Subgroups — C₁₃₉×#13797 #48008393

Order 35584.

1438 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	C₂×C₄×C₄⋊C₄×C₁₃₉	C₂	1	2	yes	maximal
2	C₂×#376×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#95×C₁₃₉	C₂	1	4	yes	maximal
4	#1434×C₁₃₉	C₂	1	8	yes	maximal
5	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃₉	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃₉	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₁₃₉	C₄	1	16	yes	complemented
10	#376×C₁₃₉	C₄	1	8	yes	complemented
11	C₂×#95×C₁₃₉	C₂²	1	16	yes	—
12	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
13	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃₉	C₂³	1	8	yes	—
15	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
16	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
17	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₉	—	2	16	no	—
19	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	16	yes	—
20	C₂×C₄²×C₁₃₉	—	2	8	no	—
21	C₂×C₄²×C₁₃₉	—	4	8	no	—
22	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₁₃₉	C₂×C₄	1	16	yes	—
24	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
26	#95×C₁₃₉	—	4	16	no	—
27	#95×C₁₃₉	C₂×C₄	1	16	yes	—
28	#95×C₁₃₉	C₂×C₄	1	16	yes	—
29	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₉	—	2	4	no	—
31	C₂²×C₄×C₁₃₉	—	2	4	no	—
32	C₂²×C₄×C₁₃₉	—	2	4	no	—
33	C₂²×C₄×C₁₃₉	—	2	4	no	—
34	C₂²×C₄×C₁₃₉	—	2	16	no	—
35	C₂²×C₄×C₁₃₉	—	2	8	no	—
36	C₂²×C₄×C₁₃₉	—	2	8	no	—
37	C₂²×C₄×C₁₃₉	—	2	16	no	—
38	C₂²×C₄×C₁₃₉	—	2	8	no	—
39	C₂²×C₄×C₁₃₉	—	2	8	no	—
40	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₁₃₉	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₃₉	#41	1	8	yes	—
48	C₄⋊C₄×C₁₃₉	—	2	32	no	—
49	C₄²×C₁₃₉	—	2	16	no	—
50	C₄²×C₁₃₉	—	4	16	no	—
51	C₄²×C₁₃₉	C₄²	1	16	yes	—
52	C₂³×C₁₃₉	C₂³×C₄	1	2	yes	—
53	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
54	C₂³×C₁₃₉	C₂×#41	1	4	yes	—
55	C₂³×C₁₃₉	H(5,𝔽₂)	1	4	yes	—
56	C₂³×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
57	C₂×C₄×C₁₃₉	—	2	4	no	—
58	C₂×C₄×C₁₃₉	—	2	4	no	—
59	C₂×C₄×C₁₃₉	—	2	8	no	—
60	C₂×C₄×C₁₃₉	—	2	8	no	—
61	C₂×C₄×C₁₃₉	—	2	8	no	—
62	C₂×C₄×C₁₃₉	—	2	16	no	—
63	C₂×C₄×C₁₃₉	—	2	16	no	—
64	C₂×C₄×C₁₃₉	—	2	16	no	—
65	C₂×C₄×C₁₃₉	—	2	16	no	—
66	C₂×C₄×C₁₃₉	—	4	4	no	—
67	C₂×C₄×C₁₃₉	—	4	16	no	—
68	C₂×C₄×C₁₃₉	—	4	4	no	—
69	C₂×C₄×C₁₃₉	—	4	8	no	—
70	C₂×C₄×C₁₃₉	—	4	16	no	—
71	C₂×C₄×C₁₃₉	C₂×C₄²	1	8	yes	—
72	C₂×C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₁₃₉	#117	1	8	yes	—
74	C₂×C₄×C₁₃₉	#117	1	16	yes	—
75	C₂²×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
76	C₂²×C₁₃₉	C₂×#117	1	2	yes	—
77	C₂²×C₁₃₉	C₂×#117	1	4	yes	—
78	C₂²×C₁₃₉	#518	1	2	yes	—
79	C₂²×C₁₃₉	#519	1	2	yes	—
80	C₂²×C₁₃₉	#520	1	4	yes	—
81	C₂²×C₁₃₉	#520	1	4	yes	—
82	C₂²×C₁₃₉	#540	1	4	yes	—
83	C₂²×C₁₃₉	#541	1	4	yes	—
84	C₂²×C₁₃₉	#542	1	8	yes	—
85	C₄×C₁₃₉	—	2	8	no	—
86	C₄×C₁₃₉	—	4	8	no	—
87	C₄×C₁₃₉	—	4	16	no	—
88	C₄×C₁₃₉	#376	1	8	yes	—
89	C₂×C₁₃₉	C₂×#376	1	1	yes	characteristic
90	C₂×C₁₃₉	#2274	1	2	yes	—
91	C₂×C₁₃₉	#2314	1	4	yes	—
92	C₂×C₁₃₉	#2319	1	4	yes	—
93	C₂×C₁₃₉	#2320	1	4	yes	—
94	#13797	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₉	#13797	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×C₄×C₄⋊C₄	C₂×C₁₃₉	1	2	yes	—
97	C₂×#376	C₂×C₁₃₉	1	1	yes	characteristic
98	C₄⋊#95	C₂×C₁₃₉	1	4	yes	—
99	#1434	C₂×C₁₃₉	1	8	yes	—
100	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
101	C₂²×C₄²	C₂²×C₁₃₉	1	2	yes	—
102	C₂²×C₄²	C₂²×C₁₃₉	1	4	yes	—
103	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
104	C₄×C₄⋊C₄	C₄×C₁₃₉	1	16	yes	complemented
105	#376	C₄×C₁₃₉	1	8	yes	complemented
106	C₂×#95	C₂²×C₁₃₉	1	16	yes	—
107	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
108	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
109	C₂³×C₄	C₂³×C₁₃₉	1	8	yes	—
110	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
111	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
112	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	16	no	—
114	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	16	yes	—
115	C₂×C₄²	—	2	8	no	—
116	C₂×C₄²	—	4	8	no	—
117	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
118	C₂×C₄²	C₂×C₄×C₁₃₉	1	16	yes	—
119	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
121	#95	—	4	16	no	—
122	#95	C₂×C₄×C₁₃₉	1	16	yes	—
123	#95	C₂×C₄×C₁₃₉	1	16	yes	—
124	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	16	no	—
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
136	C₂²×C₄	C₂²×C₄×C₁₃₉	1	16	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
142	C₂²×C₄	#41×C₁₃₉	1	8	yes	—
143	C₄⋊C₄	—	2	32	no	—
144	C₄²	—	2	16	no	—
145	C₄²	—	4	16	no	—
146	C₄²	C₄²×C₁₃₉	1	16	yes	—
147	C₂³	C₂³×C₄×C₁₃₉	1	2	yes	—
148	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
149	C₂³	C₂×#41×C₁₃₉	1	4	yes	—
150	C₂³	H(5,𝔽₂)×C₁₃₉	1	4	yes	—
151	C₂³	2₋¹⁺⁴×C₁₃₉	1	4	yes	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	2	16	no	—
159	C₂×C₄	—	2	16	no	—
160	C₂×C₄	—	2	16	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	16	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	16	no	—
166	C₂×C₄	C₂×C₄²×C₁₃₉	1	8	yes	—
167	C₂×C₄	C₂×C₄²×C₁₃₉	1	4	yes	—
168	C₂×C₄	#117×C₁₃₉	1	8	yes	—
169	C₂×C₄	#117×C₁₃₉	1	16	yes	—
170	C₂²	C₂²×C₄²×C₁₃₉	1	1	yes	characteristic
171	C₂²	C₂×#117×C₁₃₉	1	2	yes	—
172	C₂²	C₂×#117×C₁₃₉	1	4	yes	—
173	C₂²	#518×C₁₃₉	1	2	yes	—
174	C₂²	#519×C₁₃₉	1	2	yes	—
175	C₂²	#520×C₁₃₉	1	4	yes	—
176	C₂²	#520×C₁₃₉	1	4	yes	—
177	C₂²	#540×C₁₃₉	1	4	yes	—
178	C₂²	#541×C₁₃₉	1	4	yes	—
179	C₂²	#542×C₁₃₉	1	8	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	4	8	no	—
182	C₄	—	4	16	no	—
183	C₄	#376×C₁₃₉	1	8	yes	—
184	C₂	C₂×#376×C₁₃₉	1	1	yes	characteristic
185	C₂	#2274×C₁₃₉	1	2	yes	—
186	C₂	#2314×C₁₃₉	1	4	yes	—
187	C₂	#2319×C₁₃₉	1	4	yes	—
188	C₂	#2320×C₁₃₉	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄×C₄⋊C₄×C₁₃₉	C₄	semidirect
10. #376×C₁₃₉	C₄	semidirect
94. #13797	C₁₃₉	direct (N × H)
95. C₁₃₉	#13797	direct (N × H)
104. C₄×C₄⋊C₄	C₄×C₁₃₉	semidirect
105. #376	C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₁₃₉	2
C₂×#376×C₁₃₉	2
C₄⋊#95×C₁₃₉	2
#1434×C₁₃₉	2
#13797	139

Subgroups — C139×#13797 #48008393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#13797 #48008393