Subgroups — C₁₃₇×#13285 #46862671

Order 35072.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₂×C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1382×C₁₃₇	C₂	1	4	yes	maximal
5	#533×C₁₃₇	—	2	1	no	—
6	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
7	#500×C₁₃₇	C₄	1	2	yes	—
8	#499×C₁₃₇	—	2	1	no	—
9	#499×C₁₃₇	C₄	1	2	yes	—
10	#498×C₁₃₇	—	2	1	no	—
11	C₂×#108×C₁₃₇	—	2	1	no	—
12	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
13	#427×C₁₃₇	—	2	1	no	—
14	#427×C₁₃₇	C₂²	1	2	yes	—
15	#423×C₁₃₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₁₃₇	—	2	1	no	—
18	C₄×C₁₆×C₁₃₇	—	2	1	no	—
19	C₂×M₁₆×C₁₃₇	—	2	2	no	—
20	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
23	#125×C₁₃₇	—	2	2	no	—
24	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
25	C₄×Q₈×C₁₃₇	—	4	2	no	—
26	#117×C₁₃₇	—	2	2	no	—
27	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₃₇	—	2	2	no	—
29	C₂×C₁₆×C₁₃₇	—	2	2	no	—
30	C₂×C₁₆×C₁₃₇	Q₈	1	4	yes	complemented
31	#108×C₁₃₇	—	2	4	no	—
32	#108×C₁₃₇	—	4	2	no	—
33	#107×C₁₃₇	—	2	2	no	—
34	#107×C₁₃₇	C₂×C₄	1	4	yes	—
35	#106×C₁₃₇	—	2	2	no	—
36	#106×C₁₃₇	—	2	2	no	—
37	C₄⋊C₈×C₁₃₇	—	4	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	C₂×Q₈×C₁₃₇	—	4	2	no	—
43	C₂²×C₄×C₁₃₇	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₁₃₇	—	4	2	no	—
46	M₁₆×C₁₃₇	—	4	2	no	—
47	C₂×C₈×C₁₃₇	—	4	2	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	SD₁₆	1	2	yes	—
51	C₂×C₈×C₁₃₇	Q₁₆	1	2	yes	—
52	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₁₃₇	#41	1	2	yes	—
56	C₄⋊C₄×C₁₃₇	—	2	4	no	—
57	C₄⋊C₄×C₁₃₇	—	4	4	no	—
58	C₄⋊C₄×C₁₃₇	—	4	2	no	—
59	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
60	C₄²×C₁₃₇	—	4	2	no	—
61	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₃₇	D₈	1	2	yes	—
64	C₁₆×C₁₃₇	—	2	4	no	—
65	C₂³×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₁₃₇	—	8	4	no	—
67	C₂×C₄×C₁₃₇	—	2	4	no	—
68	C₂×C₄×C₁₃₇	—	4	4	no	—
69	C₂×C₄×C₁₃₇	#102	1	2	yes	—
70	C₂×C₄×C₁₃₇	#102	1	2	yes	—
71	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₁₃₇	—	8	2	no	—
78	C₈×C₁₃₇	#103	1	2	yes	—
79	C₈×C₁₃₇	#103	1	2	yes	—
80	C₈×C₁₃₇	C₄×Q₈	1	2	yes	—
81	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
82	C₂²×C₁₃₇	—	2	2	no	—
83	C₂²×C₁₃₇	C₂×#102	1	1	yes	characteristic
84	C₂²×C₁₃₇	#487	1	1	yes	characteristic
85	C₂²×C₁₃₇	#498	1	1	yes	characteristic
86	C₄×C₁₃₇	—	2	2	no	—
87	C₄×C₁₃₇	—	8	4	no	—
88	C₄×C₁₃₇	#359	1	4	yes	—
89	C₄×C₁₃₇	#391	1	1	yes	characteristic
90	C₄×C₁₃₇	C₂×#103	1	1	yes	characteristic
91	C₄×C₁₃₇	#486	1	1	yes	characteristic
92	C₄×C₁₃₇	#500	1	1	yes	characteristic
93	C₂×C₁₃₇	—	2	1	no	—
94	C₂×C₁₃₇	—	2	1	no	—
95	C₂×C₁₃₇	#1990	1	1	yes	characteristic
96	C₂×C₁₃₇	C₂×#359	1	2	yes	—
97	#13285	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₃₇	#13285	1	1	yes	characteristic, direct product factor, Sylow
99	#3160	C₂×C₁₃₇	1	1	yes	characteristic
100	C₂×C₄×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
101	#1848	C₂×C₁₃₇	1	1	yes	characteristic
102	#1382	C₂×C₁₃₇	1	4	yes	—
103	#533	—	2	1	no	—
104	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
105	#500	C₄×C₁₃₇	1	2	yes	—
106	#499	—	2	1	no	—
107	#499	C₄×C₁₃₇	1	2	yes	—
108	#498	—	2	1	no	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
111	#427	—	2	1	no	—
112	#427	C₂²×C₁₃₇	1	2	yes	—
113	#423	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Q₈×C₁₃₇	1	4	yes	complemented
129	#108	—	2	4	no	—
130	#108	—	4	2	no	—
131	#107	—	2	2	no	—
132	#107	C₂×C₄×C₁₃₇	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
138	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
139	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
140	C₂×Q₈	—	4	2	no	—
141	C₂²×C₄	SD₁₆×C₁₃₇	1	2	yes	—
142	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
148	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
149	C₂×C₈	Q₁₆×C₁₃₇	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
153	C₂×C₈	#41×C₁₃₇	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
160	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
161	C₄²	D₈×C₁₃₇	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#102×C₁₃₇	1	2	yes	—
168	C₂×C₄	#102×C₁₃₇	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
170	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
175	C₈	—	8	2	no	—
176	C₈	#103×C₁₃₇	1	2	yes	—
177	C₈	#103×C₁₃₇	1	2	yes	—
178	C₈	C₄×Q₈×C₁₃₇	1	2	yes	—
179	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#102×C₁₃₇	1	1	yes	characteristic
182	C₂²	#487×C₁₃₇	1	1	yes	characteristic
183	C₂²	#498×C₁₃₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#359×C₁₃₇	1	4	yes	—
187	C₄	#391×C₁₃₇	1	1	yes	characteristic
188	C₄	C₂×#103×C₁₃₇	1	1	yes	characteristic
189	C₄	#486×C₁₃₇	1	1	yes	characteristic
190	C₄	#500×C₁₃₇	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#1990×C₁₃₇	1	1	yes	characteristic
194	C₂	C₂×#359×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#102
prime radical	C₂³×C₁₃₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₁₃₇	Q₈	semidirect
97. #13285	C₁₃₇	direct (N × H)
98. C₁₃₇	#13285	direct (N × H)
128. C₂×C₁₆	Q₈×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
C₂×C₄×C₁₆×C₁₃₇	2
#1848×C₁₃₇	2
#1382×C₁₃₇	2
#13285	137

Subgroups — C137×#13285 #46862671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13285 #46862671