Subgroups — C₁₃₉×#33828 #48028424

Order 35584.

488 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3175×C₁₃₉	C₂	1	2	yes	maximal, complemented
2	#2227×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#2224×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#2220×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	#2216×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#2215×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#2211×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#2210×C₁₃₉	C₂	1	2	yes	maximal, complemented
9	#2151×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#542×C₁₃₉	—	2	2	no	—
11	C₂×Q₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₁₃₉	C₂²	1	2	yes	—
13	C₂×D₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
16	#500×C₁₃₉	C₂²	1	2	yes	—
17	#495×C₁₃₉	C₂²	1	2	yes	—
18	#489×C₁₃₉	—	2	2	no	—
19	#468×C₁₃₉	—	2	2	no	—
20	#467×C₁₃₉	C₂²	1	2	yes	—
21	#466×C₁₃₉	C₂²	1	2	yes	—
22	#440×C₁₃₉	—	2	2	no	—
23	C₄×Q₁₆×C₁₃₉	C₂²	1	2	yes	—
24	C₄×D₈×C₁₃₉	C₂²	1	2	yes	—
25	#405×C₁₃₉	C₂²	1	1	yes	characteristic
26	#366×C₁₃₉	C₂²	1	1	yes	characteristic
27	#366×C₁₃₉	C₂²	1	1	yes	characteristic
28	#366×C₁₃₉	C₂²	1	2	yes	—
29	C₄⋊C₁₆×C₁₃₉	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₁₃₉	C₂²	1	2	yes	—
31	#358×C₁₃₉	C₂²	1	2	yes	—
32	#358×C₁₃₉	C₂²	1	2	yes	—
33	#357×C₁₃₉	C₂²	1	2	yes	—
34	#357×C₁₃₉	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₁₃₉	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₁₃₉	—	2	2	no	—
37	C₂×D₈×C₁₃₉	C₂³	1	2	yes	—
38	C₂×M₁₆×C₁₃₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
41	#126×C₁₃₉	—	4	2	no	—
42	#124×C₁₃₉	—	2	2	no	—
43	#124×C₁₃₉	—	4	2	no	—
44	#123×C₁₃₉	—	4	2	no	—
45	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
46	C₂²⋊Q₈×C₁₃₉	—	4	2	no	—
47	#121×C₁₃₉	—	2	2	no	—
48	C₄×Q₈×C₁₃₉	—	2	2	no	—
49	C₄×D₄×C₁₃₉	—	2	2	no	—
50	#117×C₁₃₉	D₄	1	1	yes	characteristic
51	Q₃₂×C₁₃₉	—	4	1	no	—
52	SD₃₂×C₁₃₉	—	4	2	no	—
53	D₁₆×C₁₃₉	—	4	1	no	—
54	M₃₂×C₁₃₉	—	4	1	no	—
55	C₂×C₁₆×C₁₃₉	—	4	1	no	—
56	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₁₃₉	C₂³	1	2	yes	—
59	#107×C₁₃₉	—	2	2	no	—
60	#107×C₁₃₉	C₂³	1	2	yes	—
61	#107×C₁₃₉	C₂³	1	2	yes	—
62	#103×C₁₃₉	—	2	2	no	—
63	#103×C₁₃₉	—	2	2	no	—
64	#102×C₁₃₉	—	2	2	no	—
65	#102×C₁₃₉	—	2	2	no	—
66	#97×C₁₃₉	D₄	1	2	yes	—
67	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₃₉	—	2	2	no	—
69	C₂×Q₈×C₁₃₉	—	4	2	no	—
70	C₂×D₄×C₁₃₉	—	2	2	no	—
71	C₂×D₄×C₁₃₉	—	4	2	no	—
72	C₂²×C₄×C₁₃₉	—	4	2	no	—
73	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₁₃₉	—	2	2	no	—
75	Q₁₆×C₁₃₉	—	2	2	no	—
76	SD₁₆×C₁₃₉	—	4	4	no	—
77	D₈×C₁₃₉	—	2	2	no	—
78	D₈×C₁₃₉	—	2	2	no	—
79	M₁₆×C₁₃₉	—	4	1	no	—
80	C₂×C₈×C₁₃₉	—	2	1	no	—
81	C₂×C₈×C₁₃₉	—	2	1	no	—
82	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₁₃₉	—	2	2	no	—
86	C₄⋊C₄×C₁₃₉	—	2	2	no	—
87	C₄⋊C₄×C₁₃₉	—	4	2	no	—
88	C₄⋊C₄×C₁₃₉	—	4	2	no	—
89	C₄⋊C₄×C₁₃₉	—	4	2	no	—
90	C₄⋊C₄×C₁₃₉	D₈	1	1	yes	characteristic
91	C₄⋊C₄×C₁₃₉	D₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
97	C₂²⋊C₄×C₁₃₉	D₈	1	2	yes	—
98	C₄²×C₁₃₉	—	4	2	no	—
99	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
100	C₁₆×C₁₃₉	—	2	1	no	—
101	C₁₆×C₁₃₉	—	2	1	no	—
102	C₁₆×C₁₃₉	—	2	2	no	—
103	C₂³×C₁₃₉	—	4	2	no	—
104	C₂³×C₁₃₉	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₁₃₉	—	4	2	no	—
106	Q₈×C₁₃₉	—	4	2	no	—
107	Q₈×C₁₃₉	—	16	2	no	—
108	D₄×C₁₃₉	—	4	2	no	—
109	D₄×C₁₃₉	—	4	2	no	—
110	D₄×C₁₃₉	—	16	2	no	—
111	C₂×C₄×C₁₃₉	—	2	1	no	—
112	C₂×C₄×C₁₃₉	—	2	1	no	—
113	C₂×C₄×C₁₃₉	—	4	2	no	—
114	C₂×C₄×C₁₃₉	—	4	2	no	—
115	C₂×C₄×C₁₃₉	—	4	2	no	—
116	C₂×C₄×C₁₃₉	—	16	2	no	—
117	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₃₉	—	2	1	no	—
122	C₈×C₁₃₉	—	2	2	no	—
123	C₈×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂²×C₁₃₉	—	2	1	no	—
126	C₂²×C₁₃₉	—	2	1	no	—
127	C₂²×C₁₃₉	—	2	1	no	—
128	C₂²×C₁₃₉	—	8	2	no	—
129	C₂²×C₁₃₉	—	8	2	no	—
130	C₂²×C₁₃₉	—	8	2	no	—
131	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃₉	—	2	2	no	—
133	C₄×C₁₃₉	—	2	2	no	—
134	C₄×C₁₃₉	—	2	1	no	—
135	C₄×C₁₃₉	—	8	2	no	—
136	C₄×C₁₃₉	—	8	2	no	—
137	C₄×C₁₃₉	—	8	2	no	—
138	C₄×C₁₃₉	#537	1	1	yes	characteristic
139	C₄×C₁₃₉	#575	1	1	yes	characteristic
140	C₂×C₁₃₉	—	4	1	no	—
141	C₂×C₁₃₉	—	16	2	no	—
142	C₂×C₁₃₉	#3245	1	1	yes	characteristic
143	C₂×C₁₃₉	#3415	1	1	yes	characteristic
144	C₂×C₁₃₉	#3417	1	1	yes	characteristic
145	#33828	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₉	#33828	1	1	yes	characteristic, direct product factor, Sylow
147	#3175	C₂×C₁₃₉	1	2	yes	complemented
148	#2227	C₂×C₁₃₉	1	2	yes	complemented
149	#2224	C₂×C₁₃₉	1	2	yes	complemented
150	#2220	C₂×C₁₃₉	1	2	yes	complemented
151	#2216	C₂×C₁₃₉	1	1	yes	characteristic, complemented
152	#2215	C₂×C₁₃₉	1	1	yes	characteristic
153	#2211	C₂×C₁₃₉	1	2	yes	complemented
154	#2210	C₂×C₁₃₉	1	2	yes	complemented
155	#2151	C₂×C₁₃₉	1	1	yes	characteristic, complemented
156	#542	—	2	2	no	—
157	C₂×Q₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
158	C₂×SD₃₂	C₂²×C₁₃₉	1	2	yes	—
159	C₂×D₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
160	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
161	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
162	#500	C₂²×C₁₃₉	1	2	yes	—
163	#495	C₂²×C₁₃₉	1	2	yes	—
164	#489	—	2	2	no	—
165	#468	—	2	2	no	—
166	#467	C₂²×C₁₃₉	1	2	yes	—
167	#466	C₂²×C₁₃₉	1	2	yes	—
168	#440	—	2	2	no	—
169	C₄×Q₁₆	C₂²×C₁₃₉	1	2	yes	—
170	C₄×D₈	C₂²×C₁₃₉	1	2	yes	—
171	#405	C₂²×C₁₃₉	1	1	yes	characteristic
172	#366	C₂²×C₁₃₉	1	1	yes	characteristic
173	#366	C₂²×C₁₃₉	1	1	yes	characteristic
174	#366	C₂²×C₁₃₉	1	2	yes	—
175	C₄⋊C₁₆	C₂²×C₁₃₉	1	2	yes	—
176	C₄⋊C₁₆	C₂²×C₁₃₉	1	2	yes	—
177	#358	C₂²×C₁₃₉	1	2	yes	—
178	#358	C₂²×C₁₃₉	1	2	yes	—
179	#357	C₂²×C₁₃₉	1	2	yes	—
180	#357	C₂²×C₁₃₉	1	2	yes	—
181	C₂×Q₁₆	C₂³×C₁₃₉	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×D₈	C₂³×C₁₃₉	1	2	yes	—
184	C₂×M₁₆	D₄×C₁₃₉	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	2	no	—
187	#126	—	4	2	no	—
188	#124	—	2	2	no	—
189	#124	—	4	2	no	—
190	#123	—	4	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	4	2	no	—
193	#121	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	D₄×C₁₃₉	1	1	yes	characteristic
197	Q₃₂	—	4	1	no	—
198	SD₃₂	—	4	2	no	—
199	D₁₆	—	4	1	no	—
200	M₃₂	—	4	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
203	C₂×C₁₆	C₂³×C₁₃₉	1	2	yes	—
204	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	C₂³×C₁₃₉	1	2	yes	—
207	#107	C₂³×C₁₃₉	1	2	yes	—
208	#103	—	2	2	no	—
209	#103	—	2	2	no	—
210	#102	—	2	2	no	—
211	#102	—	2	2	no	—
212	#97	D₄×C₁₃₉	1	2	yes	—
213	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	D₈	—	2	2	no	—
224	D₈	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	D₈×C₁₃₉	1	1	yes	characteristic
237	C₄⋊C₄	D₈×C₁₃₉	1	1	yes	characteristic
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	D₈×C₁₃₉	1	2	yes	—
244	C₄²	—	4	2	no	—
245	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₈×C₁₃₉	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	16	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	16	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	16	2	no	—
263	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
264	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
265	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
270	C₈	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	#537×C₁₃₉	1	1	yes	characteristic
285	C₄	#575×C₁₃₉	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	16	2	no	—
288	C₂	#3245×C₁₃₉	1	1	yes	characteristic
289	C₂	#3415×C₁₃₉	1	1	yes	characteristic
290	C₂	#3417×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
prime radical	#2215×C₁₃₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ #117×C₁₃₉ ◁ #405×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3175×C₁₃₉	C₂	semidirect
2. #2227×C₁₃₉	C₂	semidirect
3. #2224×C₁₃₉	C₂	semidirect
4. #2220×C₁₃₉	C₂	semidirect
5. #2216×C₁₃₉	C₂	semidirect
7. #2211×C₁₃₉	C₂	semidirect
8. #2210×C₁₃₉	C₂	semidirect
9. #2151×C₁₃₉	C₂	semidirect
145. #33828	C₁₃₉	direct (N × H)
146. C₁₃₉	#33828	direct (N × H)
147. #3175	C₂×C₁₃₉	semidirect
148. #2227	C₂×C₁₃₉	semidirect
149. #2224	C₂×C₁₃₉	semidirect
150. #2220	C₂×C₁₃₉	semidirect
151. #2216	C₂×C₁₃₉	semidirect
153. #2211	C₂×C₁₃₉	semidirect
154. #2210	C₂×C₁₃₉	semidirect
155. #2151	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3175×C₁₃₉	2
#2227×C₁₃₉	2
#2224×C₁₃₉	2
#2220×C₁₃₉	2
#2216×C₁₃₉	2
#2215×C₁₃₉	2
#2211×C₁₃₉	2
#2210×C₁₃₉	2
#2151×C₁₃₉	2
#33828	139

Subgroups — C139×#33828 #48028424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33828 #48028424