Subgroups — #31795847

Order 17536.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984157	C₂	1	2	yes	maximal
2	C₂×C₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10587739	—	2	2	no	—
4	#17984157	—	2	2	no	—
5	#17984157	C₄	1	4	yes	complemented
6	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×#14555485	C₄	1	1	yes	characteristic, complemented
8	C₁₆×D₁₃₇	—	2	2	no	—
9	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
10	C₂²×#10587739	—	4	2	no	—
11	C₄⋊#10587739	—	2	4	no	—
12	C₄⋊#10587739	—	4	2	no	—
13	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
14	C₂×#11764040	Q₈	1	1	yes	characteristic
15	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
16	C₈×D₁₃₇	D₄	1	2	yes	—
17	#14555485	—	2	1	no	—
18	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₃₇	—	2	1	no	—
20	C₂×#10587739	—	4	4	no	—
21	C₂×#10587739	—	8	2	no	—
22	C₂×#10587739	—	8	2	no	—
23	C₂²×D₁₃₇	D₈	1	1	yes	characteristic
24	C₂×Dic₁₃₇	Q₁₆	1	1	yes	characteristic
25	C₄×D₁₃₇	C₂²⋊C₄	1	2	yes	—
26	C₄×D₁₃₇	SD₁₆	1	2	yes	—
27	#11764040	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
29	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
30	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₁₃₇	#107	1	1	yes	characteristic
32	C₂×D₁₃₇	#102	1	2	yes	—
33	C₂×D₁₃₇	D₁₆	1	1	yes	characteristic
34	C₂×D₁₃₇	SD₃₂	1	2	yes	—
35	C₂×D₁₃₇	Q₃₂	1	1	yes	characteristic
36	#10587739	—	8	4	no	—
37	C₄×C₁₃₇	#95	1	1	yes	characteristic
38	C₄×C₁₃₇	#106	1	1	yes	characteristic
39	Dic₁₃₇	#103	1	2	yes	—
40	C₂×C₁₃₇	#340	1	1	yes	characteristic
41	C₂×C₁₃₇	#366	1	1	yes	characteristic
42	C₂×C₁₃₇	#367	1	1	yes	characteristic
43	D₁₃₇	#357	1	2	yes	—
44	D₁₃₇	#358	1	2	yes	—
45	C₁₃₇	#1380	1	1	yes	characteristic, complemented, Sylow
46	#1380	—	137	1	no	maximal, Sylow
47	C₂²×C₁₆	—	137	1	no	—
48	C₂×#107	—	137	2	no	—
49	C₂²×C₈	—	137	1	no	—
50	C₂×C₄⋊C₄	—	274	2	no	—
51	C₂×C₁₆	—	274	2	no	—
52	C₂×C₁₆	—	137	1	no	—
53	C₂×C₁₆	#10587739	1	1	yes	characteristic, complemented
54	#107	—	274	2	no	—
55	#107	—	137	4	no	—
56	C₂²×C₄	—	548	2	no	—
57	C₂²×C₄	—	137	1	no	—
58	C₂×C₈	—	137	1	no	—
59	C₂×C₈	—	137	2	no	—
60	C₂×C₈	—	137	2	no	—
61	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
62	C₄⋊C₄	—	274	4	no	—
63	C₄⋊C₄	—	548	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	274	1	no	—
66	C₂³	—	137	1	no	—
67	C₂×C₄	—	548	4	no	—
68	C₂×C₄	—	1096	2	no	—
69	C₂×C₄	—	1096	2	no	—
70	C₂×C₄	—	137	1	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
74	C₈	—	137	2	no	—
75	C₈	C₄×#10587739	1	1	yes	characteristic
76	C₈	C₂²⋊#10587739	1	1	yes	characteristic
77	C₂²	—	137	2	no	—
78	C₂²	—	137	1	no	—
79	C₂²	—	137	1	no	—
80	C₂²	—	137	2	no	—
81	C₂²	#17984157	1	1	yes	characteristic
82	C₄	—	1096	4	no	—
83	C₄	—	137	2	no	—
84	C₄	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
85	C₄	#17984156	1	1	yes	characteristic
86	C₂	—	137	2	no	—
87	C₂	—	137	2	no	—
88	C₂	#22578277	1	1	yes	characteristic
89	C₂	#22578329	1	1	yes	characteristic
90	C₂	#22578330	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984157
commutator	C₈×C₁₃₇	C₄²
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#107
prime radical	C₂²×D₁₃₇	D₈
cosocle	C₂×C₈×D₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1380
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
5. #17984157	C₄	semidirect
7. C₂×#14555485	C₄	semidirect
9. C₂×C₁₆×C₁₃₇	C₄	semidirect
45. C₁₃₇	#1380	semidirect
53. C₂×C₁₆	#10587739	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984157	2
C₂×C₁₆×D₁₃₇	2
#1380	137