Subgroups — C₁₃₉×#9897 #48004493

Order 35584.

430 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2859×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1831×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1296×C₁₃₉	C₂	1	4	yes	maximal
5	C₂×#125×C₁₃₉	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃₉	—	2	1	no	—
9	#389×C₁₃₉	—	2	2	no	—
10	#389×C₁₃₉	C₄	1	4	yes	—
11	#384×C₁₃₉	C₂²	1	2	yes	—
12	#382×C₁₃₉	—	2	1	no	—
13	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
14	#336×C₁₃₉	—	2	1	no	—
15	#336×C₁₃₉	—	2	2	no	—
16	C₂²×C₈×C₁₃₉	—	2	2	no	—
17	C₂²×C₈×C₁₃₉	—	2	2	no	—
18	#125×C₁₃₉	—	4	4	no	—
19	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₁₃₉	—	2	2	no	—
29	C₄⋊C₈×C₁₃₉	—	2	4	no	—
30	C₄⋊C₈×C₁₃₉	Q₈	1	4	yes	—
31	C₄×C₈×C₁₃₉	—	4	2	no	—
32	#95×C₁₃₉	—	2	4	no	—
33	C₂²×C₄×C₁₃₉	—	2	4	no	—
34	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃₉	—	2	4	no	—
40	C₂×C₈×C₁₃₉	—	4	2	no	—
41	C₂×C₈×C₁₃₉	—	4	2	no	—
42	C₂×C₈×C₁₃₉	—	4	2	no	—
43	C₂×C₈×C₁₃₉	—	4	4	no	—
44	C₄⋊C₄×C₁₃₉	—	4	4	no	—
45	C₄⋊C₄×C₁₃₉	—	4	8	no	—
46	C₄⋊C₄×C₁₃₉	—	4	4	no	—
47	C₄⋊C₄×C₁₃₉	—	4	4	no	—
48	C₄²×C₁₃₉	—	2	4	no	—
49	C₄²×C₁₃₉	—	2	2	no	—
50	C₄²×C₁₃₉	—	4	2	no	—
51	C₄²×C₁₃₉	SD₁₆	1	2	yes	—
52	C₄²×C₁₃₉	Q₁₆	1	2	yes	—
53	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
54	C₄²×C₁₃₉	#41	1	2	yes	—
55	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	—	2	2	no	—
58	C₂×C₄×C₁₃₉	—	2	2	no	—
59	C₂×C₄×C₁₃₉	—	2	2	no	—
60	C₂×C₄×C₁₃₉	—	2	2	no	—
61	C₂×C₄×C₁₃₉	—	2	2	no	—
62	C₂×C₄×C₁₃₉	—	4	4	no	—
63	C₂×C₄×C₁₃₉	—	4	4	no	—
64	C₂×C₄×C₁₃₉	—	4	4	no	—
65	C₂×C₄×C₁₃₉	#103	1	4	yes	—
66	C₂×C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₃₉	#135	1	4	yes	—
73	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₁₃₉	—	4	4	no	—
75	C₈×C₁₃₉	—	8	2	no	—
76	C₂²×C₁₃₉	#391	1	1	yes	characteristic
77	C₂²×C₁₃₉	#413	1	2	yes	—
78	C₂²×C₁₃₉	C₂×#103	1	1	yes	characteristic
79	C₂²×C₁₃₉	#416	1	2	yes	—
80	C₂²×C₁₃₉	#418	1	1	yes	characteristic
81	C₄×C₁₃₉	—	2	4	no	—
82	C₄×C₁₃₉	—	2	2	no	—
83	C₄×C₁₃₉	—	4	2	no	—
84	C₄×C₁₃₉	—	8	4	no	—
85	C₄×C₁₃₉	#476	1	2	yes	—
86	C₄×C₁₃₉	#477	1	2	yes	—
87	C₄×C₁₃₉	#488	1	2	yes	—
88	C₄×C₁₃₉	#499	1	2	yes	—
89	C₂×C₁₃₉	#1542	1	2	yes	—
90	C₂×C₁₃₉	#1553	1	2	yes	—
91	C₂×C₁₃₉	#1987	1	1	yes	characteristic
92	C₂×C₁₃₉	#1991	1	1	yes	characteristic
93	C₂×C₁₃₉	#1998	1	1	yes	characteristic
94	#9897	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₉	#9897	1	1	yes	characteristic, direct product factor, Sylow
96	#2859	C₂×C₁₃₉	1	1	yes	characteristic
97	#1831	C₂×C₁₃₉	1	1	yes	characteristic
98	C₄×C₄⋊C₈	C₂×C₁₃₉	1	1	yes	characteristic
99	#1296	C₂×C₁₃₉	1	4	yes	—
100	C₂×#125	—	2	1	no	—
101	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
103	C₂×C₄×C₈	—	2	1	no	—
104	#389	—	2	2	no	—
105	#389	C₄×C₁₃₉	1	4	yes	—
106	#384	C₂²×C₁₃₉	1	2	yes	—
107	#382	—	2	1	no	—
108	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
109	#336	—	2	2	no	—
110	#336	—	2	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	#125	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
118	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
119	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
121	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	Q₈×C₁₃₉	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
130	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₁₃₉	1	2	yes	—
147	C₄²	Q₁₆×C₁₃₉	1	2	yes	—
148	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
149	C₄²	#41×C₁₃₉	1	2	yes	—
150	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#103×C₁₃₉	1	4	yes	—
161	C₂×C₄	C₄×Q₈×C₁₃₉	1	2	yes	—
162	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
163	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
164	C₂×C₄	C₄⋊Q₈×C₁₃₉	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₁₃₉	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₃₉	1	4	yes	—
168	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	8	2	no	—
171	C₂²	#391×C₁₃₉	1	1	yes	characteristic
172	C₂²	#413×C₁₃₉	1	2	yes	—
173	C₂²	C₂×#103×C₁₃₉	1	1	yes	characteristic
174	C₂²	#416×C₁₃₉	1	2	yes	—
175	C₂²	#418×C₁₃₉	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#476×C₁₃₉	1	2	yes	—
181	C₄	#477×C₁₃₉	1	2	yes	—
182	C₄	#488×C₁₃₉	1	2	yes	—
183	C₄	#499×C₁₃₉	1	2	yes	—
184	C₂	#1542×C₁₃₉	1	2	yes	—
185	C₂	#1553×C₁₃₉	1	2	yes	—
186	C₂	#1987×C₁₃₉	1	1	yes	characteristic
187	C₂	#1991×C₁₃₉	1	1	yes	characteristic
188	C₂	#1998×C₁₃₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
94. #9897	C₁₃₉	direct (N × H)
95. C₁₃₉	#9897	direct (N × H)

Maximal subgroups

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

subgroup	index
#2859×C₁₃₉	2
#1831×C₁₃₉	2
C₄×C₄⋊C₈×C₁₃₉	2
#1296×C₁₃₉	2
#9897	139

Subgroups — C139×#9897 #48004493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9897 #48004493