Subgroups — C₁₃₇×#12597 #46861983

Order 35072.

258 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2110×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1747×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1379×C₁₃₇	C₂	1	2	yes	maximal
5	#1378×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
7	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	#405×C₁₃₇	C₂²	1	2	yes	—
10	#405×C₁₃₇	C₂²	1	1	yes	characteristic
11	#347×C₁₃₇	C₄	1	4	yes	complemented
12	#346×C₁₃₇	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₁₃₇	C₄	1	2	yes	complemented
14	#338×C₁₃₇	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₁₃₇	C₄	1	1	yes	characteristic
16	H(3,ℤ/4)×C₁₃₇	C₄	1	2	yes	—
17	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂×M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
20	#117×C₁₃₇	C₂×C₄	1	2	yes	—
21	#117×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₁₃₇	—	2	2	no	—
23	M₃₂×C₁₃₇	—	2	2	no	—
24	M₃₂×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₁₃₇	—	2	2	no	—
26	C₂×C₁₆×C₁₃₇	—	2	2	no	—
27	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
30	#97×C₁₃₇	—	2	2	no	—
31	#97×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃₇	—	2	2	no	—
33	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₁₃₇	—	2	2	no	—
36	M₁₆×C₁₃₇	—	2	2	no	—
37	M₁₆×C₁₃₇	—	2	2	no	—
38	C₂×C₈×C₁₃₇	—	2	2	no	—
39	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
40	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃₇	#41	1	2	yes	—
43	C₂×C₈×C₁₃₇	#41	1	2	yes	—
44	C₄⋊C₄×C₁₃₇	—	2	1	no	—
45	C₄⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
47	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
48	C₄²×C₁₃₇	—	2	2	no	—
49	C₄²×C₁₃₇	C₄²	1	2	yes	—
50	C₁₆×C₁₃₇	—	2	1	no	—
51	C₁₆×C₁₃₇	—	2	1	no	—
52	C₁₆×C₁₃₇	—	2	4	no	—
53	C₁₆×C₁₃₇	—	2	2	no	—
54	C₂³×C₁₃₇	#131	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	2	2	no	—
56	C₂×C₄×C₁₃₇	—	4	2	no	—
57	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	#117	1	2	yes	—
59	C₂×C₄×C₁₃₇	#131	1	2	yes	—
60	C₂×C₄×C₁₃₇	#131	1	1	yes	characteristic
61	C₈×C₁₃₇	—	2	2	no	—
62	C₈×C₁₃₇	—	4	2	no	—
63	C₈×C₁₃₇	#117	1	1	yes	characteristic
64	C₈×C₁₃₇	#117	1	2	yes	—
65	C₈×C₁₃₇	#117	1	1	yes	characteristic
66	C₂²×C₁₃₇	—	4	1	no	—
67	C₂²×C₁₃₇	#405	1	1	yes	characteristic
68	C₂²×C₁₃₇	#433	1	2	yes	—
69	C₄×C₁₃₇	—	4	2	no	—
70	C₄×C₁₃₇	—	8	2	no	—
71	C₄×C₁₃₇	#376	1	1	yes	characteristic
72	C₄×C₁₃₇	#405	1	1	yes	characteristic
73	C₄×C₁₃₇	#433	1	2	yes	—
74	C₂×C₁₃₇	—	2	1	no	—
75	C₂×C₁₃₇	—	2	2	no	—
76	C₂×C₁₃₇	#1746	1	1	yes	characteristic
77	#12597	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₃₇	#12597	1	1	yes	characteristic, direct product factor, Sylow
79	#2110	C₂×C₁₃₇	1	1	yes	characteristic
80	#2108	C₂×C₁₃₇	1	1	yes	characteristic
81	#1747	C₂×C₁₃₇	1	1	yes	characteristic
82	#1379	C₂×C₁₃₇	1	2	yes	—
83	#1378	C₂×C₁₃₇	1	2	yes	—
84	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
85	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
86	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
87	#405	C₂²×C₁₃₇	1	2	yes	—
88	#405	C₂²×C₁₃₇	1	1	yes	characteristic
89	#347	C₄×C₁₃₇	1	4	yes	complemented
90	#346	C₄×C₁₃₇	1	2	yes	complemented
91	C₄×C₁₆	C₄×C₁₃₇	1	2	yes	complemented
92	#338	C₄×C₁₃₇	1	1	yes	characteristic
93	H(3,ℤ/4)	C₄×C₁₃₇	1	1	yes	characteristic
94	H(3,ℤ/4)	C₄×C₁₃₇	1	2	yes	—
95	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
96	C₂×M₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
98	#117	C₂×C₄×C₁₃₇	1	2	yes	—
99	#117	C₂×C₄×C₁₃₇	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	C₂×C₄×C₁₃₇	1	4	yes	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
107	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
108	#97	—	2	2	no	—
109	#97	C₂×C₄×C₁₃₇	1	2	yes	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
112	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
118	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
119	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
120	C₂×C₈	#41×C₁₃₇	1	2	yes	—
121	C₂×C₈	#41×C₁₃₇	1	2	yes	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₄²×C₁₃₇	1	2	yes	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	2	2	no	—
132	C₂³	#131×C₁₃₇	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	#117×C₁₃₇	1	2	yes	—
137	C₂×C₄	#131×C₁₃₇	1	2	yes	—
138	C₂×C₄	#131×C₁₃₇	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	#117×C₁₃₇	1	1	yes	characteristic
142	C₈	#117×C₁₃₇	1	2	yes	—
143	C₈	#117×C₁₃₇	1	1	yes	characteristic
144	C₂²	—	4	1	no	—
145	C₂²	#405×C₁₃₇	1	1	yes	characteristic
146	C₂²	#433×C₁₃₇	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	#376×C₁₃₇	1	1	yes	characteristic
150	C₄	#405×C₁₃₇	1	1	yes	characteristic
151	C₄	#433×C₁₃₇	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	#1746×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₇	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#1746
prime radical	C₂³×C₁₃₇	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #347×C₁₃₇	C₄	semidirect
12. #346×C₁₃₇	C₄	semidirect
13. C₄×C₁₆×C₁₃₇	C₄	semidirect
77. #12597	C₁₃₇	direct (N × H)
78. C₁₃₇	#12597	direct (N × H)
89. #347	C₄×C₁₃₇	semidirect
90. #346	C₄×C₁₃₇	semidirect
91. C₄×C₁₆	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2110×C₁₃₇	2
#2108×C₁₃₇	2
#1747×C₁₃₇	2
#1379×C₁₃₇	2
#1378×C₁₃₇	2
#12597	137

Subgroups — C137×#12597 #46861983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12597 #46861983