Subgroups — C₁₃₉×#9801 #48004397

Order 35584.

410 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1997×C₁₃₉	C₂	1	2	yes	maximal
3	#1567×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1549×C₁₃₉	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
7	#500×C₁₃₉	—	2	1	no	—
8	#500×C₁₃₉	—	2	1	no	—
9	#500×C₁₃₉	—	2	2	no	—
10	#431×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₃₉	C₄	1	4	yes	—
12	#423×C₁₃₉	C₂²	1	2	yes	—
13	C₄×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#343×C₁₃₉	—	2	4	no	—
16	#335×C₁₃₉	—	2	2	no	—
17	#332×C₁₃₉	—	2	4	no	—
18	#322×C₁₃₉	—	2	1	no	—
19	C₈²×C₁₃₉	—	2	1	no	—
20	C₂²×Q₈×C₁₃₉	—	2	2	no	—
21	C₂×M₁₆×C₁₃₉	—	2	2	no	—
22	C₂×M₁₆×C₁₃₉	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
24	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
27	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
31	#107×C₁₃₉	—	2	4	no	—
32	#107×C₁₃₉	—	2	4	no	—
33	C₄⋊C₈×C₁₃₉	—	2	2	no	—
34	C₄⋊C₈×C₁₃₉	—	2	2	no	—
35	C₄⋊C₈×C₁₃₉	D₄	1	2	yes	—
36	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
37	#97×C₁₃₉	—	2	1	no	—
38	C₄×C₈×C₁₃₉	—	2	1	no	—
39	C₄×C₈×C₁₃₉	—	2	1	no	—
40	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
43	C₂×Q₈×C₁₃₉	—	4	2	no	—
44	C₂×Q₈×C₁₃₉	—	4	2	no	—
45	C₂×Q₈×C₁₃₉	—	8	4	no	—
46	C₂²×C₄×C₁₃₉	—	2	4	no	—
47	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
50	M₁₆×C₁₃₉	—	2	2	no	—
51	M₁₆×C₁₃₉	—	4	4	no	—
52	M₁₆×C₁₃₉	Q₁₆	1	4	yes	—
53	C₂×C₈×C₁₃₉	—	2	2	no	—
54	C₂×C₈×C₁₃₉	—	2	4	no	—
55	C₂×C₈×C₁₃₉	—	2	2	no	—
56	C₂×C₈×C₁₃₉	—	4	2	no	—
57	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₁₃₉	—	2	4	no	—
60	C₄⋊C₄×C₁₃₉	—	2	4	no	—
61	C₄⋊C₄×C₁₃₉	—	4	2	no	—
62	C₄⋊C₄×C₁₃₉	—	4	2	no	—
63	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃₉	#41	1	1	yes	characteristic
67	C₂³×C₁₃₉	C₄×D₄	1	1	yes	characteristic
68	Q₈×C₁₃₉	—	8	4	no	—
69	Q₈×C₁₃₉	—	8	4	no	—
70	C₂×C₄×C₁₃₉	—	2	2	no	—
71	C₂×C₄×C₁₃₉	—	2	2	no	—
72	C₂×C₄×C₁₃₉	—	4	4	no	—
73	C₂×C₄×C₁₃₉	—	8	4	no	—
74	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₁₃₉	#135	1	2	yes	—
82	C₈×C₁₃₉	—	2	2	no	—
83	C₈×C₁₃₉	—	2	4	no	—
84	C₈×C₁₃₉	—	4	2	no	—
85	C₈×C₁₃₉	—	8	2	no	—
86	C₈×C₁₃₉	#101	1	4	yes	—
87	C₂²×C₁₃₉	—	2	1	no	—
88	C₂²×C₁₃₉	—	2	1	no	—
89	C₂²×C₁₃₉	#390	1	1	yes	characteristic
90	C₂²×C₁₃₉	C₄×Q₁₆	1	2	yes	—
91	C₄×C₁₃₉	—	2	1	no	—
92	C₄×C₁₃₉	—	2	1	no	—
93	C₄×C₁₃₉	—	8	4	no	—
94	C₄×C₁₃₉	C₂×#101	1	2	yes	—
95	C₄×C₁₃₉	#467	1	2	yes	—
96	C₄×C₁₃₉	#472	1	2	yes	—
97	C₄×C₁₃₉	#494	1	1	yes	characteristic
98	C₄×C₁₃₉	#495	1	1	yes	characteristic
99	C₂×C₁₃₉	—	2	2	no	—
100	C₂×C₁₃₉	#1966	1	1	yes	characteristic
101	C₂×C₁₃₉	#1969	1	1	yes	characteristic
102	C₂×C₁₃₉	#1978	1	1	yes	characteristic
103	#9801	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₉	#9801	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#500	C₂×C₁₃₉	1	1	yes	characteristic
106	#1997	C₂×C₁₃₉	1	2	yes	—
107	#1567	C₂×C₁₃₉	1	1	yes	characteristic
108	#1549	C₂×C₁₃₉	1	2	yes	—
109	C₈×M₁₆	C₂×C₁₃₉	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	—	2	2	no	—
114	#431	C₂²×C₁₃₉	1	1	yes	characteristic
115	C₂×#107	C₄×C₁₃₉	1	4	yes	—
116	#423	C₂²×C₁₃₉	1	2	yes	—
117	C₄×M₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
119	#343	—	2	4	no	—
120	#335	—	2	2	no	—
121	#332	—	2	4	no	—
122	#322	—	2	1	no	—
123	C₈²	—	2	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	D₄×C₁₃₉	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₁₃₉	1	2	yes	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
134	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	D₄×C₁₃₉	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
145	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
146	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
153	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	4	4	no	—
156	M₁₆	Q₁₆×C₁₃₉	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
162	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
170	C₄²	#41×C₁₃₉	1	1	yes	characteristic
171	C₂³	C₄×D₄×C₁₃₉	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
180	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
181	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
182	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
183	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₁₆×C₁₃₉	1	2	yes	—
185	C₂×C₄	#135×C₁₃₉	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	#101×C₁₃₉	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#390×C₁₃₉	1	1	yes	characteristic
194	C₂²	C₄×Q₁₆×C₁₃₉	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#101×C₁₃₉	1	2	yes	—
199	C₄	#467×C₁₃₉	1	2	yes	—
200	C₄	#472×C₁₃₉	1	2	yes	—
201	C₄	#494×C₁₃₉	1	1	yes	characteristic
202	C₄	#495×C₁₃₉	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#1966×C₁₃₉	1	1	yes	characteristic
205	C₂	#1969×C₁₃₉	1	1	yes	characteristic
206	C₂	#1978×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	C₂³×C₁₃₉	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
103. #9801	C₁₃₉	direct (N × H)
104. C₁₃₉	#9801	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₉	2
#1997×C₁₃₉	2
#1567×C₁₃₉	2
#1549×C₁₃₉	2
C₈×M₁₆×C₁₃₉	2
#9801	139

Subgroups — C139×#9801 #48004397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9801 #48004397