Subgroups — C₂×#2276×C₁₃₉ #48028594

Order 35584.

3654 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₃₉	C₂	1	3	yes	maximal
2	#2276×C₁₃₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊C₄×C₁₃₉	C₂	1	9	yes	maximal
4	C₂×#376×C₁₃₉	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₉	C₄	1	48	yes	complemented
7	C₂×C₄×Q₈×C₁₃₉	C₂²	1	24	yes	—
8	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	9	yes	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	6	yes	—
10	C₂²×C₄²×C₁₃₉	C₂²	1	9	yes	—
11	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	72	yes	—
12	#376×C₁₃₉	C₂²	1	24	yes	—
13	C₂×#95×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×#95×C₁₃₉	C₂²	1	9	yes	—
15	C₂²×Q₈×C₁₃₉	C₂×C₄	1	24	yes	—
16	C₂²×Q₈×C₁₃₉	C₂³	1	4	yes	—
17	C₂³×C₄×C₁₃₉	C₂³	1	9	yes	—
18	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
19	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
20	C₄×Q₈×C₁₃₉	C₂×C₄	1	192	yes	complemented
21	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	72	yes	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	36	yes	—
23	C₂×C₄²×C₁₃₉	—	4	12	no	—
24	C₂×C₄²×C₁₃₉	C₂×C₄	1	72	yes	—
25	C₂×C₄²×C₁₃₉	C₂³	1	24	yes	—
26	C₂×C₄²×C₁₃₉	C₂³	1	36	yes	—
27	#95×C₁₃₉	C₂³	1	4	yes	—
28	#95×C₁₃₉	C₂³	1	36	yes	—
29	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₉	C₄²	1	64	yes	complemented
31	C₂×Q₈×C₁₃₉	C₂²×C₄	1	48	yes	—
32	C₂²×C₄×C₁₃₉	—	2	18	no	—
33	C₂²×C₄×C₁₃₉	—	2	12	no	—
34	C₂²×C₄×C₁₃₉	—	2	18	no	—
35	C₂²×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
36	C₂²×C₄×C₁₃₉	C₂²×C₄	1	36	yes	—
37	C₂²×C₄×C₁₃₉	C₂²×C₄	1	36	yes	—
38	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
39	C₂²×C₄×C₁₃₉	C₂⁴	1	18	yes	—
40	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
41	C₄⋊C₄×C₁₃₉	C₂²×C₄	1	144	yes	—
42	C₄²×C₁₃₉	—	4	24	no	—
43	C₄²×C₁₃₉	C₂²×C₄	1	144	yes	—
44	C₂³×C₁₃₉	C₂³×C₄	1	6	yes	—
45	C₂³×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
46	C₂³×C₁₃₉	2₋¹⁺⁴	1	6	yes	—
47	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
48	Q₈×C₁₃₉	C₂×C₄²	1	64	yes	complemented
49	C₂×C₄×C₁₃₉	—	2	24	no	—
50	C₂×C₄×C₁₃₉	—	2	72	no	—
51	C₂×C₄×C₁₃₉	—	2	18	no	—
52	C₂×C₄×C₁₃₉	—	2	12	no	—
53	C₂×C₄×C₁₃₉	—	2	18	no	—
54	C₂×C₄×C₁₃₉	C₂×C₄²	1	48	yes	—
55	C₂×C₄×C₁₃₉	C₂³×C₄	1	12	yes	—
56	C₂×C₄×C₁₃₉	C₂³×C₄	1	36	yes	—
57	C₂×C₄×C₁₃₉	C₂³×C₄	1	36	yes	—
58	C₂²×C₁₃₉	C₂²×C₄²	1	4	yes	—
59	C₂²×C₁₃₉	#519	1	12	yes	—
60	C₂²×C₁₃₉	#520	1	12	yes	—
61	C₂²×C₁₃₉	C₂⁴×C₄	1	3	yes	—
62	C₂²×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²×C₁₃₉	C₂×2₋¹⁺⁴	1	3	yes	—
64	C₄×C₁₃₉	—	2	12	no	—
65	C₄×C₁₃₉	—	2	36	no	—
66	C₄×C₁₃₉	C₂²×C₄²	1	24	yes	—
67	C₂×C₁₃₉	#2276	1	8	yes	direct product factor
68	C₂×C₁₃₉	C₂³×C₄²	1	1	yes	characteristic
69	C₂×C₁₃₉	C₂×#519	1	3	yes	—
70	C₂×C₁₃₉	C₂×#520	1	3	yes	—
71	C₂×#2276	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃₉	C₂×#2276	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₄×Q₈	C₂×C₁₃₉	1	3	yes	—
74	#2276	C₂×C₁₃₉	1	16	yes	direct product factor
75	C₂×C₄×C₄⋊C₄	C₂×C₁₃₉	1	9	yes	—
76	C₂×#376	C₂×C₁₃₉	1	3	yes	—
77	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
78	C₂×C₄×Q₈	C₄×C₁₃₉	1	48	yes	complemented
79	C₂×C₄×Q₈	C₂²×C₁₃₉	1	24	yes	—
80	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	9	yes	—
81	C₂²×C₄²	C₂²×C₁₃₉	1	6	yes	—
82	C₂²×C₄²	C₂²×C₁₃₉	1	9	yes	—
83	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	72	yes	—
84	#376	C₂²×C₁₃₉	1	24	yes	—
85	C₂×#95	C₂²×C₁₃₉	1	1	yes	characteristic
86	C₂×#95	C₂²×C₁₃₉	1	9	yes	—
87	C₂²×Q₈	C₂×C₄×C₁₃₉	1	24	yes	—
88	C₂²×Q₈	C₂³×C₁₃₉	1	4	yes	—
89	C₂³×C₄	C₂³×C₁₃₉	1	9	yes	—
90	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
91	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
92	C₄×Q₈	C₂×C₄×C₁₃₉	1	192	yes	complemented
93	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	72	yes	—
94	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	36	yes	—
95	C₂×C₄²	—	4	12	no	—
96	C₂×C₄²	C₂×C₄×C₁₃₉	1	72	yes	—
97	C₂×C₄²	C₂³×C₁₃₉	1	24	yes	—
98	C₂×C₄²	C₂³×C₁₃₉	1	36	yes	—
99	#95	C₂³×C₁₃₉	1	4	yes	—
100	#95	C₂³×C₁₃₉	1	36	yes	—
101	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
102	C₂×Q₈	C₄²×C₁₃₉	1	64	yes	complemented
103	C₂×Q₈	C₂²×C₄×C₁₃₉	1	48	yes	—
104	C₂²×C₄	—	2	18	no	—
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	—	2	18	no	—
107	C₂²×C₄	C₂²×C₄×C₁₃₉	1	12	yes	—
108	C₂²×C₄	C₂²×C₄×C₁₃₉	1	36	yes	—
109	C₂²×C₄	C₂²×C₄×C₁₃₉	1	36	yes	—
110	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
111	C₂²×C₄	C₂⁴×C₁₃₉	1	18	yes	—
112	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
113	C₄⋊C₄	C₂²×C₄×C₁₃₉	1	144	yes	—
114	C₄²	—	4	24	no	—
115	C₄²	C₂²×C₄×C₁₃₉	1	144	yes	—
116	C₂³	C₂³×C₄×C₁₃₉	1	6	yes	—
117	C₂³	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
118	C₂³	2₋¹⁺⁴×C₁₃₉	1	6	yes	—
119	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
120	Q₈	C₂×C₄²×C₁₃₉	1	64	yes	complemented
121	C₂×C₄	—	2	24	no	—
122	C₂×C₄	—	2	72	no	—
123	C₂×C₄	—	2	18	no	—
124	C₂×C₄	—	2	12	no	—
125	C₂×C₄	—	2	18	no	—
126	C₂×C₄	C₂×C₄²×C₁₃₉	1	48	yes	—
127	C₂×C₄	C₂³×C₄×C₁₃₉	1	12	yes	—
128	C₂×C₄	C₂³×C₄×C₁₃₉	1	36	yes	—
129	C₂×C₄	C₂³×C₄×C₁₃₉	1	36	yes	—
130	C₂²	C₂²×C₄²×C₁₃₉	1	4	yes	—
131	C₂²	#519×C₁₃₉	1	12	yes	—
132	C₂²	#520×C₁₃₉	1	12	yes	—
133	C₂²	C₂⁴×C₄×C₁₃₉	1	3	yes	—
134	C₂²	C₂×H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
135	C₂²	C₂×2₋¹⁺⁴×C₁₃₉	1	3	yes	—
136	C₄	—	2	12	no	—
137	C₄	—	2	36	no	—
138	C₄	C₂²×C₄²×C₁₃₉	1	24	yes	—
139	C₂	#2276×C₁₃₉	1	8	yes	direct product factor
140	C₂	C₂³×C₄²×C₁₃₉	1	1	yes	characteristic
141	C₂	C₂×#519×C₁₃₉	1	3	yes	—
142	C₂	C₂×#520×C₁₃₉	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₉	C₂⁴
commutator	C₂	C₂³×C₄²×C₁₃₉
Frattini, cosocle	C₂³	C₂⁵×C₁₃₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2276×C₁₃₉	C₂	direct (N × H)
6. C₂×C₄×Q₈×C₁₃₉	C₄	semidirect
20. C₄×Q₈×C₁₃₉	C₂×C₄	semidirect
30. C₂×Q₈×C₁₃₉	C₄²	semidirect
48. Q₈×C₁₃₉	C₂×C₄²	semidirect
67. C₂×C₁₃₉	#2276	direct (N × H)
71. C₂×#2276	C₁₃₉	direct (N × H)
72. C₁₃₉	C₂×#2276	direct (N × H)
74. #2276	C₂×C₁₃₉	direct (N × H)
78. C₂×C₄×Q₈	C₄×C₁₃₉	semidirect
92. C₄×Q₈	C₂×C₄×C₁₃₉	semidirect
102. C₂×Q₈	C₄²×C₁₃₉	semidirect
120. Q₈	C₂×C₄²×C₁₃₉	semidirect
139. C₂	#2276×C₁₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₃₉	2
#2276×C₁₃₉	2
C₂×C₄×C₄⋊C₄×C₁₃₉	2
C₂×#376×C₁₃₉	2
C₂×#2276	139

Subgroups — C2×#2276×C139 #48028594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2276×C₁₃₉ #48028594