Subgroups — C₁₃₇×#8811 #46858197

Order 35072.

362 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	#1831×C₁₃₇	C₂	1	2	yes	maximal
2	#1744×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×#334×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
9	#336×C₁₃₇	—	2	4	no	—
10	#336×C₁₃₇	C₄	1	4	yes	—
11	#335×C₁₃₇	—	2	4	no	—
12	#334×C₁₃₇	—	2	4	no	—
13	C₂²×C₈×C₁₃₇	—	2	4	no	—
14	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
18	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₁₃₇	—	2	4	no	—
21	C₄⋊C₈×C₁₃₇	—	2	4	no	—
22	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
24	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
25	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
27	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₃₇	—	2	4	no	—
31	C₂×C₈×C₁₃₇	—	2	4	no	—
32	C₂×C₈×C₁₃₇	—	4	4	no	—
33	C₂×C₈×C₁₃₇	—	4	8	no	—
34	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
36	C₄²×C₁₃₇	—	2	4	no	—
37	C₄²×C₁₃₇	—	2	4	no	—
38	C₄²×C₁₃₇	—	4	2	no	—
39	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄×C₁₃₇	—	2	4	no	—
48	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	#131	1	4	yes	—
55	C₂×C₄×C₁₃₇	#135	1	4	yes	—
56	C₈×C₁₃₇	—	2	4	no	—
57	C₈×C₁₃₇	—	2	4	no	—
58	C₈×C₁₃₇	—	8	4	no	—
59	C₂²×C₁₃₇	#384	1	1	yes	characteristic
60	C₂²×C₁₃₇	#424	1	2	yes	—
61	C₂²×C₁₃₇	#427	1	2	yes	—
62	C₂²×C₁₃₇	#430	1	2	yes	—
63	C₄×C₁₃₇	—	2	1	no	—
64	C₄×C₁₃₇	—	2	1	no	—
65	C₄×C₁₃₇	—	4	4	no	—
66	C₄×C₁₃₇	#424	1	2	yes	—
67	C₄×C₁₃₇	#424	1	2	yes	—
68	C₄×C₁₃₇	#495	1	2	yes	—
69	C₄×C₁₃₇	#499	1	2	yes	—
70	C₂×C₁₃₇	#1572	1	4	yes	—
71	C₂×C₁₃₇	#1843	1	1	yes	characteristic
72	C₂×C₁₃₇	#1846	1	1	yes	characteristic
73	C₂×C₁₃₇	#1849	1	1	yes	characteristic
74	#8811	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃₇	#8811	1	1	yes	characteristic, direct product factor, Sylow
76	#1831	C₂×C₁₃₇	1	2	yes	—
77	#1744	C₂×C₁₃₇	1	1	yes	characteristic
78	C₂×#335	C₂×C₁₃₇	1	2	yes	—
79	C₂×#334	C₂×C₁₃₇	1	2	yes	—
80	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
81	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
83	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
84	#336	—	2	4	no	—
85	#336	C₄×C₁₃₇	1	4	yes	—
86	#335	—	2	4	no	—
87	#334	—	2	4	no	—
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
90	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
91	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
92	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
93	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
94	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	2	4	no	—
97	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
98	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
99	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
100	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
101	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
102	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
104	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	—	4	8	no	—
109	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
110	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
115	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
116	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
117	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
118	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
125	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
129	C₂×C₄	#131×C₁₃₇	1	4	yes	—
130	C₂×C₄	#135×C₁₃₇	1	4	yes	—
131	C₈	—	2	4	no	—
132	C₈	—	2	4	no	—
133	C₈	—	8	4	no	—
134	C₂²	#384×C₁₃₇	1	1	yes	characteristic
135	C₂²	#424×C₁₃₇	1	2	yes	—
136	C₂²	#427×C₁₃₇	1	2	yes	—
137	C₂²	#430×C₁₃₇	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#424×C₁₃₇	1	2	yes	—
142	C₄	#424×C₁₃₇	1	2	yes	—
143	C₄	#495×C₁₃₇	1	2	yes	—
144	C₄	#499×C₁₃₇	1	2	yes	—
145	C₂	#1572×C₁₃₇	1	4	yes	—
146	C₂	#1843×C₁₃₇	1	1	yes	characteristic
147	C₂	#1846×C₁₃₇	1	1	yes	characteristic
148	C₂	#1849×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
74. #8811	C₁₃₇	direct (N × H)
75. C₁₃₇	#8811	direct (N × H)

Maximal subgroups

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

subgroup	index
#1831×C₁₃₇	2
#1744×C₁₃₇	2
C₂×#335×C₁₃₇	2
C₂×#334×C₁₃₇	2
#8811	137

Subgroups — C137×#8811 #46858197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8811 #46858197