Subgroups — #31780960

Order 17536.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984023	C₂	1	4	yes	maximal
2	#22577427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	C₂²	1	2	yes	—
6	C₂²×#11764040	C₂²	1	2	yes	—
7	#17984023	—	2	8	no	—
8	C₄⋊#11764040	—	2	2	no	—
9	C₄×#11764040	C₄	1	2	yes	—
10	C₄×#11764040	C₄	1	2	yes	—
11	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₇	—	2	2	no	—
14	#14555494	—	2	4	no	—
15	C₂×#11764040	—	2	2	no	—
16	C₂×#11764040	C₂×C₄	1	4	yes	—
17	C₂×#11764040	D₄	1	4	yes	—
18	C₂×#11764040	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	M₁₆×C₁₃₇	—	2	4	no	—
22	C₂×C₈×C₁₃₇	—	2	2	no	—
23	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₇	D₄	1	2	yes	—
26	#11764040	—	4	2	no	—
27	#11764040	C₄⋊C₄	1	8	yes	—
28	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	—	2	4	no	—
30	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
36	C₈×C₁₃₇	—	4	2	no	—
37	C₂²×C₁₃₇	—	2	2	no	—
38	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
40	C₄×C₁₃₇	—	2	2	no	—
41	C₄×C₁₃₇	#108	1	4	yes	complemented
42	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₄×C₁₃₇	#127	1	1	yes	characteristic
44	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₃₇	—	2	2	no	—
46	C₂×C₁₃₇	#384	1	1	yes	characteristic
47	C₂×C₁₃₇	C₂×#108	1	2	yes	—
48	C₁₃₇	#1848	1	1	yes	characteristic, complemented, Sylow
49	#1848	—	137	1	no	maximal, Sylow
50	C₂×#108	—	137	4	no	—
51	#423	—	137	1	no	—
52	#423	D₁₃₇	1	1	yes	characteristic
53	C₂×C₄×C₈	—	137	1	no	—
54	C₂×M₁₆	—	137	2	no	—
55	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
56	C₂²×C₈	—	137	2	no	—
57	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
58	#108	—	274	8	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	274	2	no	—
61	C₄×C₈	—	137	2	no	—
62	C₄×C₈	—	137	2	no	—
63	C₂²×C₄	Dic₂₇₄	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	274	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	274	2	no	—
69	C₂×C₈	—	137	4	no	—
70	C₂×C₈	—	137	4	no	—
71	C₂×C₈	—	137	4	no	—
72	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
73	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
74	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
75	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	#14555489	1	2	yes	—
78	C₂×C₄	#14555489	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
80	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
81	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
82	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
83	C₈	—	4	2	no	—
84	C₈	—	548	2	no	—
85	C₈	—	137	8	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂×#14555489	1	1	yes	characteristic
88	C₂²	#17984087	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	#17984023	1	4	yes	complemented
91	C₄	C₄⋊C₄×D₁₃₇	1	2	yes	—
92	C₄	#17984084	1	1	yes	characteristic
93	C₄	#17984086	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#22577272	1	1	yes	characteristic
96	C₂	C₂×#17984023	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#423×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#423	D₁₃₇
hypocenter	C₁₃₇	#1848
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
41. C₄×C₁₃₇	#108	semidirect
48. C₁₃₇	#1848	semidirect
90. C₄	#17984023	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984023	2
#22577427	2
C₂×C₄×#11764040	2
#423×C₁₃₇	2
#1848	137