Subgroups — C₁₃₉×#10705 #48005301

Order 35584.

870 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₉	C₂	1	2	yes	maximal
2	C₂²×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₁₃₉	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₁₃₉	C₂²	1	4	yes	—
5	C₂³×C₈×C₁₃₉	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₁₃₉	—	2	4	no	—
8	C₄×M₁₆×C₁₃₉	C₄	1	8	yes	—
9	C₂×#97×C₁₃₉	—	2	4	no	—
10	C₂×#97×C₁₃₉	—	2	2	no	—
11	C₂×#97×C₁₃₉	C₄	1	4	yes	—
12	C₂×C₄×C₈×C₁₃₉	—	2	2	no	—
13	#341×C₁₃₉	C₄	1	16	yes	complemented
14	C₂³×C₄×C₁₃₉	Q₈	1	2	yes	—
15	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₉	—	2	4	no	—
17	C₂×M₁₆×C₁₃₉	—	2	8	no	—
18	C₂×M₁₆×C₁₃₉	C₂×C₄	1	8	yes	—
19	C₂×M₁₆×C₁₃₉	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₁₃₉	—	2	4	no	—
21	C₂²×C₈×C₁₃₉	—	2	4	no	—
22	C₂²×C₈×C₁₃₉	—	2	4	no	—
23	C₂²×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃₉	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃₉	D₄	1	4	yes	—
28	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
29	#97×C₁₃₉	—	2	8	no	—
30	#97×C₁₃₉	—	2	8	no	—
31	C₄×C₈×C₁₃₉	—	2	8	no	—
32	C₂⁴×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₉	—	4	4	no	—
34	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	8	yes	—
36	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
42	M₁₆×C₁₃₉	—	2	8	no	—
43	M₁₆×C₁₃₉	—	2	16	no	—
44	M₁₆×C₁₃₉	C₄²	1	16	yes	—
45	C₂×C₈×C₁₃₉	—	2	8	no	—
46	C₂×C₈×C₁₃₉	—	2	4	no	—
47	C₂×C₈×C₁₃₉	—	2	4	no	—
48	C₂×C₈×C₁₃₉	—	2	16	no	—
49	C₂×C₈×C₁₃₉	—	2	16	no	—
50	C₂×C₈×C₁₃₉	C₄²	1	8	yes	—
51	C₄²×C₁₃₉	C₄²	1	4	yes	—
52	C₄²×C₁₃₉	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₁₃₉	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₁₃₉	C₂²⋊C₄	1	4	yes	—
55	C₂³×C₁₃₉	—	2	4	no	—
56	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₁₃₉	C₄×Q₈	1	4	yes	—
59	C₂×C₄×C₁₃₉	—	2	8	no	—
60	C₂×C₄×C₁₃₉	—	4	8	no	—
61	C₂×C₄×C₁₃₉	#95	1	8	yes	—
62	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
64	C₂×C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
65	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₃₉	#117	1	2	yes	—
68	C₂×C₄×C₁₃₉	#117	1	4	yes	—
69	C₂×C₄×C₁₃₉	C₄×D₄	1	4	yes	—
70	C₂×C₄×C₁₃₉	C₄×D₄	1	8	yes	—
71	C₈×C₁₃₉	—	2	8	no	—
72	C₈×C₁₃₉	—	2	16	no	—
73	C₂²×C₁₃₉	—	2	4	no	—
74	C₂²×C₁₃₉	—	2	8	no	—
75	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
76	C₂²×C₁₃₉	C₄×C₄⋊C₄	1	4	yes	—
77	C₂²×C₁₃₉	C₄×C₄⋊C₄	1	2	yes	—
78	C₂²×C₁₃₉	#430	1	4	yes	—
79	C₄×C₁₃₉	—	2	8	no	—
80	C₄×C₁₃₉	C₄³	1	1	yes	characteristic
81	C₄×C₁₃₉	#376	1	1	yes	characteristic
82	C₄×C₁₃₉	C₄×C₂²⋊C₄	1	2	yes	—
83	C₄×C₁₃₉	C₄×C₂²⋊C₄	1	4	yes	—
84	C₂×C₁₃₉	—	2	4	no	—
85	C₂×C₁₃₉	C₄×#95	1	1	yes	characteristic
86	C₂×C₁₃₉	#1738	1	2	yes	—
87	C₂×C₁₃₉	#1778	1	4	yes	—
88	#10705	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₃₉	#10705	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×C₄×M₁₆	C₂×C₁₃₉	1	2	yes	—
91	C₂²×#97	C₂×C₁₃₉	1	1	yes	characteristic
92	C₂×#341	C₂×C₁₃₉	1	4	yes	—
93	C₂²×M₁₆	C₂²×C₁₃₉	1	4	yes	—
94	C₂³×C₈	C₂²×C₁₃₉	1	2	yes	—
95	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
96	C₄×M₁₆	—	2	4	no	—
97	C₄×M₁₆	C₄×C₁₃₉	1	8	yes	—
98	C₂×#97	—	2	4	no	—
99	C₂×#97	—	2	2	no	—
100	C₂×#97	C₄×C₁₃₉	1	4	yes	—
101	C₂×C₄×C₈	—	2	2	no	—
102	#341	C₄×C₁₃₉	1	16	yes	complemented
103	C₂³×C₄	Q₈×C₁₃₉	1	2	yes	—
104	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	—	2	8	no	—
107	C₂×M₁₆	C₂×C₄×C₁₃₉	1	16	yes	—
108	C₂×M₁₆	C₂×C₄×C₁₃₉	1	8	yes	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	—	2	4	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₂×C₄×C₁₃₉	1	8	yes	—
113	C₂²×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
114	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
116	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
117	C₂×C₄²	D₄×C₁₃₉	1	4	yes	—
118	#97	—	2	8	no	—
119	#97	—	2	8	no	—
120	C₄×C₈	—	2	8	no	—
121	C₂⁴	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
124	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	8	yes	—
125	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
128	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
130	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
131	M₁₆	—	2	8	no	—
132	M₁₆	—	2	16	no	—
133	M₁₆	C₄²×C₁₃₉	1	16	yes	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	16	no	—
138	C₂×C₈	—	2	16	no	—
139	C₂×C₈	C₄²×C₁₃₉	1	8	yes	—
140	C₄²	C₄²×C₁₃₉	1	4	yes	—
141	C₄²	C₂²⋊C₄×C₁₃₉	1	4	yes	—
142	C₄²	C₂²⋊C₄×C₁₃₉	1	4	yes	—
143	C₄²	C₂²⋊C₄×C₁₃₉	1	4	yes	—
144	C₂³	—	2	4	no	—
145	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
147	C₂³	C₄×Q₈×C₁₃₉	1	4	yes	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	#95×C₁₃₉	1	8	yes	—
151	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄²×C₁₃₉	1	2	yes	—
153	C₂×C₄	C₂×C₄²×C₁₃₉	1	4	yes	—
154	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
156	C₂×C₄	#117×C₁₃₉	1	2	yes	—
157	C₂×C₄	#117×C₁₃₉	1	4	yes	—
158	C₂×C₄	C₄×D₄×C₁₃₉	1	4	yes	—
159	C₂×C₄	C₄×D₄×C₁₃₉	1	8	yes	—
160	C₈	—	2	8	no	—
161	C₈	—	2	16	no	—
162	C₂²	—	2	8	no	—
163	C₂²	—	2	4	no	—
164	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
165	C₂²	C₄×C₄⋊C₄×C₁₃₉	1	4	yes	—
166	C₂²	C₄×C₄⋊C₄×C₁₃₉	1	2	yes	—
167	C₂²	#430×C₁₃₉	1	4	yes	—
168	C₄	—	2	8	no	—
169	C₄	C₄³×C₁₃₉	1	1	yes	characteristic
170	C₄	#376×C₁₃₉	1	1	yes	characteristic
171	C₄	C₄×C₂²⋊C₄×C₁₃₉	1	2	yes	—
172	C₄	C₄×C₂²⋊C₄×C₁₃₉	1	4	yes	—
173	C₂	—	2	4	no	—
174	C₂	C₄×#95×C₁₃₉	1	1	yes	characteristic
175	C₂	#1738×C₁₃₉	1	2	yes	—
176	C₂	#1778×C₁₃₉	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₉	C₂×D₄
commutator	C₄	C₄³×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
socle	C₂³×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₉	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. #341×C₁₃₉	C₄	semidirect
88. #10705	C₁₃₉	direct (N × H)
89. C₁₃₉	#10705	direct (N × H)
102. #341	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×M₁₆×C₁₃₉	2
C₂²×#97×C₁₃₉	2
C₂×#341×C₁₃₉	2
#10705	139

Subgroups — C139×#10705 #48005301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#10705 #48005301