Subgroups — C₁₃₇×#8786 #46858172

Order 35072.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#1277×C₁₃₇	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
6	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×#97×C₁₃₇	C₄	1	2	yes	—
9	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
10	C₂×#97×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
12	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
13	#336×C₁₃₇	—	2	2	no	—
14	C₂²×C₈×C₁₃₇	—	2	2	no	—
15	C₂²×C₈×C₁₃₇	—	2	2	no	—
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
22	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₁₃₇	—	2	4	no	—
24	C₄⋊C₈×C₁₃₇	—	2	4	no	—
25	C₄⋊C₈×C₁₃₇	—	4	4	no	—
26	#97×C₁₃₇	—	2	4	no	—
27	#97×C₁₃₇	—	2	2	no	—
28	#97×C₁₃₇	C₈	1	4	yes	complemented
29	C₄×C₈×C₁₃₇	—	2	2	no	—
30	C₄×C₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃₇	—	2	2	no	—
37	C₂×C₈×C₁₃₇	—	2	2	no	—
38	C₂×C₈×C₁₃₇	—	2	2	no	—
39	C₂×C₈×C₁₃₇	—	2	2	no	—
40	C₂×C₈×C₁₃₇	—	4	2	no	—
41	C₂×C₈×C₁₃₇	—	4	2	no	—
42	C₂×C₈×C₁₃₇	—	4	4	no	—
43	C₂×C₈×C₁₃₇	—	4	2	no	—
44	C₂×C₈×C₁₃₇	—	4	2	no	—
45	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
47	C₄²×C₁₃₇	—	2	2	no	—
48	C₄²×C₁₃₇	—	2	2	no	—
49	C₄²×C₁₃₇	—	2	2	no	—
50	C₄²×C₁₃₇	—	4	2	no	—
51	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
52	C₄²×C₁₃₇	M₁₆	1	2	yes	—
53	C₄²×C₁₃₇	#41	1	2	yes	—
54	C₄²×C₁₃₇	#41	1	2	yes	—
55	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄×C₁₃₇	—	2	2	no	—
61	C₂×C₄×C₁₃₇	—	2	2	no	—
62	C₂×C₄×C₁₃₇	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₁₃₇	#117	1	2	yes	—
64	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	#125	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₁₃₇	#137	1	2	yes	—
72	C₈×C₁₃₇	—	2	4	no	—
73	C₈×C₁₃₇	—	2	4	no	—
74	C₈×C₁₃₇	—	8	2	no	—
75	C₈×C₁₃₇	—	8	2	no	—
76	C₂²×C₁₃₇	#382	1	1	yes	characteristic
77	C₂²×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₁₃₇	#423	1	1	yes	characteristic
79	C₂²×C₁₃₇	#428	1	2	yes	—
80	C₂²×C₁₃₇	#430	1	2	yes	—
81	C₄×C₁₃₇	—	2	4	no	—
82	C₄×C₁₃₇	—	2	2	no	—
83	C₄×C₁₃₇	—	4	2	no	—
84	C₄×C₁₃₇	#431	1	2	yes	—
85	C₄×C₁₃₇	#432	1	2	yes	—
86	C₄×C₁₃₇	#489	1	2	yes	—
87	C₄×C₁₃₇	#490	1	1	yes	characteristic
88	C₄×C₁₃₇	#491	1	1	yes	characteristic
89	C₂×C₁₃₇	#1565	1	2	yes	—
90	C₂×C₁₃₇	#1569	1	2	yes	—
91	C₂×C₁₃₇	#1831	1	1	yes	characteristic
92	C₂×C₁₃₇	#1836	1	1	yes	characteristic
93	C₂×C₁₃₇	#1838	1	1	yes	characteristic
94	#8786	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₇	#8786	1	1	yes	characteristic, direct product factor, Sylow
96	#1842	C₂×C₁₃₇	1	1	yes	characteristic
97	#1831	C₂×C₁₃₇	1	1	yes	characteristic
98	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
99	#1277	C₂×C₁₃₇	1	4	yes	—
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
103	C₂×#97	C₄×C₁₃₇	1	2	yes	—
104	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
105	C₂×#97	C₄×C₁₃₇	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
107	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
112	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
115	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
116	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
117	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	4	4	no	—
121	#97	—	2	4	no	—
122	#97	—	2	2	no	—
123	#97	C₈×C₁₃₇	1	4	yes	complemented
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
127	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
141	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
147	C₄²	M₁₆×C₁₃₇	1	2	yes	—
148	C₄²	#41×C₁₃₇	1	2	yes	—
149	C₄²	#41×C₁₃₇	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₄⋊C₈×C₁₃₇	1	4	yes	—
158	C₂×C₄	#117×C₁₃₇	1	2	yes	—
159	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
161	C₂×C₄	#125×C₁₃₇	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
164	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
165	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
166	C₂×C₄	#137×C₁₃₇	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	2	4	no	—
169	C₈	—	8	2	no	—
170	C₈	—	8	2	no	—
171	C₂²	#382×C₁₃₇	1	1	yes	characteristic
172	C₂²	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
173	C₂²	#423×C₁₃₇	1	1	yes	characteristic
174	C₂²	#428×C₁₃₇	1	2	yes	—
175	C₂²	#430×C₁₃₇	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	#431×C₁₃₇	1	2	yes	—
180	C₄	#432×C₁₃₇	1	2	yes	—
181	C₄	#489×C₁₃₇	1	2	yes	—
182	C₄	#490×C₁₃₇	1	1	yes	characteristic
183	C₄	#491×C₁₃₇	1	1	yes	characteristic
184	C₂	#1565×C₁₃₇	1	2	yes	—
185	C₂	#1569×C₁₃₇	1	2	yes	—
186	C₂	#1831×C₁₃₇	1	1	yes	characteristic
187	C₂	#1836×C₁₃₇	1	1	yes	characteristic
188	C₂	#1838×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
28. #97×C₁₃₇	C₈	semidirect
94. #8786	C₁₃₇	direct (N × H)
95. C₁₃₇	#8786	direct (N × H)
123. #97	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
#1277×C₁₃₇	2
#8786	137

Subgroups — C137×#8786 #46858172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8786 #46858172