristic
13	#22580960	C₂	1	1	yes	maximal, characteristic
14	#103×D₁₃₉	C₂	1	1	yes	maximal, characteristic
15	#475×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
19	#17984841	C₂²	1	1	yes	characteristic
20	#17984840	—	2	1	no	—
21	#17984834	C₂²	1	1	yes	characteristic
22	#17984833	C₂²	1	1	yes	characteristic
23	#17984832	—	2	1	no	—
24	#17984820	—	2	1	no	—
25	#17984818	—	2	1	no	—
26	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₇₈	—	2	1	no	—
28	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
29	#17984807	—	2	1	no	—
30	#17984807	C₂²	1	1	yes	characteristic
31	#17984800	—	2	1	no	—
32	#17984800	C₂²	1	1	yes	characteristic
33	#17984800	C₂²	1	1	yes	characteristic
34	#17984799	—	2	1	no	—
35	#17984799	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984774	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	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	#17984766	C₂²	1	1	yes	characteristic
49	#17984766	C₂²	1	1	yes	characteristic
50	#17984765	C₂²	1	1	yes	characteristic
51	#17984765	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#106×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	—	2	1	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	Q₈×D₁₃₉	—	2	2	no	—
63	Q₈×D₁₃₉	Q₈	1	4	yes	complemented
64	C₂×C₄×D₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
67	#14555768	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₁₃₉	—	2	2	no	—
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	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	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₉	—	2	1	no	—
96	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	4	no	—
109	Dic₂₇₈	—	8	1	no	—
110	Dic₂₇₈	C₂×Q₈	1	2	yes	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₃₉	—	2	4	no	—
119	C₄×D₁₃₉	—	8	1	no	—
120	C₄×D₁₃₉	C₂×Q₈	1	2	yes	—
121	C₄×D₁₃₉	#41	1	2	yes	—
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₁₃₉⋊C₈	—	2	1	no	—
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	—	8	1	no	—
126	Q₈×C₁₃₉	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
131	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
134	C₈×C₁₃₉	—	2	1	no	—
135	C₈×C₁₃₉	—	2	1	no	—
136	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₃₉	—	2	1	no	—
138	C₂×D₁₃₉	—	2	1	no	—
139	C₂×D₁₃₉	C₂²⋊Q₈	1	2	yes	—
140	C₄×C₁₃₉	—	2	2	no	—
141	C₄×C₁₃₉	—	2	1	no	—
142	C₄×C₁₃₉	—	2	1	no	—
143	C₄×C₁₃₉	—	4	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	4	1	no	—
146	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
148	Dic₁₃₉	—	2	2	no	—
149	Dic₁₃₉	—	2	1	no	—
150	Dic₁₃₉	—	2	1	no	—
151	Dic₁₃₉	—	4	1	no	—
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	—	4	1	no	—
154	Dic₁₃₉	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₁₃₉	#577	1	1	yes	characteristic
157	C₂×C₁₃₉	#578	1	1	yes	characteristic
158	D₁₃₉	—	2	2	no	—
159	C₁₃₉	#3082	1	1	yes	characteristic, complemented, Sylow
160	#3082	—	139	1	no	maximal, Sylow
161	#544	—	139	1	no	—
162	#519	—	139	1	no	—
163	#479	—	139	1	no	—
164	#479	—	139	1	no	—
165	#479	—	139	1	no	—
166	#479	—	139	1	no	—
167	#477	—	139	1	no	—
168	#477	—	139	1	no	—
169	#475	—	139	1	no	—
170	#475	D₁₃₉	1	1	yes	characteristic, complemented
171	#428	—	139	1	no	—
172	#427	—	139	1	no	—
173	#424	—	139	1	no	—
174	#419	—	139	1	no	—
175	C₂×#103	—	139	1	no	—
176	C₂²×Q₈	—	139	1	no	—
177	C₂×M₁₆	—	139	1	no	—
178	C₂²×C₈	—	139	1	no	—
179	C₄⋊Q₈	—	278	1	no	—
180	C₄⋊Q₈	—	139	1	no	—
181	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
182	#125	—	278	1	no	—
183	#125	—	139	1	no	—
184	#125	—	139	1	no	—
185	C₂²⋊Q₈	—	278	1	no	—
186	C₂²⋊Q₈	—	278	1	no	—
187	C₄×Q₈	—	278	1	no	—
188	C₄×Q₈	—	278	1	no	—
189	C₄×Q₈	—	139	1	no	—
190	C₄×Q₈	—	139	1	no	—
191	C₄×Q₈	—	139	1	no	—
192	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
193	#117	—	278	1	no	—
194	#117	—	278	1	no	—
195	#117	—	139	1	no	—
196	#117	—	139	1	no	—
197	#117	—	139	1	no	—
198	C₂×C₄⋊C₄	—	139	1	no	—
199	#107	—	139	1	no	—
200	#107	—	139	1	no	—
201	#107	—	139	1	no	—
202	#107	—	139	1	no	—
203	#106	—	139	1	no	—
204	#106	—	139	1	no	—
205	#106	C₂×D₁₃₉	1	1	yes	characteristic
206	#106	C₂×D₁₃₉	1	1	yes	characteristic
207	C₄⋊C₈	—	139	1	no	—
208	C₄⋊C₈	—	139	1	no	—
209	C₄⋊C₈	—	139	1	no	—
210	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
211	#103	—	139	1	no	—
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	C₂×D₁₃₉	1	1	yes	characteristic
218	#103	C₂×D₁₃₉	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	278	1	no	—
221	C₂×Q₈	—	278	1	no	—
222	C₂×Q₈	—	278	2	no	—
223	C₂×Q₈	—	139	4	no	—
224	C₂×Q₈	—	139	1	no	—
225	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
226	C₂²×C₄	—	278	1	no	—
227	C₂²×C₄	—	278	1	no	—
228	C₂²×C₄	—	139	1	no	—
229	M₁₆	—	278	2	no	—
230	C₂×C₈	—	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	—	278	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	—	139	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	—	139	1	no	—
257	C₂²⋊C₄	—	139	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	278	1	no	—
260	C₄²	—	278	1	no	—
261	C₄²	—	278	1	no	—
262	C₄²	—	278	1	no	—
263	C₄²	—	278	1	no	—
264	C₄²	—	139	1	no	—
265	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂³	—	139	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	8	1	no	—
269	Q₈	—	278	4	no	—
270	Q₈	—	278	1	no	—
271	Q₈	—	1112	1	no	—
272	Q₈	—	139	2	no	—
273	Q₈	Q₈×D₁₃₉	1	2	yes	—
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	4	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	—	1112	1	no	—
284	C₂×C₄	—	139	1	no	—
285	C₂×C₄	—	139	1	no	—
286	C₂×C₄	—	139	1	no	—
287	C₂×C₄	—	139	2	no	—
288	C₂×C₄	—	139	2	no	—
289	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	278	1	no	—
295	C₈	—	278	1	no	—
296	C₂²	—	278	1	no	—
297	C₂²	—	278	1	no	—
298	C₂²	—	139	2	no	—
299	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	278	1	no	—
307	C₄	—	278	2	no	—
308	C₄	—	278	1	no	—
309	C₄	—	556	1	no	—
310	C₄	—	556	1	no	—
311	C₄	—	556	1	no	—
312	C₄	—	139	2	no	—
313	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
314	C₄	#41×D₁₃₉	1	1	yes	characteristic
315	C₂	—	278	2	no	—
316	C₂	C₂²⋊Q₈×D₁₃₉	1	1	yes	characteristic
317	C₂	#22581759	1	1	yes	characteristic
318	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	#475×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#475	D₁₃₉
hypocenter	C₁₃₉	#3082
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
3. #22581263	C₂	semidirect
4. #22581261	C₂	semidirect
5. #22581260	C₂	semidirect
6. #22581258	C₂	semidirect
7. #22581246	C₂	semidirect
8. #22581245	C₂	semidirect
9. #22581237	C₂	semidirect
15. #475×C₁₃₉	C₂	semidirect
63. Q₈×D₁₃₉	Q₈	semidirect
159. C₁₃₉	#3082	semidirect
170. #475	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581580	2
#22581429	2
#22581263	2
#22581261	2
#22581260	2
#22581258	2
#22581246	2
#22581245	2
#22581237	2
#22580992	2
#22580988	2
#22580978	2
#22580960	2
#103×D₁₃₉	2
#475×C₁₃₉	2
#3082	139