Subgroups — C₁₃₇×#8266 #46857652

Order 35072.

414 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	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1564×C₁₃₇	C₂	1	2	yes	maximal
3	#1450×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	2	no	—
14	C₈²×C₁₃₇	—	2	2	no	—
15	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃₇	—	2	2	no	—
19	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
22	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
23	C₄×C₈×C₁₃₇	—	2	4	no	—
24	C₄×C₈×C₁₃₇	—	2	4	no	—
25	C₄×C₈×C₁₃₇	C₈	1	8	yes	complemented
26	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
31	C₂²×C₄×C₁₃₇	—	2	2	no	—
32	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
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	2	yes	—
40	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
44	C₄²×C₁₃₇	—	2	8	no	—
45	C₄²×C₁₃₇	—	2	2	no	—
46	C₄²×C₁₃₇	C₂×C₈	1	4	yes	—
47	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₁₃₇	#117	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	—	2	8	no	—
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	—	2	4	no	—
54	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
60	C₈×C₁₃₇	—	2	2	no	—
61	C₈×C₁₃₇	—	4	8	no	—
62	C₈×C₁₃₇	C₄×C₈	1	2	yes	—
63	C₈×C₁₃₇	C₄×C₈	1	2	yes	—
64	C₈×C₁₃₇	C₄⋊C₈	1	4	yes	—
65	C₈×C₁₃₇	C₄⋊C₈	1	4	yes	—
66	C₂²×C₁₃₇	—	2	2	no	—
67	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₃₇	#431	1	2	yes	—
69	C₄×C₁₃₇	—	2	8	no	—
70	C₄×C₁₃₇	—	2	2	no	—
71	C₄×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
73	C₄×C₁₃₇	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₁₃₇	#443	1	4	yes	—
75	C₂×C₁₃₇	—	2	2	no	—
76	C₂×C₁₃₇	C₄×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₁₃₇	#1772	1	2	yes	—
78	#8266	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₁₃₇	#8266	1	1	yes	characteristic, direct product factor, Sylow
80	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
81	#1564	C₂×C₁₃₇	1	2	yes	—
82	#1450	C₂×C₁₃₇	1	2	yes	—
83	#1276	C₂×C₁₃₇	1	2	yes	—
84	#431	C₂²×C₁₃₇	1	4	yes	—
85	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	—
86	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
89	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
90	#335	C₄×C₁₃₇	1	4	yes	complemented
91	#334	C₄×C₁₃₇	1	4	yes	complemented
92	#322	—	2	2	no	—
93	C₈²	—	2	2	no	—
94	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
96	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
99	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
100	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
101	C₂²⋊C₈	—	2	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	C₈×C₁₃₇	1	8	yes	complemented
105	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
106	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
107	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
108	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
112	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	8	no	—
115	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
116	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
117	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
118	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
119	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
120	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
121	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
122	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×C₈×C₁₃₇	1	4	yes	—
126	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
127	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
128	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
129	C₂³	#117×C₁₃₇	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
136	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
137	C₂×C₄	C₂²×C₈×C₁₃₇	1	2	yes	—
138	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	8	no	—
141	C₈	C₄×C₈×C₁₃₇	1	2	yes	—
142	C₈	C₄×C₈×C₁₃₇	1	2	yes	—
143	C₈	C₄⋊C₈×C₁₃₇	1	4	yes	—
144	C₈	C₄⋊C₈×C₁₃₇	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
147	C₂²	#431×C₁₃₇	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×C₈×C₁₃₇	1	1	yes	characteristic
151	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
152	C₄	C₂×C₄⋊C₈×C₁₃₇	1	2	yes	—
153	C₄	#443×C₁₃₇	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
156	C₂	#1772×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₃₇	D₄
commutator	C₄	C₂×C₄×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃₇	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×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
25. C₄×C₈×C₁₃₇	C₈	semidirect
78. #8266	C₁₃₇	direct (N × H)
79. C₁₃₇	#8266	direct (N × H)
90. #335	C₄×C₁₃₇	semidirect
91. #334	C₄×C₁₃₇	semidirect
104. C₄×C₈	C₈×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
#1450×C₁₃₇	2
#1276×C₁₃₇	2
#8266	137

Subgroups — C137×#8266 #46857652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8266 #46857652