Subgroups — #31795907

Order 17536.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578335	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984138	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577821	C₂	1	1	yes	maximal, characteristic
4	#17984138	—	2	1	no	—
5	#17984138	—	2	1	no	—
6	#17984094	C₄	1	1	yes	characteristic
7	#17984027	—	2	1	no	—
8	C₂×#14555485	C₂²	1	1	yes	characteristic
9	C₁₆×D₁₃₇	—	2	1	no	—
10	M₃₂×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555510	C₈	1	1	yes	characteristic
12	#14555494	C₈	1	1	yes	characteristic
13	#14555493	—	2	1	no	—
14	C₈×D₁₃₇	—	2	1	no	—
15	#14555485	D₄	1	1	yes	characteristic
16	#14555485	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	—	2	1	no	—
19	Dic₂₇₄	C₁₆	1	1	yes	characteristic
20	D₅₄₈	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₃₇	—	2	1	no	—
22	C₄×D₁₃₇	—	2	1	no	—
23	#11764040	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	1	no	—
29	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₃₇	—	2	1	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₃₇	—	4	1	no	—
35	C₁₃₇	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	137	1	no	maximal, Sylow
37	#504	—	137	1	no	—
38	M₆₄	—	137	1	no	—
39	C₂×C₃₂	—	137	1	no	—
40	#131	—	137	1	no	—
41	M₃₂	—	274	1	no	—
42	M₃₂	#10587739	1	1	yes	characteristic
43	C₂×C₁₆	—	274	1	no	—
44	C₂×C₁₆	—	137	1	no	—
45	C₃₂	—	274	1	no	—
46	C₃₂	—	274	1	no	—
47	#41	—	137	1	no	—
48	M₁₆	—	274	1	no	—
49	M₁₆	—	137	1	no	—
50	C₂×C₈	—	274	1	no	—
51	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	137	1	no	—
54	C₁₆	—	137	1	no	—
55	Q₈	—	137	1	no	—
56	D₄	—	274	1	no	—
57	D₄	—	137	1	no	—
58	C₂×C₄	—	274	1	no	—
59	C₂×C₄	#14555518	1	1	yes	characteristic
60	C₈	—	274	1	no	—
61	C₈	C₂²⋊#10587739	1	1	yes	characteristic
62	C₈	#14555520	1	1	yes	characteristic
63	C₂²	—	274	1	no	—
64	C₂²	#17984163	1	1	yes	characteristic
65	C₄	—	274	1	no	—
66	C₄	#17984145	1	1	yes	characteristic
67	C₄	#17984161	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	548	1	no	—
70	C₂	#22578298	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10587739
Fitting	M₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	#14555510	C₈
cosocle	C₂×#14555485	C₂²
hypercenter	M₃₂	#10587739
hypocenter	C₁₃₇	#1401
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #22578335	C₂	semidirect
2. C₂×#17984138	C₂	semidirect
35. C₁₃₇	#1401	semidirect

Maximal subgroups

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

subgroup	index
#22578335	2
C₂×#17984138	2
#22577821	2
#1401	137