Subgroups — #31779091

Order 17536.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555493	C₂	1	3	yes	maximal
2	#22577124	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14555493	C₄	1	6	yes	—
5	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
6	#17983998	C₄	1	6	yes	—
7	C₄×#11764040	C₂²	1	3	yes	—
8	C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
9	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
10	#14555493	—	2	12	no	—
11	C₂×#11764040	C₂×C₄	1	6	yes	—
12	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
14	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
16	C₂×Dic₁₃₇	C₄²	1	3	yes	—
17	C₄×D₁₃₇	—	2	6	no	—
18	#11764040	—	2	6	no	—
19	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
20	C₈×C₁₃₇	M₁₆	1	12	yes	complemented
21	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₁₃₇	—	2	3	no	—
23	C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
24	Dic₁₃₇	—	2	3	no	—
25	C₂×C₁₃₇	C₄×M₁₆	1	3	yes	—
26	D₁₃₇	—	4	1	no	—
27	C₁₃₇	#1455	1	1	yes	characteristic, complemented, Sylow
28	#1455	—	137	1	no	maximal, Sylow
29	C₄×M₁₆	—	137	3	no	—
30	#322	—	137	3	no	—
31	C₈²	D₁₃₇	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	137	6	no	—
33	C₂×C₄²	—	137	1	no	—
34	#97	—	137	6	no	—
35	C₄×C₈	—	137	3	no	—
36	C₄×C₈	C₂×D₁₃₇	1	3	yes	—
37	C₂²×C₄	—	137	3	no	—
38	M₁₆	—	274	12	no	—
39	C₂×C₈	—	137	6	no	—
40	C₂×C₈	C₄×D₁₃₇	1	6	yes	—
41	C₄²	—	137	3	no	—
42	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
43	C₂³	—	137	1	no	—
44	C₂×C₄	—	274	6	no	—
45	C₂×C₄	—	137	3	no	—
46	C₂×C₄	C₂×C₄×D₁₃₇	1	3	yes	—
47	C₈	—	274	6	no	—
48	C₈	#14555493	1	12	yes	complemented
49	C₂²	—	274	3	no	—
50	C₂²	C₄²×D₁₃₇	1	1	yes	characteristic
51	C₄	—	274	3	no	—
52	C₄	C₂×#14555493	1	6	yes	—
53	C₂	—	548	1	no	—
54	C₂	C₄×#14555493	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₈²×C₁₃₇	C₂
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	C₈²	D₁₃₇
hypocenter	C₁₃₇	#1455
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #22577124	C₂	semidirect
3. C₈²×C₁₃₇	C₂	semidirect
20. C₈×C₁₃₇	M₁₆	semidirect
27. C₁₃₇	#1455	semidirect
31. C₈²	D₁₃₇	semidirect
48. C₈	#14555493	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555493	2
#22577124	2
C₈²×C₁₃₇	2
#1455	137