Subgroups — C₁₃₇×#8819 #46858205

Order 35072.

438 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1844×C₁₃₇	C₂	1	2	yes	maximal
4	#1834×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₁₃₇	—	2	2	no	—
7	C₂×#106×C₁₃₇	—	2	4	no	—
8	C₂×#106×C₁₃₇	—	2	2	no	—
9	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	#384×C₁₃₇	C₂²	1	1	yes	characteristic
12	#384×C₁₃₇	C₂²	1	2	yes	—
13	#384×C₁₃₇	C₂²	1	1	yes	characteristic
14	#340×C₁₃₇	—	2	4	no	—
15	#322×C₁₃₇	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
17	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
25	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
26	#107×C₁₃₇	—	4	4	no	—
27	#106×C₁₃₇	—	4	8	no	—
28	#106×C₁₃₇	—	4	4	no	—
29	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
32	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₃₇	—	4	1	no	—
34	C₂²×C₄×C₁₃₇	—	4	1	no	—
35	C₂²×C₄×C₁₃₇	—	4	2	no	—
36	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₃₇	—	2	4	no	—
39	C₂×C₈×C₁₃₇	—	2	4	no	—
40	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
41	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
43	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
44	C₄⋊C₄×C₁₃₇	—	4	4	no	—
45	C₄⋊C₄×C₁₃₇	—	4	2	no	—
46	C₄⋊C₄×C₁₃₇	—	4	2	no	—
47	C₄⋊C₄×C₁₃₇	—	4	4	no	—
48	C₄⋊C₄×C₁₃₇	—	4	4	no	—
49	C₄⋊C₄×C₁₃₇	—	4	4	no	—
50	C₄⋊C₄×C₁₃₇	—	4	4	no	—
51	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₁₃₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	8	2	no	—
56	C₂×C₄×C₁₃₇	—	8	1	no	—
57	C₂×C₄×C₁₃₇	—	8	1	no	—
58	C₂×C₄×C₁₃₇	—	8	2	no	—
59	C₂×C₄×C₁₃₇	—	8	2	no	—
60	C₂×C₄×C₁₃₇	—	8	2	no	—
61	C₂×C₄×C₁₃₇	—	8	2	no	—
62	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
69	C₂×C₄×C₁₃₇	#137	1	4	yes	—
70	C₈×C₁₃₇	—	2	8	no	—
71	C₈×C₁₃₇	#106	1	8	yes	complemented
72	C₂²×C₁₃₇	#384	1	1	yes	characteristic
73	C₂²×C₁₃₇	#492	1	1	yes	characteristic
74	C₂²×C₁₃₇	#492	1	1	yes	characteristic
75	C₂²×C₁₃₇	#496	1	2	yes	—
76	C₂²×C₁₃₇	#497	1	2	yes	—
77	C₄×C₁₃₇	—	16	2	no	—
78	C₄×C₁₃₇	—	16	1	no	—
79	C₄×C₁₃₇	—	16	1	no	—
80	C₄×C₁₃₇	C₂×#106	1	2	yes	—
81	C₄×C₁₃₇	#428	1	4	yes	—
82	C₄×C₁₃₇	#498	1	2	yes	—
83	C₄×C₁₃₇	#501	1	4	yes	—
84	C₂×C₁₃₇	#1714	1	2	yes	—
85	C₂×C₁₃₇	#1715	1	1	yes	characteristic
86	C₂×C₁₃₇	#1716	1	1	yes	characteristic
87	C₂×C₁₃₇	#1844	1	1	yes	characteristic
88	C₂×C₁₃₇	#1847	1	2	yes	—
89	#8819	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₁₃₇	#8819	1	1	yes	characteristic, direct product factor, Sylow
91	#1845	C₂×C₁₃₇	1	1	yes	characteristic
92	#1844	C₂×C₁₃₇	1	1	yes	characteristic
93	#1844	C₂×C₁₃₇	1	2	yes	—
94	#1834	C₂×C₁₃₇	1	2	yes	—
95	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
96	C₂×#107	—	2	2	no	—
97	C₂×#106	—	2	4	no	—
98	C₂×#106	—	2	2	no	—
99	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
101	#384	C₂²×C₁₃₇	1	1	yes	characteristic
102	#384	C₂²×C₁₃₇	1	2	yes	—
103	#384	C₂²×C₁₃₇	1	1	yes	characteristic
104	#340	—	2	4	no	—
105	#322	C₄×C₁₃₇	1	4	yes	complemented
106	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
107	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
116	#107	—	4	4	no	—
117	#106	—	4	8	no	—
118	#106	—	4	4	no	—
119	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
120	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
121	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
122	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
123	C₂²×C₄	—	4	1	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
131	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
132	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
133	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
142	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
143	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
144	C₂³	#127×C₁₃₇	1	1	yes	characteristic
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
154	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
156	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
157	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
158	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
159	C₂×C₄	#137×C₁₃₇	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#106×C₁₃₇	1	8	yes	complemented
162	C₂²	#384×C₁₃₇	1	1	yes	characteristic
163	C₂²	#492×C₁₃₇	1	1	yes	characteristic
164	C₂²	#492×C₁₃₇	1	1	yes	characteristic
165	C₂²	#496×C₁₃₇	1	2	yes	—
166	C₂²	#497×C₁₃₇	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	16	1	no	—
169	C₄	—	16	1	no	—
170	C₄	C₂×#106×C₁₃₇	1	2	yes	—
171	C₄	#428×C₁₃₇	1	4	yes	—
172	C₄	#498×C₁₃₇	1	2	yes	—
173	C₄	#501×C₁₃₇	1	4	yes	—
174	C₂	#1714×C₁₃₇	1	2	yes	—
175	C₂	#1715×C₁₃₇	1	1	yes	characteristic
176	C₂	#1716×C₁₃₇	1	1	yes	characteristic
177	C₂	#1844×C₁₃₇	1	1	yes	characteristic
178	C₂	#1847×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #322×C₁₃₇	C₄	semidirect
71. C₈×C₁₃₇	#106	semidirect
89. #8819	C₁₃₇	direct (N × H)
90. C₁₃₇	#8819	direct (N × H)
105. #322	C₄×C₁₃₇	semidirect
161. C₈	#106×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1845×C₁₃₇	2
#1844×C₁₃₇	2
#1834×C₁₃₇	2
C₂×#322×C₁₃₇	2
#8819	137

Subgroups — C137×#8819 #46858205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8819 #46858205