Subgroups — C₁₃₉×#17981 #48012577

Order 35584.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2931×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#1586×C₁₃₉	C₂	1	8	yes	maximal, complemented
4	#1566×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1492×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#131×C₁₃₉	—	2	2	no	—
8	#532×C₁₃₉	C₂²	1	2	yes	—
9	#500×C₁₃₉	C₄	1	1	yes	characteristic
10	#493×C₁₃₉	C₄	1	1	yes	characteristic
11	#486×C₁₃₉	C₄	1	2	yes	—
12	#466×C₁₃₉	C₄	1	4	yes	—
13	C₄×D₈×C₁₃₉	C₂²	1	4	yes	—
14	#436×C₁₃₉	—	2	2	no	—
15	#436×C₁₃₉	C₂²	1	4	yes	—
16	C₈×D₄×C₁₃₉	—	2	2	no	—
17	C₈×D₄×C₁₃₉	C₂²	1	4	yes	—
18	#423×C₁₃₉	C₂²	1	2	yes	—
19	#408×C₁₃₉	—	2	2	no	—
20	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	#334×C₁₃₉	C₂²	1	4	yes	—
24	D₄⋊C₈×C₁₃₉	C₂²	1	8	yes	—
25	#322×C₁₃₉	C₂²	1	4	yes	—
26	C₂×#41×C₁₃₉	—	2	2	no	—
27	C₂×D₈×C₁₃₉	C₂×C₄	1	4	yes	—
28	#131×C₁₃₉	—	4	8	no	—
29	C₂×M₁₆×C₁₃₉	—	2	2	no	—
30	C₂×M₁₆×C₁₃₉	—	2	4	no	—
31	C₂²×C₈×C₁₃₉	—	2	1	no	—
32	C₂²×C₈×C₁₃₉	—	2	4	no	—
33	C₂²×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
35	C₄⋊Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
36	#127×C₁₃₉	C₂×C₄	1	2	yes	—
37	#124×C₁₃₉	—	2	2	no	—
38	#121×C₁₃₉	—	2	2	no	—
39	#121×C₁₃₉	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₁₃₉	—	2	2	no	—
41	C₄×D₄×C₁₃₉	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
43	#107×C₁₃₉	C₂×C₄	1	4	yes	—
44	C₄⋊C₈×C₁₃₉	—	2	2	no	—
45	C₄⋊C₈×C₁₃₉	C₂³	1	4	yes	—
46	#102×C₁₃₉	C₂×C₄	1	8	yes	—
47	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
48	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
49	C₄×C₈×C₁₃₉	—	2	1	no	—
50	C₄×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
57	#41×C₁₃₉	—	4	4	no	—
58	#41×C₁₃₉	—	4	4	no	—
59	C₂×Q₈×C₁₃₉	—	2	2	no	—
60	C₂×D₄×C₁₃₉	—	2	4	no	—
61	C₂×D₄×C₁₃₉	—	2	2	no	—
62	C₂×D₄×C₁₃₉	—	2	2	no	—
63	C₂×D₄×C₁₃₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃₉	—	2	4	no	—
65	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
68	D₈×C₁₃₉	—	4	4	no	—
69	M₁₆×C₁₃₉	—	4	8	no	—
70	M₁₆×C₁₃₉	—	4	4	no	—
71	C₂×C₈×C₁₃₉	—	2	4	no	—
72	C₂×C₈×C₁₃₉	—	2	4	no	—
73	C₂×C₈×C₁₃₉	—	2	2	no	—
74	C₂×C₈×C₁₃₉	—	2	4	no	—
75	C₂×C₈×C₁₃₉	—	4	2	no	—
76	C₂×C₈×C₁₃₉	—	4	8	no	—
77	C₂×C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₁₃₉	—	2	2	no	—
82	C₄⋊C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
84	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
85	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃₉	#41	1	1	yes	characteristic
88	C₄²×C₁₃₉	C₂⁴	1	1	yes	characteristic
89	C₂³×C₁₃₉	—	2	4	no	—
90	C₂³×C₁₃₉	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₁₃₉	—	4	4	no	—
92	D₄×C₁₃₉	—	2	8	no	—
93	D₄×C₁₃₉	—	4	8	no	—
94	D₄×C₁₃₉	—	4	4	no	—
95	D₄×C₁₃₉	—	4	4	no	—
96	C₂×C₄×C₁₃₉	—	2	2	no	—
97	C₂×C₄×C₁₃₉	—	2	2	no	—
98	C₂×C₄×C₁₃₉	—	2	4	no	—
99	C₂×C₄×C₁₃₉	—	4	4	no	—
100	C₂×C₄×C₁₃₉	—	4	4	no	—
101	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
108	C₈×C₁₃₉	—	2	4	no	—
109	C₈×C₁₃₉	—	2	2	no	—
110	C₈×C₁₃₉	—	2	2	no	—
111	C₈×C₁₃₉	—	2	2	no	—
112	C₈×C₁₃₉	—	2	2	no	—
113	C₈×C₁₃₉	—	4	4	no	—
114	C₂²×C₁₃₉	—	2	1	no	—
115	C₂²×C₁₃₉	—	2	1	no	—
116	C₂²×C₁₃₉	—	4	4	no	—
117	C₂²×C₁₃₉	—	4	4	no	—
118	C₂²×C₁₃₉	—	4	4	no	—
119	C₂²×C₁₃₉	#442	1	2	yes	—
120	C₂²×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₃₉	—	2	1	no	—
122	C₄×C₁₃₉	—	2	1	no	—
123	C₄×C₁₃₉	—	4	4	no	—
124	C₄×C₁₃₉	#442	1	2	yes	—
125	C₄×C₁₃₉	#443	1	2	yes	—
126	C₄×C₁₃₉	#568	1	2	yes	—
127	C₄×C₁₃₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₁₃₉	—	2	2	no	—
129	C₂×C₁₃₉	—	8	4	no	—
130	C₂×C₁₃₉	#2932	1	1	yes	characteristic
131	C₂×C₁₃₉	C₂×#442	1	1	yes	characteristic
132	C₂×C₁₃₉	C₂×#443	1	1	yes	characteristic
133	#17981	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₉	#17981	1	1	yes	characteristic, direct product factor, Sylow
135	#3169	C₂×C₁₃₉	1	1	yes	characteristic
136	#2931	C₂×C₁₃₉	1	2	yes	complemented
137	#1586	C₂×C₁₃₉	1	8	yes	complemented
138	#1566	C₂×C₁₃₉	1	1	yes	characteristic, complemented
139	#1492	C₂×C₁₃₉	1	2	yes	complemented
140	C₂×#322	C₂×C₁₃₉	1	1	yes	characteristic, complemented
141	C₂×#131	—	2	2	no	—
142	#532	C₂²×C₁₃₉	1	2	yes	—
143	#500	C₄×C₁₃₉	1	1	yes	characteristic
144	#493	C₄×C₁₃₉	1	1	yes	characteristic
145	#486	C₄×C₁₃₉	1	2	yes	—
146	#466	C₄×C₁₃₉	1	4	yes	—
147	C₄×D₈	C₂²×C₁₃₉	1	4	yes	—
148	#436	—	2	2	no	—
149	#436	C₂²×C₁₃₉	1	4	yes	—
150	C₈×D₄	—	2	2	no	—
151	C₈×D₄	C₂²×C₁₃₉	1	4	yes	—
152	#423	C₂²×C₁₃₉	1	2	yes	—
153	#408	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
157	#334	C₂²×C₁₃₉	1	4	yes	—
158	D₄⋊C₈	C₂²×C₁₃₉	1	8	yes	—
159	#322	C₂²×C₁₃₉	1	4	yes	—
160	C₂×#41	—	2	2	no	—
161	C₂×D₈	C₂×C₄×C₁₃₉	1	4	yes	—
162	#131	—	4	8	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	C₂×C₄×C₁₃₉	1	2	yes	—
168	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
169	C₄⋊Q₈	C₂×C₄×C₁₃₉	1	2	yes	—
170	#127	C₂×C₄×C₁₃₉	1	2	yes	—
171	#124	—	2	2	no	—
172	#121	—	2	2	no	—
173	#121	C₂×C₄×C₁₃₉	1	4	yes	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	C₂³×C₁₃₉	1	4	yes	—
176	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
177	#107	C₂×C₄×C₁₃₉	1	4	yes	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂³×C₁₃₉	1	4	yes	—
180	#102	C₂×C₄×C₁₃₉	1	8	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
186	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
187	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
189	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
190	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
191	#41	—	4	4	no	—
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂²×C₄×C₁₃₉	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
201	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
202	D₈	—	4	4	no	—
203	M₁₆	—	4	8	no	—
204	M₁₆	—	4	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	C₂²×C₄×C₁₃₉	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₁₃₉	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
221	C₄²	#41×C₁₃₉	1	1	yes	characteristic
222	C₄²	C₂⁴×C₁₃₉	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	C₄×D₄×C₁₃₉	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
238	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	4	yes	—
239	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
241	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	4	4	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	#442×C₁₃₉	1	2	yes	—
254	C₂²	C₂×C₄×D₄×C₁₃₉	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	#442×C₁₃₉	1	2	yes	—
259	C₄	#443×C₁₃₉	1	2	yes	—
260	C₄	#568×C₁₃₉	1	2	yes	—
261	C₄	C₂×AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	#2932×C₁₃₉	1	1	yes	characteristic
265	C₂	C₂×#442×C₁₃₉	1	1	yes	characteristic
266	C₂	C₂×#443×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₄²	C₂⁴×C₁₃₉
socle	C₂²×C₁₃₉	C₂×C₄×D₄
prime radical	#3169×C₁₃₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2931×C₁₃₉	C₂	semidirect
3. #1586×C₁₃₉	C₂	semidirect
4. #1566×C₁₃₉	C₂	semidirect
5. #1492×C₁₃₉	C₂	semidirect
6. C₂×#322×C₁₃₉	C₂	semidirect
133. #17981	C₁₃₉	direct (N × H)
134. C₁₃₉	#17981	direct (N × H)
136. #2931	C₂×C₁₃₉	semidirect
137. #1586	C₂×C₁₃₉	semidirect
138. #1566	C₂×C₁₃₉	semidirect
139. #1492	C₂×C₁₃₉	semidirect
140. C₂×#322	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₉	2
#2931×C₁₃₉	2
#1586×C₁₃₉	2
#1566×C₁₃₉	2
#1492×C₁₃₉	2
C₂×#322×C₁₃₉	2
#17981	139

Subgroups — C139×#17981 #48012577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#17981 #48012577