Subgroups — #31796730

Order 17536.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984159	C₂	1	4	yes	maximal
2	#22578407	C₂	1	2	yes	maximal
3	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555520	C₂²	1	4	yes	—
5	#17984159	—	2	8	no	—
6	C₄⋊#11764047	—	2	4	no	—
7	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
8	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
9	C₄×#11764040	C₄	1	1	yes	characteristic
10	C₈×Dic₁₃₇	C₄	1	2	yes	—
11	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555520	—	2	8	no	—
13	C₂×#11764047	—	2	4	no	—
14	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
15	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
16	C₂×#11764040	C₂×C₄	1	2	yes	—
17	C₈×D₁₃₇	D₄	1	4	yes	—
18	C₈×D₁₃₇	Q₈	1	4	yes	—
19	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₁₃₇	D₄	1	2	yes	—
21	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	#11764047	—	4	4	no	—
24	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
26	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
27	C₄×D₁₃₇	—	2	4	no	—
28	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
29	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
30	#11764040	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
34	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₁₃₇	—	2	2	no	—
36	C₂×D₁₃₇	C₄⋊Q₈	1	2	yes	—
37	C₄×C₁₃₇	#108	1	4	yes	complemented
38	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
39	Dic₁₃₇	—	2	2	no	—
40	Dic₁₃₇	#127	1	1	yes	characteristic
41	Dic₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₁₃₇	#384	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂×#108	1	2	yes	—
44	D₁₃₇	—	2	2	no	—
45	C₁₃₇	#1848	1	1	yes	characteristic, complemented, Sylow
46	#1848	—	137	1	no	maximal, Sylow
47	C₂×#108	—	137	4	no	—
48	#423	—	137	2	no	—
49	C₂×C₄×C₈	—	137	1	no	—
50	C₂×M₁₆	—	137	4	no	—
51	C₂²×C₈	—	137	2	no	—
52	C₂×C₄²	—	137	1	no	—
53	#108	—	274	8	no	—
54	C₄⋊C₈	—	274	4	no	—
55	C₄×C₈	—	137	2	no	—
56	C₄×C₈	—	137	1	no	—
57	C₄×C₈	#10587739	1	1	yes	characteristic
58	C₂²×C₄	—	137	2	no	—
59	C₂²×C₄	—	137	1	no	—
60	M₁₆	—	274	8	no	—
61	C₂×C₈	—	274	4	no	—
62	C₂×C₈	—	137	2	no	—
63	C₂×C₈	—	137	4	no	—
64	C₂×C₈	—	137	4	no	—
65	C₂×C₈	C₂×#10587739	1	2	yes	—
66	C₄²	—	137	2	no	—
67	C₄²	—	137	1	no	—
68	C₄²	C₂×#10587739	1	1	yes	characteristic
69	C₂³	—	137	1	no	—
70	C₂×C₄	—	274	4	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	—	137	1	no	—
73	C₂×C₄	—	137	2	no	—
74	C₂×C₄	—	137	2	no	—
75	C₂×C₄	C₄⋊#10587739	1	2	yes	—
76	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
77	C₈	—	548	4	no	—
78	C₈	—	137	4	no	—
79	C₈	C₄⋊#10587739	1	4	yes	—
80	C₂²	—	274	2	no	—
81	C₂²	—	137	2	no	—
82	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
83	C₄	—	274	2	no	—
84	C₄	—	137	1	no	—
85	C₄	—	137	1	no	—
86	C₄	#17984159	1	4	yes	complemented
87	C₄	C₂×C₄⋊#10587739	1	2	yes	—
88	C₂	—	274	2	no	—
89	C₂	#22578349	1	1	yes	characteristic
90	C₂	C₂×#17984159	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1848
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
37. C₄×C₁₃₇	#108	semidirect
45. C₁₃₇	#1848	semidirect
86. C₄	#17984159	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984159	2
#22578407	2
C₄×C₈×D₁₃₇	2
#1848	137