Subgroups — C₁₃₉×#9872 #48004468

Order 35584.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1842×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1744×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1279×C₁₃₉	C₂	1	4	yes	maximal
5	C₂×#125×C₁₃₉	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₁₃₉	—	2	2	no	—
7	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
9	#389×C₁₃₉	C₄	1	2	yes	—
10	#387×C₁₃₉	—	2	2	no	—
11	#382×C₁₃₉	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
14	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
15	#336×C₁₃₉	—	2	2	no	—
16	C₂²×C₈×C₁₃₉	—	2	2	no	—
17	C₂²×C₈×C₁₃₉	—	2	2	no	—
18	#125×C₁₃₉	—	2	2	no	—
19	#125×C₁₃₉	C₈	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃₉	—	2	2	no	—
25	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₁₃₉	—	2	4	no	—
31	C₄⋊C₈×C₁₃₉	—	4	4	no	—
32	C₄×C₈×C₁₃₉	—	2	2	no	—
33	C₄×C₈×C₁₃₉	Q₈	1	4	yes	—
34	#95×C₁₃₉	—	2	2	no	—
35	#95×C₁₃₉	—	2	2	no	—
36	#95×C₁₃₉	—	2	2	no	—
37	C₂²×C₄×C₁₃₉	—	2	2	no	—
38	C₂²×C₄×C₁₃₉	—	2	2	no	—
39	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₉	—	2	4	no	—
45	C₂×C₈×C₁₃₉	—	4	4	no	—
46	C₂×C₈×C₁₃₉	—	4	2	no	—
47	C₂×C₈×C₁₃₉	—	4	4	no	—
48	C₄⋊C₄×C₁₃₉	—	2	8	no	—
49	C₄⋊C₄×C₁₃₉	—	2	2	no	—
50	C₄⋊C₄×C₁₃₉	—	2	4	no	—
51	C₄⋊C₄×C₁₃₉	—	4	4	no	—
52	C₄⋊C₄×C₁₃₉	C₂×C₈	1	4	yes	—
53	C₄²×C₁₃₉	—	2	2	no	—
54	C₄²×C₁₃₉	—	2	2	no	—
55	C₄²×C₁₃₉	—	2	2	no	—
56	C₄²×C₁₃₉	—	4	2	no	—
57	C₄²×C₁₃₉	—	4	2	no	—
58	C₄²×C₁₃₉	—	4	2	no	—
59	C₄²×C₁₃₉	C₂×C₈	1	2	yes	—
60	C₄²×C₁₃₉	M₁₆	1	2	yes	—
61	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
62	C₄²×C₁₃₉	#41	1	2	yes	—
63	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₉	—	2	2	no	—
65	C₂×C₄×C₁₃₉	—	2	2	no	—
66	C₂×C₄×C₁₃₉	—	2	2	no	—
67	C₂×C₄×C₁₃₉	—	2	4	no	—
68	C₂×C₄×C₁₃₉	—	2	2	no	—
69	C₂×C₄×C₁₃₉	—	2	2	no	—
70	C₂×C₄×C₁₃₉	—	2	2	no	—
71	C₂×C₄×C₁₃₉	—	4	2	no	—
72	C₂×C₄×C₁₃₉	—	4	2	no	—
73	C₂×C₄×C₁₃₉	—	4	2	no	—
74	C₂×C₄×C₁₃₉	—	4	2	no	—
75	C₂×C₄×C₁₃₉	—	4	2	no	—
76	C₂×C₄×C₁₃₉	C₂²⋊C₈	1	4	yes	—
77	C₂×C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
78	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃₉	#131	1	2	yes	—
84	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₃₉	#137	1	2	yes	—
86	C₈×C₁₃₉	—	4	4	no	—
87	C₈×C₁₃₉	—	8	2	no	—
88	C₂²×C₁₃₉	#391	1	1	yes	characteristic
89	C₂²×C₁₃₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂²×C₁₃₉	#408	1	1	yes	characteristic
91	C₂²×C₁₃₉	#417	1	2	yes	—
92	C₂²×C₁₃₉	#421	1	2	yes	—
93	C₄×C₁₃₉	—	2	4	no	—
94	C₄×C₁₃₉	—	2	2	no	—
95	C₄×C₁₃₉	—	4	4	no	—
96	C₄×C₁₃₉	—	4	2	no	—
97	C₄×C₁₃₉	—	8	2	no	—
98	C₄×C₁₃₉	C₈×Q₈	1	2	yes	—
99	C₄×C₁₃₉	#446	1	2	yes	—
100	C₄×C₁₃₉	#490	1	1	yes	characteristic
101	C₄×C₁₃₉	#491	1	1	yes	characteristic
102	C₄×C₁₃₉	#501	1	2	yes	—
103	C₂×C₁₃₉	#1477	1	2	yes	—
104	C₂×C₁₃₉	#1495	1	2	yes	—
105	C₂×C₁₃₉	#1984	1	1	yes	characteristic
106	C₂×C₁₃₉	#1992	1	1	yes	characteristic
107	C₂×C₁₃₉	#1995	1	1	yes	characteristic
108	#9872	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₃₉	#9872	1	1	yes	characteristic, direct product factor, Sylow
110	C₄×#125	C₂×C₁₃₉	1	1	yes	characteristic
111	#1842	C₂×C₁₃₉	1	1	yes	characteristic
112	#1744	C₂×C₁₃₉	1	1	yes	characteristic
113	#1279	C₂×C₁₃₉	1	4	yes	—
114	C₂×#125	C₄×C₁₃₉	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	2	2	no	—
116	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
118	#389	C₄×C₁₃₉	1	2	yes	—
119	#387	—	2	2	no	—
120	#382	C₄×C₁₃₉	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
123	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
124	#336	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	#125	—	2	2	no	—
128	#125	C₈×C₁₃₉	1	4	yes	complemented
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
136	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
137	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
138	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	4	4	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	Q₈×C₁₃₉	1	4	yes	—
143	#95	—	2	2	no	—
144	#95	—	2	2	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
149	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	C₂×C₈×C₁₃₉	1	4	yes	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₈×C₁₃₉	1	2	yes	—
169	C₄²	M₁₆×C₁₃₉	1	2	yes	—
170	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
171	C₄²	#41×C₁₃₉	1	2	yes	—
172	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂²⋊C₈×C₁₃₉	1	4	yes	—
186	C₂×C₄	C₄×Q₈×C₁₃₉	1	2	yes	—
187	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
188	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
189	C₂×C₄	C₄⋊Q₈×C₁₃₉	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₈×C₁₃₉	1	1	yes	characteristic
191	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
192	C₂×C₄	#131×C₁₃₉	1	2	yes	—
193	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
194	C₂×C₄	#137×C₁₃₉	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	2	no	—
197	C₂²	#391×C₁₃₉	1	1	yes	characteristic
198	C₂²	C₂×C₂²⋊C₈×C₁₃₉	1	1	yes	characteristic
199	C₂²	#408×C₁₃₉	1	1	yes	characteristic
200	C₂²	#417×C₁₃₉	1	2	yes	—
201	C₂²	#421×C₁₃₉	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	4	no	—
206	C₄	—	8	2	no	—
207	C₄	C₈×Q₈×C₁₃₉	1	2	yes	—
208	C₄	#446×C₁₃₉	1	2	yes	—
209	C₄	#490×C₁₃₉	1	1	yes	characteristic
210	C₄	#491×C₁₃₉	1	1	yes	characteristic
211	C₄	#501×C₁₃₉	1	2	yes	—
212	C₂	#1477×C₁₃₉	1	2	yes	—
213	C₂	#1495×C₁₃₉	1	2	yes	—
214	C₂	#1984×C₁₃₉	1	1	yes	characteristic
215	C₂	#1992×C₁₃₉	1	1	yes	characteristic
216	C₂	#1995×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
socle, prime radical	C₂³×C₁₃₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₉ ◁ C₄³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
19. #125×C₁₃₉	C₈	semidirect
108. #9872	C₁₃₉	direct (N × H)
109. C₁₃₉	#9872	direct (N × H)
128. #125	C₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₁₃₉	2
#1842×C₁₃₉	2
#1744×C₁₃₉	2
#1279×C₁₃₉	2
#9872	139

Subgroups — C139×#9872 #48004468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9872 #48004468