Subgroups — C₄×#117×C₁₃₇ #31777281

Order 17536.

752 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₁₃₇	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₄×C₂²⋊C₄×C₁₃₇	C₂	1	4	yes	maximal
5	#376×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₄³×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
8	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
9	#117×C₁₃₇	C₄	1	16	yes	direct product factor
10	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃₇	C₄	1	8	yes	complemented
13	C₂×C₄²×C₁₃₇	C₂²	1	8	yes	—
14	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
18	#95×C₁₃₇	C₂²	1	8	yes	—
19	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	—	2	8	no	—
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	8	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₄⋊C₄×C₁₃₇	C₂×C₄	1	16	yes	complemented
30	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	16	yes	—
31	C₄²×C₁₃₇	—	2	16	no	—
32	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
34	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₄²×C₁₃₇	C₂×C₄	1	16	yes	complemented
36	C₂³×C₁₃₇	—	2	4	no	—
37	C₂³×C₁₃₇	C₂²×C₄	1	4	yes	—
38	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
39	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	—	2	16	no	—
41	C₂×C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄×C₁₃₇	—	2	16	no	—
43	C₂×C₄×C₁₃₇	C₄²	1	16	yes	complemented
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	16	yes	—
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₃₇	#41	1	8	yes	—
52	C₂²×C₁₃₇	—	2	4	no	—
53	C₂²×C₁₃₇	—	2	8	no	—
54	C₂²×C₁₃₇	C₂×C₄²	1	4	yes	—
55	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
56	C₂²×C₁₃₇	C₂³×C₄	1	2	yes	—
57	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
58	C₄×C₁₃₇	—	2	4	no	—
59	C₄×C₁₃₇	—	2	16	no	—
60	C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
61	C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
62	C₄×C₁₃₇	#117	1	8	yes	direct product factor
63	C₂×C₁₃₇	—	2	4	no	—
64	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
65	C₂×C₁₃₇	C₂×#117	1	2	yes	—
66	C₂×C₁₃₇	C₄×#41	1	4	yes	—
67	C₁₃₇	C₄×#117	1	1	yes	characteristic, direct product factor, Sylow
68	C₄×#117	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×#117	C₂×C₁₃₇	1	2	yes	—
70	C₂²×C₄²	C₂×C₁₃₇	1	1	yes	characteristic
71	C₄×C₄⋊C₄	C₂×C₁₃₇	1	4	yes	complemented
72	C₄×C₂²⋊C₄	C₂×C₁₃₇	1	4	yes	—
73	#376	C₂×C₁₃₇	1	2	yes	complemented
74	C₄³	C₂×C₁₃₇	1	2	yes	complemented
75	C₂³×C₄	C₂²×C₁₃₇	1	2	yes	—
76	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
77	#117	C₄×C₁₃₇	1	16	yes	direct product factor
78	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
80	C₂×C₄²	C₄×C₁₃₇	1	8	yes	complemented
81	C₂×C₄²	C₂²×C₁₃₇	1	8	yes	—
82	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
84	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
85	C₂×C₄²	C₂²×C₁₃₇	1	4	yes	—
86	#95	C₂²×C₁₃₇	1	8	yes	—
87	C₂⁴	C₂³×C₁₃₇	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₂×C₄×C₁₃₇	1	8	yes	—
90	C₂²×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
91	C₂²×C₄	C₂×C₄×C₁₃₇	1	8	yes	—
92	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
93	C₂²×C₄	C₂³×C₁₃₇	1	8	yes	—
94	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
97	C₄⋊C₄	C₂×C₄×C₁₃₇	1	16	yes	complemented
98	C₂²⋊C₄	C₂×C₄×C₁₃₇	1	16	yes	—
99	C₄²	—	2	16	no	—
100	C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
101	C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
102	C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
103	C₄²	C₂×C₄×C₁₃₇	1	16	yes	complemented
104	C₂³	—	2	4	no	—
105	C₂³	C₂²×C₄×C₁₃₇	1	4	yes	—
106	C₂³	C₂²×C₄×C₁₃₇	1	2	yes	—
107	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
108	C₂×C₄	—	2	16	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	C₄²×C₁₃₇	1	16	yes	complemented
112	C₂×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
113	C₂×C₄	C₂²×C₄×C₁₃₇	1	16	yes	—
114	C₂×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
115	C₂×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
116	C₂×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
117	C₂×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
118	C₂×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
119	C₂×C₄	#41×C₁₃₇	1	8	yes	—
120	C₂²	—	2	4	no	—
121	C₂²	—	2	8	no	—
122	C₂²	C₂×C₄²×C₁₃₇	1	4	yes	—
123	C₂²	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
124	C₂²	C₂³×C₄×C₁₃₇	1	2	yes	—
125	C₂²	C₂×#41×C₁₃₇	1	4	yes	—
126	C₄	—	2	4	no	—
127	C₄	—	2	16	no	—
128	C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
129	C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
130	C₄	#117×C₁₃₇	1	8	yes	direct product factor
131	C₂	—	2	4	no	—
132	C₂	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
133	C₂	C₂×#117×C₁₃₇	1	2	yes	—
134	C₂	C₄×#41×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂²
commutator	C₂	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂³	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂⁴×C₁₃₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊C₄×C₁₃₇	C₂	semidirect
5. #376×C₁₃₇	C₂	semidirect
6. C₄³×C₁₃₇	C₂	semidirect
9. #117×C₁₃₇	C₄	direct (N × H)
12. C₂×C₄²×C₁₃₇	C₄	semidirect
29. C₄⋊C₄×C₁₃₇	C₂×C₄	semidirect
35. C₄²×C₁₃₇	C₂×C₄	semidirect
43. C₂×C₄×C₁₃₇	C₄²	semidirect
62. C₄×C₁₃₇	#117	direct (N × H)
67. C₁₃₇	C₄×#117	direct (N × H)
68. C₄×#117	C₁₃₇	direct (N × H)
71. C₄×C₄⋊C₄	C₂×C₁₃₇	semidirect
73. #376	C₂×C₁₃₇	semidirect
74. C₄³	C₂×C₁₃₇	semidirect
77. #117	C₄×C₁₃₇	direct (N × H)
80. C₂×C₄²	C₄×C₁₃₇	semidirect
97. C₄⋊C₄	C₂×C₄×C₁₃₇	semidirect
103. C₄²	C₂×C₄×C₁₃₇	semidirect
111. C₂×C₄	C₄²×C₁₃₇	semidirect
130. C₄	#117×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
C₂×#117×C₁₃₇	2
C₂²×C₄²×C₁₃₇	2
C₄×C₄⋊C₄×C₁₃₇	2
C₄×C₂²⋊C₄×C₁₃₇	2
#376×C₁₃₇	2
C₄³×C₁₃₇	2
C₄×#117	137

Subgroups — C4×#117×C137 #31777281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#117×C₁₃₇ #31777281