Subgroups — C₁₃₉×#14664 #48009260

Order 35584.

1334 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₁₃₉	C₂	1	2	yes	maximal
3	#1445×C₁₃₉	C₂	1	4	yes	maximal
4	#1441×C₁₃₉	C₂	1	8	yes	maximal
5	C₂×#125×C₁₃₉	—	2	8	no	—
6	C₂×#125×C₁₃₉	C₄	1	8	yes	complemented
7	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
8	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
10	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
11	#382×C₁₃₉	—	2	8	no	—
12	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
13	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
14	C₂×#95×C₁₃₉	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₃₉	C₂³	1	8	yes	—
16	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
17	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
18	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
19	#125×C₁₃₉	—	2	32	no	—
20	C₂×C₄⋊C₄×C₁₃₉	—	2	8	no	—
21	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₃₉	—	2	8	no	—
24	C₂×C₄⋊C₄×C₁₃₉	—	2	8	no	—
25	C₂×C₄⋊C₄×C₁₃₉	—	4	8	no	—
26	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	16	yes	—
27	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃₉	D₄	1	8	yes	—
30	#95×C₁₃₉	—	2	8	no	—
31	#95×C₁₃₉	—	2	8	no	—
32	#95×C₁₃₉	—	4	16	no	—
33	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₉	—	2	4	no	—
35	C₂²×C₄×C₁₃₉	—	2	4	no	—
36	C₂²×C₄×C₁₃₉	—	2	4	no	—
37	C₂²×C₄×C₁₃₉	—	2	4	no	—
38	C₂²×C₄×C₁₃₉	—	2	4	no	—
39	C₂²×C₄×C₁₃₉	—	2	8	no	—
40	C₂²×C₄×C₁₃₉	—	2	8	no	—
41	C₂²×C₄×C₁₃₉	—	2	8	no	—
42	C₂²×C₄×C₁₃₉	—	2	8	no	—
43	C₂²×C₄×C₁₃₉	—	2	16	no	—
44	C₂²×C₄×C₁₃₉	—	2	16	no	—
45	C₂²×C₄×C₁₃₉	—	2	8	no	—
46	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
47	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
52	C₄⋊C₄×C₁₃₉	—	2	16	no	—
53	C₄⋊C₄×C₁₃₉	—	2	32	no	—
54	C₄⋊C₄×C₁₃₉	—	4	16	no	—
55	C₄²×C₁₃₉	C₂²⋊C₄	1	16	yes	—
56	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
57	C₂³×C₁₃₉	C₂²×D₄	1	2	yes	—
58	C₂³×C₁₃₉	H(5,𝔽₂)	1	4	yes	—
59	C₂³×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
60	C₂³×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
61	C₂×C₄×C₁₃₉	—	2	4	no	—
62	C₂×C₄×C₁₃₉	—	2	8	no	—
63	C₂×C₄×C₁₃₉	—	2	4	no	—
64	C₂×C₄×C₁₃₉	—	2	8	no	—
65	C₂×C₄×C₁₃₉	—	2	4	no	—
66	C₂×C₄×C₁₃₉	—	2	8	no	—
67	C₂×C₄×C₁₃₉	—	2	8	no	—
68	C₂×C₄×C₁₃₉	—	2	16	no	—
69	C₂×C₄×C₁₃₉	—	4	4	no	—
70	C₂×C₄×C₁₃₉	—	4	4	no	—
71	C₂×C₄×C₁₃₉	—	4	8	no	—
72	C₂×C₄×C₁₃₉	—	4	8	no	—
73	C₂×C₄×C₁₃₉	—	4	8	no	—
74	C₂×C₄×C₁₃₉	—	4	8	no	—
75	C₂×C₄×C₁₃₉	—	4	16	no	—
76	C₂×C₄×C₁₃₉	—	4	16	no	—
77	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	8	yes	—
79	C₂²×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₁₃₉	#519	1	2	yes	—
81	C₂²×C₁₃₉	#520	1	4	yes	—
82	C₂²×C₁₃₉	#535	1	2	yes	—
83	C₂²×C₁₃₉	#536	1	4	yes	—
84	C₂²×C₁₃₉	#536	1	2	yes	—
85	C₂²×C₁₃₉	#537	1	4	yes	—
86	C₂²×C₁₃₉	#562	1	4	yes	—
87	C₂²×C₁₃₉	#563	1	8	yes	—
88	C₂²×C₁₃₉	#564	1	4	yes	—
89	C₄×C₁₃₉	—	2	4	no	—
90	C₄×C₁₃₉	—	2	8	no	—
91	C₄×C₁₃₉	—	4	8	no	—
92	C₄×C₁₃₉	—	8	8	no	—
93	C₂×C₁₃₉	#2310	1	1	yes	characteristic
94	C₂×C₁₃₉	#2311	1	2	yes	—
95	C₂×C₁₃₉	#2382	1	4	yes	—
96	C₂×C₁₃₉	#2820	1	4	yes	—
97	C₂×C₁₃₉	#2822	1	4	yes	—
98	#14664	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₃₉	#14664	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×#125	C₂×C₁₃₉	1	1	yes	characteristic
101	C₂×#382	C₂×C₁₃₉	1	2	yes	—
102	#1445	C₂×C₁₃₉	1	4	yes	—
103	#1441	C₂×C₁₃₉	1	8	yes	—
104	C₂×#125	—	2	8	no	—
105	C₂×#125	C₄×C₁₃₉	1	8	yes	complemented
106	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
107	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
108	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
109	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
110	#382	—	2	8	no	—
111	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
112	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
113	C₂×#95	C₂²×C₁₃₉	1	16	yes	—
114	C₂³×C₄	C₂³×C₁₃₉	1	8	yes	—
115	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
116	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
117	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
118	#125	—	2	32	no	—
119	C₂×C₄⋊C₄	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	8	no	—
123	C₂×C₄⋊C₄	—	2	8	no	—
124	C₂×C₄⋊C₄	—	4	8	no	—
125	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	16	yes	—
126	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	8	yes	—
127	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
128	C₂×C₄²	D₄×C₁₃₉	1	8	yes	—
129	#95	—	2	8	no	—
130	#95	—	2	8	no	—
131	#95	—	4	16	no	—
132	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	2	16	no	—
143	C₂²×C₄	—	2	16	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
148	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
149	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
150	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
151	C₄⋊C₄	—	2	16	no	—
152	C₄⋊C₄	—	2	32	no	—
153	C₄⋊C₄	—	4	16	no	—
154	C₄²	C₂²⋊C₄×C₁₃₉	1	16	yes	—
155	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
156	C₂³	C₂²×D₄×C₁₃₉	1	2	yes	—
157	C₂³	H(5,𝔽₂)×C₁₃₉	1	4	yes	—
158	C₂³	2₋¹⁺⁴×C₁₃₉	1	4	yes	—
159	C₂³	2₋¹⁺⁴×C₁₃₉	1	4	yes	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	4	16	no	—
175	C₂×C₄	—	4	16	no	—
176	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	8	yes	—
178	C₂²	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
179	C₂²	#519×C₁₃₉	1	2	yes	—
180	C₂²	#520×C₁₃₉	1	4	yes	—
181	C₂²	#535×C₁₃₉	1	2	yes	—
182	C₂²	#536×C₁₃₉	1	4	yes	—
183	C₂²	#536×C₁₃₉	1	2	yes	—
184	C₂²	#537×C₁₃₉	1	4	yes	—
185	C₂²	#562×C₁₃₉	1	4	yes	—
186	C₂²	#563×C₁₃₉	1	8	yes	—
187	C₂²	#564×C₁₃₉	1	4	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	2	8	no	—
190	C₄	—	4	8	no	—
191	C₄	—	8	8	no	—
192	C₂	#2310×C₁₃₉	1	1	yes	characteristic
193	C₂	#2311×C₁₃₉	1	2	yes	—
194	C₂	#2382×C₁₃₉	1	4	yes	—
195	C₂	#2820×C₁₃₉	1	4	yes	—
196	C₂	#2822×C₁₃₉	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₂×#125×C₁₃₉	C₄	semidirect
98. #14664	C₁₃₉	direct (N × H)
99. C₁₃₉	#14664	direct (N × H)
105. C₂×#125	C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₃₉	2
C₂×#382×C₁₃₉	2
#1445×C₁₃₉	2
#1441×C₁₃₉	2
#14664	139

Subgroups — C139×#14664 #48009260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#14664 #48009260