Subgroups — C₁₃₇×#8808 #46858194

Order 35072.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1831×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×#334×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
7	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#97×C₁₃₇	C₄	1	2	yes	—
11	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
15	#336×C₁₃₇	—	2	2	no	—
16	#335×C₁₃₇	—	2	4	no	—
17	#334×C₁₃₇	—	2	4	no	—
18	C₂²×C₈×C₁₃₇	—	2	2	no	—
19	C₂²×C₈×C₁₃₇	—	2	1	no	—
20	C₂²×C₈×C₁₃₇	—	2	1	no	—
21	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
27	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₁₃₇	—	2	2	no	—
29	C₄⋊C₈×C₁₃₇	—	2	2	no	—
30	C₄⋊C₈×C₁₃₇	—	2	2	no	—
31	C₄⋊C₈×C₁₃₇	—	2	2	no	—
32	C₄⋊C₈×C₁₃₇	—	4	4	no	—
33	#97×C₁₃₇	—	2	4	no	—
34	#97×C₁₃₇	—	2	2	no	—
35	#97×C₁₃₇	C₈	1	4	yes	complemented
36	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
37	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
38	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₃₇	—	2	2	no	—
44	C₂×C₈×C₁₃₇	—	2	2	no	—
45	C₂×C₈×C₁₃₇	—	4	2	no	—
46	C₂×C₈×C₁₃₇	—	4	1	no	—
47	C₂×C₈×C₁₃₇	—	4	1	no	—
48	C₂×C₈×C₁₃₇	—	4	4	no	—
49	C₂×C₈×C₁₃₇	—	4	2	no	—
50	C₂×C₈×C₁₃₇	—	4	2	no	—
51	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
55	C₄²×C₁₃₇	—	2	2	no	—
56	C₄²×C₁₃₇	—	2	2	no	—
57	C₄²×C₁₃₇	—	2	2	no	—
58	C₄²×C₁₃₇	—	4	2	no	—
59	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
60	C₄²×C₁₃₇	M₁₆	1	2	yes	—
61	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
63	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	—	2	2	no	—
65	C₂×C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₁₃₇	#137	1	2	yes	—
80	C₈×C₁₃₇	—	2	4	no	—
81	C₈×C₁₃₇	—	8	2	no	—
82	C₈×C₁₃₇	—	8	1	no	—
83	C₈×C₁₃₇	—	8	1	no	—
84	C₈×C₁₃₇	C₄⋊C₈	1	8	yes	complemented
85	C₂²×C₁₃₇	#384	1	1	yes	characteristic
86	C₂²×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₁₃₇	#423	1	1	yes	characteristic
88	C₂²×C₁₃₇	#428	1	2	yes	—
89	C₂²×C₁₃₇	#430	1	2	yes	—
90	C₄×C₁₃₇	—	2	4	no	—
91	C₄×C₁₃₇	—	2	2	no	—
92	C₄×C₁₃₇	—	4	2	no	—
93	C₄×C₁₃₇	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₁₃₇	#423	1	2	yes	—
95	C₄×C₁₃₇	#496	1	1	yes	characteristic
96	C₄×C₁₃₇	#497	1	1	yes	characteristic
97	C₄×C₁₃₇	#501	1	2	yes	—
98	C₂×C₁₃₇	#1565	1	2	yes	—
99	C₂×C₁₃₇	#1569	1	2	yes	—
100	C₂×C₁₃₇	#1842	1	1	yes	characteristic
101	C₂×C₁₃₇	#1847	1	1	yes	characteristic
102	C₂×C₁₃₇	#1849	1	1	yes	characteristic
103	#8808	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₇	#8808	1	1	yes	characteristic, direct product factor, Sylow
105	#1842	C₂×C₁₃₇	1	1	yes	characteristic
106	#1831	C₂×C₁₃₇	1	1	yes	characteristic
107	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
108	C₂×#335	C₂×C₁₃₇	1	2	yes	—
109	C₂×#334	C₂×C₁₃₇	1	2	yes	—
110	C₂×C₄⋊C₈	—	2	2	no	—
111	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
114	C₂×#97	C₄×C₁₃₇	1	2	yes	—
115	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
116	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
118	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#335	—	2	4	no	—
121	#334	—	2	4	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
126	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
129	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
130	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
131	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	4	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	#97	C₈×C₁₃₇	1	4	yes	complemented
140	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
141	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
142	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
143	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
156	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
157	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
158	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
164	C₄²	M₁₆×C₁₃₇	1	2	yes	—
165	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
166	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
167	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₄⋊C₈×C₁₃₇	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
176	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
178	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
181	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
183	C₂×C₄	#137×C₁₃₇	1	2	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	8	2	no	—
186	C₈	—	8	1	no	—
187	C₈	—	8	1	no	—
188	C₈	C₄⋊C₈×C₁₃₇	1	8	yes	complemented
189	C₂²	#384×C₁₃₇	1	1	yes	characteristic
190	C₂²	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
191	C₂²	#423×C₁₃₇	1	1	yes	characteristic
192	C₂²	#428×C₁₃₇	1	2	yes	—
193	C₂²	#430×C₁₃₇	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₄⋊C₈×C₁₃₇	1	2	yes	—
198	C₄	#423×C₁₃₇	1	2	yes	—
199	C₄	#496×C₁₃₇	1	1	yes	characteristic
200	C₄	#497×C₁₃₇	1	1	yes	characteristic
201	C₄	#501×C₁₃₇	1	2	yes	—
202	C₂	#1565×C₁₃₇	1	2	yes	—
203	C₂	#1569×C₁₃₇	1	2	yes	—
204	C₂	#1842×C₁₃₇	1	1	yes	characteristic
205	C₂	#1847×C₁₃₇	1	1	yes	characteristic
206	C₂	#1849×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
35. #97×C₁₃₇	C₈	semidirect
84. C₈×C₁₃₇	C₄⋊C₈	semidirect
103. #8808	C₁₃₇	direct (N × H)
104. C₁₃₇	#8808	direct (N × H)
139. #97	C₈×C₁₃₇	semidirect
188. C₈	C₄⋊C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#1842×C₁₃₇	2
#1831×C₁₃₇	2
C₄×#97×C₁₃₇	2
C₂×#335×C₁₃₇	2
C₂×#334×C₁₃₇	2
#8808	137

Subgroups — C137×#8808 #46858194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8808 #46858194