Subgroups — C₁₃₉×#29294 #48023890

Order 35584.

630 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2330×C₁₃₉	C₂	1	2	yes	maximal
2	#2087×C₁₃₉	C₂	1	2	yes	maximal
3	#2086×C₁₃₉	C₂	1	2	yes	maximal
4	#1989×C₁₃₉	C₂	1	4	yes	maximal
5	#1947×C₁₃₉	C₂	1	4	yes	maximal
6	#1843×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₁₃₉	C₂²	1	2	yes	—
8	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
9	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#389×C₁₃₉	—	2	4	no	—
14	#389×C₁₃₉	C₂²	1	4	yes	—
15	#387×C₁₃₉	—	2	4	no	—
16	#384×C₁₃₉	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
18	#376×C₁₃₉	C₂²	1	1	yes	characteristic
19	#340×C₁₃₉	C₂²	1	4	yes	—
20	#340×C₁₃₉	C₂²	1	2	yes	—
21	#340×C₁₃₉	C₂²	1	4	yes	—
22	#340×C₁₃₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
24	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
25	#125×C₁₃₉	—	2	4	no	—
26	#125×C₁₃₉	Q₈	1	8	yes	—
27	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃₉	—	2	4	no	—
33	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
35	#107×C₁₃₉	—	4	2	no	—
36	#107×C₁₃₉	—	4	2	no	—
37	C₄⋊C₈×C₁₃₉	—	2	8	no	—
38	C₄⋊C₈×C₁₃₉	—	4	1	no	—
39	C₄⋊C₈×C₁₃₉	—	4	1	no	—
40	#95×C₁₃₉	—	2	4	no	—
41	#95×C₁₃₉	—	2	4	no	—
42	#95×C₁₃₉	D₄	1	2	yes	—
43	#95×C₁₃₉	D₄	1	2	yes	—
44	C₂²×C₄×C₁₃₉	—	2	4	no	—
45	C₂²×C₄×C₁₃₉	—	2	4	no	—
46	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₁₃₉	—	2	4	no	—
51	C₂×C₈×C₁₃₉	—	2	4	no	—
52	C₂×C₈×C₁₃₉	—	2	2	no	—
53	C₂×C₈×C₁₃₉	—	2	2	no	—
54	C₄⋊C₄×C₁₃₉	—	2	16	no	—
55	C₄⋊C₄×C₁₃₉	—	2	4	no	—
56	C₄⋊C₄×C₁₃₉	—	2	4	no	—
57	C₄⋊C₄×C₁₃₉	—	2	4	no	—
58	C₄⋊C₄×C₁₃₉	—	8	4	no	—
59	C₄⋊C₄×C₁₃₉	—	8	4	no	—
60	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	8	yes	—
61	C₄²×C₁₃₉	—	2	1	no	—
62	C₄²×C₁₃₉	—	2	1	no	—
63	C₄²×C₁₃₉	—	8	4	no	—
64	C₄²×C₁₃₉	C₂×Q₈	1	4	yes	—
65	C₄²×C₁₃₉	#41	1	4	yes	—
66	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃₉	—	2	4	no	—
68	C₂×C₄×C₁₃₉	—	2	4	no	—
69	C₂×C₄×C₁₃₉	—	2	8	no	—
70	C₂×C₄×C₁₃₉	—	2	1	no	—
71	C₂×C₄×C₁₃₉	—	2	1	no	—
72	C₂×C₄×C₁₃₉	—	2	2	no	—
73	C₂×C₄×C₁₃₉	—	2	2	no	—
74	C₂×C₄×C₁₃₉	—	4	4	no	—
75	C₂×C₄×C₁₃₉	—	4	4	no	—
76	C₂×C₄×C₁₃₉	—	4	4	no	—
77	C₂×C₄×C₁₃₉	—	4	4	no	—
78	C₂×C₄×C₁₃₉	—	4	4	no	—
79	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	8	yes	—
80	C₂×C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
82	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
84	C₈×C₁₃₉	—	4	2	no	—
85	C₈×C₁₃₉	—	4	2	no	—
86	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	2	yes	—
87	C₂²×C₁₃₉	#537	1	1	yes	characteristic
88	C₂²×C₁₃₉	#576	1	2	yes	—
89	C₂²×C₁₃₉	#578	1	2	yes	—
90	C₄×C₁₃₉	—	2	8	no	—
91	C₄×C₁₃₉	—	4	8	no	—
92	C₄×C₁₃₉	—	4	1	no	—
93	C₄×C₁₃₉	—	4	1	no	—
94	C₄×C₁₃₉	—	8	4	no	—
95	C₄×C₁₃₉	#557	1	2	yes	—
96	C₄×C₁₃₉	Q₈²	1	1	yes	characteristic
97	C₄×C₁₃₉	#559	1	1	yes	characteristic
98	C₂×C₁₃₉	#2582	1	1	yes	characteristic
99	C₂×C₁₃₉	#3083	1	4	yes	—
100	C₂×C₁₃₉	#3259	1	1	yes	characteristic
101	C₂×C₁₃₉	#3261	1	1	yes	characteristic
102	#29294	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₁₃₉	#29294	1	1	yes	characteristic, direct product factor, Sylow
104	#2330	C₂×C₁₃₉	1	2	yes	—
105	#2087	C₂×C₁₃₉	1	2	yes	—
106	#2086	C₂×C₁₃₉	1	2	yes	—
107	#1989	C₂×C₁₃₉	1	4	yes	—
108	#1947	C₂×C₁₃₉	1	4	yes	—
109	#1843	C₂×C₁₃₉	1	1	yes	characteristic
110	C₂×#125	C₂²×C₁₃₉	1	2	yes	—
111	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
112	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
114	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
116	#389	—	2	4	no	—
117	#389	C₂²×C₁₃₉	1	4	yes	—
118	#387	—	2	4	no	—
119	#384	C₂²×C₁₃₉	1	2	yes	—
120	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
121	#376	C₂²×C₁₃₉	1	1	yes	characteristic
122	#340	C₂²×C₁₃₉	1	4	yes	—
123	#340	C₂²×C₁₃₉	1	2	yes	—
124	#340	C₂²×C₁₃₉	1	4	yes	—
125	#340	C₂²×C₁₃₉	1	2	yes	—
126	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
127	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
128	#125	—	2	4	no	—
129	#125	Q₈×C₁₃₉	1	8	yes	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
134	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
135	C₂×C₄²	—	2	4	no	—
136	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
138	#107	—	4	2	no	—
139	#107	—	4	2	no	—
140	C₄⋊C₈	—	2	8	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄⋊C₈	—	4	1	no	—
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	#95	D₄×C₁₃₉	1	2	yes	—
146	#95	D₄×C₁₃₉	1	2	yes	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
152	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₄⋊C₄	—	2	16	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	8	4	no	—
162	C₄⋊C₄	—	8	4	no	—
163	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	8	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	8	4	no	—
167	C₄²	C₂×Q₈×C₁₃₉	1	4	yes	—
168	C₄²	#41×C₁₃₉	1	4	yes	—
169	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	8	yes	—
183	C₂×C₄	C₂²×Q₈×C₁₃₉	1	2	yes	—
184	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
185	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
186	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	—	4	2	no	—
189	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	2	yes	—
190	C₂²	#537×C₁₃₉	1	1	yes	characteristic
191	C₂²	#576×C₁₃₉	1	2	yes	—
192	C₂²	#578×C₁₃₉	1	2	yes	—
193	C₄	—	2	8	no	—
194	C₄	—	4	8	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	1	no	—
197	C₄	—	8	4	no	—
198	C₄	#557×C₁₃₉	1	2	yes	—
199	C₄	Q₈²×C₁₃₉	1	1	yes	characteristic
200	C₄	#559×C₁₃₉	1	1	yes	characteristic
201	C₂	#2582×C₁₃₉	1	1	yes	characteristic
202	C₂	#3083×C₁₃₉	1	4	yes	—
203	C₂	#3259×C₁₃₉	1	1	yes	characteristic
204	C₂	#3261×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ #376×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
102. #29294	C₁₃₉	direct (N × H)
103. C₁₃₉	#29294	direct (N × H)

Maximal subgroups

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

subgroup	index
#2330×C₁₃₉	2
#2087×C₁₃₉	2
#2086×C₁₃₉	2
#1989×C₁₃₉	2
#1947×C₁₃₉	2
#1843×C₁₃₉	2
#29294	139

Subgroups — C139×#29294 #48023890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29294 #48023890