Subgroups — C₁₃₇×#8781 #46858167

Order 35072.

310 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1726×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1564×C₁₃₇	C₂	1	2	yes	maximal
3	#1281×C₁₃₇	C₂	1	2	yes	maximal
4	#1274×C₁₃₇	C₂	1	2	yes	maximal
5	#431×C₁₃₇	C₂²	1	4	yes	—
6	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#376×C₁₃₇	C₂²	1	1	yes	characteristic
13	#335×C₁₃₇	—	2	2	no	—
14	#334×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	1	yes	characteristic
18	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₇	—	2	4	no	—
23	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
24	#97×C₁₃₇	—	2	2	no	—
25	#97×C₁₃₇	—	2	2	no	—
26	#97×C₁₃₇	C₈	1	4	yes	complemented
27	C₄×C₈×C₁₃₇	—	2	2	no	—
28	C₄×C₈×C₁₃₇	—	2	2	no	—
29	C₄×C₈×C₁₃₇	—	2	4	no	—
30	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
31	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
32	#95×C₁₃₇	—	2	2	no	—
33	C₂²×C₄×C₁₃₇	—	2	1	no	—
34	C₂²×C₄×C₁₃₇	—	2	1	no	—
35	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
37	C₂×C₈×C₁₃₇	—	2	1	no	—
38	C₂×C₈×C₁₃₇	—	2	1	no	—
39	C₂×C₈×C₁₃₇	—	2	1	no	—
40	C₂×C₈×C₁₃₇	—	2	1	no	—
41	C₂×C₈×C₁₃₇	—	2	1	no	—
42	C₂×C₈×C₁₃₇	—	2	1	no	—
43	C₂×C₈×C₁₃₇	—	4	4	no	—
44	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
50	C₄²×C₁₃₇	—	2	1	no	—
51	C₄²×C₁₃₇	—	2	1	no	—
52	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
53	C₄²×C₁₃₇	M₁₆	1	2	yes	—
54	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₃₇	#41	1	2	yes	—
57	C₂³×C₁₃₇	#117	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄×C₁₃₇	—	4	2	no	—
61	C₂×C₄×C₁₃₇	—	4	2	no	—
62	C₂×C₄×C₁₃₇	—	4	2	no	—
63	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	#124	1	2	yes	—
66	C₂×C₄×C₁₃₇	#125	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₁₃₇	—	2	2	no	—
72	C₈×C₁₃₇	—	2	2	no	—
73	C₈×C₁₃₇	—	2	2	no	—
74	C₈×C₁₃₇	—	8	4	no	—
75	C₈×C₁₃₇	C₄⋊C₈	1	4	yes	—
76	C₂²×C₁₃₇	—	2	2	no	—
77	C₂²×C₁₃₇	#382	1	1	yes	characteristic
78	C₂²×C₁₃₇	#431	1	1	yes	characteristic
79	C₂²×C₁₃₇	#432	1	1	yes	characteristic
80	C₄×C₁₃₇	—	2	2	no	—
81	C₄×C₁₃₇	—	2	2	no	—
82	C₄×C₁₃₇	—	4	2	no	—
83	C₄×C₁₃₇	—	4	2	no	—
84	C₄×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₁₃₇	#423	1	1	yes	characteristic
86	C₄×C₁₃₇	#431	1	1	yes	characteristic
87	C₄×C₁₃₇	#432	1	1	yes	characteristic
88	C₂×C₁₃₇	—	2	1	no	—
89	C₂×C₁₃₇	—	2	1	no	—
90	C₂×C₁₃₇	#1831	1	1	yes	characteristic
91	C₂×C₁₃₇	#1833	1	2	yes	—
92	#8781	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₁₃₇	#8781	1	1	yes	characteristic, direct product factor, Sylow
94	#1726	C₂×C₁₃₇	1	1	yes	characteristic
95	#1564	C₂×C₁₃₇	1	2	yes	—
96	#1281	C₂×C₁₃₇	1	2	yes	—
97	#1274	C₂×C₁₃₇	1	2	yes	—
98	#431	C₂²×C₁₃₇	1	4	yes	—
99	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
100	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
101	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
105	#376	C₂²×C₁₃₇	1	1	yes	characteristic
106	#335	—	2	2	no	—
107	#334	—	2	2	no	—
108	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₂²⋊C₈	—	2	4	no	—
117	#97	—	2	2	no	—
118	#97	—	2	2	no	—
119	#97	C₈×C₁₃₇	1	4	yes	complemented
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
124	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
138	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
139	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
140	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
141	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
142	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
146	C₄²	M₁₆×C₁₃₇	1	2	yes	—
147	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
148	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
149	C₄²	#41×C₁₃₇	1	2	yes	—
150	C₂³	#117×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄	#124×C₁₃₇	1	2	yes	—
159	C₂×C₄	#125×C₁₃₇	1	2	yes	—
160	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
161	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	8	4	no	—
168	C₈	C₄⋊C₈×C₁₃₇	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	#382×C₁₃₇	1	1	yes	characteristic
171	C₂²	#431×C₁₃₇	1	1	yes	characteristic
172	C₂²	#432×C₁₃₇	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
178	C₄	#423×C₁₃₇	1	1	yes	characteristic
179	C₄	#431×C₁₃₇	1	1	yes	characteristic
180	C₄	#432×C₁₃₇	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#1831×C₁₃₇	1	1	yes	characteristic
184	C₂	#1833×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#117
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#382
prime radical	C₂³×C₁₃₇	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ #376×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
26. #97×C₁₃₇	C₈	semidirect
92. #8781	C₁₃₇	direct (N × H)
93. C₁₃₇	#8781	direct (N × H)
119. #97	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1726×C₁₃₇	2
#1564×C₁₃₇	2
#1281×C₁₃₇	2
#1274×C₁₃₇	2
#8781	137

Subgroups — C137×#8781 #46858167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8781 #46858167