Subgroups — #31780963

Order 17536.

212 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×#17983999	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	#17983999	C₄	1	4	yes	—
10	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	—	2	2	no	—
13	#14555494	—	2	4	no	—
14	C₂×#11764040	—	2	2	no	—
15	C₂×#11764040	C₂×C₄	1	4	yes	—
16	C₂×#11764040	D₄	1	4	yes	—
17	C₂×#11764040	Q₈	1	4	yes	—
18	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₃₇	—	2	4	no	—
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₇	D₄	1	2	yes	—
24	#11764040	—	4	2	no	—
25	#11764040	C₄⋊C₄	1	8	yes	—
26	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	—	4	2	no	—
28	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
33	C₈×C₁₃₇	—	4	2	no	—
34	C₂²×C₁₃₇	—	2	2	no	—
35	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
37	C₄×C₁₃₇	—	2	4	no	—
38	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
39	C₄×C₁₃₇	#127	1	1	yes	characteristic
40	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₃₇	—	2	2	no	—
42	C₂×C₁₃₇	#384	1	1	yes	characteristic
43	C₂×C₁₃₇	#430	1	2	yes	—
44	C₁₃₇	#1849	1	1	yes	characteristic, complemented, Sylow
45	#1849	—	137	1	no	maximal, Sylow
46	C₂×#108	—	137	4	no	—
47	#423	—	137	1	no	—
48	#423	D₁₃₇	1	1	yes	characteristic
49	C₂×#97	—	137	1	no	—
50	C₂×M₁₆	—	137	2	no	—
51	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
52	C₂²×C₈	—	137	2	no	—
53	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
54	#108	—	274	8	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	274	2	no	—
57	#97	—	137	4	no	—
58	C₂²×C₄	Dic₂₇₄	1	2	yes	—
59	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	—	274	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	274	2	no	—
64	C₂×C₈	—	137	4	no	—
65	C₂×C₈	—	137	4	no	—
66	C₂×C₈	—	137	4	no	—
67	C₄²	C₄×D₁₃₇	1	2	yes	—
68	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
69	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#14555489	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
73	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
74	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
75	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
76	C₈	—	4	2	no	—
77	C₈	—	548	2	no	—
78	C₈	—	137	8	no	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×#14555489	1	1	yes	characteristic
81	C₂²	#17984087	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₄⋊C₄×D₁₃₇	1	2	yes	—
84	C₄	#17984084	1	1	yes	characteristic
85	C₄	#17984086	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	#22577272	1	1	yes	characteristic
88	C₂	#22577451	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	#1849
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
44. C₁₃₇	#1849	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×#17983999	2
#423×C₁₃₇	2
#1849	137