Subgroups — C₁₃₇×#8824 #46858210

Order 35072.

322 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	#1837×C₁₃₇	C₂	1	2	yes	maximal
2	#1567×C₁₃₇	C₂	1	2	yes	maximal
3	#1566×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#423×C₁₃₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	#341×C₁₃₇	—	2	4	no	—
10	#335×C₁₃₇	—	2	4	no	—
11	#334×C₁₃₇	—	2	4	no	—
12	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
13	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
14	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
15	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₇	—	2	4	no	—
17	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
18	C₂²×C₈×C₁₃₇	Q₈	1	2	yes	—
19	C₂²×C₈×C₁₃₇	Q₈	1	2	yes	—
20	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	—	2	4	no	—
22	C₄⋊C₈×C₁₃₇	—	2	4	no	—
23	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
30	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
31	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
35	M₁₆×C₁₃₇	—	4	8	no	—
36	C₂×C₈×C₁₃₇	—	2	4	no	—
37	C₂×C₈×C₁₃₇	—	2	4	no	—
38	C₂×C₈×C₁₃₇	—	2	4	no	—
39	C₂×C₈×C₁₃₇	—	4	4	no	—
40	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	#135	1	4	yes	—
60	C₈×C₁₃₇	—	2	2	no	—
61	C₈×C₁₃₇	—	2	2	no	—
62	C₈×C₁₃₇	—	2	2	no	—
63	C₈×C₁₃₇	—	2	2	no	—
64	C₈×C₁₃₇	—	8	4	no	—
65	C₈×C₁₃₇	#108	1	4	yes	—
66	C₈×C₁₃₇	#108	1	4	yes	—
67	C₂²×C₁₃₇	—	2	1	no	—
68	C₂²×C₁₃₇	—	2	1	no	—
69	C₂²×C₁₃₇	#384	1	1	yes	characteristic
70	C₂²×C₁₃₇	#499	1	2	yes	—
71	C₄×C₁₃₇	—	2	1	no	—
72	C₄×C₁₃₇	—	2	1	no	—
73	C₄×C₁₃₇	#427	1	2	yes	—
74	C₄×C₁₃₇	C₂×#108	1	2	yes	—
75	C₄×C₁₃₇	C₂×#108	1	2	yes	—
76	C₄×C₁₃₇	#495	1	2	yes	—
77	C₂×C₁₃₇	—	2	2	no	—
78	C₂×C₁₃₇	#1846	1	1	yes	characteristic
79	C₂×C₁₃₇	#1848	1	1	yes	characteristic
80	C₂×C₁₃₇	#1848	1	1	yes	characteristic
81	#8824	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₁₃₇	#8824	1	1	yes	characteristic, direct product factor, Sylow
83	#1837	C₂×C₁₃₇	1	2	yes	—
84	#1567	C₂×C₁₃₇	1	2	yes	—
85	#1566	C₂×C₁₃₇	1	2	yes	—
86	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
87	#423	C₂²×C₁₃₇	1	4	yes	—
88	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
91	#341	—	2	4	no	—
92	#335	—	2	4	no	—
93	#334	—	2	4	no	—
94	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
95	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
96	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
97	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
100	C₂²×C₈	Q₈×C₁₃₇	1	2	yes	—
101	C₂²×C₈	Q₈×C₁₃₇	1	2	yes	—
102	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
106	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
108	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
111	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
112	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
113	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
114	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
115	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
117	M₁₆	—	4	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	4	4	no	—
122	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
123	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
124	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
125	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
126	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
127	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
128	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
129	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
130	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
131	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
132	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
141	C₂×C₄	#135×C₁₃₇	1	4	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	8	4	no	—
147	C₈	#108×C₁₃₇	1	4	yes	—
148	C₈	#108×C₁₃₇	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#384×C₁₃₇	1	1	yes	characteristic
152	C₂²	#499×C₁₃₇	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#427×C₁₃₇	1	2	yes	—
156	C₄	C₂×#108×C₁₃₇	1	2	yes	—
157	C₄	C₂×#108×C₁₃₇	1	2	yes	—
158	C₄	#495×C₁₃₇	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#1846×C₁₃₇	1	1	yes	characteristic
161	C₂	#1848×C₁₃₇	1	1	yes	characteristic
162	C₂	#1848×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#384
prime radical	C₂³×C₁₃₇	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
81. #8824	C₁₃₇	direct (N × H)
82. C₁₃₇	#8824	direct (N × H)

Maximal subgroups

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

subgroup	index
#1837×C₁₃₇	2
#1567×C₁₃₇	2
#1566×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#8824	137

Subgroups — C137×#8824 #46858210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8824 #46858210