Subgroups — C₁₃₇×#9755 #46859141

Order 35072.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1518×C₁₃₇	C₂	1	4	yes	maximal
4	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#533×C₁₃₇	C₂²	1	2	yes	—
6	#500×C₁₃₇	C₄	1	2	yes	complemented
7	#499×C₁₃₇	—	2	1	no	—
8	#499×C₁₃₇	—	2	1	no	—
9	#498×C₁₃₇	C₄	1	2	yes	complemented
10	#431×C₁₃₇	C₂²	1	2	yes	—
11	#427×C₁₃₇	—	2	1	no	—
12	#427×C₁₃₇	—	2	1	no	—
13	C₂×#107×C₁₃₇	—	2	1	no	—
14	C₂×#106×C₁₃₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#382×C₁₃₇	C₂²	1	2	yes	—
17	#328×C₁₃₇	—	2	8	no	—
18	#322×C₁₃₇	—	2	1	no	—
19	C₈²×C₁₃₇	—	2	1	no	—
20	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
23	#125×C₁₃₇	—	2	2	no	—
24	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
25	C₄×Q₈×C₁₃₇	—	2	4	no	—
26	#117×C₁₃₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
29	#107×C₁₃₇	—	2	2	no	—
30	#107×C₁₃₇	—	2	2	no	—
31	#107×C₁₃₇	—	4	2	no	—
32	#106×C₁₃₇	—	2	2	no	—
33	#106×C₁₃₇	—	2	2	no	—
34	#106×C₁₃₇	—	4	2	no	—
35	C₄⋊C₈×C₁₃₇	—	2	2	no	—
36	C₂²⋊C₈×C₁₃₇	D₄	1	4	yes	—
37	C₄×C₈×C₁₃₇	—	2	2	no	—
38	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
42	#95×C₁₃₇	—	2	4	no	—
43	C₂×Q₈×C₁₃₇	—	2	4	no	—
44	C₂²×C₄×C₁₃₇	—	4	2	no	—
45	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	—	2	4	no	—
48	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₃₇	Q₁₆	1	2	yes	—
55	C₂×C₈×C₁₃₇	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₁₃₇	—	2	4	no	—
57	C₄⋊C₄×C₁₃₇	—	2	4	no	—
58	C₄⋊C₄×C₁₃₇	—	4	4	no	—
59	C₄⋊C₄×C₁₃₇	—	4	2	no	—
60	C₄⋊C₄×C₁₃₇	—	4	2	no	—
61	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
62	C₄²×C₁₃₇	—	4	2	no	—
63	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₃₇	#41	1	2	yes	—
66	C₂³×C₁₃₇	#127	1	1	yes	characteristic
67	Q₈×C₁₃₇	—	8	4	no	—
68	C₂×C₄×C₁₃₇	—	4	2	no	—
69	C₂×C₄×C₁₃₇	—	4	4	no	—
70	C₂×C₄×C₁₃₇	—	8	4	no	—
71	C₂×C₄×C₁₃₇	—	8	2	no	—
72	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₁₃₇	#121	1	2	yes	—
75	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₈×C₁₃₇	—	4	4	no	—
81	C₈×C₁₃₇	#103	1	2	yes	—
82	C₈×C₁₃₇	#103	1	2	yes	—
83	C₈×C₁₃₇	#103	1	2	yes	—
84	C₈×C₁₃₇	#103	1	2	yes	—
85	C₂²×C₁₃₇	—	2	2	no	—
86	C₂²×C₁₃₇	#390	1	1	yes	characteristic
87	C₂²×C₁₃₇	#492	1	1	yes	characteristic
88	C₂²×C₁₃₇	#494	1	1	yes	characteristic
89	C₄×C₁₃₇	—	2	2	no	—
90	C₄×C₁₃₇	—	2	2	no	—
91	C₄×C₁₃₇	—	8	4	no	—
92	C₄×C₁₃₇	—	16	2	no	—
93	C₄×C₁₃₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₁₃₇	C₂×#103	1	1	yes	characteristic
95	C₄×C₁₃₇	#487	1	1	yes	characteristic
96	C₄×C₁₃₇	#487	1	1	yes	characteristic
97	C₂×C₁₃₇	—	2	1	no	—
98	C₂×C₁₃₇	—	2	1	no	—
99	C₂×C₁₃₇	#1956	1	1	yes	characteristic
100	C₂×C₁₃₇	#1976	1	2	yes	—
101	#9755	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₃₇	#9755	1	1	yes	characteristic, direct product factor, Sylow
103	#3160	C₂×C₁₃₇	1	1	yes	characteristic
104	#1846	C₂×C₁₃₇	1	1	yes	characteristic
105	#1518	C₂×C₁₃₇	1	4	yes	—
106	#1450	C₂×C₁₃₇	1	1	yes	characteristic
107	#533	C₂²×C₁₃₇	1	2	yes	—
108	#500	C₄×C₁₃₇	1	2	yes	complemented
109	#499	—	2	1	no	—
110	#499	—	2	1	no	—
111	#498	C₄×C₁₃₇	1	2	yes	complemented
112	#431	C₂²×C₁₃₇	1	2	yes	—
113	#427	—	2	1	no	—
114	#427	—	2	1	no	—
115	C₂×#107	—	2	1	no	—
116	C₂×#106	—	2	1	no	—
117	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
118	#382	C₂²×C₁₃₇	1	2	yes	—
119	#328	—	2	8	no	—
120	#322	—	2	1	no	—
121	C₈²	—	2	1	no	—
122	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
123	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	#117	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	4	2	no	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	4	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	D₄×C₁₃₇	1	4	yes	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
141	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
142	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
143	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
144	#95	—	2	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
148	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
154	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
155	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
156	C₂×C₈	Q₁₆×C₁₃₇	1	2	yes	—
157	C₂×C₈	Q₁₆×C₁₃₇	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
167	C₄²	#41×C₁₃₇	1	2	yes	—
168	C₂³	#127×C₁₃₇	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
176	C₂×C₄	#121×C₁₃₇	1	2	yes	—
177	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
182	C₈	—	4	4	no	—
183	C₈	#103×C₁₃₇	1	2	yes	—
184	C₈	#103×C₁₃₇	1	2	yes	—
185	C₈	#103×C₁₃₇	1	2	yes	—
186	C₈	#103×C₁₃₇	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#390×C₁₃₇	1	1	yes	characteristic
189	C₂²	#492×C₁₃₇	1	1	yes	characteristic
190	C₂²	#494×C₁₃₇	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	16	2	no	—
195	C₄	C₂×#103×C₁₃₇	1	1	yes	characteristic
196	C₄	C₂×#103×C₁₃₇	1	1	yes	characteristic
197	C₄	#487×C₁₃₇	1	1	yes	characteristic
198	C₄	#487×C₁₃₇	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#1956×C₁₃₇	1	1	yes	characteristic
202	C₂	#1976×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. #500×C₁₃₇	C₄	semidirect
9. #498×C₁₃₇	C₄	semidirect
101. #9755	C₁₃₇	direct (N × H)
102. C₁₃₇	#9755	direct (N × H)
108. #500	C₄×C₁₃₇	semidirect
111. #498	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₃₇	2
#1846×C₁₃₇	2
#1518×C₁₃₇	2
#1450×C₁₃₇	2
#9755	137

Subgroups — C137×#9755 #46859141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9755 #46859141