Subgroups — C₁₃₇×#8780 #46858166

Order 35072.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 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	—	2	2	no	—
27	C₄×C₈×C₁₃₇	—	2	2	no	—
28	C₄×C₈×C₁₃₇	—	2	4	no	—
29	C₄×C₈×C₁₃₇	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
34	#95×C₁₃₇	—	2	2	no	—
35	C₂²×C₄×C₁₃₇	—	2	1	no	—
36	C₂²×C₄×C₁₃₇	—	2	1	no	—
37	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
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	—	2	1	no	—
44	C₂×C₈×C₁₃₇	—	2	1	no	—
45	C₂×C₈×C₁₃₇	—	4	4	no	—
46	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
54	C₄²×C₁₃₇	—	2	1	no	—
55	C₄²×C₁₃₇	—	2	1	no	—
56	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
57	C₄²×C₁₃₇	M₁₆	1	2	yes	—
58	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₃₇	#41	1	2	yes	—
61	C₂³×C₁₃₇	#117	1	1	yes	characteristic
62	C₂×C₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄×C₁₃₇	—	4	2	no	—
65	C₂×C₄×C₁₃₇	—	4	2	no	—
66	C₂×C₄×C₁₃₇	—	4	2	no	—
67	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₁₃₇	#124	1	2	yes	—
70	C₂×C₄×C₁₃₇	#125	1	2	yes	—
71	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₁₃₇	—	2	1	no	—
76	C₈×C₁₃₇	—	2	1	no	—
77	C₈×C₁₃₇	—	2	2	no	—
78	C₈×C₁₃₇	—	2	2	no	—
79	C₈×C₁₃₇	—	8	4	no	—
80	C₈×C₁₃₇	C₄⋊C₈	1	2	yes	—
81	C₈×C₁₃₇	C₄⋊C₈	1	2	yes	—
82	C₂²×C₁₃₇	—	2	2	no	—
83	C₂²×C₁₃₇	#382	1	1	yes	characteristic
84	C₂²×C₁₃₇	#431	1	1	yes	characteristic
85	C₂²×C₁₃₇	#432	1	1	yes	characteristic
86	C₄×C₁₃₇	—	2	2	no	—
87	C₄×C₁₃₇	—	2	2	no	—
88	C₄×C₁₃₇	—	4	2	no	—
89	C₄×C₁₃₇	—	4	2	no	—
90	C₄×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₁₃₇	#423	1	1	yes	characteristic
92	C₄×C₁₃₇	#431	1	1	yes	characteristic
93	C₄×C₁₃₇	#432	1	1	yes	characteristic
94	C₂×C₁₃₇	—	2	1	no	—
95	C₂×C₁₃₇	—	2	1	no	—
96	C₂×C₁₃₇	#1831	1	1	yes	characteristic
97	C₂×C₁₃₇	#1833	1	2	yes	—
98	#8780	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₃₇	#8780	1	1	yes	characteristic, direct product factor, Sylow
100	#1726	C₂×C₁₃₇	1	1	yes	characteristic
101	#1564	C₂×C₁₃₇	1	2	yes	—
102	#1281	C₂×C₁₃₇	1	2	yes	—
103	#1274	C₂×C₁₃₇	1	2	yes	—
104	#431	C₂²×C₁₃₇	1	4	yes	—
105	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
106	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
107	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
109	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
111	#376	C₂²×C₁₃₇	1	1	yes	characteristic
112	#335	—	2	2	no	—
113	#334	—	2	2	no	—
114	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
118	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
121	C₄⋊C₈	—	2	4	no	—
122	C₂²⋊C₈	—	2	4	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	C₈×C₁₃₇	1	4	yes	complemented
129	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
130	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
131	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
132	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
137	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
146	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
148	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
149	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
150	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
151	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
152	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
156	C₄²	M₁₆×C₁₃₇	1	2	yes	—
157	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
158	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
159	C₄²	#41×C₁₃₇	1	2	yes	—
160	C₂³	#117×C₁₃₇	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
167	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
168	C₂×C₄	#124×C₁₃₇	1	2	yes	—
169	C₂×C₄	#125×C₁₃₇	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	8	4	no	—
179	C₈	C₄⋊C₈×C₁₃₇	1	2	yes	—
180	C₈	C₄⋊C₈×C₁₃₇	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#382×C₁₃₇	1	1	yes	characteristic
183	C₂²	#431×C₁₃₇	1	1	yes	characteristic
184	C₂²	#432×C₁₃₇	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
190	C₄	#423×C₁₃₇	1	1	yes	characteristic
191	C₄	#431×C₁₃₇	1	1	yes	characteristic
192	C₄	#432×C₁₃₇	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#1831×C₁₃₇	1	1	yes	characteristic
196	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
29. C₄×C₈×C₁₃₇	C₈	semidirect
98. #8780	C₁₃₇	direct (N × H)
99. C₁₃₇	#8780	direct (N × H)
128. 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
#1726×C₁₃₇	2
#1564×C₁₃₇	2
#1281×C₁₃₇	2
#1274×C₁₃₇	2
#8780	137

Subgroups — C137×#8780 #46858166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8780 #46858166