Subgroups — C₁₃₇×#12033 #46861419

Order 35072.

222 conjugacy classes of subgroups.

The subgroup lattice has 164 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	#1567×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1372×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#423×C₁₃₇	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₃₇	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
9	C₂²⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₁₃₇	C₂²	1	2	yes	—
11	#335×C₁₃₇	—	2	1	no	—
12	#335×C₁₃₇	—	2	1	no	—
13	#335×C₁₃₇	C₂²	1	2	yes	—
14	#335×C₁₃₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₃₇	—	4	1	no	—
16	C₂²×C₈×C₁₃₇	—	4	1	no	—
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
21	C₄⋊C₈×C₁₃₇	—	2	2	no	—
22	C₄⋊C₈×C₁₃₇	—	2	2	no	—
23	C₄⋊C₈×C₁₃₇	—	4	1	no	—
24	C₄⋊C₈×C₁₃₇	—	4	1	no	—
25	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₁₃₇	—	8	2	no	—
33	C₂×C₈×C₁₃₇	—	2	2	no	—
34	C₂×C₈×C₁₃₇	—	4	1	no	—
35	C₂×C₈×C₁₃₇	—	4	2	no	—
36	C₂×C₈×C₁₃₇	—	4	2	no	—
37	C₂×C₈×C₁₃₇	—	8	2	no	—
38	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
44	C₄²×C₁₃₇	Q₁₆	1	1	yes	characteristic
45	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₁₃₇	—	2	4	no	—
47	C₂³×C₁₃₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	—	2	1	no	—
49	C₂×C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄×C₁₃₇	—	4	1	no	—
51	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	SD₃₂	1	2	yes	—
55	C₂×C₄×C₁₃₇	SD₃₂	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₃₇	—	2	1	no	—
60	C₈×C₁₃₇	—	2	1	no	—
61	C₈×C₁₃₇	—	2	1	no	—
62	C₈×C₁₃₇	—	8	2	no	—
63	C₈×C₁₃₇	—	8	2	no	—
64	C₈×C₁₃₇	#131	1	2	yes	—
65	C₂²×C₁₃₇	—	2	1	no	—
66	C₂²×C₁₃₇	—	2	1	no	—
67	C₂²×C₁₃₇	#367	1	2	yes	—
68	C₂²×C₁₃₇	C₂×#107	1	1	yes	characteristic
69	C₄×C₁₃₇	—	2	1	no	—
70	C₄×C₁₃₇	—	2	1	no	—
71	C₄×C₁₃₇	—	2	1	no	—
72	C₄×C₁₃₇	#367	1	2	yes	—
73	C₄×C₁₃₇	#424	1	1	yes	characteristic
74	C₄×C₁₃₇	#430	1	1	yes	characteristic
75	C₄×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
76	C₄×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
77	C₂×C₁₃₇	—	2	2	no	—
78	C₂×C₁₃₇	#1571	1	1	yes	characteristic
79	C₂×C₁₃₇	C₂×#367	1	1	yes	characteristic
80	C₂×C₁₃₇	#2161	1	1	yes	characteristic
81	#12033	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₁₃₇	#12033	1	1	yes	characteristic, direct product factor, Sylow
83	#2162	C₂×C₁₃₇	1	1	yes	characteristic
84	#1567	C₂×C₁₃₇	1	1	yes	characteristic
85	C₂×#335	C₂×C₁₃₇	1	1	yes	characteristic
86	#1372	C₂×C₁₃₇	1	4	yes	complemented
87	#423	—	2	1	no	—
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
90	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
91	C₂²⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
92	C₄×C₁₆	C₂²×C₁₃₇	1	2	yes	—
93	#335	—	2	1	no	—
94	#335	—	2	1	no	—
95	#335	C₂²×C₁₃₇	1	2	yes	—
96	#335	C₂²×C₁₃₇	1	2	yes	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
100	C₂²×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
108	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
112	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
113	C₂²×C₄	Q₁₆×C₁₃₇	1	1	yes	characteristic
114	M₁₆	—	8	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
124	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
125	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
126	C₄²	Q₁₆×C₁₃₇	1	1	yes	characteristic
127	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₂³	#107×C₁₃₇	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
134	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	SD₃₂×C₁₃₇	1	2	yes	—
137	C₂×C₄	SD₃₂×C₁₃₇	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	2	no	—
146	C₈	#131×C₁₃₇	1	2	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#367×C₁₃₇	1	2	yes	—
150	C₂²	C₂×#107×C₁₃₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#367×C₁₃₇	1	2	yes	—
155	C₄	#424×C₁₃₇	1	1	yes	characteristic
156	C₄	#430×C₁₃₇	1	1	yes	characteristic
157	C₄	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
158	C₄	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#1571×C₁₃₇	1	1	yes	characteristic
161	C₂	C₂×#367×C₁₃₇	1	1	yes	characteristic
162	C₂	#2161×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#107
prime radical	C₂³×C₁₃₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1372×C₁₃₇	C₂	semidirect
81. #12033	C₁₃₇	direct (N × H)
82. C₁₃₇	#12033	direct (N × H)
86. #1372	C₂×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
#1567×C₁₃₇	2
C₂×#335×C₁₃₇	2
#1372×C₁₃₇	2
#12033	137

Subgroups — C137×#12033 #46861419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12033 #46861419