Subgroups — C₁₃₇×#8790 #46858176

Order 35072.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1286×C₁₃₇	C₂	1	4	yes	maximal
5	#431×C₁₃₇	C₂²	1	2	yes	—
6	C₂×#108×C₁₃₇	—	2	1	no	—
7	C₂×#108×C₁₃₇	—	2	1	no	—
8	C₂×#107×C₁₃₇	—	2	1	no	—
9	C₂×#106×C₁₃₇	—	2	1	no	—
10	#423×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#382×C₁₃₇	C₂²	1	2	yes	—
13	#322×C₁₃₇	C₄	1	2	yes	complemented
14	C₈²×C₁₃₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₃₇	—	2	2	no	—
16	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
19	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
20	#108×C₁₃₇	—	4	2	no	—
21	#108×C₁₃₇	—	4	2	no	—
22	#107×C₁₃₇	—	4	2	no	—
23	#106×C₁₃₇	—	4	2	no	—
24	C₄⋊C₈×C₁₃₇	—	2	2	no	—
25	C₄⋊C₈×C₁₃₇	—	2	4	no	—
26	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
27	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
32	#95×C₁₃₇	—	2	4	no	—
33	C₂²×C₄×C₁₃₇	—	4	2	no	—
34	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
36	M₁₆×C₁₃₇	—	4	2	no	—
37	M₁₆×C₁₃₇	—	4	2	no	—
38	C₂×C₈×C₁₃₇	—	2	4	no	—
39	C₂×C₈×C₁₃₇	—	4	2	no	—
40	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₇	D₈	1	2	yes	—
45	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₃₇	Q₁₆	1	2	yes	—
48	C₄⋊C₄×C₁₃₇	—	4	2	no	—
49	C₄⋊C₄×C₁₃₇	—	4	2	no	—
50	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₃₇	#41	1	2	yes	—
53	C₂³×C₁₃₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	—	4	2	no	—
55	C₂×C₄×C₁₃₇	—	8	4	no	—
56	C₂×C₄×C₁₃₇	—	8	2	no	—
57	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	#117	1	2	yes	—
59	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₃₇	#125	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₃₇	—	4	4	no	—
66	C₈×C₁₃₇	—	8	2	no	—
67	C₈×C₁₃₇	#106	1	2	yes	—
68	C₈×C₁₃₇	#106	1	2	yes	—
69	C₈×C₁₃₇	#107	1	2	yes	—
70	C₈×C₁₃₇	#107	1	2	yes	—
71	C₂²×C₁₃₇	—	2	2	no	—
72	C₂²×C₁₃₇	#382	1	1	yes	characteristic
73	C₂²×C₁₃₇	#486	1	1	yes	characteristic
74	C₂²×C₁₃₇	#487	1	1	yes	characteristic
75	C₄×C₁₃₇	—	2	2	no	—
76	C₄×C₁₃₇	—	2	2	no	—
77	C₄×C₁₃₇	—	16	2	no	—
78	C₄×C₁₃₇	C₂×#106	1	1	yes	characteristic
79	C₄×C₁₃₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₁₃₇	#486	1	1	yes	characteristic
81	C₄×C₁₃₇	#487	1	1	yes	characteristic
82	C₂×C₁₃₇	—	2	1	no	—
83	C₂×C₁₃₇	—	2	1	no	—
84	C₂×C₁₃₇	#1833	1	2	yes	—
85	C₂×C₁₃₇	#1834	1	1	yes	characteristic
86	#8790	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₁₃₇	#8790	1	1	yes	characteristic, direct product factor, Sylow
88	#1848	C₂×C₁₃₇	1	1	yes	characteristic
89	#1846	C₂×C₁₃₇	1	1	yes	characteristic
90	#1450	C₂×C₁₃₇	1	1	yes	characteristic
91	#1286	C₂×C₁₃₇	1	4	yes	—
92	#431	C₂²×C₁₃₇	1	2	yes	—
93	C₂×#108	—	2	1	no	—
94	C₂×#108	—	2	1	no	—
95	C₂×#107	—	2	1	no	—
96	C₂×#106	—	2	1	no	—
97	#423	C₂²×C₁₃₇	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
99	#382	C₂²×C₁₃₇	1	2	yes	—
100	#322	C₄×C₁₃₇	1	2	yes	complemented
101	C₈²	C₄×C₁₃₇	1	2	yes	complemented
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
104	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
107	#108	—	4	2	no	—
108	#108	—	4	2	no	—
109	#107	—	4	2	no	—
110	#106	—	4	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
115	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
116	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
119	#95	—	2	4	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
122	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
128	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
131	C₂×C₈	D₈×C₁₃₇	1	2	yes	—
132	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
133	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
134	C₂×C₈	Q₁₆×C₁₃₇	1	2	yes	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
138	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
139	C₄²	#41×C₁₃₇	1	2	yes	—
140	C₂³	#124×C₁₃₇	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
145	C₂×C₄	#117×C₁₃₇	1	2	yes	—
146	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
147	C₂×C₄	#125×C₁₃₇	1	2	yes	—
148	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
149	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
150	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	8	2	no	—
154	C₈	#106×C₁₃₇	1	2	yes	—
155	C₈	#106×C₁₃₇	1	2	yes	—
156	C₈	#107×C₁₃₇	1	2	yes	—
157	C₈	#107×C₁₃₇	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#382×C₁₃₇	1	1	yes	characteristic
160	C₂²	#486×C₁₃₇	1	1	yes	characteristic
161	C₂²	#487×C₁₃₇	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	C₂×#106×C₁₃₇	1	1	yes	characteristic
166	C₄	C₂×#107×C₁₃₇	1	1	yes	characteristic
167	C₄	#486×C₁₃₇	1	1	yes	characteristic
168	C₄	#487×C₁₃₇	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#1833×C₁₃₇	1	2	yes	—
172	C₂	#1834×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#382
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#124
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
13. #322×C₁₃₇	C₄	semidirect
14. C₈²×C₁₃₇	C₄	semidirect
86. #8790	C₁₃₇	direct (N × H)
87. C₁₃₇	#8790	direct (N × H)
100. #322	C₄×C₁₃₇	semidirect
101. C₈²	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1848×C₁₃₇	2
#1846×C₁₃₇	2
#1450×C₁₃₇	2
#1286×C₁₃₇	2
#8790	137

Subgroups — C137×#8790 #46858176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8790 #46858176