Subgroups — C₁₃₇×#7153 #46856539

Order 35072.

218 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	#2150×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₃₇	—	2	1	no	—
5	C₂²×C₁₆×C₁₃₇	—	2	1	no	—
6	#431×C₁₃₇	C₄	1	1	yes	characteristic
7	#431×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
10	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
12	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
13	#322×C₁₃₇	—	2	1	no	—
14	C₈²×C₁₃₇	—	2	1	no	—
15	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₁₃₇	—	4	1	no	—
19	M₃₂×C₁₃₇	—	4	1	no	—
20	C₂×C₁₆×C₁₃₇	—	2	2	no	—
21	C₂×C₁₆×C₁₃₇	—	2	2	no	—
22	C₂×C₁₆×C₁₃₇	—	4	1	no	—
23	C₂×C₁₆×C₁₃₇	—	4	1	no	—
24	C₄⋊C₈×C₁₃₇	—	2	1	no	—
25	C₄⋊C₈×C₁₃₇	C₈	1	2	yes	—
26	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
27	C₂²⋊C₈×C₁₃₇	C₈	1	2	yes	—
28	C₄×C₈×C₁₃₇	—	2	1	no	—
29	C₄×C₈×C₁₃₇	—	2	1	no	—
30	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃₇	—	2	1	no	—
38	C₂×C₈×C₁₃₇	—	2	1	no	—
39	C₂×C₈×C₁₃₇	—	2	2	no	—
40	C₂×C₈×C₁₃₇	—	2	2	no	—
41	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
49	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
50	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₁₃₇	M₁₆	1	1	yes	characteristic
53	C₁₆×C₁₃₇	—	4	2	no	—
54	C₁₆×C₁₃₇	—	4	2	no	—
55	C₂³×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄×C₁₃₇	—	4	1	no	—
58	C₂×C₄×C₁₃₇	#95	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	#106	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	#108	1	1	yes	characteristic
67	C₂×C₄×C₁₃₇	#108	1	1	yes	characteristic
68	C₈×C₁₃₇	—	2	1	no	—
69	C₈×C₁₃₇	—	2	1	no	—
70	C₈×C₁₃₇	—	4	2	no	—
71	C₈×C₁₃₇	—	4	2	no	—
72	C₈×C₁₃₇	#102	1	1	yes	characteristic
73	C₈×C₁₃₇	#102	1	1	yes	characteristic
74	C₈×C₁₃₇	#103	1	1	yes	characteristic
75	C₈×C₁₃₇	#103	1	1	yes	characteristic
76	C₂²×C₁₃₇	—	2	1	no	—
77	C₂²×C₁₃₇	—	2	1	no	—
78	C₂²×C₁₃₇	#334	1	1	yes	characteristic
79	C₂²×C₁₃₇	#335	1	1	yes	characteristic
80	C₂²×C₁₃₇	#336	1	1	yes	characteristic
81	C₄×C₁₃₇	—	2	1	no	—
82	C₄×C₁₃₇	—	2	1	no	—
83	C₄×C₁₃₇	—	2	1	no	—
84	C₄×C₁₃₇	#334	1	1	yes	characteristic
85	C₄×C₁₃₇	#335	1	1	yes	characteristic
86	C₄×C₁₃₇	#340	1	1	yes	characteristic
87	C₄×C₁₃₇	#341	1	1	yes	characteristic
88	C₄×C₁₃₇	#350	1	2	yes	—
89	C₂×C₁₃₇	—	2	1	no	—
90	C₂×C₁₃₇	—	2	1	no	—
91	C₂×C₁₃₇	#1277	1	1	yes	characteristic
92	C₂×C₁₃₇	#1378	1	1	yes	characteristic
93	C₂×C₁₃₇	#1379	1	1	yes	characteristic
94	#7153	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₇	#7153	1	1	yes	characteristic, direct product factor, Sylow
96	#2150	C₂×C₁₃₇	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
98	#1450	C₂×C₁₃₇	1	1	yes	characteristic
99	C₂×M₃₂	—	2	1	no	—
100	C₂²×C₁₆	—	2	1	no	—
101	#431	C₄×C₁₃₇	1	1	yes	characteristic
102	#431	C₄×C₁₃₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
104	C₄⋊C₁₆	—	2	1	no	—
105	C₄⋊C₁₆	—	2	1	no	—
106	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
107	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
108	#322	—	2	1	no	—
109	C₈²	—	2	1	no	—
110	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
111	C₂²×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
113	M₃₂	—	4	1	no	—
114	M₃₂	—	4	1	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	4	1	no	—
118	C₂×C₁₆	—	4	1	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	C₈×C₁₃₇	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	C₈×C₁₃₇	1	2	yes	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
127	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
128	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
131	C₂²×C₄	M₁₆×C₁₃₇	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
137	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
138	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
140	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
141	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
142	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
143	C₂×C₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
144	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
145	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
146	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
147	C₄²	M₁₆×C₁₃₇	1	1	yes	characteristic
148	C₁₆	—	4	2	no	—
149	C₁₆	—	4	2	no	—
150	C₂³	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	#95×C₁₃₇	1	1	yes	characteristic
154	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄	#97×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
159	C₂×C₄	#106×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
161	C₂×C₄	#108×C₁₃₇	1	1	yes	characteristic
162	C₂×C₄	#108×C₁₃₇	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	2	no	—
167	C₈	#102×C₁₃₇	1	1	yes	characteristic
168	C₈	#102×C₁₃₇	1	1	yes	characteristic
169	C₈	#103×C₁₃₇	1	1	yes	characteristic
170	C₈	#103×C₁₃₇	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#334×C₁₃₇	1	1	yes	characteristic
174	C₂²	#335×C₁₃₇	1	1	yes	characteristic
175	C₂²	#336×C₁₃₇	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#334×C₁₃₇	1	1	yes	characteristic
180	C₄	#335×C₁₃₇	1	1	yes	characteristic
181	C₄	#340×C₁₃₇	1	1	yes	characteristic
182	C₄	#341×C₁₃₇	1	1	yes	characteristic
183	C₄	#350×C₁₃₇	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#1277×C₁₃₇	1	1	yes	characteristic
187	C₂	#1378×C₁₃₇	1	1	yes	characteristic
188	C₂	#1379×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₇
socle	C₂²×C₁₃₇	#336
prime radical	C₂³×C₁₃₇	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
94. #7153	C₁₃₇	direct (N × H)
95. C₁₃₇	#7153	direct (N × H)

Maximal subgroups

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

subgroup	index
#2150×C₁₃₇	2
C₂×C₄⋊C₁₆×C₁₃₇	2
#1450×C₁₃₇	2
#7153	137

Subgroups — C137×#7153 #46856539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7153 #46856539