Subgroups — #31795983

Order 17536.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555518	C₂	1	2	yes	maximal
2	#22578244	C₂	1	4	yes	maximal, complemented
3	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555518	C₄	1	4	yes	—
5	C₄×#11764047	C₄	1	4	yes	—
6	C₄×#11764047	C₂²	1	4	yes	—
7	C₂×C₈×D₁₃₇	C₄	1	2	yes	—
8	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
9	C₄×#11764040	C₄	1	1	yes	characteristic
10	C₈×Dic₁₃₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555518	—	2	8	no	—
13	C₂×#11764047	C₂×C₄	1	8	yes	—
14	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
15	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#11764040	C₂×C₄	1	2	yes	—
17	C₈×D₁₃₇	C₈	1	8	yes	complemented
18	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	#11764047	—	2	8	no	—
23	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
24	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
25	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
26	C₄×D₁₃₇	C₂×C₈	1	4	yes	—
27	C₄×D₁₃₇	M₁₆	1	8	yes	—
28	#11764040	C₂×C₈	1	4	yes	complemented
29	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
30	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₃₇	C₂×C₈	1	4	yes	complemented
32	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
33	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
34	C₂×D₁₃₇	#97	1	4	yes	—
35	C₄×C₁₃₇	#97	1	4	yes	—
36	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
37	Dic₁₃₇	C₄×C₈	1	2	yes	—
38	Dic₁₃₇	C₂×M₁₆	1	4	yes	—
39	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×#97	1	2	yes	—
41	D₁₃₇	#322	1	4	yes	complemented
42	C₁₃₇	C₂×#322	1	1	yes	characteristic, complemented, Sylow
43	C₂×#322	—	137	1	no	maximal, Sylow
44	C₂×C₄×C₈	—	137	2	no	—
45	C₂×C₄×C₈	—	137	1	no	—
46	#322	—	137	4	no	—
47	C₂²×C₈	—	137	4	no	—
48	C₂²×C₈	—	137	2	no	—
49	C₂×C₄²	—	137	1	no	—
50	C₄×C₈	—	137	4	no	—
51	C₄×C₈	—	137	4	no	—
52	C₄×C₈	—	137	2	no	—
53	C₄×C₈	—	137	1	no	—
54	C₄×C₈	#10587739	1	1	yes	characteristic
55	C₂²×C₄	—	137	2	no	—
56	C₂²×C₄	—	137	1	no	—
57	C₂×C₈	—	274	8	no	—
58	C₂×C₈	—	137	8	no	—
59	C₂×C₈	—	137	2	no	—
60	C₂×C₈	—	137	8	no	—
61	C₂×C₈	C₂×#10587739	1	2	yes	—
62	C₄²	—	137	2	no	—
63	C₄²	—	137	1	no	—
64	C₄²	C₂×#10587739	1	1	yes	characteristic
65	C₂³	—	137	1	no	—
66	C₂×C₄	—	137	2	no	—
67	C₂×C₄	—	137	1	no	—
68	C₂×C₄	—	137	8	no	—
69	C₂×C₄	—	137	4	no	—
70	C₂×C₄	C₄×#10587739	1	2	yes	—
71	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
72	C₈	—	274	8	no	—
73	C₈	—	137	4	no	—
74	C₈	#14555518	1	4	yes	complemented
75	C₂²	—	137	2	no	—
76	C₂²	—	137	4	no	—
77	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
78	C₄	—	137	2	no	—
79	C₄	—	137	4	no	—
80	C₄	#17984142	1	4	yes	—
81	C₄	C₂×#14555518	1	2	yes	—
82	C₂	—	137	4	no	—
83	C₂	C₄×#14555518	1	1	yes	characteristic
84	C₂	C₂×#17984142	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	C₂×#322
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22578244	C₂	semidirect
17. C₈×D₁₃₇	C₈	semidirect
28. #11764040	C₂×C₈	semidirect
31. C₈×C₁₃₇	C₂×C₈	semidirect
41. D₁₃₇	#322	semidirect
42. C₁₃₇	C₂×#322	semidirect
74. C₈	#14555518	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555518	2
#22578244	2
C₄×C₈×D₁₃₇	2
C₂×#322	137