Subgroups — C₂×C₄²×Q₈×C₁₃₉ #48028588

Order 35584.

3966 conjugacy classes of subgroups.

The subgroup lattice has 148 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	C₄²×Q₈×C₁₃₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊C₄×C₁₃₉	C₂	1	9	yes	maximal
4	C₂×C₄³×C₁₃₉	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₉	C₄	1	48	yes	direct product factor
7	C₂×C₄×Q₈×C₁₃₉	C₂²	1	24	yes	—
8	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	9	yes	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	9	yes	—
10	C₂²×C₄²×C₁₃₉	C₂²	1	9	yes	—
11	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	72	yes	—
13	C₂×#95×C₁₃₉	C₂²	1	6	yes	—
14	C₄³×C₁₃₉	C₂²	1	24	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	direct product factor
21	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	72	yes	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	36	yes	—
23	C₂×C₄²×C₁₃₉	—	2	36	no	—
24	C₂×C₄²×C₁₃₉	C₂×C₄	1	72	yes	—
25	C₂×C₄²×C₁₃₉	Q₈	1	8	yes	direct product factor
26	C₂×C₄²×C₁₃₉	C₂³	1	36	yes	—
27	C₂×C₄²×C₁₃₉	C₂³	1	36	yes	—
28	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
29	#95×C₁₃₉	C₂³	1	24	yes	—
30	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
31	C₂×Q₈×C₁₃₉	C₄²	1	64	yes	direct product factor
32	C₂×Q₈×C₁₃₉	C₂²×C₄	1	48	yes	—
33	C₂²×C₄×C₁₃₉	—	2	36	no	—
34	C₂²×C₄×C₁₃₉	C₂²×C₄	1	36	yes	—
35	C₂²×C₄×C₁₃₉	C₂²×C₄	1	36	yes	—
36	C₂²×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
37	C₂²×C₄×C₁₃₉	C₂×Q₈	1	12	yes	—
38	C₂²×C₄×C₁₃₉	#41	1	12	yes	—
39	C₂²×C₄×C₁₃₉	C₂⁴	1	18	yes	—
40	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
41	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
42	C₄⋊C₄×C₁₃₉	C₂²×C₄	1	144	yes	—
43	C₄²×C₁₃₉	—	2	72	no	—
44	C₄²×C₁₃₉	C₂²×C₄	1	144	yes	—
45	C₄²×C₁₃₉	C₂×Q₈	1	16	yes	direct product factor
46	C₂³×C₁₃₉	C₂³×C₄	1	6	yes	—
47	C₂³×C₁₃₉	C₂²×Q₈	1	2	yes	—
48	C₂³×C₁₃₉	C₂×#41	1	6	yes	—
49	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
50	Q₈×C₁₃₉	C₂×C₄²	1	64	yes	direct product factor
51	C₂×C₄×C₁₃₉	—	2	72	no	—
52	C₂×C₄×C₁₃₉	—	2	36	no	—
53	C₂×C₄×C₁₃₉	C₂×C₄²	1	48	yes	—
54	C₂×C₄×C₁₃₉	C₄×Q₈	1	48	yes	direct product factor
55	C₂×C₄×C₁₃₉	C₂³×C₄	1	36	yes	—
56	C₂×C₄×C₁₃₉	C₂³×C₄	1	12	yes	—
57	C₂×C₄×C₁₃₉	C₂³×C₄	1	36	yes	—
58	C₂×C₄×C₁₃₉	C₂²×Q₈	1	12	yes	—
59	C₂×C₄×C₁₃₉	C₂×#41	1	12	yes	—
60	C₂²×C₁₃₉	C₂²×C₄²	1	4	yes	—
61	C₂²×C₁₃₉	C₂×C₄×Q₈	1	12	yes	—
62	C₂²×C₁₃₉	C₄×#41	1	12	yes	—
63	C₂²×C₁₃₉	C₂⁴×C₄	1	3	yes	—
64	C₂²×C₁₃₉	C₂³×Q₈	1	1	yes	characteristic
65	C₂²×C₁₃₉	C₂²×#41	1	3	yes	—
66	C₄×C₁₃₉	—	2	36	no	—
67	C₄×C₁₃₉	C₂²×C₄²	1	24	yes	—
68	C₄×C₁₃₉	C₂×C₄×Q₈	1	24	yes	direct product factor
69	C₂×C₁₃₉	C₄²×Q₈	1	8	yes	direct product factor
70	C₂×C₁₃₉	C₂³×C₄²	1	1	yes	characteristic
71	C₂×C₁₃₉	C₂²×C₄×Q₈	1	3	yes	—
72	C₂×C₁₃₉	C₂×C₄×#41	1	3	yes	—
73	C₂×C₄²×Q₈	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃₉	C₂×C₄²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄×Q₈	C₂×C₁₃₉	1	3	yes	—
76	C₄²×Q₈	C₂×C₁₃₉	1	16	yes	direct product factor
77	C₂×C₄×C₄⋊C₄	C₂×C₁₃₉	1	9	yes	—
78	C₂×C₄³	C₂×C₁₃₉	1	3	yes	—
79	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
80	C₂×C₄×Q₈	C₄×C₁₃₉	1	48	yes	direct product factor
81	C₂×C₄×Q₈	C₂²×C₁₃₉	1	24	yes	—
82	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	9	yes	—
83	C₂²×C₄²	C₂²×C₁₃₉	1	9	yes	—
84	C₂²×C₄²	C₂²×C₁₃₉	1	9	yes	—
85	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	72	yes	—
87	C₂×#95	C₂²×C₁₃₉	1	6	yes	—
88	C₄³	C₂²×C₁₃₉	1	24	yes	—
89	C₂²×Q₈	C₂×C₄×C₁₃₉	1	24	yes	—
90	C₂²×Q₈	C₂³×C₁₃₉	1	4	yes	—
91	C₂³×C₄	C₂³×C₁₃₉	1	9	yes	—
92	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
93	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
94	C₄×Q₈	C₂×C₄×C₁₃₉	1	192	yes	direct product factor
95	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	72	yes	—
96	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	36	yes	—
97	C₂×C₄²	—	2	36	no	—
98	C₂×C₄²	C₂×C₄×C₁₃₉	1	72	yes	—
99	C₂×C₄²	Q₈×C₁₃₉	1	8	yes	direct product factor
100	C₂×C₄²	C₂³×C₁₃₉	1	36	yes	—
101	C₂×C₄²	C₂³×C₁₃₉	1	36	yes	—
102	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
103	#95	C₂³×C₁₃₉	1	24	yes	—
104	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
105	C₂×Q₈	C₄²×C₁₃₉	1	64	yes	direct product factor
106	C₂×Q₈	C₂²×C₄×C₁₃₉	1	48	yes	—
107	C₂²×C₄	—	2	36	no	—
108	C₂²×C₄	C₂²×C₄×C₁₃₉	1	36	yes	—
109	C₂²×C₄	C₂²×C₄×C₁₃₉	1	36	yes	—
110	C₂²×C₄	C₂²×C₄×C₁₃₉	1	12	yes	—
111	C₂²×C₄	C₂×Q₈×C₁₃₉	1	12	yes	—
112	C₂²×C₄	#41×C₁₃₉	1	12	yes	—
113	C₂²×C₄	C₂⁴×C₁₃₉	1	18	yes	—
114	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
115	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
116	C₄⋊C₄	C₂²×C₄×C₁₃₉	1	144	yes	—
117	C₄²	—	2	72	no	—
118	C₄²	C₂²×C₄×C₁₃₉	1	144	yes	—
119	C₄²	C₂×Q₈×C₁₃₉	1	16	yes	direct product factor
120	C₂³	C₂³×C₄×C₁₃₉	1	6	yes	—
121	C₂³	C₂²×Q₈×C₁₃₉	1	2	yes	—
122	C₂³	C₂×#41×C₁₃₉	1	6	yes	—
123	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
124	Q₈	C₂×C₄²×C₁₃₉	1	64	yes	direct product factor
125	C₂×C₄	—	2	72	no	—
126	C₂×C₄	—	2	36	no	—
127	C₂×C₄	C₂×C₄²×C₁₃₉	1	48	yes	—
128	C₂×C₄	C₄×Q₈×C₁₃₉	1	48	yes	direct product factor
129	C₂×C₄	C₂³×C₄×C₁₃₉	1	36	yes	—
130	C₂×C₄	C₂³×C₄×C₁₃₉	1	36	yes	—
131	C₂×C₄	C₂³×C₄×C₁₃₉	1	12	yes	—
132	C₂×C₄	C₂²×Q₈×C₁₃₉	1	12	yes	—
133	C₂×C₄	C₂×#41×C₁₃₉	1	12	yes	—
134	C₂²	C₂²×C₄²×C₁₃₉	1	4	yes	—
135	C₂²	C₂×C₄×Q₈×C₁₃₉	1	12	yes	—
136	C₂²	C₄×#41×C₁₃₉	1	12	yes	—
137	C₂²	C₂⁴×C₄×C₁₃₉	1	3	yes	—
138	C₂²	C₂³×Q₈×C₁₃₉	1	1	yes	characteristic
139	C₂²	C₂²×#41×C₁₃₉	1	3	yes	—
140	C₄	—	2	36	no	—
141	C₄	C₂²×C₄²×C₁₃₉	1	24	yes	—
142	C₄	C₂×C₄×Q₈×C₁₃₉	1	24	yes	direct product factor
143	C₂	C₄²×Q₈×C₁₃₉	1	8	yes	direct product factor
144	C₂	C₂³×C₄²×C₁₃₉	1	1	yes	characteristic
145	C₂	C₂²×C₄×Q₈×C₁₃₉	1	3	yes	—
146	C₂	C₂×C₄×#41×C₁₃₉	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₁₃₉	C₂²
commutator	C₂	C₂³×C₄²×C₁₃₉
Frattini, cosocle	C₂³	C₂⁵×C₁₃₉
socle, prime radical	C₂⁴×C₁₃₉	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈×C₁₃₉	C₂	direct (N × H)
6. C₂×C₄×Q₈×C₁₃₉	C₄	direct (N × H)
20. C₄×Q₈×C₁₃₉	C₂×C₄	direct (N × H)
25. C₂×C₄²×C₁₃₉	Q₈	direct (N × H)
31. C₂×Q₈×C₁₃₉	C₄²	direct (N × H)
45. C₄²×C₁₃₉	C₂×Q₈	direct (N × H)
50. Q₈×C₁₃₉	C₂×C₄²	direct (N × H)
54. C₂×C₄×C₁₃₉	C₄×Q₈	direct (N × H)
68. C₄×C₁₃₉	C₂×C₄×Q₈	direct (N × H)
69. C₂×C₁₃₉	C₄²×Q₈	direct (N × H)
73. C₂×C₄²×Q₈	C₁₃₉	direct (N × H)
74. C₁₃₉	C₂×C₄²×Q₈	direct (N × H)
76. C₄²×Q₈	C₂×C₁₃₉	direct (N × H)
80. C₂×C₄×Q₈	C₄×C₁₃₉	direct (N × H)
94. C₄×Q₈	C₂×C₄×C₁₃₉	direct (N × H)
99. C₂×C₄²	Q₈×C₁₃₉	direct (N × H)
105. C₂×Q₈	C₄²×C₁₃₉	direct (N × H)
119. C₄²	C₂×Q₈×C₁₃₉	direct (N × H)
124. Q₈	C₂×C₄²×C₁₃₉	direct (N × H)
128. C₂×C₄	C₄×Q₈×C₁₃₉	direct (N × H)
142. C₄	C₂×C₄×Q₈×C₁₃₉	direct (N × H)
143. C₂	C₄²×Q₈×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
C₄²×Q₈×C₁₃₉	2
C₂×C₄×C₄⋊C₄×C₁₃₉	2
C₂×C₄³×C₁₃₉	2
C₂×C₄²×Q₈	139

Subgroups — C2×C42×Q8×C139 #48028588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Q₈×C₁₃₉ #48028588