Subgroups — #31815277

Order 17792.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984862	C₂	1	1	yes	maximal, characteristic
2	#22581433	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581053	C₂	1	2	yes	maximal, complemented
4	#22581051	C₂	1	2	yes	maximal, complemented
5	C₄×#14555775	C₂	1	2	yes	maximal, complemented
6	C₄×#14555770	C₂	1	2	yes	maximal, complemented
7	C₂×#17984775	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580954	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580952	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580909	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	1	yes	characteristic
13	#17984862	C₄	1	8	yes	complemented
14	C₂×#14555775	C₂²	1	1	yes	characteristic
15	C₂×#14555773	C₂²	1	1	yes	characteristic
16	C₂×#14555772	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	C₂×#14555769	C₂²	1	1	yes	characteristic
19	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₇₈	C₂²	1	2	yes	—
21	C₄×D₅₅₆	C₂²	1	2	yes	—
22	(C₂²⋊D₁₃₉)⋊C₄	—	2	2	no	—
23	#17984800	—	2	1	no	—
24	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₃₉	—	2	1	no	—
26	#17984775	C₂²	1	2	yes	—
27	#17984775	C₂²	1	2	yes	—
28	#17984766	C₂²	1	2	yes	—
29	#17984765	C₂²	1	2	yes	—
30	#17984763	C₂²	1	2	yes	—
31	#17984762	C₂²	1	2	yes	—
32	#17984753	C₂²	1	2	yes	—
33	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
36	C₄×D₄×C₁₃₉	—	2	2	no	—
37	C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
38	#117×C₁₃₉	—	2	1	no	—
39	#117×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
42	#14555785	—	2	2	no	—
43	#14555785	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
45	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
47	C₂²×Dic₁₃₉	—	2	1	no	—
48	C₂×C₄×D₁₃₉	—	2	1	no	—
49	C₂×C₄×D₁₃₉	—	2	2	no	—
50	#14555775	C₂×C₄	1	4	yes	complemented
51	#14555773	C₂×C₄	1	4	yes	complemented
52	#14555772	C₂×C₄	1	4	yes	complemented
53	#14555770	C₂×C₄	1	4	yes	complemented
54	#14555769	—	2	2	no	—
55	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
57	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₉	—	2	1	no	—
60	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
61	Dic₁₃₉⋊C₄	—	2	2	no	—
62	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
63	C₄×Dic₁₃₉	—	2	2	no	—
64	#41×C₁₃₉	—	2	2	no	—
65	#41×C₁₃₉	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₃₉	—	2	1	no	—
68	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	—	2	1	no	—
70	C₂²×C₄×C₁₃₉	—	2	1	no	—
71	C₂²×C₄×C₁₃₉	—	2	2	no	—
72	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₉	—	2	1	no	—
74	C₄⋊C₄×C₁₃₉	—	2	2	no	—
75	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
80	C₄²×C₁₃₉	—	2	2	no	—
81	C₄²×C₁₃₉	C₂³	1	2	yes	—
82	C₂²×D₁₃₉	—	2	1	no	—
83	Dic₂₇₈	—	2	1	no	—
84	Dic₂₇₈	C₂²×C₄	1	2	yes	—
85	D₅₅₆	—	2	1	no	—
86	D₅₅₆	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₃₉	—	2	4	no	—
88	C₂×Dic₁₃₉	—	2	1	no	—
89	C₂×Dic₁₃₉	—	2	2	no	—
90	C₂×Dic₁₃₉	—	8	1	no	—
91	C₄×D₁₃₉	—	2	4	no	—
92	C₄×D₁₃₉	—	8	2	no	—
93	C₁₃₉⋊C₈	—	2	2	no	—
94	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
95	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂³×C₁₃₉	—	2	1	no	—
97	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₉	—	2	1	no	—
99	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
100	D₄×C₁₃₉	—	2	4	no	—
101	D₄×C₁₃₉	—	2	1	no	—
102	D₄×C₁₃₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₃₉	—	2	1	no	—
104	C₂×C₄×C₁₃₉	—	2	4	no	—
105	C₂×C₄×C₁₃₉	—	2	2	no	—
106	C₂×C₄×C₁₃₉	—	8	2	no	—
107	C₂×C₄×C₁₃₉	—	8	1	no	—
108	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	#41	1	2	yes	—
113	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
114	C₂²×C₁₃₉	—	2	1	no	—
115	C₂²×C₁₃₉	—	2	1	no	—
116	C₂²×C₁₃₉	—	2	2	no	—
117	C₂²×C₁₃₉	—	4	1	no	—
118	C₂²×C₁₃₉	—	4	1	no	—
119	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
120	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₃₉	—	2	2	no	—
122	C₂×D₁₃₉	—	4	1	no	—
123	C₂×D₁₃₉	—	4	1	no	—
124	C₄×C₁₃₉	—	2	2	no	—
125	C₄×C₁₃₉	—	2	2	no	—
126	C₄×C₁₃₉	—	2	2	no	—
127	C₄×C₁₃₉	—	4	2	no	—
128	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
129	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
131	Dic₁₃₉	—	2	2	no	—
132	Dic₁₃₉	—	4	2	no	—
133	C₂×C₁₃₉	—	2	2	no	—
134	C₂×C₁₃₉	—	4	2	no	—
135	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₃₉	#576	1	1	yes	characteristic
137	C₂×C₁₃₉	#578	1	1	yes	characteristic
138	D₁₃₉	—	4	2	no	—
139	C₁₃₉	#2951	1	1	yes	characteristic, complemented, Sylow
140	#2951	—	139	1	no	maximal, Sylow
141	C₂×#135	—	139	1	no	—
142	#520	—	139	1	no	—
143	#520	D₁₃₉	1	1	yes	characteristic, complemented
144	#440	—	139	2	no	—
145	#440	—	139	2	no	—
146	C₄×Q₁₆	—	139	2	no	—
147	C₄×D₈	—	139	2	no	—
148	C₂×#107	—	139	1	no	—
149	#417	—	139	1	no	—
150	#417	—	139	1	no	—
151	#405	—	139	1	no	—
152	C₂×#41	—	139	1	no	—
153	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
154	#135	—	139	8	no	—
155	C₂×Q₁₆	—	139	1	no	—
156	C₂×SD₁₆	—	139	1	no	—
157	C₂×SD₁₆	—	139	1	no	—
158	C₂×D₈	—	139	1	no	—
159	C₂×M₁₆	—	139	1	no	—
160	C₂²×C₈	—	139	1	no	—
161	C₄×Q₈	—	139	2	no	—
162	C₄×Q₈	C₂×D₁₃₉	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	278	2	no	—
165	C₄×D₄	—	139	2	no	—
166	C₄×D₄	C₂×D₁₃₉	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	278	1	no	—
169	#117	C₂×D₁₃₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	278	1	no	—
172	C₂×C₄⋊C₄	—	139	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
174	#107	—	139	2	no	—
175	#107	—	139	2	no	—
176	#103	—	139	2	no	—
177	#103	—	139	2	no	—
178	#102	—	139	2	no	—
179	#102	—	139	2	no	—
180	#97	—	139	2	no	—
181	C₄×C₈	—	139	2	no	—
182	#41	—	2	2	no	—
183	#41	—	278	2	no	—
184	#41	—	139	4	no	—
185	#41	C₄×D₁₃₉	1	4	yes	complemented
186	C₂×Q₈	—	139	1	no	—
187	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	278	1	no	—
190	C₂×D₄	—	139	1	no	—
191	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	278	1	no	—
196	C₂²×C₄	—	278	1	no	—
197	C₂²×C₄	—	278	2	no	—
198	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
199	Q₁₆	—	139	4	no	—
200	SD₁₆	—	139	4	no	—
201	SD₁₆	—	139	4	no	—
202	D₈	—	139	4	no	—
203	M₁₆	—	278	2	no	—
204	C₂×C₈	—	139	2	no	—
205	C₂×C₈	—	139	4	no	—
206	C₂×C₈	—	139	1	no	—
207	C₂×C₈	—	139	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	278	1	no	—
211	C₄⋊C₄	—	278	2	no	—
212	C₄⋊C₄	—	139	2	no	—
213	C₄⋊C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	278	2	no	—
218	C₂²⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	278	2	no	—
221	C₄²	C₂²×D₁₃₉	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	278	1	no	—
224	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	278	1	no	—
227	Q₈	—	139	2	no	—
228	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	278	4	no	—
232	D₄	—	278	1	no	—
233	D₄	—	139	2	no	—
234	D₄	C₂×C₄×D₁₃₉	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	278	1	no	—
241	C₂×C₄	—	278	4	no	—
242	C₂×C₄	—	278	2	no	—
243	C₂×C₄	—	1112	2	no	—
244	C₂×C₄	—	1112	1	no	—
245	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
249	C₂×C₄	#14555785	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
251	C₈	—	278	2	no	—
252	C₈	—	139	2	no	—
253	C₈	—	139	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	278	2	no	—
260	C₂²	—	556	1	no	—
261	C₂²	—	556	1	no	—
262	C₂²	C₄×C₂²⋊D₁₃₉	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	278	2	no	—
269	C₄	—	556	2	no	—
270	C₄	C₄×C₂²⋊D₁₃₉	1	2	yes	—
271	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
272	C₄	C₂×#14555785	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	556	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₁₃₉	1	1	yes	characteristic
277	C₂	#22581756	1	1	yes	characteristic
278	C₂	#22581768	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#520×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984862	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#520	D₁₃₉
hypocenter	C₁₃₉	#2951
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #22581433	C₂	semidirect
3. #22581053	C₂	semidirect
4. #22581051	C₂	semidirect
5. C₄×#14555775	C₂	semidirect
6. C₄×#14555770	C₂	semidirect
7. C₂×#17984775	C₂	semidirect
8. #22580954	C₂	semidirect
9. #22580952	C₂	semidirect
10. #22580909	C₂	semidirect
11. #520×C₁₃₉	C₂	semidirect
13. #17984862	C₄	semidirect
43. #14555785	C₂×C₄	semidirect
50. #14555775	C₂×C₄	semidirect
51. #14555773	C₂×C₄	semidirect
52. #14555772	C₂×C₄	semidirect
53. #14555770	C₂×C₄	semidirect
55. C₂×C₁₃₉⋊C₈	C₂×C₄	semidirect
65. #41×C₁₃₉	C₂×C₄	semidirect
139. C₁₃₉	#2951	semidirect
143. #520	D₁₃₉	semidirect
185. #41	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984862	2
#22581433	2
#22581053	2
#22581051	2
C₄×#14555775	2
C₄×#14555770	2
C₂×#17984775	2
#22580954	2
#22580952	2
#22580909	2
#520×C₁₃₉	2
#2951	139