Subgroups — C₁₃₉×#11657 #48006253

Order 35584.

430 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#2162×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#1567×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#1331×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈×C₁₃₉	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₁₃₉	—	2	1	no	—
7	C₄×D₈×C₁₃₉	—	2	1	no	—
8	C₄×D₈×C₁₃₉	—	2	1	no	—
9	C₄×D₈×C₁₃₉	C₄	1	2	yes	—
10	C₄×D₈×C₁₃₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₉	C₄	1	1	yes	characteristic
12	#423×C₁₃₉	—	2	1	no	—
13	C₂×#102×C₁₃₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₃₉	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₃₉	—	2	1	no	—
17	C₄×C₁₆×C₁₃₉	C₂²	1	2	yes	complemented
18	#335×C₁₃₉	—	2	1	no	—
19	#335×C₁₃₉	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃₉	—	2	1	no	—
21	C₂³×C₄×C₁₃₉	—	4	1	no	—
22	C₂×D₈×C₁₃₉	—	2	4	no	—
23	C₂×D₈×C₁₃₉	—	2	1	no	—
24	C₂×D₈×C₁₃₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₃₉	—	4	1	no	—
26	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
28	C₄×D₄×C₁₃₉	—	2	2	no	—
29	C₄×D₄×C₁₃₉	—	2	2	no	—
30	C₄×D₄×C₁₃₉	—	4	1	no	—
31	C₄×D₄×C₁₃₉	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₁₃₉	—	4	1	no	—
34	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₃₉	—	2	2	no	—
36	#107×C₁₃₉	—	2	1	no	—
37	#107×C₁₃₉	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₁₃₉	—	2	2	no	—
39	C₄⋊C₈×C₁₃₉	—	4	1	no	—
40	#102×C₁₃₉	—	2	2	no	—
41	#102×C₁₃₉	—	2	2	no	—
42	C₄×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂⁴×C₁₃₉	—	4	1	no	—
47	C₂×D₄×C₁₃₉	—	2	4	no	—
48	C₂×D₄×C₁₃₉	—	2	2	no	—
49	C₂×D₄×C₁₃₉	—	4	2	no	—
50	C₂×D₄×C₁₃₉	—	4	1	no	—
51	C₂²×C₄×C₁₃₉	—	4	1	no	—
52	C₂²×C₄×C₁₃₉	—	4	2	no	—
53	C₂²×C₄×C₁₃₉	—	4	2	no	—
54	C₂²×C₄×C₁₃₉	—	8	2	no	—
55	C₂²×C₄×C₁₃₉	—	8	2	no	—
56	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₉	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₉	SD₁₆	1	1	yes	characteristic
59	D₈×C₁₃₉	—	2	4	no	—
60	D₈×C₁₃₉	—	2	2	no	—
61	D₈×C₁₃₉	M₁₆	1	4	yes	complemented
62	M₁₆×C₁₃₉	—	8	2	no	—
63	C₂×C₈×C₁₃₉	—	2	2	no	—
64	C₂×C₈×C₁₃₉	—	4	1	no	—
65	C₂×C₈×C₁₃₉	—	4	2	no	—
66	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₉	—	2	2	no	—
71	C₄⋊C₄×C₁₃₉	—	4	1	no	—
72	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
73	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
74	C₄²×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃₉	D₈	1	1	yes	characteristic
76	C₄²×C₁₃₉	SD₁₆	1	1	yes	characteristic
77	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₁₃₉	—	4	2	no	—
79	C₂³×C₁₃₉	—	4	4	no	—
80	C₂³×C₁₃₉	—	4	2	no	—
81	C₂³×C₁₃₉	—	8	2	no	—
82	C₂³×C₁₃₉	—	8	2	no	—
83	C₂³×C₁₃₉	#102	1	1	yes	characteristic
84	D₄×C₁₃₉	—	2	4	no	—
85	D₄×C₁₃₉	—	4	4	no	—
86	D₄×C₁₃₉	—	4	2	no	—
87	C₂×C₄×C₁₃₉	—	2	1	no	—
88	C₂×C₄×C₁₃₉	—	2	1	no	—
89	C₂×C₄×C₁₃₉	—	4	1	no	—
90	C₂×C₄×C₁₃₉	—	4	2	no	—
91	C₂×C₄×C₁₃₉	—	8	2	no	—
92	C₂×C₄×C₁₃₉	—	8	4	no	—
93	C₂×C₄×C₁₃₉	—	8	2	no	—
94	C₂×C₄×C₁₃₉	—	8	2	no	—
95	C₂×C₄×C₁₃₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₁₃₉	#102	1	1	yes	characteristic
97	C₂×C₄×C₁₃₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₃₉	D₁₆	1	2	yes	—
99	C₂×C₄×C₁₃₉	SD₃₂	1	2	yes	—
100	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₁₃₉	—	2	1	no	—
104	C₈×C₁₃₉	—	2	1	no	—
105	C₈×C₁₃₉	—	2	1	no	—
106	C₈×C₁₃₉	—	8	2	no	—
107	C₈×C₁₃₉	C₂×M₁₆	1	2	yes	—
108	C₂²×C₁₃₉	—	2	1	no	—
109	C₂²×C₁₃₉	—	2	1	no	—
110	C₂²×C₁₃₉	—	4	4	no	—
111	C₂²×C₁₃₉	—	8	8	no	—
112	C₂²×C₁₃₉	—	8	2	no	—
113	C₂²×C₁₃₉	—	8	2	no	—
114	C₂²×C₁₃₉	#357	1	2	yes	—
115	C₂²×C₁₃₉	C₂×#102	1	1	yes	characteristic
116	C₄×C₁₃₉	—	2	1	no	—
117	C₄×C₁₃₉	—	2	1	no	—
118	C₄×C₁₃₉	—	2	1	no	—
119	C₄×C₁₃₉	—	8	2	no	—
120	C₄×C₁₃₉	#357	1	2	yes	—
121	C₄×C₁₃₉	#407	1	1	yes	characteristic
122	C₄×C₁₃₉	#421	1	1	yes	characteristic
123	C₄×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
124	C₄×C₁₃₉	C₂×SD₃₂	1	1	yes	characteristic
125	C₂×C₁₃₉	—	2	2	no	—
126	C₂×C₁₃₉	—	8	4	no	—
127	C₂×C₁₃₉	#1486	1	1	yes	characteristic
128	C₂×C₁₃₉	C₂×#357	1	1	yes	characteristic
129	C₂×C₁₃₉	#2143	1	1	yes	characteristic
130	#11657	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₉	#11657	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×C₄×D₈	C₂×C₁₃₉	1	1	yes	characteristic
133	#2162	C₂×C₁₃₉	1	1	yes	characteristic, complemented
134	#1567	C₂×C₁₃₉	1	1	yes	characteristic, complemented
135	#1331	C₂×C₁₃₉	1	4	yes	complemented
136	C₂²×D₈	C₄×C₁₃₉	1	1	yes	characteristic
137	C₂×C₄×D₄	—	2	1	no	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	—	2	1	no	—
140	C₄×D₈	C₄×C₁₃₉	1	2	yes	—
141	C₄×D₈	C₂²×C₁₃₉	1	2	yes	—
142	C₂×#107	C₄×C₁₃₉	1	1	yes	characteristic
143	#423	—	2	1	no	—
144	C₂×#102	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
146	C₄⋊C₁₆	—	2	1	no	—
147	C₂²⋊C₁₆	—	2	1	no	—
148	C₄×C₁₆	C₂²×C₁₃₉	1	2	yes	complemented
149	#335	—	2	1	no	—
150	#335	C₂²×C₁₃₉	1	2	yes	complemented
151	C₂²×D₄	—	2	1	no	—
152	C₂³×C₄	—	4	1	no	—
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	C₂×C₄×C₁₃₉	1	2	yes	—
156	C₂×M₁₆	—	4	1	no	—
157	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	4	1	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₂²⋊C₄	—	4	1	no	—
165	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
166	C₂×C₁₆	—	2	2	no	—
167	#107	—	2	1	no	—
168	#107	C₂×C₄×C₁₃₉	1	2	yes	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	4	1	no	—
171	#102	—	2	2	no	—
172	#102	—	2	2	no	—
173	C₄×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
174	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
175	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
177	C₂⁴	—	4	1	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	8	2	no	—
186	C₂²×C₄	—	8	2	no	—
187	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
188	C₂²×C₄	D₈×C₁₃₉	1	1	yes	characteristic
189	C₂²×C₄	SD₁₆×C₁₃₉	1	1	yes	characteristic
190	D₈	—	2	4	no	—
191	D₈	—	2	2	no	—
192	D₈	M₁₆×C₁₃₉	1	4	yes	complemented
193	M₁₆	—	8	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
206	C₄²	D₈×C₁₃₉	1	1	yes	characteristic
207	C₄²	SD₁₆×C₁₃₉	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	#102×C₁₃₉	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	#102×C₁₃₉	1	1	yes	characteristic
227	C₂×C₄	#102×C₁₃₉	1	1	yes	characteristic
228	C₂×C₄	#102×C₁₃₉	1	1	yes	characteristic
229	C₂×C₄	D₁₆×C₁₃₉	1	2	yes	—
230	C₂×C₄	SD₃₂×C₁₃₉	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	8	2	no	—
238	C₈	C₂×M₁₆×C₁₃₉	1	2	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	8	8	no	—
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	#357×C₁₃₉	1	2	yes	—
246	C₂²	C₂×#102×C₁₃₉	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	2	no	—
251	C₄	#357×C₁₃₉	1	2	yes	—
252	C₄	#407×C₁₃₉	1	1	yes	characteristic
253	C₄	#421×C₁₃₉	1	1	yes	characteristic
254	C₄	C₂×D₁₆×C₁₃₉	1	1	yes	characteristic
255	C₄	C₂×SD₃₂×C₁₃₉	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	4	no	—
258	C₂	#1486×C₁₃₉	1	1	yes	characteristic
259	C₂	C₂×#357×C₁₃₉	1	1	yes	characteristic
260	C₂	#2143×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₉
socle	C₂²×C₁₃₉	C₂×#102
prime radical	C₂²×D₈×C₁₃₉	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2162×C₁₃₉	C₂	semidirect
3. #1567×C₁₃₉	C₂	semidirect
4. #1331×C₁₃₉	C₂	semidirect
17. C₄×C₁₆×C₁₃₉	C₂²	semidirect
19. #335×C₁₃₉	C₂²	semidirect
61. D₈×C₁₃₉	M₁₆	semidirect
130. #11657	C₁₃₉	direct (N × H)
131. C₁₃₉	#11657	direct (N × H)
133. #2162	C₂×C₁₃₉	semidirect
134. #1567	C₂×C₁₃₉	semidirect
135. #1331	C₂×C₁₃₉	semidirect
148. C₄×C₁₆	C₂²×C₁₃₉	semidirect
150. #335	C₂²×C₁₃₉	semidirect
192. D₈	M₁₆×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
#2162×C₁₃₉	2
#1567×C₁₃₉	2
#1331×C₁₃₉	2
#11657	139

Subgroups — C139×#11657 #48006253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#11657 #48006253