Subgroups — C₁₃₉×#20360 #48014956

Order 35584.

1996 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#568×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#3428×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₁₃₉	C₂	1	12	yes	maximal, complemented
5	C₂²×#41×C₁₃₉	C₂²	1	1	yes	characteristic
6	#568×C₁₃₉	—	2	4	no	—
7	#568×C₁₃₉	—	2	4	no	—
8	#568×C₁₃₉	—	2	4	no	—
9	#568×C₁₃₉	C₄	1	8	yes	complemented
10	C₂×#131×C₁₃₉	—	2	4	no	—
11	C₂×#131×C₁₃₉	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₁₃₉	C₂²	1	6	yes	—
13	C₂²×M₁₆×C₁₃₉	C₂²	1	6	yes	—
14	#520×C₁₃₉	—	2	4	no	—
15	#519×C₁₃₉	C₄	1	2	yes	—
16	#518×C₁₃₉	C₄	1	6	yes	—
17	C₄×#41×C₁₃₉	—	2	4	no	—
18	C₂×#117×C₁₃₉	C₂²	1	6	yes	—
19	#344×C₁₃₉	C₂²	1	16	yes	—
20	C₂×#41×C₁₃₉	C₂×C₄	1	4	yes	—
21	C₂×#41×C₁₃₉	C₂×C₄	1	4	yes	—
22	C₂×#41×C₁₃₉	D₄	1	4	yes	—
23	C₂×#41×C₁₃₉	D₄	1	4	yes	—
24	C₂×#41×C₁₃₉	D₄	1	4	yes	—
25	C₂×#41×C₁₃₉	Q₈	1	4	yes	—
26	C₂²×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃₉	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
29	#131×C₁₃₉	—	2	32	no	—
30	#131×C₁₃₉	—	2	32	no	—
31	C₂×M₁₆×C₁₃₉	—	2	6	no	—
32	C₂×M₁₆×C₁₃₉	—	2	12	no	—
33	C₂×M₁₆×C₁₃₉	—	2	12	no	—
34	C₂×M₁₆×C₁₃₉	—	2	12	no	—
35	C₂×M₁₆×C₁₃₉	C₂×C₄	1	24	yes	complemented
36	C₂×M₁₆×C₁₃₉	C₂×C₄	1	12	yes	complemented
37	C₂×M₁₆×C₁₃₉	C₂³	1	4	yes	—
38	C₂×M₁₆×C₁₃₉	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₃₉	—	2	6	no	—
40	C₂²×C₈×C₁₃₉	C₂×C₄	1	12	yes	complemented
41	C₄×Q₈×C₁₃₉	—	2	8	no	—
42	C₄×D₄×C₁₃₉	—	2	24	no	—
43	#117×C₁₃₉	—	2	6	no	—
44	#117×C₁₃₉	—	2	6	no	—
45	#117×C₁₃₉	C₂×C₄	1	12	yes	—
46	#117×C₁₃₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	—	2	6	no	—
48	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	12	yes	—
49	C₂×C₄²×C₁₃₉	—	2	6	no	—
50	C₂⁴×C₁₃₉	C₂²×C₄	1	3	yes	—
51	#41×C₁₃₉	C₂²⋊C₄	1	16	yes	—
52	#41×C₁₃₉	C₂²⋊C₄	1	16	yes	—
53	#41×C₁₃₉	C₄⋊C₄	1	16	yes	—
54	#41×C₁₃₉	C₄⋊C₄	1	16	yes	—
55	C₂×Q₈×C₁₃₉	C₄²	1	4	yes	—
56	C₂×Q₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
57	C₂×Q₈×C₁₃₉	C₄⋊C₄	1	4	yes	—
58	C₂×D₄×C₁₃₉	C₄²	1	12	yes	—
59	C₂×D₄×C₁₃₉	C₂²⋊C₄	1	12	yes	—
60	C₂×D₄×C₁₃₉	C₄⋊C₄	1	12	yes	—
61	C₂²×C₄×C₁₃₉	—	2	12	no	—
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₁₃₉	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₁₃₉	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₁₃₉	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₁₃₉	C₂×Q₈	1	6	yes	—
69	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₁₃₉	—	2	24	no	—
71	M₁₆×C₁₃₉	—	2	24	no	—
72	M₁₆×C₁₃₉	—	2	4	no	—
73	M₁₆×C₁₃₉	—	2	4	no	—
74	M₁₆×C₁₃₉	—	2	4	no	—
75	M₁₆×C₁₃₉	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₁₃₉	—	2	24	no	—
77	C₂×C₈×C₁₃₉	—	2	24	no	—
78	C₂×C₈×C₁₃₉	—	2	4	no	—
79	C₂×C₈×C₁₃₉	C₂²×C₄	1	8	yes	—
80	C₄⋊C₄×C₁₃₉	—	2	12	no	—
81	C₄⋊C₄×C₁₃₉	—	2	4	no	—
82	C₂²⋊C₄×C₁₃₉	—	2	12	no	—
83	C₂²⋊C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
84	C₄²×C₁₃₉	—	2	12	no	—
85	C₄²×C₁₃₉	—	2	4	no	—
86	C₂³×C₁₃₉	—	8	3	no	—
87	C₂³×C₁₃₉	C₂×C₄²	1	6	yes	—
88	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	6	yes	—
90	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
91	Q₈×C₁₃₉	#95	1	16	yes	—
92	D₄×C₁₃₉	#95	1	48	yes	—
93	C₂×C₄×C₁₃₉	—	2	8	no	—
94	C₂×C₄×C₁₃₉	—	8	12	no	—
95	C₂×C₄×C₁₃₉	C₂×C₄²	1	6	yes	—
96	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	12	yes	—
97	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	12	yes	—
98	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
99	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	12	yes	—
101	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₁₃₉	—	2	8	no	—
109	C₈×C₁₃₉	—	2	8	no	—
110	C₂²×C₁₃₉	—	4	6	no	—
111	C₂²×C₁₃₉	—	4	1	no	—
112	C₂²×C₁₃₉	—	4	6	no	—
113	C₂²×C₁₃₉	—	4	6	no	—
114	C₂²×C₁₃₉	C₂×#95	1	12	yes	—
115	C₂²×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₁₃₉	—	4	8	no	—
119	C₄×C₁₃₉	C₂×#95	1	12	yes	—
120	C₄×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₁₃₉	—	2	1	no	—
125	C₂×C₁₃₉	—	2	1	no	—
126	C₂×C₁₃₉	—	2	1	no	—
127	C₂×C₁₃₉	—	2	12	no	—
128	C₂×C₁₃₉	C₂²×#95	1	1	yes	characteristic
129	#20360	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₉	#20360	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#568	C₂×C₁₃₉	1	1	yes	characteristic
132	C₂×#568	C₂×C₁₃₉	1	1	yes	characteristic
133	#3428	C₂×C₁₃₉	1	1	yes	characteristic
134	C₂×#344	C₂×C₁₃₉	1	12	yes	complemented
135	C₂²×#41	C₂²×C₁₃₉	1	1	yes	characteristic
136	#568	—	2	4	no	—
137	#568	—	2	4	no	—
138	#568	—	2	4	no	—
139	#568	C₄×C₁₃₉	1	8	yes	complemented
140	C₂×#131	—	2	4	no	—
141	C₂×#131	C₄×C₁₃₉	1	8	yes	complemented
142	C₂²×M₁₆	C₂²×C₁₃₉	1	6	yes	—
143	C₂²×M₁₆	C₂²×C₁₃₉	1	6	yes	—
144	#520	—	2	4	no	—
145	#519	C₄×C₁₃₉	1	2	yes	—
146	#518	C₄×C₁₃₉	1	6	yes	—
147	C₄×#41	—	2	4	no	—
148	C₂×#117	C₂²×C₁₃₉	1	6	yes	—
149	#344	C₂²×C₁₃₉	1	16	yes	—
150	C₂×#41	C₂×C₄×C₁₃₉	1	4	yes	—
151	C₂×#41	C₂×C₄×C₁₃₉	1	4	yes	—
152	C₂×#41	D₄×C₁₃₉	1	4	yes	—
153	C₂×#41	D₄×C₁₃₉	1	4	yes	—
154	C₂×#41	D₄×C₁₃₉	1	4	yes	—
155	C₂×#41	Q₈×C₁₃₉	1	4	yes	—
156	C₂²×Q₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
157	C₂²×D₄	C₂×C₄×C₁₃₉	1	3	yes	—
158	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
159	#131	—	2	32	no	—
160	#131	—	2	32	no	—
161	C₂×M₁₆	—	2	6	no	—
162	C₂×M₁₆	—	2	12	no	—
163	C₂×M₁₆	—	2	12	no	—
164	C₂×M₁₆	—	2	12	no	—
165	C₂×M₁₆	C₂×C₄×C₁₃₉	1	24	yes	complemented
166	C₂×M₁₆	C₂×C₄×C₁₃₉	1	12	yes	complemented
167	C₂×M₁₆	C₂³×C₁₃₉	1	4	yes	—
168	C₂×M₁₆	C₂³×C₁₃₉	1	4	yes	—
169	C₂²×C₈	—	2	6	no	—
170	C₂²×C₈	C₂×C₄×C₁₃₉	1	12	yes	complemented
171	C₄×Q₈	—	2	8	no	—
172	C₄×D₄	—	2	24	no	—
173	#117	—	2	6	no	—
174	#117	—	2	6	no	—
175	#117	C₂×C₄×C₁₃₉	1	12	yes	—
176	#117	C₂³×C₁₃₉	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	6	no	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	12	yes	—
179	C₂×C₄²	—	2	6	no	—
180	C₂⁴	C₂²×C₄×C₁₃₉	1	3	yes	—
181	#41	C₂²⋊C₄×C₁₃₉	1	16	yes	—
182	#41	C₂²⋊C₄×C₁₃₉	1	16	yes	—
183	#41	C₄⋊C₄×C₁₃₉	1	16	yes	—
184	#41	C₄⋊C₄×C₁₃₉	1	16	yes	—
185	C₂×Q₈	C₄²×C₁₃₉	1	4	yes	—
186	C₂×Q₈	C₂²⋊C₄×C₁₃₉	1	4	yes	—
187	C₂×Q₈	C₄⋊C₄×C₁₃₉	1	4	yes	—
188	C₂×D₄	C₄²×C₁₃₉	1	12	yes	—
189	C₂×D₄	C₂²⋊C₄×C₁₃₉	1	12	yes	—
190	C₂×D₄	C₄⋊C₄×C₁₃₉	1	12	yes	—
191	C₂²×C₄	—	2	12	no	—
192	C₂²×C₄	C₂²×C₄×C₁₃₉	1	3	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₉	1	6	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₃₉	1	6	yes	—
195	C₂²×C₄	C₂×D₄×C₁₃₉	1	6	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₉	1	6	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₉	1	6	yes	—
198	C₂²×C₄	C₂×Q₈×C₁₃₉	1	6	yes	—
199	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
200	M₁₆	—	2	24	no	—
201	M₁₆	—	2	24	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂²×C₄×C₁₃₉	1	8	yes	—
206	C₂×C₈	—	2	24	no	—
207	C₂×C₈	—	2	24	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂²×C₄×C₁₃₉	1	8	yes	—
210	C₄⋊C₄	—	2	12	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	12	no	—
213	C₂²⋊C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
214	C₄²	—	2	12	no	—
215	C₄²	—	2	4	no	—
216	C₂³	—	8	3	no	—
217	C₂³	C₂×C₄²×C₁₃₉	1	6	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
219	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	6	yes	—
220	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
221	Q₈	#95×C₁₃₉	1	16	yes	—
222	D₄	#95×C₁₃₉	1	48	yes	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	8	12	no	—
225	C₂×C₄	C₂×C₄²×C₁₃₉	1	6	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	12	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	12	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	12	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	12	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	6	yes	—
232	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
238	C₈	—	2	8	no	—
239	C₈	—	2	8	no	—
240	C₂²	—	4	6	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	6	no	—
243	C₂²	—	4	6	no	—
244	C₂²	C₂×#95×C₁₃₉	1	12	yes	—
245	C₂²	C₂²×C₄²×C₁₃₉	1	1	yes	characteristic
246	C₂²	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
247	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
248	C₄	—	4	8	no	—
249	C₄	C₂×#95×C₁₃₉	1	12	yes	—
250	C₄	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
251	C₄	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
252	C₄	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
253	C₄	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	12	no	—
258	C₂	C₂²×#95×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₉	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₁₃₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
socle	C₂×C₁₃₉	C₂²×#95
prime radical	C₂²×#41×C₁₃₉	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₉ ◁ C₂²×#41×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#344×C₁₃₉	C₂	semidirect
9. #568×C₁₃₉	C₄	semidirect
11. C₂×#131×C₁₃₉	C₄	semidirect
35. C₂×M₁₆×C₁₃₉	C₂×C₄	semidirect
36. C₂×M₁₆×C₁₃₉	C₂×C₄	semidirect
40. C₂²×C₈×C₁₃₉	C₂×C₄	semidirect
129. #20360	C₁₃₉	direct (N × H)
130. C₁₃₉	#20360	direct (N × H)
134. C₂×#344	C₂×C₁₃₉	semidirect
139. #568	C₄×C₁₃₉	semidirect
141. C₂×#131	C₄×C₁₃₉	semidirect
165. C₂×M₁₆	C₂×C₄×C₁₃₉	semidirect
166. C₂×M₁₆	C₂×C₄×C₁₃₉	semidirect
170. C₂²×C₈	C₂×C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₁₃₉	2
#3428×C₁₃₉	2
C₂×#344×C₁₃₉	2
#20360	139

Subgroups — C139×#20360 #48014956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#20360 #48014956