Subgroups — C₁₃₇×#8812 #46858198

Order 35072.

382 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#1744×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×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
12	#336×C₁₃₇	—	2	2	no	—
13	#336×C₁₃₇	C₄	1	2	yes	—
14	#336×C₁₃₇	C₄	1	2	yes	—
15	#335×C₁₃₇	—	2	4	no	—
16	#334×C₁₃₇	—	2	4	no	—
17	C₂²×C₈×C₁₃₇	—	2	2	no	—
18	C₂²×C₈×C₁₃₇	—	2	1	no	—
19	C₂²×C₈×C₁₃₇	—	2	1	no	—
20	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₁₃₇	—	2	2	no	—
27	C₄⋊C₈×C₁₃₇	—	2	2	no	—
28	C₄⋊C₈×C₁₃₇	—	2	2	no	—
29	C₄⋊C₈×C₁₃₇	—	2	2	no	—
30	C₄⋊C₈×C₁₃₇	—	4	4	no	—
31	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
32	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
33	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₁₃₇	—	2	4	no	—
39	C₂×C₈×C₁₃₇	—	2	4	no	—
40	C₂×C₈×C₁₃₇	—	4	2	no	—
41	C₂×C₈×C₁₃₇	—	4	1	no	—
42	C₂×C₈×C₁₃₇	—	4	1	no	—
43	C₂×C₈×C₁₃₇	—	4	4	no	—
44	C₂×C₈×C₁₃₇	—	4	2	no	—
45	C₂×C₈×C₁₃₇	—	4	2	no	—
46	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
47	C₄²×C₁₃₇	—	2	4	no	—
48	C₄²×C₁₃₇	—	2	2	no	—
49	C₄²×C₁₃₇	—	2	2	no	—
50	C₄²×C₁₃₇	—	2	2	no	—
51	C₄²×C₁₃₇	—	2	2	no	—
52	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
53	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	2	2	no	—
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	#108	1	4	yes	—
62	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	#131	1	2	yes	—
67	C₂×C₄×C₁₃₇	#131	1	2	yes	—
68	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₁₃₇	#137	1	2	yes	—
70	C₈×C₁₃₇	—	2	8	no	—
71	C₈×C₁₃₇	—	8	2	no	—
72	C₈×C₁₃₇	—	8	1	no	—
73	C₈×C₁₃₇	—	8	1	no	—
74	C₂²×C₁₃₇	#384	1	1	yes	characteristic
75	C₂²×C₁₃₇	#424	1	1	yes	characteristic
76	C₂²×C₁₃₇	#424	1	1	yes	characteristic
77	C₂²×C₁₃₇	#428	1	2	yes	—
78	C₂²×C₁₃₇	C₂×#108	1	2	yes	—
79	C₄×C₁₃₇	—	2	4	no	—
80	C₄×C₁₃₇	—	2	2	no	—
81	C₄×C₁₃₇	—	4	2	no	—
82	C₄×C₁₃₇	#424	1	4	yes	—
83	C₄×C₁₃₇	#496	1	1	yes	characteristic
84	C₄×C₁₃₇	#497	1	1	yes	characteristic
85	C₄×C₁₃₇	#501	1	2	yes	—
86	C₂×C₁₃₇	#1573	1	2	yes	—
87	C₂×C₁₃₇	#1574	1	2	yes	—
88	C₂×C₁₃₇	#1843	1	1	yes	characteristic
89	C₂×C₁₃₇	#1847	1	1	yes	characteristic
90	C₂×C₁₃₇	#1848	1	1	yes	characteristic
91	#8812	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₁₃₇	#8812	1	1	yes	characteristic, direct product factor, Sylow
93	#1842	C₂×C₁₃₇	1	1	yes	characteristic
94	#1831	C₂×C₁₃₇	1	1	yes	characteristic
95	#1744	C₂×C₁₃₇	1	1	yes	characteristic
96	C₂×#335	C₂×C₁₃₇	1	2	yes	—
97	C₂×#334	C₂×C₁₃₇	1	2	yes	—
98	C₂×C₄⋊C₈	—	2	2	no	—
99	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
100	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
103	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
104	#336	—	2	2	no	—
105	#336	C₄×C₁₃₇	1	2	yes	—
106	#336	C₄×C₁₃₇	1	2	yes	—
107	#335	—	2	4	no	—
108	#334	—	2	4	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	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	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	4	4	no	—
123	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
124	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
125	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
126	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
145	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#108×C₁₃₇	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
155	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
157	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄	#131×C₁₃₇	1	2	yes	—
159	C₂×C₄	#131×C₁₃₇	1	2	yes	—
160	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
161	C₂×C₄	#137×C₁₃₇	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	8	2	no	—
164	C₈	—	8	1	no	—
165	C₈	—	8	1	no	—
166	C₂²	#384×C₁₃₇	1	1	yes	characteristic
167	C₂²	#424×C₁₃₇	1	1	yes	characteristic
168	C₂²	#424×C₁₃₇	1	1	yes	characteristic
169	C₂²	#428×C₁₃₇	1	2	yes	—
170	C₂²	C₂×#108×C₁₃₇	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#424×C₁₃₇	1	4	yes	—
175	C₄	#496×C₁₃₇	1	1	yes	characteristic
176	C₄	#497×C₁₃₇	1	1	yes	characteristic
177	C₄	#501×C₁₃₇	1	2	yes	—
178	C₂	#1573×C₁₃₇	1	2	yes	—
179	C₂	#1574×C₁₃₇	1	2	yes	—
180	C₂	#1843×C₁₃₇	1	1	yes	characteristic
181	C₂	#1847×C₁₃₇	1	1	yes	characteristic
182	C₂	#1848×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×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
91. #8812	C₁₃₇	direct (N × H)
92. C₁₃₇	#8812	direct (N × H)

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
#1744×C₁₃₇	2
C₂×#335×C₁₃₇	2
C₂×#334×C₁₃₇	2
#8812	137

Subgroups — C137×#8812 #46858198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8812 #46858198