Subgroups — C₁₃₇×#13553 #46862939

Order 35072.

330 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₃₂×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₃₇	—	2	4	no	—
6	C₂²×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₃₇	C₄	1	2	yes	—
8	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	M₆₄×C₁₃₇	—	2	4	no	—
11	C₂×C₃₂×C₁₃₇	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₁₃₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₇	—	2	2	no	—
14	C₂²×C₈×C₁₃₇	—	2	2	no	—
15	C₂²×C₈×C₁₃₇	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₇	C₈	1	2	yes	—
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₃₇	—	2	2	no	—
20	C₂×C₁₆×C₁₃₇	—	2	4	no	—
21	C₂×C₁₆×C₁₃₇	—	2	4	no	—
22	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₃₇	D₄	1	4	yes	—
25	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₃₂×C₁₃₇	—	2	4	no	—
27	C₂⁴×C₁₃₇	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	—	2	2	no	—
29	C₂²×C₄×C₁₃₇	—	2	2	no	—
30	C₂²×C₄×C₁₃₇	C₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	C₁₆	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃₇	—	2	2	no	—
35	C₂×C₈×C₁₃₇	—	2	4	no	—
36	C₂×C₈×C₁₃₇	—	2	2	no	—
37	C₂×C₈×C₁₃₇	—	2	2	no	—
38	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₁₃₇	—	2	2	no	—
44	C₁₆×C₁₃₇	C₂²⋊C₄	1	2	yes	—
45	C₁₆×C₁₃₇	C₂×D₄	1	2	yes	—
46	C₂³×C₁₃₇	—	2	2	no	—
47	C₂³×C₁₃₇	—	2	4	no	—
48	C₂³×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
49	C₂³×C₁₃₇	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₃₇	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	M₃₂	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₁₃₇	—	2	2	no	—
60	C₈×C₁₃₇	C₂²⋊C₈	1	2	yes	—
61	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₈×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₁₃₇	—	2	8	no	—
64	C₂²×C₁₃₇	—	2	2	no	—
65	C₂²×C₁₃₇	—	2	2	no	—
66	C₂²×C₁₃₇	—	2	2	no	—
67	C₂²×C₁₃₇	C₂²⋊C₁₆	1	2	yes	—
68	C₂²×C₁₃₇	M₆₄	1	4	yes	complemented
69	C₂²×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₁₃₇	—	2	2	no	—
71	C₄×C₁₃₇	C₂²⋊C₁₆	1	2	yes	—
72	C₄×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₄×C₁₃₇	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₁₃₇	—	2	2	no	—
75	C₂×C₁₃₇	—	2	4	no	—
76	C₂×C₁₃₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₂×C₁₃₇	C₂×M₆₄	1	2	yes	—
78	#13553	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₁₃₇	#13553	1	1	yes	characteristic, direct product factor, Sylow
80	C₂³×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
81	C₂×M₆₄	C₂×C₁₃₇	1	2	yes	complemented
82	C₂²⋊C₃₂	C₂×C₁₃₇	1	4	yes	complemented
83	C₂³×C₈	C₄×C₁₃₇	1	1	yes	characteristic
84	C₂²×C₁₆	—	2	4	no	—
85	C₂²×C₁₆	C₄×C₁₃₇	1	1	yes	characteristic
86	C₂²×C₁₆	C₄×C₁₃₇	1	2	yes	—
87	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
89	M₆₄	—	2	4	no	—
90	C₂×C₃₂	C₂²×C₁₃₇	1	4	yes	complemented
91	C₂³×C₄	C₈×C₁₃₇	1	1	yes	characteristic
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	C₈×C₁₃₇	1	1	yes	characteristic
95	C₂²×C₈	C₈×C₁₃₇	1	2	yes	—
96	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
97	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	4	no	—
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
102	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
103	C₂×C₁₆	D₄×C₁₃₇	1	4	yes	—
104	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
105	C₃₂	—	2	4	no	—
106	C₂⁴	C₁₆×C₁₃₇	1	1	yes	characteristic
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₁₆×C₁₃₇	1	1	yes	characteristic
110	C₂²×C₄	C₁₆×C₁₃₇	1	2	yes	—
111	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
112	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
118	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
119	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
120	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
121	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	C₂²⋊C₄×C₁₃₇	1	2	yes	—
124	C₁₆	C₂×D₄×C₁₃₇	1	2	yes	—
125	C₂³	—	2	4	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₂×C₁₆×C₁₃₇	1	2	yes	—
128	C₂³	C₂×C₁₆×C₁₃₇	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	2	yes	—
134	C₂×C₄	C₂×C₁₆×C₁₃₇	1	2	yes	—
135	C₂×C₄	C₂×C₁₆×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	M₃₂×C₁₃₇	1	2	yes	—
137	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	C₂²⋊C₈×C₁₃₇	1	2	yes	—
140	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
141	C₈	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
142	C₂²	—	2	8	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²⋊C₁₆×C₁₃₇	1	2	yes	—
147	C₂²	M₆₄×C₁₃₇	1	4	yes	complemented
148	C₂²	C₂²×C₁₆×C₁₃₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	C₂²⋊C₁₆×C₁₃₇	1	2	yes	—
151	C₄	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
152	C₄	C₂×M₃₂×C₁₃₇	1	1	yes	characteristic
153	C₂	—	2	4	no	—
154	C₂	—	2	2	no	—
155	C₂	C₂×C₂²⋊C₁₆×C₁₃₇	1	1	yes	characteristic
156	C₂	C₂×M₆₄×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₁₃₇	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂²×C₁₆
prime radical	C₂⁴×C₁₃₇	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₁₃₇	C₂	semidirect
3. C₂²⋊C₃₂×C₁₃₇	C₂	semidirect
11. C₂×C₃₂×C₁₃₇	C₂²	semidirect
68. C₂²×C₁₃₇	M₆₄	semidirect
78. #13553	C₁₃₇	direct (N × H)
79. C₁₃₇	#13553	direct (N × H)
81. C₂×M₆₄	C₂×C₁₃₇	semidirect
82. C₂²⋊C₃₂	C₂×C₁₃₇	semidirect
90. C₂×C₃₂	C₂²×C₁₃₇	semidirect
147. C₂²	M₆₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₃₇	2
C₂×M₆₄×C₁₃₇	2
C₂²⋊C₃₂×C₁₃₇	2
#13553	137

Subgroups — C137×#13553 #46862939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13553 #46862939