Subgroups — C₁₃₉×#12678 #48007274

Order 35584.

292 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	#2161×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2155×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#347×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1778×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1773×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#1389×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#1388×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#108×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×#108×C₁₃₉	C₂²	1	1	yes	characteristic
13	#427×C₁₃₉	C₂²	1	1	yes	characteristic
14	#405×C₁₃₉	—	2	1	no	—
15	C₄×M₁₆×C₁₃₉	—	2	1	no	—
16	C₂×#97×C₁₃₉	C₂²	1	1	yes	characteristic
17	#368×C₁₃₉	C₄	1	4	yes	complemented
18	#365×C₁₃₉	C₄	1	4	yes	complemented
19	#347×C₁₃₉	C₄	1	4	yes	complemented
20	#341×C₁₃₉	—	2	1	no	—
21	#339×C₁₃₉	—	2	1	no	—
22	C₂×M₁₆×C₁₃₉	—	2	1	no	—
23	C₂×M₁₆×C₁₃₉	—	2	1	no	—
24	C₂×M₁₆×C₁₃₉	—	2	1	no	—
25	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
27	#117×C₁₃₉	—	2	1	no	—
28	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
29	M₃₂×C₁₃₉	—	2	1	no	—
30	M₃₂×C₁₃₉	C₂×C₄	1	2	yes	—
31	M₃₂×C₁₃₉	C₂×C₄	1	2	yes	—
32	M₃₂×C₁₃₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₁₃₉	—	2	1	no	—
34	C₂×C₁₆×C₁₃₉	C₂×C₄	1	2	yes	—
35	#108×C₁₃₉	—	2	1	no	—
36	#108×C₁₃₉	—	2	1	no	—
37	#108×C₁₃₉	—	2	1	no	—
38	#108×C₁₃₉	C₂×C₄	1	2	yes	—
39	#108×C₁₃₉	C₂×C₄	1	2	yes	—
40	#108×C₁₃₉	C₂×C₄	1	2	yes	—
41	#107×C₁₃₉	—	2	1	no	—
42	#106×C₁₃₉	C₂×C₄	1	2	yes	—
43	#97×C₁₃₉	—	4	1	no	—
44	#97×C₁₃₉	—	4	1	no	—
45	#97×C₁₃₉	C₂×C₄	1	2	yes	—
46	#97×C₁₃₉	Q₈	1	2	yes	—
47	C₄×C₈×C₁₃₉	—	4	1	no	—
48	C₄×C₈×C₁₃₉	—	4	1	no	—
49	C₂²×C₄×C₁₃₉	—	2	1	no	—
50	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₁₃₉	—	2	2	no	—
52	M₁₆×C₁₃₉	—	2	2	no	—
53	M₁₆×C₁₃₉	—	2	2	no	—
54	M₁₆×C₁₃₉	—	4	1	no	—
55	M₁₆×C₁₃₉	—	4	1	no	—
56	M₁₆×C₁₃₉	—	4	1	no	—
57	C₂×C₈×C₁₃₉	—	2	1	no	—
58	C₂×C₈×C₁₃₉	—	2	1	no	—
59	C₂×C₈×C₁₃₉	—	4	1	no	—
60	C₂×C₈×C₁₃₉	—	4	1	no	—
61	C₂×C₈×C₁₃₉	—	4	1	no	—
62	C₂×C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₁₃₉	—	2	2	no	—
70	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
71	C₄²×C₁₃₉	—	2	1	no	—
72	C₄²×C₁₃₉	—	4	1	no	—
73	C₄²×C₁₃₉	C₄⋊C₄	1	2	yes	—
74	C₁₆×C₁₃₉	—	2	2	no	—
75	C₁₆×C₁₃₉	C₄²	1	4	yes	—
76	C₂³×C₁₃₉	#137	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	—	2	2	no	—
78	C₂×C₄×C₁₃₉	—	4	2	no	—
79	C₂×C₄×C₁₃₉	—	8	1	no	—
80	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₁₃₉	#137	1	1	yes	characteristic
86	C₈×C₁₃₉	—	2	2	no	—
87	C₈×C₁₃₉	—	4	2	no	—
88	C₈×C₁₃₉	—	4	2	no	—
89	C₈×C₁₃₉	—	4	2	no	—
90	C₈×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
91	C₈×C₁₃₉	#117	1	1	yes	characteristic
92	C₈×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
94	C₂²×C₁₃₉	—	4	1	no	—
95	C₂²×C₁₃₉	#428	1	1	yes	characteristic
96	C₂²×C₁₃₉	#440	1	1	yes	characteristic
97	C₂²×C₁₃₉	#441	1	1	yes	characteristic
98	C₄×C₁₃₉	—	4	2	no	—
99	C₄×C₁₃₉	—	8	2	no	—
100	C₄×C₁₃₉	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₁₃₉	#428	1	1	yes	characteristic
102	C₄×C₁₃₉	#440	1	1	yes	characteristic
103	C₄×C₁₃₉	#442	1	1	yes	characteristic
104	C₂×C₁₃₉	—	2	1	no	—
105	C₂×C₁₃₉	—	2	1	no	—
106	C₂×C₁₃₉	—	2	1	no	—
107	C₂×C₁₃₉	#1776	1	1	yes	characteristic
108	#12678	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₃₉	#12678	1	1	yes	characteristic, direct product factor, Sylow
110	#2161	C₂×C₁₃₉	1	1	yes	characteristic
111	#2155	C₂×C₁₃₉	1	1	yes	characteristic
112	C₂×#347	C₂×C₁₃₉	1	1	yes	characteristic
113	#1778	C₂×C₁₃₉	1	1	yes	characteristic
114	#1773	C₂×C₁₃₉	1	1	yes	characteristic
115	#1389	C₂×C₁₃₉	1	1	yes	characteristic
116	#1388	C₂×C₁₃₉	1	1	yes	characteristic
117	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
118	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
119	C₂×#108	C₂²×C₁₃₉	1	1	yes	characteristic
120	C₂×#108	C₂²×C₁₃₉	1	1	yes	characteristic
121	C₂×#108	C₂²×C₁₃₉	1	1	yes	characteristic
122	#427	C₂²×C₁₃₉	1	1	yes	characteristic
123	#405	—	2	1	no	—
124	C₄×M₁₆	—	2	1	no	—
125	C₂×#97	C₂²×C₁₃₉	1	1	yes	characteristic
126	#368	C₄×C₁₃₉	1	4	yes	complemented
127	#365	C₄×C₁₃₉	1	4	yes	complemented
128	#347	C₄×C₁₃₉	1	4	yes	complemented
129	#341	—	2	1	no	—
130	#339	—	2	1	no	—
131	C₂×M₁₆	—	2	1	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	2	1	no	—
134	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
136	#117	—	2	1	no	—
137	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
138	M₃₂	—	2	1	no	—
139	M₃₂	C₂×C₄×C₁₃₉	1	2	yes	—
140	M₃₂	C₂×C₄×C₁₃₉	1	2	yes	—
141	M₃₂	C₂×C₄×C₁₃₉	1	2	yes	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	C₂×C₄×C₁₃₉	1	2	yes	—
144	#108	—	2	1	no	—
145	#108	—	2	1	no	—
146	#108	—	2	1	no	—
147	#108	C₂×C₄×C₁₃₉	1	2	yes	—
148	#108	C₂×C₄×C₁₃₉	1	2	yes	—
149	#108	C₂×C₄×C₁₃₉	1	2	yes	—
150	#107	—	2	1	no	—
151	#106	C₂×C₄×C₁₃₉	1	2	yes	—
152	#97	—	4	1	no	—
153	#97	—	4	1	no	—
154	#97	C₂×C₄×C₁₃₉	1	2	yes	—
155	#97	Q₈×C₁₃₉	1	2	yes	—
156	C₄×C₈	—	4	1	no	—
157	C₄×C₈	—	4	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
160	M₁₆	—	2	2	no	—
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	4	1	no	—
164	M₁₆	—	4	1	no	—
165	M₁₆	—	4	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₄⋊C₄×C₁₃₉	1	2	yes	—
172	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
175	C₂×C₈	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
176	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
177	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	4	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	4	1	no	—
182	C₄²	C₄⋊C₄×C₁₃₉	1	2	yes	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₄²×C₁₃₉	1	4	yes	—
185	C₂³	#137×C₁₃₉	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	8	1	no	—
189	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
190	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
192	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
193	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
194	C₂×C₄	#137×C₁₃₉	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	4	2	no	—
197	C₈	—	4	2	no	—
198	C₈	—	4	2	no	—
199	C₈	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
200	C₈	#117×C₁₃₉	1	1	yes	characteristic
201	C₈	C₄×Q₈×C₁₃₉	1	1	yes	characteristic
202	C₈	C₄×Q₈×C₁₃₉	1	1	yes	characteristic
203	C₂²	—	4	1	no	—
204	C₂²	#428×C₁₃₉	1	1	yes	characteristic
205	C₂²	#440×C₁₃₉	1	1	yes	characteristic
206	C₂²	#441×C₁₃₉	1	1	yes	characteristic
207	C₄	—	4	2	no	—
208	C₄	—	8	2	no	—
209	C₄	C₄×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
210	C₄	#428×C₁₃₉	1	1	yes	characteristic
211	C₄	#440×C₁₃₉	1	1	yes	characteristic
212	C₄	#442×C₁₃₉	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#1776×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₉	#442
commutator	C₈	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
socle	C₂×C₁₃₉	#1776
prime radical	C₂³×C₁₃₉	#137
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×#97×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. #368×C₁₃₉	C₄	semidirect
18. #365×C₁₃₉	C₄	semidirect
19. #347×C₁₃₉	C₄	semidirect
108. #12678	C₁₃₉	direct (N × H)
109. C₁₃₉	#12678	direct (N × H)
126. #368	C₄×C₁₃₉	semidirect
127. #365	C₄×C₁₃₉	semidirect
128. #347	C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2161×C₁₃₉	2
#2155×C₁₃₉	2
C₂×#347×C₁₃₉	2
#1778×C₁₃₉	2
#1773×C₁₃₉	2
#1389×C₁₃₉	2
#1388×C₁₃₉	2
#12678	139

Subgroups — C139×#12678 #48007274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#12678 #48007274