Subgroups — C₁₃₉×#22739 #48017335

Order 35584.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2355×C₁₃₉	C₂	1	2	yes	maximal
3	#1949×C₁₃₉	C₂	1	4	yes	maximal
4	#1947×C₁₃₉	C₂	1	2	yes	maximal
5	#1835×C₁₃₉	C₂	1	2	yes	maximal
6	#1834×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₁₃₉	C₂	1	2	yes	maximal
8	C₄×#106×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃₉	C₄	1	8	yes	complemented
11	C₂×#107×C₁₃₉	C₂²	1	4	yes	—
12	C₂×#106×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
17	#391×C₁₃₉	—	2	2	no	—
18	#387×C₁₃₉	—	2	4	no	—
19	#384×C₁₃₉	C₂²	1	2	yes	—
20	#382×C₁₃₉	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
24	#340×C₁₃₉	C₂²	1	1	yes	characteristic
25	#340×C₁₃₉	C₂²	1	1	yes	characteristic
26	#340×C₁₃₉	C₂²	1	2	yes	—
27	#340×C₁₃₉	C₂²	1	2	yes	—
28	#340×C₁₃₉	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
33	C₄×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₁₃₉	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₉	—	2	2	no	—
43	C₂×C₄²×C₁₃₉	—	2	4	no	—
44	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
46	#107×C₁₃₉	—	2	8	no	—
47	#106×C₁₃₉	—	2	4	no	—
48	C₄⋊C₈×C₁₃₉	C₂×C₄	1	4	yes	—
49	C₄⋊C₈×C₁₃₉	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₃₉	C₂×C₄	1	8	yes	—
52	#95×C₁₃₉	—	2	4	no	—
53	#95×C₁₃₉	—	2	2	no	—
54	#95×C₁₃₉	—	2	4	no	—
55	#95×C₁₃₉	—	2	4	no	—
56	C₂×Q₈×C₁₃₉	—	2	4	no	—
57	C₂×Q₈×C₁₃₉	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃₉	—	2	2	no	—
59	C₂²×C₄×C₁₃₉	—	2	2	no	—
60	C₂²×C₄×C₁₃₉	—	2	4	no	—
61	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃₉	—	2	2	no	—
67	C₂×C₈×C₁₃₉	—	2	4	no	—
68	C₂×C₈×C₁₃₉	—	2	2	no	—
69	C₂×C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
72	C₄⋊C₄×C₁₃₉	—	2	2	no	—
73	C₄⋊C₄×C₁₃₉	—	2	2	no	—
74	C₄⋊C₄×C₁₃₉	—	2	4	no	—
75	C₄⋊C₄×C₁₃₉	—	2	2	no	—
76	C₄⋊C₄×C₁₃₉	—	2	4	no	—
77	C₄⋊C₄×C₁₃₉	—	4	4	no	—
78	C₄⋊C₄×C₁₃₉	—	8	2	no	—
79	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	8	yes	—
80	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₁₃₉	#41	1	4	yes	—
82	C₄²×C₁₃₉	—	4	4	no	—
83	C₄²×C₁₃₉	—	8	4	no	—
84	C₄²×C₁₃₉	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃₉	C₂×Q₈	1	4	yes	—
87	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
88	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₃₉	—	2	8	no	—
90	C₂×C₄×C₁₃₉	—	2	4	no	—
91	C₂×C₄×C₁₃₉	—	2	2	no	—
92	C₂×C₄×C₁₃₉	—	2	1	no	—
93	C₂×C₄×C₁₃₉	—	2	2	no	—
94	C₂×C₄×C₁₃₉	—	2	1	no	—
95	C₂×C₄×C₁₃₉	—	2	4	no	—
96	C₂×C₄×C₁₃₉	—	4	2	no	—
97	C₂×C₄×C₁₃₉	—	4	2	no	—
98	C₂×C₄×C₁₃₉	—	4	4	no	—
99	C₂×C₄×C₁₃₉	—	4	2	no	—
100	C₂×C₄×C₁₃₉	—	4	2	no	—
101	C₂×C₄×C₁₃₉	—	4	4	no	—
102	C₂×C₄×C₁₃₉	—	4	4	no	—
103	C₂×C₄×C₁₃₉	—	4	2	no	—
104	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃₉	#135	1	4	yes	—
109	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
112	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
113	C₈×C₁₃₉	—	2	2	no	—
114	C₈×C₁₃₉	—	2	4	no	—
115	C₈×C₁₃₉	—	2	2	no	—
116	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃₉	#548	1	2	yes	—
118	C₂²×C₁₃₉	C₂×#135	1	2	yes	—
119	C₂²×C₁₃₉	#576	1	1	yes	characteristic
120	C₂²×C₁₃₉	#578	1	1	yes	characteristic
121	C₄×C₁₃₉	—	2	4	no	—
122	C₄×C₁₃₉	—	2	4	no	—
123	C₄×C₁₃₉	—	2	2	no	—
124	C₄×C₁₃₉	—	4	4	no	—
125	C₄×C₁₃₉	—	8	4	no	—
126	C₄×C₁₃₉	—	8	2	no	—
127	C₄×C₁₃₉	#427	1	4	yes	—
128	C₄×C₁₃₉	C₄×#41	1	1	yes	characteristic
129	C₄×C₁₃₉	#519	1	1	yes	characteristic
130	C₄×C₁₃₉	#557	1	2	yes	—
131	C₂×C₁₃₉	#2351	1	1	yes	characteristic
132	C₂×C₁₃₉	C₂×#427	1	1	yes	characteristic
133	C₂×C₁₃₉	#2913	1	1	yes	characteristic
134	C₂×C₁₃₉	#3361	1	2	yes	—
135	C₂×C₁₃₉	#3364	1	2	yes	—
136	#22739	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₉	#22739	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×C₈×Q₈	C₂×C₁₃₉	1	1	yes	characteristic
139	#2355	C₂×C₁₃₉	1	2	yes	—
140	#1949	C₂×C₁₃₉	1	4	yes	—
141	#1947	C₂×C₁₃₉	1	2	yes	—
142	#1835	C₂×C₁₃₉	1	2	yes	—
143	#1834	C₂×C₁₃₉	1	1	yes	characteristic
144	C₄×#107	C₂×C₁₃₉	1	2	yes	—
145	C₄×#106	C₂×C₁₃₉	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
147	C₈×Q₈	C₄×C₁₃₉	1	8	yes	complemented
148	C₂×#107	C₂²×C₁₃₉	1	4	yes	—
149	C₂×#106	C₂²×C₁₃₉	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
154	#391	—	2	2	no	—
155	#387	—	2	4	no	—
156	#384	C₂²×C₁₃₉	1	2	yes	—
157	#382	C₂²×C₁₃₉	1	4	yes	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
161	#340	C₂²×C₁₃₉	1	2	yes	—
162	#340	C₂²×C₁₃₉	1	2	yes	—
163	#340	C₂²×C₁₃₉	1	1	yes	characteristic
164	#340	C₂²×C₁₃₉	1	4	yes	—
165	#340	C₂²×C₁₃₉	1	1	yes	characteristic
166	C₂²×Q₈	D₄×C₁₃₉	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
170	C₄×Q₈	C₂×C₄×C₁₃₉	1	4	yes	—
171	C₄×Q₈	Q₈×C₁₃₉	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#106	—	2	4	no	—
185	C₄⋊C₈	C₂×C₄×C₁₃₉	1	4	yes	—
186	C₄⋊C₈	C₂×C₄×C₁₃₉	1	8	yes	—
187	C₄×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
188	C₄×C₈	C₂×C₄×C₁₃₉	1	8	yes	—
189	#95	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	4	no	—
192	#95	—	2	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊C₄×C₁₃₉	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²×C₄×C₁₃₉	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₁₃₉	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₃₉	1	4	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	8	yes	—
217	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
218	C₄⋊C₄	#41×C₁₃₉	1	4	yes	—
219	C₄²	—	4	4	no	—
220	C₄²	—	8	4	no	—
221	C₄²	C₂²×C₄×C₁₃₉	1	4	yes	—
222	C₄²	C₂²×C₄×C₁₃₉	1	2	yes	—
223	C₄²	C₂×Q₈×C₁₃₉	1	4	yes	—
224	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
225	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
245	C₂×C₄	#135×C₁₃₉	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
249	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	—	2	2	no	—
253	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
254	C₂²	#548×C₁₃₉	1	2	yes	—
255	C₂²	C₂×#135×C₁₃₉	1	2	yes	—
256	C₂²	#576×C₁₃₉	1	1	yes	characteristic
257	C₂²	#578×C₁₃₉	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	#427×C₁₃₉	1	4	yes	—
265	C₄	C₄×#41×C₁₃₉	1	1	yes	characteristic
266	C₄	#519×C₁₃₉	1	1	yes	characteristic
267	C₄	#557×C₁₃₉	1	2	yes	—
268	C₂	#2351×C₁₃₉	1	1	yes	characteristic
269	C₂	C₂×#427×C₁₃₉	1	1	yes	characteristic
270	C₂	#2913×C₁₃₉	1	1	yes	characteristic
271	C₂	#3361×C₁₃₉	1	2	yes	—
272	C₂	#3364×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂×C₄×Q₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₁₃₉	C₄	semidirect
136. #22739	C₁₃₉	direct (N × H)
137. C₁₃₉	#22739	direct (N × H)
147. C₈×Q₈	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×C₈×Q₈×C₁₃₉	2
#2355×C₁₃₉	2
#1949×C₁₃₉	2
#1947×C₁₃₉	2
#1835×C₁₃₉	2
#1834×C₁₃₉	2
C₄×#107×C₁₃₉	2
C₄×#106×C₁₃₉	2
#22739	139

Subgroups — C139×#22739 #48017335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#22739 #48017335