Subgroups — C₁₃₇×#8267 #46857653

Order 35072.

390 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1564×C₁₃₇	C₂	1	2	yes	maximal
3	#1452×C₁₃₇	C₂	1	2	yes	maximal
4	#1276×C₁₃₇	C₂	1	2	yes	maximal
5	#431×C₁₃₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
11	#335×C₁₃₇	C₄	1	4	yes	complemented
12	#334×C₁₃₇	C₄	1	4	yes	complemented
13	#322×C₁₃₇	—	2	4	no	—
14	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃₇	—	2	2	no	—
18	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
21	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
22	C₄×C₈×C₁₃₇	—	2	4	no	—
23	C₄×C₈×C₁₃₇	—	2	4	no	—
24	C₄×C₈×C₁₃₇	—	2	4	no	—
25	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₃₇	—	2	2	no	—
30	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
32	C₂×C₈×C₁₃₇	—	2	2	no	—
33	C₂×C₈×C₁₃₇	—	2	2	no	—
34	C₂×C₈×C₁₃₇	—	2	2	no	—
35	C₂×C₈×C₁₃₇	—	2	8	no	—
36	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
37	C₂×C₈×C₁₃₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₃₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
41	C₄²×C₁₃₇	—	2	8	no	—
42	C₄²×C₁₃₇	—	2	2	no	—
43	C₄²×C₁₃₇	M₁₆	1	4	yes	—
44	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
45	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
47	C₂³×C₁₃₇	#117	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	—	2	8	no	—
49	C₂×C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
57	C₈×C₁₃₇	—	2	2	no	—
58	C₈×C₁₃₇	—	2	4	no	—
59	C₈×C₁₃₇	—	4	8	no	—
60	C₈×C₁₃₇	#97	1	2	yes	—
61	C₈×C₁₃₇	#97	1	2	yes	—
62	C₂²×C₁₃₇	—	2	2	no	—
63	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₃₇	#432	1	2	yes	—
65	C₄×C₁₃₇	—	2	8	no	—
66	C₄×C₁₃₇	—	2	2	no	—
67	C₄×C₁₃₇	C₂×#97	1	1	yes	characteristic
68	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
69	C₄×C₁₃₇	#423	1	2	yes	—
70	C₄×C₁₃₇	#443	1	4	yes	—
71	C₂×C₁₃₇	—	2	2	no	—
72	C₂×C₁₃₇	#1767	1	1	yes	characteristic
73	C₂×C₁₃₇	#1772	1	2	yes	—
74	#8267	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃₇	#8267	1	1	yes	characteristic, direct product factor, Sylow
76	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
77	#1564	C₂×C₁₃₇	1	2	yes	—
78	#1452	C₂×C₁₃₇	1	2	yes	—
79	#1276	C₂×C₁₃₇	1	2	yes	—
80	#431	C₂²×C₁₃₇	1	4	yes	—
81	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	—
82	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
85	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
86	#335	C₄×C₁₃₇	1	4	yes	complemented
87	#334	C₄×C₁₃₇	1	4	yes	complemented
88	#322	—	2	4	no	—
89	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
94	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
95	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
96	C₂²⋊C₈	—	2	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
101	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
102	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
103	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
106	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
112	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
114	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
115	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	M₁₆×C₁₃₇	1	4	yes	—
119	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
120	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
121	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
122	C₂³	#117×C₁₃₇	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
129	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
130	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
131	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	2	4	no	—
134	C₈	—	4	8	no	—
135	C₈	#97×C₁₃₇	1	2	yes	—
136	C₈	#97×C₁₃₇	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
139	C₂²	#432×C₁₃₇	1	2	yes	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×#97×C₁₃₇	1	1	yes	characteristic
143	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
144	C₄	#423×C₁₃₇	1	2	yes	—
145	C₄	#443×C₁₃₇	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#1767×C₁₃₇	1	1	yes	characteristic
148	C₂	#1772×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃₇	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #335×C₁₃₇	C₄	semidirect
12. #334×C₁₃₇	C₄	semidirect
74. #8267	C₁₃₇	direct (N × H)
75. C₁₃₇	#8267	direct (N × H)
86. #335	C₄×C₁₃₇	semidirect
87. #334	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₁₃₇	2
#1564×C₁₃₇	2
#1452×C₁₃₇	2
#1276×C₁₃₇	2
#8267	137

Subgroups — C137×#8267 #46857653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8267 #46857653