Subgroups — C₁₃₇×#7844 #46857230

Order 35072.

518 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	#1744×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#322×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#97×C₁₃₇	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
12	#336×C₁₃₇	C₄	1	8	yes	—
13	#322×C₁₃₇	C₄	1	8	yes	complemented
14	#322×C₁₃₇	C₄	1	4	yes	complemented
15	C₈²×C₁₃₇	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
23	#97×C₁₃₇	—	2	8	no	—
24	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
32	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
34	C₂×C₈×C₁₃₇	—	2	4	no	—
35	C₂×C₈×C₁₃₇	—	2	8	no	—
36	C₂×C₈×C₁₃₇	—	2	8	no	—
37	C₂×C₈×C₁₃₇	C₄²	1	4	yes	—
38	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
39	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
40	C₂×C₈×C₁₃₇	C₄²	1	4	yes	—
41	C₂×C₈×C₁₃₇	M₁₆	1	8	yes	—
42	C₄²×C₁₃₇	—	2	4	no	—
43	C₄²×C₁₃₇	—	2	8	no	—
44	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₄×C₁₃₇	—	2	8	no	—
50	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
55	C₂×C₄×C₁₃₇	#131	1	8	yes	—
56	C₈×C₁₃₇	—	2	4	no	—
57	C₈×C₁₃₇	—	2	8	no	—
58	C₈×C₁₃₇	—	2	8	no	—
59	C₈×C₁₃₇	#97	1	8	yes	—
60	C₂²×C₁₃₇	C₄³	1	1	yes	characteristic
61	C₂²×C₁₃₇	C₄×M₁₆	1	2	yes	—
62	C₂²×C₁₃₇	#405	1	4	yes	—
63	C₄×C₁₃₇	—	4	4	no	—
64	C₄×C₁₃₇	C₂×#97	1	2	yes	—
65	C₄×C₁₃₇	C₄×M₁₆	1	2	yes	—
66	C₄×C₁₃₇	#405	1	4	yes	—
67	C₄×C₁₃₇	#405	1	4	yes	—
68	C₂×C₁₃₇	#1453	1	4	yes	—
69	C₂×C₁₃₇	C₄×#97	1	1	yes	characteristic
70	C₂×C₁₃₇	#1726	1	2	yes	—
71	#7844	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃₇	#7844	1	1	yes	characteristic, direct product factor, Sylow
73	#1744	C₂×C₁₃₇	1	2	yes	—
74	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
75	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
76	C₂×#322	C₂×C₁₃₇	1	2	yes	—
77	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
78	C₂×#97	C₄×C₁₃₇	1	4	yes	—
79	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
83	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
84	#336	C₄×C₁₃₇	1	8	yes	—
85	#322	C₄×C₁₃₇	1	8	yes	complemented
86	#322	C₄×C₁₃₇	1	4	yes	complemented
87	C₈²	C₄×C₁₃₇	1	4	yes	complemented
88	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
89	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
90	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
92	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
93	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
94	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
95	#97	—	2	8	no	—
96	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
97	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
98	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
99	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
100	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
101	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
102	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
103	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
104	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	8	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	C₄²×C₁₃₇	1	4	yes	—
110	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
111	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
112	C₂×C₈	C₄²×C₁₃₇	1	4	yes	—
113	C₂×C₈	M₁₆×C₁₃₇	1	8	yes	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	8	no	—
116	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
117	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
118	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
119	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
125	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
126	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
127	C₂×C₄	#131×C₁₃₇	1	8	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	8	no	—
130	C₈	—	2	8	no	—
131	C₈	#97×C₁₃₇	1	8	yes	—
132	C₂²	C₄³×C₁₃₇	1	1	yes	characteristic
133	C₂²	C₄×M₁₆×C₁₃₇	1	2	yes	—
134	C₂²	#405×C₁₃₇	1	4	yes	—
135	C₄	—	4	4	no	—
136	C₄	C₂×#97×C₁₃₇	1	2	yes	—
137	C₄	C₄×M₁₆×C₁₃₇	1	2	yes	—
138	C₄	#405×C₁₃₇	1	4	yes	—
139	C₄	#405×C₁₃₇	1	4	yes	—
140	C₂	#1453×C₁₃₇	1	4	yes	—
141	C₂	C₄×#97×C₁₃₇	1	1	yes	characteristic
142	C₂	#1726×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂³
commutator	C₂²	C₄³×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×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
13. #322×C₁₃₇	C₄	semidirect
14. #322×C₁₃₇	C₄	semidirect
15. C₈²×C₁₃₇	C₄	semidirect
71. #7844	C₁₃₇	direct (N × H)
72. C₁₃₇	#7844	direct (N × H)
85. #322	C₄×C₁₃₇	semidirect
86. #322	C₄×C₁₃₇	semidirect
87. C₈²	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1744×C₁₃₇	2
C₄×#97×C₁₃₇	2
C₂×#322×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#7844	137

Subgroups — C137×#7844 #46857230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7844 #46857230