Subgroups — C₁₃₉×#20817 #48015413

Order 35584.

982 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	C₂²×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2285×C₁₃₉	C₂	1	2	yes	maximal
3	#1845×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#1834×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	#1808×C₁₃₉	C₂	1	4	yes	maximal
6	C₄×#107×C₁₃₉	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₃₉	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₉	—	2	2	no	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₃₉	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₁₃₉	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#384×C₁₃₉	C₂²	1	4	yes	—
16	#382×C₁₃₉	—	2	2	no	—
17	#380×C₁₃₉	—	2	2	no	—
18	#380×C₁₃₉	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
20	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
21	#340×C₁₃₉	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₉	—	2	8	no	—
26	C₂²×C₈×C₁₃₉	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
29	#117×C₁₃₉	—	4	8	no	—
30	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
34	C₂×C₄²×C₁₃₉	—	2	4	no	—
35	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₉	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
41	#107×C₁₃₉	—	2	16	no	—
42	C₄×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₃₉	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
45	#95×C₁₃₉	—	2	4	no	—
46	#95×C₁₃₉	—	2	4	no	—
47	C₂⁴×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₉	—	2	4	no	—
49	C₂²×C₄×C₁₃₉	—	2	4	no	—
50	C₂²×C₄×C₁₃₉	—	2	8	no	—
51	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃₉	D₈	1	4	yes	—
54	C₂²×C₄×C₁₃₉	Q₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₁₃₉	—	2	4	no	—
64	C₂×C₈×C₁₃₉	—	2	16	no	—
65	C₂×C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₃₉	#41	1	8	yes	—
68	C₄⋊C₄×C₁₃₉	—	2	16	no	—
69	C₄⋊C₄×C₁₃₉	—	4	8	no	—
70	C₄⋊C₄×C₁₃₉	—	4	8	no	—
71	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
72	C₄²×C₁₃₉	—	4	8	no	—
73	C₄²×C₁₃₉	C₄⋊C₄	1	4	yes	—
74	C₄²×C₁₃₉	C₄⋊C₄	1	4	yes	—
75	C₄²×C₁₃₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃₉	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
78	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
79	C₂³×C₁₃₉	—	2	2	no	—
80	C₂³×C₁₃₉	—	2	2	no	—
81	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₁₃₉	C₂×D₈	1	2	yes	—
83	C₂³×C₁₃₉	C₂×Q₁₆	1	2	yes	—
84	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₉	—	2	2	no	—
86	C₂×C₄×C₁₃₉	—	2	2	no	—
87	C₂×C₄×C₁₃₉	—	2	4	no	—
88	C₂×C₄×C₁₃₉	—	2	4	no	—
89	C₂×C₄×C₁₃₉	—	2	8	no	—
90	C₂×C₄×C₁₃₉	—	4	8	no	—
91	C₂×C₄×C₁₃₉	—	4	8	no	—
92	C₂×C₄×C₁₃₉	—	4	8	no	—
93	C₂×C₄×C₁₃₉	#107	1	16	yes	complemented
94	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₁₃₉	#135	1	4	yes	—
104	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂×#41	1	4	yes	—
107	C₈×C₁₃₉	—	2	4	no	—
108	C₈×C₁₃₉	#117	1	8	yes	complemented
109	C₂²×C₁₃₉	—	2	2	no	—
110	C₂²×C₁₃₉	—	2	4	no	—
111	C₂²×C₁₃₉	—	2	4	no	—
112	C₂²×C₁₃₉	—	2	2	no	—
113	C₂²×C₁₃₉	C₂×#107	1	4	yes	—
114	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₃₉	#532	1	2	yes	—
116	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
117	C₂²×C₁₃₉	C₂²×Q₁₆	1	1	yes	characteristic
118	C₂²×C₁₃₉	C₂×#135	1	2	yes	—
119	C₄×C₁₃₉	—	2	2	no	—
120	C₄×C₁₃₉	—	2	2	no	—
121	C₄×C₁₃₉	—	8	8	no	—
122	C₄×C₁₃₉	C₂×#107	1	4	yes	—
123	C₄×C₁₃₉	C₂×#107	1	4	yes	—
124	C₄×C₁₃₉	#427	1	4	yes	—
125	C₄×C₁₃₉	C₂×#117	1	2	yes	—
126	C₄×C₁₃₉	#533	1	2	yes	—
127	C₂×C₁₃₉	—	2	4	no	—
128	C₂×C₁₃₉	#2285	1	1	yes	characteristic
129	C₂×C₁₃₉	C₂²×#107	1	1	yes	characteristic
130	C₂×C₁₃₉	C₂×#427	1	1	yes	characteristic
131	C₂×C₁₃₉	#3169	1	2	yes	—
132	C₂×C₁₃₉	#3170	1	2	yes	—
133	#20817	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₉	#20817	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×C₄×C₈	C₂×C₁₃₉	1	1	yes	characteristic
136	#2285	C₂×C₁₃₉	1	2	yes	—
137	#1845	C₂×C₁₃₉	1	2	yes	complemented
138	#1834	C₂×C₁₃₉	1	2	yes	complemented
139	#1808	C₂×C₁₃₉	1	4	yes	—
140	C₄×#107	C₂×C₁₃₉	1	4	yes	complemented
141	C₂³×C₈	C₂²×C₁₃₉	1	2	yes	—
142	C₂×#117	—	2	2	no	—
143	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
144	C₂×#107	C₂²×C₁₃₉	1	8	yes	—
145	C₂×C₄×C₈	C₄×C₁₃₉	1	8	yes	complemented
146	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
149	#384	C₂²×C₁₃₉	1	4	yes	—
150	#382	—	2	2	no	—
151	#380	—	2	2	no	—
152	#380	C₂²×C₁₃₉	1	4	yes	—
153	C₄×C₄⋊C₄	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
155	#340	C₂²×C₁₃₉	1	8	yes	—
156	C₂³×C₄	D₄×C₁₃₉	1	1	yes	characteristic
157	C₂³×C₄	D₄×C₁₃₉	1	1	yes	characteristic
158	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂×C₄×C₁₃₉	1	8	yes	—
161	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
162	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
163	#117	—	4	8	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	8	yes	—
167	C₂×C₂²⋊C₄	—	2	4	no	—
168	C₂×C₄²	—	2	4	no	—
169	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
170	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
172	C₂×C₄²	Q₈×C₁₃₉	1	4	yes	—
173	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
175	#107	—	2	16	no	—
176	C₄×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
177	C₄×C₈	C₂×C₄×C₁₃₉	1	8	yes	—
178	C₄×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
179	#95	—	2	4	no	—
180	#95	—	2	4	no	—
181	C₂⁴	C₂×D₄×C₁₃₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
186	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
187	C₂²×C₄	D₈×C₁₃₉	1	4	yes	—
188	C₂²×C₄	Q₁₆×C₁₃₉	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	16	no	—
199	C₂×C₈	C₂²×C₄×C₁₃₉	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₁₃₉	1	4	yes	—
201	C₂×C₈	#41×C₁₃₉	1	8	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	4	8	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₄²	—	4	8	no	—
207	C₄²	C₄⋊C₄×C₁₃₉	1	4	yes	—
208	C₄²	C₄⋊C₄×C₁₃₉	1	4	yes	—
209	C₄²	C₂²×C₄×C₁₃₉	1	2	yes	—
210	C₄²	C₂²×C₄×C₁₃₉	1	2	yes	—
211	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
212	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
216	C₂³	C₂×D₈×C₁₃₉	1	2	yes	—
217	C₂³	C₂×Q₁₆×C₁₃₉	1	2	yes	—
218	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	#107×C₁₃₉	1	16	yes	complemented
228	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
235	C₂×C₄	C₂×Q₁₆×C₁₃₉	1	2	yes	—
236	C₂×C₄	C₂×Q₁₆×C₁₃₉	1	2	yes	—
237	C₂×C₄	#135×C₁₃₉	1	4	yes	—
238	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
239	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
240	C₂×C₄	C₂×#41×C₁₃₉	1	4	yes	—
241	C₈	—	2	4	no	—
242	C₈	#117×C₁₃₉	1	8	yes	complemented
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	C₂×#107×C₁₃₉	1	4	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
249	C₂²	#532×C₁₃₉	1	2	yes	—
250	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
251	C₂²	C₂²×Q₁₆×C₁₃₉	1	1	yes	characteristic
252	C₂²	C₂×#135×C₁₃₉	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	8	8	no	—
256	C₄	C₂×#107×C₁₃₉	1	4	yes	—
257	C₄	C₂×#107×C₁₃₉	1	4	yes	—
258	C₄	#427×C₁₃₉	1	4	yes	—
259	C₄	C₂×#117×C₁₃₉	1	2	yes	—
260	C₄	#533×C₁₃₉	1	2	yes	—
261	C₂	—	2	4	no	—
262	C₂	#2285×C₁₃₉	1	1	yes	characteristic
263	C₂	C₂²×#107×C₁₃₉	1	1	yes	characteristic
264	C₂	C₂×#427×C₁₃₉	1	1	yes	characteristic
265	C₂	#3169×C₁₃₉	1	2	yes	—
266	C₂	#3170×C₁₃₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #1845×C₁₃₉	C₂	semidirect
4. #1834×C₁₃₉	C₂	semidirect
6. C₄×#107×C₁₃₉	C₂	semidirect
11. C₂×C₄×C₈×C₁₃₉	C₄	semidirect
93. C₂×C₄×C₁₃₉	#107	semidirect
108. C₈×C₁₃₉	#117	semidirect
133. #20817	C₁₃₉	direct (N × H)
134. C₁₃₉	#20817	direct (N × H)
137. #1845	C₂×C₁₃₉	semidirect
138. #1834	C₂×C₁₃₉	semidirect
140. C₄×#107	C₂×C₁₃₉	semidirect
145. C₂×C₄×C₈	C₄×C₁₃₉	semidirect
227. C₂×C₄	#107×C₁₃₉	semidirect
242. C₈	#117×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃₉	2
#2285×C₁₃₉	2
#1845×C₁₃₉	2
#1834×C₁₃₉	2
#1808×C₁₃₉	2
C₄×#107×C₁₃₉	2
#20817	139

Subgroups — C139×#20817 #48015413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#20817 #48015413