Subgroups — C₂×#559×C₁₃₇ #31778490

Order 17536.

1240 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₁₃₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#124×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂×#123×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₈×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×D₄×C₁₃₇	C₂²	1	4	yes	—
9	C₂²×D₄×C₁₃₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₃₇	C₂²	1	4	yes	—
11	#127×C₁₃₇	C₂²	1	8	yes	—
12	#124×C₁₃₇	C₂²	1	32	yes	complemented
13	#123×C₁₃₇	C₂²	1	32	yes	complemented
14	C₄×Q₈×C₁₃₇	C₂²	1	16	yes	complemented
15	C₄×D₄×C₁₃₇	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₁₃₇	C₂³	1	4	yes	—
23	C₂×Q₈×C₁₃₇	—	2	8	no	—
24	C₂×Q₈×C₁₃₇	C₂³	1	8	yes	—
25	C₂×D₄×C₁₃₇	—	2	16	no	—
26	C₂×D₄×C₁₃₇	—	4	4	no	—
27	C₂×D₄×C₁₃₇	C₂³	1	16	yes	—
28	C₂×D₄×C₁₃₇	C₂³	1	8	yes	—
29	C₂²×C₄×C₁₃₇	—	4	8	no	—
30	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂³	1	16	yes	—
34	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
36	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
37	C₂²⋊C₄×C₁₃₇	C₂³	1	32	yes	—
38	C₂²⋊C₄×C₁₃₇	C₂³	1	16	yes	—
39	C₄²×C₁₃₇	C₂³	1	16	yes	—
40	C₄²×C₁₃₇	C₂³	1	4	yes	—
41	C₂³×C₁₃₇	—	2	16	no	—
42	C₂³×C₁₃₇	—	2	8	no	—
43	C₂³×C₁₃₇	C₂⁴	1	8	yes	—
44	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₃₇	—	2	16	no	—
46	D₄×C₁₃₇	—	2	32	no	—
47	D₄×C₁₃₇	—	4	8	no	—
48	C₂×C₄×C₁₃₇	—	2	8	no	—
49	C₂×C₄×C₁₃₇	—	2	8	no	—
50	C₂×C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₄×C₁₃₇	—	4	16	no	—
52	C₂×C₄×C₁₃₇	#41	1	8	yes	—
53	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
54	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
55	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
57	C₂²×C₁₃₇	—	2	16	no	—
58	C₂²×C₁₃₇	—	2	8	no	—
59	C₂²×C₁₃₇	—	4	16	no	—
60	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
61	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
63	C₄×C₁₃₇	—	2	8	no	—
64	C₄×C₁₃₇	—	2	8	no	—
65	C₄×C₁₃₇	—	2	2	no	—
66	C₄×C₁₃₇	C₂×#41	1	8	yes	—
67	C₂×C₁₃₇	—	4	8	no	—
68	C₂×C₁₃₇	#559	1	4	yes	direct product factor
69	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
70	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
71	C₁₃₇	C₂×#559	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×#559	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	#559	C₂×C₁₃₇	1	16	yes	direct product factor
74	C₂×#127	C₂×C₁₃₇	1	1	yes	characteristic
75	C₂×#124	C₂×C₁₃₇	1	4	yes	complemented
76	C₂×#123	C₂×C₁₃₇	1	4	yes	complemented
77	C₂×C₄×Q₈	C₂×C₁₃₇	1	2	yes	complemented
78	C₂×C₄×D₄	C₂×C₁₃₇	1	4	yes	complemented
79	C₂²×Q₈	C₂²×C₁₃₇	1	2	yes	—
80	C₂²×D₄	C₂²×C₁₃₇	1	4	yes	—
81	C₂²×D₄	C₂²×C₁₃₇	1	2	yes	—
82	C₂³×C₄	C₂²×C₁₃₇	1	4	yes	—
83	#127	C₂²×C₁₃₇	1	8	yes	—
84	#124	C₂²×C₁₃₇	1	32	yes	complemented
85	#123	C₂²×C₁₃₇	1	32	yes	complemented
86	C₄×Q₈	C₂²×C₁₃₇	1	16	yes	complemented
87	C₄×D₄	C₂²×C₁₃₇	1	32	yes	complemented
88	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	8	yes	—
91	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
92	C₂×C₄²	C₂²×C₁₃₇	1	4	yes	—
93	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
94	C₂⁴	C₂³×C₁₃₇	1	4	yes	—
95	C₂×Q₈	—	2	8	no	—
96	C₂×Q₈	C₂³×C₁₃₇	1	8	yes	—
97	C₂×D₄	—	2	16	no	—
98	C₂×D₄	—	4	4	no	—
99	C₂×D₄	C₂³×C₁₃₇	1	16	yes	—
100	C₂×D₄	C₂³×C₁₃₇	1	8	yes	—
101	C₂²×C₄	—	4	8	no	—
102	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
103	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
104	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₁₃₇	1	16	yes	—
106	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
107	C₄⋊C₄	C₂³×C₁₃₇	1	8	yes	—
108	C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
109	C₂²⋊C₄	C₂³×C₁₃₇	1	32	yes	—
110	C₂²⋊C₄	C₂³×C₁₃₇	1	16	yes	—
111	C₄²	C₂³×C₁₃₇	1	16	yes	—
112	C₄²	C₂³×C₁₃₇	1	4	yes	—
113	C₂³	—	2	16	no	—
114	C₂³	—	2	8	no	—
115	C₂³	C₂⁴×C₁₃₇	1	8	yes	—
116	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
117	Q₈	—	2	16	no	—
118	D₄	—	2	32	no	—
119	D₄	—	4	8	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	16	no	—
124	C₂×C₄	#41×C₁₃₇	1	8	yes	—
125	C₂×C₄	C₂⁴×C₁₃₇	1	8	yes	—
126	C₂×C₄	C₂⁴×C₁₃₇	1	8	yes	—
127	C₂×C₄	C₂⁴×C₁₃₇	1	2	yes	—
128	C₂×C₄	C₂⁴×C₁₃₇	1	4	yes	—
129	C₂²	—	2	16	no	—
130	C₂²	—	2	8	no	—
131	C₂²	—	4	16	no	—
132	C₂²	C₂×#41×C₁₃₇	1	4	yes	—
133	C₂²	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
134	C₂²	C₂⁵×C₁₃₇	1	1	yes	characteristic
135	C₄	—	2	8	no	—
136	C₄	—	2	8	no	—
137	C₄	—	2	2	no	—
138	C₄	C₂×#41×C₁₃₇	1	8	yes	—
139	C₂	—	4	8	no	—
140	C₂	#559×C₁₃₇	1	4	yes	direct product factor
141	C₂	C₂²×#41×C₁₃₇	1	2	yes	—
142	C₂	C₂×H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₁₃₇	C₂	direct (N × H)
3. C₂×#124×C₁₃₇	C₂	semidirect
4. C₂×#123×C₁₃₇	C₂	semidirect
5. C₂×C₄×Q₈×C₁₃₇	C₂	semidirect
6. C₂×C₄×D₄×C₁₃₇	C₂	semidirect
12. #124×C₁₃₇	C₂²	semidirect
13. #123×C₁₃₇	C₂²	semidirect
14. C₄×Q₈×C₁₃₇	C₂²	semidirect
15. C₄×D₄×C₁₃₇	C₂²	semidirect
68. C₂×C₁₃₇	#559	direct (N × H)
71. C₁₃₇	C₂×#559	direct (N × H)
72. C₂×#559	C₁₃₇	direct (N × H)
73. #559	C₂×C₁₃₇	direct (N × H)
75. C₂×#124	C₂×C₁₃₇	semidirect
76. C₂×#123	C₂×C₁₃₇	semidirect
77. C₂×C₄×Q₈	C₂×C₁₃₇	semidirect
78. C₂×C₄×D₄	C₂×C₁₃₇	semidirect
84. #124	C₂²×C₁₃₇	semidirect
85. #123	C₂²×C₁₃₇	semidirect
86. C₄×Q₈	C₂²×C₁₃₇	semidirect
87. C₄×D₄	C₂²×C₁₃₇	semidirect
140. C₂	#559×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#559×C₁₃₇	2
C₂×#127×C₁₃₇	2
C₂×#124×C₁₃₇	2
C₂×#123×C₁₃₇	2
C₂×C₄×Q₈×C₁₃₇	2
C₂×C₄×D₄×C₁₃₇	2
C₂×#559	137

Subgroups — C2×#559×C137 #31778490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#559×C₁₃₇ #31778490