Subgroups — C₁₃₉×#9665 #48004261

Order 35584.

586 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	C₂×C₄×D₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1954×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₂×#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₂×#322×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
8	C₄×D₈×C₁₃₉	—	2	2	no	—
9	C₄×D₈×C₁₃₉	—	2	2	no	—
10	C₂×#107×C₁₃₉	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
12	C₂×#102×C₁₃₉	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₃₉	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	#334×C₁₃₉	—	2	2	no	—
18	D₄⋊C₈×C₁₃₉	—	2	4	no	—
19	#322×C₁₃₉	—	2	2	no	—
20	C₂²×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃₉	—	2	2	no	—
22	C₂×D₈×C₁₃₉	—	2	4	no	—
23	C₂×D₈×C₁₃₉	—	2	2	no	—
24	C₂²×C₈×C₁₃₉	—	2	1	no	—
25	C₂²×C₈×C₁₃₉	—	2	2	no	—
26	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
29	C₄×D₄×C₁₃₉	—	2	2	no	—
30	C₄×D₄×C₁₃₉	—	2	2	no	—
31	C₄×D₄×C₁₃₉	—	2	2	no	—
32	C₄×D₄×C₁₃₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
36	#107×C₁₃₉	—	2	2	no	—
37	C₄⋊C₈×C₁₃₉	—	2	2	no	—
38	C₄⋊C₈×C₁₃₉	—	2	2	no	—
39	#102×C₁₃₉	—	2	4	no	—
40	C₂²⋊C₈×C₁₃₉	—	4	8	no	—
41	C₄×C₈×C₁₃₉	—	2	1	no	—
42	C₄×C₈×C₁₃₉	—	2	1	no	—
43	C₄×C₈×C₁₃₉	—	2	1	no	—
44	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
45	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
46	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
47	C₂⁴×C₁₃₉	—	2	2	no	—
48	C₂×D₄×C₁₃₉	—	2	2	no	—
49	C₂×D₄×C₁₃₉	—	2	2	no	—
50	C₂×D₄×C₁₃₉	—	2	2	no	—
51	C₂×D₄×C₁₃₉	—	2	2	no	—
52	C₂×D₄×C₁₃₉	—	2	2	no	—
53	C₂×D₄×C₁₃₉	M₁₆	1	4	yes	—
54	C₂²×C₄×C₁₃₉	—	2	2	no	—
55	C₂²×C₄×C₁₃₉	—	4	2	no	—
56	C₂²×C₄×C₁₃₉	—	4	2	no	—
57	C₂²×C₄×C₁₃₉	—	4	4	no	—
58	C₂²×C₄×C₁₃₉	—	4	4	no	—
59	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
62	D₈×C₁₃₉	—	2	8	no	—
63	C₂×C₈×C₁₃₉	—	2	4	no	—
64	C₂×C₈×C₁₃₉	—	2	2	no	—
65	C₂×C₈×C₁₃₉	—	2	2	no	—
66	C₂×C₈×C₁₃₉	—	2	2	no	—
67	C₂×C₈×C₁₃₉	—	2	2	no	—
68	C₂×C₈×C₁₃₉	—	2	4	no	—
69	C₂×C₈×C₁₃₉	—	4	2	no	—
70	C₂×C₈×C₁₃₉	—	4	4	no	—
71	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₁₃₉	—	2	2	no	—
74	C₄⋊C₄×C₁₃₉	—	2	2	no	—
75	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
76	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
77	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃₉	#41	1	1	yes	characteristic
81	C₂³×C₁₃₉	—	2	4	no	—
82	C₂³×C₁₃₉	—	4	2	no	—
83	C₂³×C₁₃₉	—	4	2	no	—
84	C₂³×C₁₃₉	—	4	4	no	—
85	C₂³×C₁₃₉	—	4	4	no	—
86	C₂³×C₁₃₉	C₄×D₄	1	1	yes	characteristic
87	D₄×C₁₃₉	—	2	4	no	—
88	D₄×C₁₃₉	—	2	4	no	—
89	D₄×C₁₃₉	—	4	2	no	—
90	D₄×C₁₃₉	—	4	2	no	—
91	C₂×C₄×C₁₃₉	—	2	2	no	—
92	C₂×C₄×C₁₃₉	—	4	2	no	—
93	C₂×C₄×C₁₃₉	—	4	4	no	—
94	C₂×C₄×C₁₃₉	—	8	4	no	—
95	C₂×C₄×C₁₃₉	—	8	2	no	—
96	C₂×C₄×C₁₃₉	—	8	2	no	—
97	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₁₃₉	#131	1	2	yes	—
105	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
106	C₈×C₁₃₉	—	2	4	no	—
107	C₈×C₁₃₉	—	2	2	no	—
108	C₈×C₁₃₉	—	2	2	no	—
109	C₈×C₁₃₉	—	4	2	no	—
110	C₈×C₁₃₉	—	8	2	no	—
111	C₂²×C₁₃₉	—	4	8	no	—
112	C₂²×C₁₃₉	—	4	4	no	—
113	C₂²×C₁₃₉	—	8	4	no	—
114	C₂²×C₁₃₉	—	8	2	no	—
115	C₂²×C₁₃₉	—	8	2	no	—
116	C₂²×C₁₃₉	#390	1	1	yes	characteristic
117	C₂²×C₁₃₉	#435	1	2	yes	—
118	C₂²×C₁₃₉	#442	1	2	yes	—
119	C₂²×C₁₃₉	#443	1	2	yes	—
120	C₄×C₁₃₉	—	2	1	no	—
121	C₄×C₁₃₉	—	2	1	no	—
122	C₄×C₁₃₉	—	8	2	no	—
123	C₄×C₁₃₉	#407	1	1	yes	characteristic
124	C₄×C₁₃₉	#408	1	1	yes	characteristic
125	C₄×C₁₃₉	#435	1	2	yes	—
126	C₄×C₁₃₉	#469	1	2	yes	—
127	C₄×C₁₃₉	#496	1	2	yes	—
128	C₂×C₁₃₉	—	8	4	no	—
129	C₂×C₁₃₉	#1586	1	4	yes	—
130	C₂×C₁₃₉	#1955	1	1	yes	characteristic
131	C₂×C₁₃₉	#1972	1	1	yes	characteristic
132	C₂×C₁₃₉	#1974	1	1	yes	characteristic
133	#9665	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₉	#9665	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₄×D₈	C₂×C₁₃₉	1	1	yes	characteristic
136	#1954	C₂×C₁₃₉	1	2	yes	complemented
137	C₂×#334	C₂×C₁₃₉	1	1	yes	characteristic, complemented
138	C₂×D₄⋊C₈	C₂×C₁₃₉	1	2	yes	complemented
139	C₂×#322	C₂×C₁₃₉	1	1	yes	characteristic, complemented
140	C₂²×D₈	C₄×C₁₃₉	1	1	yes	characteristic
141	C₂×C₄×D₄	C₂²×C₁₃₉	1	2	yes	—
142	C₄×D₈	—	2	2	no	—
143	C₄×D₈	—	2	2	no	—
144	C₂×#107	C₄×C₁₃₉	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
146	C₂×#102	C₄×C₁₃₉	1	2	yes	—
147	C₂×C₂²⋊C₈	—	2	4	no	—
148	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
151	#334	—	2	2	no	—
152	D₄⋊C₈	—	2	4	no	—
153	#322	—	2	2	no	—
154	C₂²×D₄	C₂×C₄×C₁₃₉	1	2	yes	—
155	C₂³×C₄	—	2	2	no	—
156	C₂×D₈	—	2	4	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
168	C₂×C₂²⋊C₄	—	2	2	no	—
169	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
170	#107	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#102	—	2	4	no	—
174	C₂²⋊C₈	—	4	8	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
179	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
180	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
181	C₂⁴	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	M₁₆×C₁₃₉	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
195	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
196	D₈	—	2	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
206	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
213	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
214	C₄²	#41×C₁₃₉	1	1	yes	characteristic
215	C₂³	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₄×D₄×C₁₃₉	1	1	yes	characteristic
221	D₄	—	2	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
233	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
234	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
235	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
236	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₁₃₉	1	2	yes	—
238	C₂×C₄	#131×C₁₃₉	1	2	yes	—
239	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	4	yes	—
240	C₈	—	2	4	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₂²	—	4	8	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	#390×C₁₃₉	1	1	yes	characteristic
251	C₂²	#435×C₁₃₉	1	2	yes	—
252	C₂²	#442×C₁₃₉	1	2	yes	—
253	C₂²	#443×C₁₃₉	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	#407×C₁₃₉	1	1	yes	characteristic
258	C₄	#408×C₁₃₉	1	1	yes	characteristic
259	C₄	#435×C₁₃₉	1	2	yes	—
260	C₄	#469×C₁₃₉	1	2	yes	—
261	C₄	#496×C₁₃₉	1	2	yes	—
262	C₂	—	8	4	no	—
263	C₂	#1586×C₁₃₉	1	4	yes	—
264	C₂	#1955×C₁₃₉	1	1	yes	characteristic
265	C₂	#1972×C₁₃₉	1	1	yes	characteristic
266	C₂	#1974×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
socle	C₂³×C₁₃₉	C₄×D₄
prime radical	C₂²×D₈×C₁₃₉	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1954×C₁₃₉	C₂	semidirect
3. C₂×#334×C₁₃₉	C₂	semidirect
4. C₂×D₄⋊C₈×C₁₃₉	C₂	semidirect
5. C₂×#322×C₁₃₉	C₂	semidirect
133. #9665	C₁₃₉	direct (N × H)
134. C₁₃₉	#9665	direct (N × H)
136. #1954	C₂×C₁₃₉	semidirect
137. C₂×#334	C₂×C₁₃₉	semidirect
138. C₂×D₄⋊C₈	C₂×C₁₃₉	semidirect
139. 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
C₂×C₄×D₈×C₁₃₉	2
#1954×C₁₃₉	2
C₂×#334×C₁₃₉	2
C₂×D₄⋊C₈×C₁₃₉	2
C₂×#322×C₁₃₉	2
#9665	139

Subgroups — C139×#9665 #48004261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9665 #48004261