Subgroups — C₂×#1454×C₁₃₇ #46866708

Order 35072.

802 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	3	yes	maximal
2	#1454×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×#322×C₁₃₇	C₂	1	3	yes	maximal, complemented
4	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
6	#431×C₁₃₇	C₄	1	12	yes	—
7	#431×C₁₃₇	C₂²	1	12	yes	—
8	C₂×C₄⋊C₈×C₁₃₇	C₄	1	6	yes	—
9	C₂×C₂²⋊C₈×C₁₃₇	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
12	#322×C₁₃₇	C₂²	1	12	yes	complemented
13	C₈²×C₁₃₇	C₂²	1	4	yes	complemented
14	C₂³×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
18	C₂×C₄²×C₁₃₇	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
20	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂×C₄	1	24	yes	—
22	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	24	yes	—
23	C₄×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
24	C₄×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
25	C₄×C₈×C₁₃₇	C₂³	1	6	yes	—
26	C₄×C₈×C₁₃₇	C₂³	1	6	yes	—
27	C₂⁴×C₁₃₇	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	—	2	12	no	—
29	C₂²×C₄×C₁₃₇	C₄²	1	3	yes	—
30	C₂²×C₄×C₁₃₇	C₄²	1	6	yes	—
31	C₂²×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
32	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
33	C₂×C₈×C₁₃₇	—	2	12	no	—
34	C₂×C₈×C₁₃₇	—	2	12	no	—
35	C₂×C₈×C₁₃₇	C₂²×C₄	1	12	yes	—
36	C₂×C₈×C₁₃₇	C₂²×C₄	1	12	yes	—
37	C₄²×C₁₃₇	C₄²	1	6	yes	—
38	C₄²×C₁₃₇	C₂²×C₄	1	3	yes	—
39	C₄²×C₁₃₇	C₂²×C₄	1	6	yes	—
40	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₂³×C₁₃₇	—	2	6	no	—
42	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
43	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	—	2	6	no	—
45	C₂×C₄×C₁₃₇	—	2	12	no	—
46	C₂×C₄×C₁₃₇	—	2	12	no	—
47	C₂×C₄×C₁₃₇	C₂×C₄²	1	6	yes	—
48	C₂×C₄×C₁₃₇	C₂×C₄²	1	3	yes	—
49	C₂×C₄×C₁₃₇	#131	1	12	yes	—
50	C₂×C₄×C₁₃₇	C₂³×C₄	1	3	yes	—
51	C₈×C₁₃₇	—	2	12	no	—
52	C₈×C₁₃₇	—	2	12	no	—
53	C₂²×C₁₃₇	—	2	6	no	—
54	C₂²×C₁₃₇	—	4	4	no	—
55	C₂²×C₁₃₇	#405	1	6	yes	—
56	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
57	C₄×C₁₃₇	—	2	6	no	—
58	C₄×C₁₃₇	#405	1	6	yes	—
59	C₄×C₁₃₇	C₂×#131	1	6	yes	—
60	C₂×C₁₃₇	—	4	2	no	—
61	C₂×C₁₃₇	#1454	1	4	yes	direct product factor
62	C₂×C₁₃₇	C₂×#405	1	3	yes	—
63	C₂×#1454	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₃₇	C₂×#1454	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×#431	C₂×C₁₃₇	1	3	yes	—
66	#1454	C₂×C₁₃₇	1	8	yes	direct product factor
67	C₂×#322	C₂×C₁₃₇	1	3	yes	complemented
68	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic, complemented
69	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
70	#431	C₄×C₁₃₇	1	12	yes	—
71	#431	C₂²×C₁₃₇	1	12	yes	—
72	C₂×C₄⋊C₈	C₄×C₁₃₇	1	6	yes	—
73	C₂×C₂²⋊C₈	C₄×C₁₃₇	1	6	yes	—
74	C₂×C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
75	C₂×C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
76	#322	C₂²×C₁₃₇	1	12	yes	complemented
77	C₈²	C₂²×C₁₃₇	1	4	yes	complemented
78	C₂³×C₄	C₂×C₄×C₁₃₇	1	3	yes	—
79	C₂²×C₈	C₂×C₄×C₁₃₇	1	6	yes	—
80	C₂²×C₈	C₂×C₄×C₁₃₇	1	6	yes	—
81	C₂×C₄²	C₂×C₄×C₁₃₇	1	3	yes	—
82	C₂×C₄²	C₂×C₄×C₁₃₇	1	6	yes	—
83	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
84	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
85	C₄⋊C₈	C₂×C₄×C₁₃₇	1	24	yes	—
86	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	24	yes	—
87	C₄×C₈	C₂×C₄×C₁₃₇	1	6	yes	—
88	C₄×C₈	C₂×C₄×C₁₃₇	1	6	yes	—
89	C₄×C₈	C₂³×C₁₃₇	1	6	yes	—
90	C₄×C₈	C₂³×C₁₃₇	1	6	yes	—
91	C₂⁴	C₄²×C₁₃₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	C₄²×C₁₃₇	1	3	yes	—
94	C₂²×C₄	C₄²×C₁₃₇	1	6	yes	—
95	C₂²×C₄	C₂²×C₄×C₁₃₇	1	6	yes	—
96	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
97	C₂×C₈	—	2	12	no	—
98	C₂×C₈	—	2	12	no	—
99	C₂×C₈	C₂²×C₄×C₁₃₇	1	12	yes	—
100	C₂×C₈	C₂²×C₄×C₁₃₇	1	12	yes	—
101	C₄²	C₄²×C₁₃₇	1	6	yes	—
102	C₄²	C₂²×C₄×C₁₃₇	1	3	yes	—
103	C₄²	C₂²×C₄×C₁₃₇	1	6	yes	—
104	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
105	C₂³	—	2	6	no	—
106	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
107	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	2	12	no	—
111	C₂×C₄	C₂×C₄²×C₁₃₇	1	6	yes	—
112	C₂×C₄	C₂×C₄²×C₁₃₇	1	3	yes	—
113	C₂×C₄	#131×C₁₃₇	1	12	yes	—
114	C₂×C₄	C₂³×C₄×C₁₃₇	1	3	yes	—
115	C₈	—	2	12	no	—
116	C₈	—	2	12	no	—
117	C₂²	—	2	6	no	—
118	C₂²	—	4	4	no	—
119	C₂²	#405×C₁₃₇	1	6	yes	—
120	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
121	C₄	—	2	6	no	—
122	C₄	#405×C₁₃₇	1	6	yes	—
123	C₄	C₂×#131×C₁₃₇	1	6	yes	—
124	C₂	—	4	2	no	—
125	C₂	#1454×C₁₃₇	1	4	yes	direct product factor
126	C₂	C₂×#405×C₁₃₇	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂×C₄²
prime radical	C₂⁴×C₁₃₇	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1454×C₁₃₇	C₂	direct (N × H)
3. C₂×#322×C₁₃₇	C₂	semidirect
4. C₂×C₈²×C₁₃₇	C₂	semidirect
12. #322×C₁₃₇	C₂²	semidirect
13. C₈²×C₁₃₇	C₂²	semidirect
61. C₂×C₁₃₇	#1454	direct (N × H)
63. C₂×#1454	C₁₃₇	direct (N × H)
64. C₁₃₇	C₂×#1454	direct (N × H)
66. #1454	C₂×C₁₃₇	direct (N × H)
67. C₂×#322	C₂×C₁₃₇	semidirect
68. C₂×C₈²	C₂×C₁₃₇	semidirect
76. #322	C₂²×C₁₃₇	semidirect
77. C₈²	C₂²×C₁₃₇	semidirect
125. C₂	#1454×C₁₃₇	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×#431×C₁₃₇	2
#1454×C₁₃₇	2
C₂×#322×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
C₂×#1454	137

Subgroups — C2×#1454×C137 #46866708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1454×C₁₃₇ #46866708