Subgroups — #31797932

Order 17536.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₂×#11764049)⋊C₄	C₂	1	6	yes	maximal
2	C₄²×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764049	C₂²	1	4	yes	—
4	C₂×C₄×#10587739	C₄	1	3	yes	—
5	C₂×C₄×#10587739	C₂²	1	3	yes	—
6	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764049	—	2	12	no	—
8	C₂²×#10587739	C₂×C₄	1	3	yes	—
9	C₂²×#10587739	C₂³	1	1	yes	characteristic
10	C₄×#10587739	—	2	6	no	—
11	C₄×#10587739	—	2	6	no	—
12	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
13	C₄×Dic₁₃₇	C₈	1	3	yes	complemented
14	C₄²×C₁₃₇	C₈	1	1	yes	characteristic, complemented
15	#11764049	—	4	4	no	—
16	C₂×#10587739	—	2	3	no	—
17	C₂×#10587739	—	2	3	no	—
18	C₂×#10587739	M₁₆	1	6	yes	—
19	C₂×#10587739	#41	1	6	yes	—
20	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×Dic₁₃₇	C₂×C₈	1	3	yes	—
22	C₄×D₁₃₇	—	2	3	no	—
23	C₄×D₁₃₇	—	2	3	no	—
24	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
25	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×D₁₃₇	#117	1	3	yes	—
27	C₂×D₁₃₇	C₂×M₁₆	1	3	yes	—
28	#10587739	—	2	6	no	—
29	#10587739	#126	1	4	yes	—
30	C₄×C₁₃₇	—	2	3	no	—
31	Dic₁₃₇	—	2	3	no	—
32	C₂×C₁₃₇	#431	1	3	yes	—
33	D₁₃₇	#383	1	1	yes	characteristic
34	D₁₃₇	#432	1	3	yes	—
35	C₁₃₇	#1839	1	1	yes	characteristic, complemented, Sylow
36	#1839	—	137	1	no	maximal, Sylow
37	C₄³	—	137	1	no	—
38	#336	—	137	6	no	—
39	C₂²×C₈	—	137	4	no	—
40	C₂×C₄²	—	137	3	no	—
41	C₂×C₄²	—	137	3	no	—
42	C₂×C₄²	—	137	1	no	—
43	C₂²×C₄	—	137	3	no	—
44	C₂²×C₄	—	137	3	no	—
45	C₂²×C₄	—	137	1	no	—
46	C₂×C₈	—	274	12	no	—
47	C₄²	—	274	6	no	—
48	C₄²	—	274	6	no	—
49	C₄²	—	137	3	no	—
50	C₄²	#11764049	1	1	yes	characteristic, complemented
51	C₂³	—	137	1	no	—
52	C₂×C₄	—	274	3	no	—
53	C₂×C₄	—	274	3	no	—
54	C₂×C₄	—	274	3	no	—
55	C₂×C₄	—	274	3	no	—
56	C₂×C₄	—	137	6	no	—
57	C₂×C₄	—	137	3	no	—
58	C₂×C₄	—	137	6	no	—
59	C₂×C₄	C₂×#11764049	1	3	yes	—
60	C₈	—	548	4	no	—
61	C₂²	—	137	3	no	—
62	C₂²	—	137	3	no	—
63	C₂²	C₂²×#11764049	1	1	yes	characteristic
64	C₄	—	2	3	no	—
65	C₄	—	274	6	no	—
66	C₄	—	274	3	no	—
67	C₄	—	137	4	no	—
68	C₂	—	137	3	no	—
69	C₂	—	137	1	no	—
70	C₂	#22578516	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₄²×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂²×#10587739	C₂³
hypercenter	C₄²	#11764049
hypocenter	C₁₃₇	#1839
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₁₃₇	C₈	semidirect
14. C₄²×C₁₃₇	C₈	semidirect
35. C₁₃₇	#1839	semidirect
50. C₄²	#11764049	semidirect

Maximal subgroups

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

subgroup	index
(C₂×#11764049)⋊C₄	2
C₄²×#10587739	2
#1839	137