Subgroups — C₂×#1746×C₁₃₇ #46869749

Order 35072.

1134 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1746×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#336×C₁₃₇	C₂	1	4	yes	maximal
6	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#97×C₁₃₇	C₄	1	8	yes	complemented
11	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₃₇	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
14	#376×C₁₃₇	C₄	1	4	yes	—
15	#376×C₁₃₇	C₂²	1	4	yes	—
16	C₂×#95×C₁₃₇	C₄	1	4	yes	—
17	#336×C₁₃₇	C₂²	1	16	yes	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₇	—	2	8	no	—
22	C₂²×C₈×C₁₃₇	—	2	8	no	—
23	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₁₃₇	—	2	8	no	—
28	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
34	#97×C₁₃₇	C₂×C₄	1	16	yes	complemented
35	C₄×C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
36	#95×C₁₃₇	C₂×C₄	1	16	yes	—
37	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	—	2	8	no	—
39	C₂²×C₄×C₁₃₇	—	2	8	no	—
40	C₂²×C₄×C₁₃₇	—	2	16	no	—
41	C₂²×C₄×C₁₃₇	C₄²	1	8	yes	—
42	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₃₇	#41	1	8	yes	—
48	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₁₃₇	—	2	16	no	—
50	C₂×C₈×C₁₃₇	—	2	16	no	—
51	C₂×C₈×C₁₃₇	—	2	8	no	—
52	C₂×C₈×C₁₃₇	—	2	8	no	—
53	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
54	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
55	C₄²×C₁₃₇	—	2	8	no	—
56	C₄²×C₁₃₇	—	2	8	no	—
57	C₄²×C₁₃₇	C₄²	1	8	yes	—
58	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
59	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
61	C₂³×C₁₃₇	#117	1	4	yes	—
62	C₂³×C₁₃₇	#131	1	8	yes	—
63	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	—	2	8	no	—
65	C₂×C₄×C₁₃₇	—	2	16	no	—
66	C₂×C₄×C₁₃₇	—	2	8	no	—
67	C₂×C₄×C₁₃₇	—	2	16	no	—
68	C₂×C₄×C₁₃₇	—	4	16	no	—
69	C₂×C₄×C₁₃₇	C₂×C₄²	1	8	yes	—
70	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₁₃₇	#117	1	8	yes	—
72	C₂×C₄×C₁₃₇	#117	1	8	yes	—
73	C₂×C₄×C₁₃₇	#117	1	4	yes	—
74	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
77	C₈×C₁₃₇	—	2	8	no	—
78	C₈×C₁₃₇	—	2	8	no	—
79	C₂²×C₁₃₇	#376	1	4	yes	—
80	C₂²×C₁₃₇	#405	1	8	yes	—
81	C₂²×C₁₃₇	#433	1	16	yes	—
82	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
84	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
85	C₄×C₁₃₇	—	2	8	no	—
86	C₄×C₁₃₇	—	4	8	no	—
87	C₄×C₁₃₇	#376	1	4	yes	—
88	C₄×C₁₃₇	C₂×#117	1	2	yes	—
89	C₄×C₁₃₇	C₂×#117	1	2	yes	—
90	C₂×C₁₃₇	#1746	1	8	yes	direct product factor
91	C₂×C₁₃₇	C₂×#376	1	1	yes	characteristic
92	C₂×C₁₃₇	C₂×#405	1	2	yes	—
93	C₂×C₁₃₇	C₂×#433	1	4	yes	—
94	C₂×#1746	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₇	C₂×#1746	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×#97	C₂×C₁₃₇	1	1	yes	characteristic
97	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
98	C₂×#376	C₂×C₁₃₇	1	1	yes	characteristic
99	#1746	C₂×C₁₃₇	1	8	yes	direct product factor
100	C₂×#336	C₂×C₁₃₇	1	4	yes	—
101	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
102	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
103	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
104	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
105	C₂×#97	C₄×C₁₃₇	1	8	yes	complemented
106	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
107	C₂×C₄×C₈	C₄×C₁₃₇	1	8	yes	complemented
108	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
109	#376	C₄×C₁₃₇	1	4	yes	—
110	#376	C₂²×C₁₃₇	1	4	yes	—
111	C₂×#95	C₄×C₁₃₇	1	4	yes	—
112	#336	C₂²×C₁₃₇	1	16	yes	—
113	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
114	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
115	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
116	C₂²×C₈	—	2	8	no	—
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
119	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
120	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
121	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
122	C₂×C₄²	—	2	8	no	—
123	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
125	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
126	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
127	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
128	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
129	#97	C₂×C₄×C₁₃₇	1	16	yes	complemented
130	C₄×C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
131	#95	C₂×C₄×C₁₃₇	1	16	yes	—
132	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	2	16	no	—
136	C₂²×C₄	C₄²×C₁₃₇	1	8	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
142	C₂²×C₄	#41×C₁₃₇	1	8	yes	—
143	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
144	C₂×C₈	—	2	16	no	—
145	C₂×C₈	—	2	16	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
149	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
150	C₄²	—	2	8	no	—
151	C₄²	—	2	8	no	—
152	C₄²	C₄²×C₁₃₇	1	8	yes	—
153	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
154	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
155	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
156	C₂³	#117×C₁₃₇	1	4	yes	—
157	C₂³	#131×C₁₃₇	1	8	yes	—
158	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
159	C₂×C₄	—	2	16	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	4	16	no	—
164	C₂×C₄	C₂×C₄²×C₁₃₇	1	8	yes	—
165	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
166	C₂×C₄	#117×C₁₃₇	1	4	yes	—
167	C₂×C₄	#117×C₁₃₇	1	8	yes	—
168	C₂×C₄	#117×C₁₃₇	1	8	yes	—
169	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
170	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
172	C₈	—	2	8	no	—
173	C₈	—	2	8	no	—
174	C₂²	#376×C₁₃₇	1	4	yes	—
175	C₂²	#405×C₁₃₇	1	8	yes	—
176	C₂²	#433×C₁₃₇	1	16	yes	—
177	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
178	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
179	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	4	8	no	—
182	C₄	#376×C₁₃₇	1	4	yes	—
183	C₄	C₂×#117×C₁₃₇	1	2	yes	—
184	C₄	C₂×#117×C₁₃₇	1	2	yes	—
185	C₂	#1746×C₁₃₇	1	8	yes	direct product factor
186	C₂	C₂×#376×C₁₃₇	1	1	yes	characteristic
187	C₂	C₂×#405×C₁₃₇	1	2	yes	—
188	C₂	C₂×#433×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃₇	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle, prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1746×C₁₃₇	C₂	direct (N × H)
10. C₂×#97×C₁₃₇	C₄	semidirect
12. C₂×C₄×C₈×C₁₃₇	C₄	semidirect
34. #97×C₁₃₇	C₂×C₄	semidirect
35. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
90. C₂×C₁₃₇	#1746	direct (N × H)
94. C₂×#1746	C₁₃₇	direct (N × H)
95. C₁₃₇	C₂×#1746	direct (N × H)
99. #1746	C₂×C₁₃₇	direct (N × H)
105. C₂×#97	C₄×C₁₃₇	semidirect
107. C₂×C₄×C₈	C₄×C₁₃₇	semidirect
129. #97	C₂×C₄×C₁₃₇	semidirect
130. C₄×C₈	C₂×C₄×C₁₃₇	semidirect
185. C₂	#1746×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#97×C₁₃₇	2
C₂²×C₄×C₈×C₁₃₇	2
C₂×#376×C₁₃₇	2
#1746×C₁₃₇	2
C₂×#336×C₁₃₇	2
C₂×#1746	137

Subgroups — C2×#1746×C137 #46869749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1746×C₁₃₇ #46869749