Subgroups — #31780961

Order 17536.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984024	C₂	1	4	yes	maximal
2	#22577427	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555494	C₂²	1	4	yes	—
5	#17984024	—	2	8	no	—
6	C₄⋊#11764040	—	2	4	no	—
7	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₃₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₃₇	C₄	1	2	yes	—
11	#14555494	—	2	8	no	—
12	C₂×#11764040	—	2	4	no	—
13	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
14	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
17	C₂×C₈×C₁₃₇	Q₈	1	4	yes	—
18	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₃₇	D₄	1	2	yes	—
21	#11764040	—	4	4	no	—
22	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	—	2	4	no	—
24	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
30	C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
31	C₂²×C₁₃₇	—	2	2	no	—
32	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
34	C₄×C₁₃₇	—	2	2	no	—
35	C₄×C₁₃₇	#108	1	4	yes	complemented
36	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₁₃₇	#127	1	1	yes	characteristic
38	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₁₃₇	—	2	2	no	—
40	C₂×C₁₃₇	#384	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂×#108	1	2	yes	—
42	C₁₃₇	#1848	1	1	yes	characteristic, complemented, Sylow
43	#1848	—	137	1	no	maximal, Sylow
44	C₂×#108	—	137	4	no	—
45	#423	—	137	2	no	—
46	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
47	C₂×M₁₆	—	137	4	no	—
48	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
49	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
50	#108	—	274	8	no	—
51	C₄⋊C₈	—	274	4	no	—
52	C₄×C₈	Dic₁₃₇	1	2	yes	—
53	C₄×C₈	Dic₁₃₇	1	2	yes	—
54	C₂²×C₄	Dic₂₇₄	1	2	yes	—
55	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
56	M₁₆	—	274	8	no	—
57	C₂×C₈	—	274	4	no	—
58	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
59	C₂×C₈	D₅₄₈	1	4	yes	—
60	C₂×C₈	Dic₂₇₄	1	4	yes	—
61	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
62	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
63	C₄²	D₅₄₈	1	2	yes	—
64	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
68	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
70	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
71	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
72	C₈	—	548	4	no	—
73	C₈	C₄⋊Dic₁₃₇	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
76	C₂²	#17984088	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#17984024	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
80	C₄	#17984085	1	1	yes	characteristic
81	C₄	#17984088	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#22577273	1	1	yes	characteristic
84	C₂	C₂×#17984024	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#1848
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₁₃₇	#108	semidirect
42. C₁₃₇	#1848	semidirect
78. C₄	#17984024	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984024	2
#22577427	2
C₂×C₄×C₈×C₁₃₇	2
#1848	137