Subgroups — C₁₃₇×#19653 #46869039

Order 35072.

730 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	#3229×C₁₃₇	C₂	1	4	yes	maximal
2	#3160×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1716×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1707×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₁₃₇	—	2	2	no	—
8	#533×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	#500×C₁₃₇	C₂²	1	2	yes	—
11	#499×C₁₃₇	—	2	2	no	—
12	#498×C₁₃₇	C₂²	1	2	yes	—
13	#498×C₁₃₇	C₂²	1	8	yes	—
14	#494×C₁₃₇	—	2	2	no	—
15	#494×C₁₃₇	C₂²	1	2	yes	—
16	#494×C₁₃₇	C₂²	1	2	yes	—
17	#487×C₁₃₇	C₂²	1	8	yes	—
18	#483×C₁₃₇	—	2	8	no	—
19	#475×C₁₃₇	—	2	8	no	—
20	#431×C₁₃₇	C₂²	1	2	yes	—
21	#427×C₁₃₇	—	2	2	no	—
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	#322×C₁₃₇	C₂²	1	2	yes	—
24	C₈²×C₁₃₇	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₃₇	—	2	2	no	—
26	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
27	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
28	C₂×Q₁₆×C₁₃₇	—	4	8	no	—
29	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
32	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
34	#125×C₁₃₇	—	2	2	no	—
35	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
36	C₄×Q₈×C₁₃₇	—	2	4	no	—
37	#117×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	#107×C₁₃₇	—	2	4	no	—
41	#106×C₁₃₇	—	2	4	no	—
42	#106×C₁₃₇	—	2	16	no	—
43	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	—
44	#103×C₁₃₇	—	2	16	no	—
45	C₂²⋊C₈×C₁₃₇	D₄	1	4	yes	—
46	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
47	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
50	C₂×Q₈×C₁₃₇	—	2	4	no	—
51	C₂×Q₈×C₁₃₇	—	2	4	no	—
52	C₂×Q₈×C₁₃₇	—	4	2	no	—
53	C₂×Q₈×C₁₃₇	—	4	8	no	—
54	C₂²×C₄×C₁₃₇	—	4	2	no	—
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	Q₁₆×C₁₃₇	—	4	8	no	—
58	Q₁₆×C₁₃₇	—	4	8	no	—
59	C₂×C₈×C₁₃₇	SD₁₆	1	8	yes	—
60	C₂×C₈×C₁₃₇	C₂×D₄	1	8	yes	—
61	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₁₃₇	—	2	8	no	—
64	C₄⋊C₄×C₁₃₇	—	2	4	no	—
65	C₄⋊C₄×C₁₃₇	—	2	4	no	—
66	C₄⋊C₄×C₁₃₇	—	4	4	no	—
67	C₄⋊C₄×C₁₃₇	—	4	4	no	—
68	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
69	C₄²×C₁₃₇	—	4	2	no	—
70	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
73	C₂³×C₁₃₇	#127	1	1	yes	characteristic
74	Q₈×C₁₃₇	—	4	8	no	—
75	Q₈×C₁₃₇	—	4	8	no	—
76	Q₈×C₁₃₇	—	8	4	no	—
77	C₂×C₄×C₁₃₇	—	2	4	no	—
78	C₂×C₄×C₁₃₇	—	4	4	no	—
79	C₂×C₄×C₁₃₇	—	4	4	no	—
80	C₂×C₄×C₁₃₇	—	8	4	no	—
81	C₂×C₄×C₁₃₇	#127	1	2	yes	—
82	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
86	C₈×C₁₃₇	—	2	8	no	—
87	C₈×C₁₃₇	C₂×SD₁₆	1	4	yes	—
88	C₈×C₁₃₇	C₂×SD₁₆	1	4	yes	—
89	C₂²×C₁₃₇	—	2	2	no	—
90	C₂²×C₁₃₇	#492	1	2	yes	—
91	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
92	C₄×C₁₃₇	—	2	2	no	—
93	C₄×C₁₃₇	—	8	4	no	—
94	C₄×C₁₃₇	—	8	4	no	—
95	C₄×C₁₃₇	#492	1	2	yes	—
96	C₄×C₁₃₇	C₂²×SD₁₆	1	2	yes	—
97	C₄×C₁₃₇	#578	1	4	yes	—
98	C₂×C₁₃₇	—	2	2	no	—
99	C₂×C₁₃₇	C₂×#492	1	1	yes	characteristic
100	C₂×C₁₃₇	#3156	1	2	yes	—
101	#19653	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₃₇	#19653	1	1	yes	characteristic, direct product factor, Sylow
103	#3229	C₂×C₁₃₇	1	4	yes	—
104	#3160	C₂×C₁₃₇	1	1	yes	characteristic
105	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic
106	#1716	C₂×C₁₃₇	1	4	yes	complemented
107	#1707	C₂×C₁₃₇	1	4	yes	complemented
108	#1450	C₂×C₁₃₇	1	1	yes	characteristic
109	C₂²×Q₁₆	—	2	2	no	—
110	#533	C₂²×C₁₃₇	1	2	yes	—
111	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
112	#500	C₂²×C₁₃₇	1	2	yes	—
113	#499	—	2	2	no	—
114	#498	C₂²×C₁₃₇	1	2	yes	—
115	#498	C₂²×C₁₃₇	1	8	yes	—
116	#494	—	2	2	no	—
117	#494	C₂²×C₁₃₇	1	2	yes	—
118	#494	C₂²×C₁₃₇	1	2	yes	—
119	#487	C₂²×C₁₃₇	1	8	yes	—
120	#483	—	2	8	no	—
121	#475	—	2	8	no	—
122	#431	C₂²×C₁₃₇	1	2	yes	—
123	#427	—	2	2	no	—
124	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
125	#322	C₂²×C₁₃₇	1	2	yes	—
126	C₈²	C₂²×C₁₃₇	1	2	yes	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	—	4	8	no	—
131	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
135	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
136	#125	—	2	2	no	—
137	C₂²⋊Q₈	—	2	4	no	—
138	C₄×Q₈	—	2	4	no	—
139	#117	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
142	#107	—	2	4	no	—
143	#106	—	2	4	no	—
144	#106	—	2	16	no	—
145	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	—
146	#103	—	2	16	no	—
147	C₂²⋊C₈	D₄×C₁₃₇	1	4	yes	—
148	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
149	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
150	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
151	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	8	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
159	Q₁₆	—	4	8	no	—
160	Q₁₆	—	4	8	no	—
161	C₂×C₈	SD₁₆×C₁₃₇	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₁₃₇	1	8	yes	—
163	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
173	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
174	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
175	C₂³	#127×C₁₃₇	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	4	8	no	—
178	Q₈	—	8	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	4	no	—
183	C₂×C₄	#127×C₁₃₇	1	2	yes	—
184	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
186	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
188	C₈	—	2	8	no	—
189	C₈	C₂×SD₁₆×C₁₃₇	1	4	yes	—
190	C₈	C₂×SD₁₆×C₁₃₇	1	4	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	#492×C₁₃₇	1	2	yes	—
193	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	—	8	4	no	—
197	C₄	#492×C₁₃₇	1	2	yes	—
198	C₄	C₂²×SD₁₆×C₁₃₇	1	2	yes	—
199	C₄	#578×C₁₃₇	1	4	yes	—
200	C₂	—	2	2	no	—
201	C₂	C₂×#492×C₁₃₇	1	1	yes	characteristic
202	C₂	#3156×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×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
4. #1716×C₁₃₇	C₂	semidirect
5. #1707×C₁₃₇	C₂	semidirect
101. #19653	C₁₃₇	direct (N × H)
102. C₁₃₇	#19653	direct (N × H)
106. #1716	C₂×C₁₃₇	semidirect
107. #1707	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3229×C₁₃₇	2
#3160×C₁₃₇	2
C₂×#494×C₁₃₇	2
#1716×C₁₃₇	2
#1707×C₁₃₇	2
#1450×C₁₃₇	2
#19653	137

Subgroups — C137×#19653 #46869039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19653 #46869039