Subgroups — C₁₃₇×#11409 #46860795

Order 35072.

284 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1312×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1312×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1311×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₃₂×C₁₃₇	C₄	1	2	yes	—
9	#431×C₁₃₇	C₄	1	1	yes	characteristic
10	C₄×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
13	C₂²⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
14	#346×C₁₃₇	C₄	1	2	yes	—
15	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
19	#322×C₁₃₇	C₄	1	1	yes	characteristic
20	#322×C₁₃₇	C₄	1	1	yes	characteristic
21	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₃₇	C₈	1	2	yes	—
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₁₃₇	—	2	4	no	—
28	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄⋊C₈×C₁₃₇	C₈	1	2	yes	—
33	C₂²⋊C₈×C₁₃₇	C₈	1	2	yes	—
34	#97×C₁₃₇	C₈	1	2	yes	—
35	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
46	M₁₆×C₁₃₇	—	2	4	no	—
47	C₂×C₈×C₁₃₇	—	2	2	no	—
48	C₂×C₈×C₁₃₇	—	2	2	no	—
49	C₂×C₈×C₁₃₇	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₃₇	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
58	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₁₆×C₁₃₇	—	2	2	no	—
62	C₁₆×C₁₃₇	—	2	2	no	—
63	C₁₆×C₁₃₇	—	2	2	no	—
64	C₁₆×C₁₃₇	M₁₆	1	4	yes	complemented
65	C₂³×C₁₃₇	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	—	2	1	no	—
67	C₂×C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
76	C₈×C₁₃₇	—	2	1	no	—
77	C₈×C₁₃₇	—	2	1	no	—
78	C₈×C₁₃₇	—	2	2	no	—
79	C₈×C₁₃₇	—	2	2	no	—
80	C₈×C₁₃₇	—	2	2	no	—
81	C₈×C₁₃₇	M₃₂	1	4	yes	complemented
82	C₈×C₁₃₇	C₂×M₁₆	1	2	yes	—
83	C₈×C₁₃₇	#131	1	2	yes	—
84	C₂²×C₁₃₇	—	2	1	no	—
85	C₂²×C₁₃₇	—	2	1	no	—
86	C₂²×C₁₃₇	—	2	1	no	—
87	C₂²×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₁₃₇	—	2	1	no	—
89	C₄×C₁₃₇	—	2	1	no	—
90	C₄×C₁₃₇	—	2	1	no	—
91	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₁₃₇	#405	1	1	yes	characteristic
93	C₄×C₁₃₇	C₂×M₃₂	1	2	yes	—
94	C₄×C₁₃₇	#504	1	2	yes	—
95	C₂×C₁₃₇	—	4	1	no	—
96	C₂×C₁₃₇	C₈×M₁₆	1	1	yes	characteristic
97	C₂×C₁₃₇	C₄×M₃₂	1	1	yes	characteristic
98	C₂×C₁₃₇	#2108	1	1	yes	characteristic
99	#11409	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₃₇	#11409	1	1	yes	characteristic, direct product factor, Sylow
101	#2162	C₂×C₁₃₇	1	1	yes	characteristic
102	C₄×M₃₂	C₂×C₁₃₇	1	1	yes	characteristic
103	C₈×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
104	#1312	C₂×C₁₃₇	1	1	yes	characteristic, complemented
105	#1312	C₂×C₁₃₇	1	1	yes	characteristic, complemented
106	#1311	C₂×C₁₃₇	1	1	yes	characteristic, complemented
107	C₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
108	C₂×M₃₂	C₄×C₁₃₇	1	2	yes	—
109	#431	C₄×C₁₃₇	1	1	yes	characteristic
110	C₄×M₁₆	C₄×C₁₃₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
112	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
113	C₂²⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
114	#346	C₄×C₁₃₇	1	2	yes	—
115	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
116	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
117	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
118	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
119	#322	C₄×C₁₃₇	1	1	yes	characteristic
120	#322	C₄×C₁₃₇	1	1	yes	characteristic
121	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
122	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
123	C₂×M₁₆	C₈×C₁₃₇	1	2	yes	—
124	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
127	M₃₂	—	2	4	no	—
128	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
129	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
130	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
131	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
132	C₄⋊C₈	C₈×C₁₃₇	1	2	yes	—
133	C₂²⋊C₈	C₈×C₁₃₇	1	2	yes	—
134	#97	C₈×C₁₃₇	1	2	yes	—
135	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
138	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
140	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
141	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
142	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
143	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
150	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
152	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
153	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
154	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
157	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
158	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
159	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
161	C₁₆	—	2	2	no	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	M₁₆×C₁₃₇	1	4	yes	complemented
165	C₂³	C₄×C₈×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	M₃₂×C₁₃₇	1	4	yes	complemented
182	C₈	C₂×M₁₆×C₁₃₇	1	2	yes	—
183	C₈	#131×C₁₃₇	1	2	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₂×C₄×C₈×C₁₃₇	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
192	C₄	#405×C₁₃₇	1	1	yes	characteristic
193	C₄	C₂×M₃₂×C₁₃₇	1	2	yes	—
194	C₄	#504×C₁₃₇	1	2	yes	—
195	C₂	—	4	1	no	—
196	C₂	C₈×M₁₆×C₁₃₇	1	1	yes	characteristic
197	C₂	C₄×M₃₂×C₁₃₇	1	1	yes	characteristic
198	C₂	#2108×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #1312×C₁₃₇	C₂	semidirect
5. #1312×C₁₃₇	C₂	semidirect
6. #1311×C₁₃₇	C₂	semidirect
7. C₈×C₁₆×C₁₃₇	C₂	semidirect
64. C₁₆×C₁₃₇	M₁₆	semidirect
81. C₈×C₁₃₇	M₃₂	semidirect
99. #11409	C₁₃₇	direct (N × H)
100. C₁₃₇	#11409	direct (N × H)
104. #1312	C₂×C₁₃₇	semidirect
105. #1312	C₂×C₁₃₇	semidirect
106. #1311	C₂×C₁₃₇	semidirect
107. C₈×C₁₆	C₂×C₁₃₇	semidirect
164. C₁₆	M₁₆×C₁₃₇	semidirect
181. C₈	M₃₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2162×C₁₃₇	2
C₄×M₃₂×C₁₃₇	2
C₈×M₁₆×C₁₃₇	2
#1312×C₁₃₇	2
#1311×C₁₃₇	2
C₈×C₁₆×C₁₃₇	2
#11409	137

Subgroups — C137×#11409 #46860795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11409 #46860795