Subgroups — #22578418

Order 8768.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555520	C₂	1	1	yes	maximal, characteristic
2	D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#17984159	C₂	1	2	yes	maximal, complemented
4	#17984158	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#14555520	—	2	1	no	—
7	#14555520	C₂²	1	1	yes	characteristic
8	#14555520	C₂²	1	2	yes	—
9	#14555520	C₂²	1	1	yes	characteristic
10	#14555519	—	2	1	no	—
11	#14555518	—	2	1	no	—
12	C₂×#11764047	—	2	1	no	—
13	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	#14555494	C₄	1	1	yes	characteristic
15	#14555493	C₄	1	2	yes	—
16	C₈×D₁₃₇	C₂²	1	2	yes	—
17	M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
18	#11764047	—	2	2	no	—
19	#11764047	—	2	1	no	—
20	#11764047	—	2	1	no	—
21	C₂²×D₁₃₇	Q₈	1	1	yes	characteristic
22	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
23	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₃₇	D₄	1	1	yes	characteristic
25	C₄×D₁₃₇	Q₈	1	1	yes	characteristic
26	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
27	#11764040	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₁₃₇	—	4	1	no	—
32	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
37	Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₁₃₇	—	2	1	no	—
39	C₂×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	D₁₃₇	—	2	1	no	—
41	D₁₃₇	—	2	1	no	—
42	C₁₃₇	#430	1	1	yes	characteristic, complemented, Sylow
43	#430	—	137	1	no	maximal, Sylow
44	C₂×M₁₆	—	137	1	no	—
45	C₂×M₁₆	—	137	1	no	—
46	C₂×M₁₆	—	137	1	no	—
47	#108	—	137	2	no	—
48	#108	—	137	2	no	—
49	C₂²×C₄	—	137	1	no	—
50	M₁₆	—	274	1	no	—
51	M₁₆	—	274	1	no	—
52	M₁₆	—	137	1	no	—
53	M₁₆	—	137	2	no	—
54	M₁₆	—	137	1	no	—
55	M₁₆	—	137	1	no	—
56	M₁₆	—	137	2	no	—
57	M₁₆	#10587739	1	1	yes	characteristic
58	C₂×C₈	—	274	1	no	—
59	C₂×C₈	—	274	1	no	—
60	C₂×C₈	—	137	2	no	—
61	C₂³	—	137	1	no	—
62	C₂×C₄	—	137	1	no	—
63	C₂×C₄	—	137	1	no	—
64	C₂×C₄	—	137	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	—	137	1	no	—
67	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
68	C₈	—	274	2	no	—
69	C₈	—	274	1	no	—
70	C₈	—	274	1	no	—
71	C₈	—	137	2	no	—
72	C₈	C₂×#10587739	1	2	yes	—
73	C₂²	—	548	1	no	—
74	C₂²	—	137	1	no	—
75	C₂²	—	137	1	no	—
76	C₂²	C₄⋊#10587739	1	1	yes	characteristic
77	C₄	—	137	1	no	—
78	C₄	—	137	1	no	—
79	C₄	C₄⋊#10587739	1	1	yes	characteristic
80	C₄	C₂²×#10587739	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	274	1	no	—
83	C₂	—	274	1	no	—
84	C₂	C₂×C₄⋊#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#10587739
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₄	C₂²×#10587739
Fitting	M₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄⋊C₄
hypercenter	M₁₆	#10587739
hypocenter	C₁₃₇	#430
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #17984159	C₂	semidirect
4. #17984158	C₂	semidirect
42. C₁₃₇	#430	semidirect