Subgroups — C₁₃₇×#8785 #46858171

Order 35072.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1831×C₁₃₇	C₂	1	2	yes	maximal
2	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1277×C₁₃₇	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₁₃₇	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
10	#336×C₁₃₇	—	2	4	no	—
11	C₂²×C₈×C₁₃₇	—	2	4	no	—
12	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₁₃₇	—	2	8	no	—
20	C₄×C₈×C₁₃₇	—	2	4	no	—
21	C₄×C₈×C₁₃₇	—	2	2	no	—
22	C₄×C₈×C₁₃₇	—	2	4	no	—
23	C₄×C₈×C₁₃₇	C₈	1	4	yes	complemented
24	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	—	2	2	no	—
30	C₂×C₈×C₁₃₇	—	2	2	no	—
31	C₂×C₈×C₁₃₇	—	2	2	no	—
32	C₂×C₈×C₁₃₇	—	2	2	no	—
33	C₂×C₈×C₁₃₇	—	4	4	no	—
34	C₂×C₈×C₁₃₇	—	4	8	no	—
35	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
37	C₄²×C₁₃₇	—	2	4	no	—
38	C₄²×C₁₃₇	—	2	2	no	—
39	C₄²×C₁₃₇	—	2	4	no	—
40	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
41	C₄²×C₁₃₇	M₁₆	1	2	yes	—
42	C₄²×C₁₃₇	#41	1	4	yes	—
43	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄×C₁₃₇	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₁₃₇	#108	1	4	yes	—
51	C₂×C₄×C₁₃₇	#117	1	2	yes	—
52	C₂×C₄×C₁₃₇	#124	1	2	yes	—
53	C₂×C₄×C₁₃₇	#125	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	#135	1	4	yes	—
58	C₈×C₁₃₇	—	2	4	no	—
59	C₈×C₁₃₇	—	2	4	no	—
60	C₈×C₁₃₇	—	8	4	no	—
61	C₂²×C₁₃₇	#382	1	1	yes	characteristic
62	C₂²×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₁₃₇	#423	1	1	yes	characteristic
64	C₂²×C₁₃₇	#427	1	2	yes	—
65	C₂²×C₁₃₇	C₂×#108	1	2	yes	—
66	C₄×C₁₃₇	—	2	4	no	—
67	C₄×C₁₃₇	—	2	4	no	—
68	C₄×C₁₃₇	—	2	2	no	—
69	C₄×C₁₃₇	#431	1	2	yes	—
70	C₄×C₁₃₇	#432	1	2	yes	—
71	C₄×C₁₃₇	#488	1	4	yes	—
72	C₂×C₁₃₇	#1564	1	2	yes	—
73	C₂×C₁₃₇	#1568	1	2	yes	—
74	C₂×C₁₃₇	#1831	1	1	yes	characteristic
75	C₂×C₁₃₇	#1835	1	1	yes	characteristic
76	C₂×C₁₃₇	#1837	1	1	yes	characteristic
77	#8785	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₃₇	#8785	1	1	yes	characteristic, direct product factor, Sylow
79	#1831	C₂×C₁₃₇	1	2	yes	—
80	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
81	#1277	C₂×C₁₃₇	1	4	yes	—
82	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
83	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	—
84	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
85	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
87	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
88	#336	—	2	4	no	—
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
92	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
93	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
94	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
95	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
96	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
97	C₄⋊C₈	—	2	8	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	4	no	—
101	C₄×C₈	C₈×C₁₃₇	1	4	yes	complemented
102	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
103	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
104	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	4	no	—
112	C₂×C₈	—	4	8	no	—
113	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
114	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
119	C₄²	M₁₆×C₁₃₇	1	2	yes	—
120	C₄²	#41×C₁₃₇	1	4	yes	—
121	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊C₈×C₁₃₇	1	4	yes	—
128	C₂×C₄	#108×C₁₃₇	1	4	yes	—
129	C₂×C₄	#117×C₁₃₇	1	2	yes	—
130	C₂×C₄	#124×C₁₃₇	1	2	yes	—
131	C₂×C₄	#125×C₁₃₇	1	2	yes	—
132	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
133	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
134	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄	#135×C₁₃₇	1	4	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	8	4	no	—
139	C₂²	#382×C₁₃₇	1	1	yes	characteristic
140	C₂²	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
141	C₂²	#423×C₁₃₇	1	1	yes	characteristic
142	C₂²	#427×C₁₃₇	1	2	yes	—
143	C₂²	C₂×#108×C₁₃₇	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₄	#431×C₁₃₇	1	2	yes	—
148	C₄	#432×C₁₃₇	1	2	yes	—
149	C₄	#488×C₁₃₇	1	4	yes	—
150	C₂	#1564×C₁₃₇	1	2	yes	—
151	C₂	#1568×C₁₃₇	1	2	yes	—
152	C₂	#1831×C₁₃₇	1	1	yes	characteristic
153	C₂	#1835×C₁₃₇	1	1	yes	characteristic
154	C₂	#1837×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
23. C₄×C₈×C₁₃₇	C₈	semidirect
77. #8785	C₁₃₇	direct (N × H)
78. C₁₃₇	#8785	direct (N × H)
101. C₄×C₈	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1831×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
#1277×C₁₃₇	2
#8785	137

Subgroups — C137×#8785 #46858171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8785 #46858171