Subgroups — #22577880

Order 8768.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₇₄	C₂	1	4	yes	maximal, complemented
3	#17984053	C₂	1	4	yes	maximal, complemented
4	#17984047	C₂	1	1	yes	maximal, characteristic
5	#17984044	C₂	1	4	yes	maximal
6	#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₇₄	C₄	1	8	yes	complemented
8	C₂×Dic₂₇₄	C₂²	1	2	yes	—
9	C₂×Dic₂₇₄	C₂²	1	2	yes	—
10	C₂²×Dic₁₃₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
12	#14555489	C₂²	1	8	yes	—
13	C₂²⋊Dic₁₃₇	C₂²	1	2	yes	—
14	C₄×Dic₁₃₇	C₂²	1	8	yes	—
15	C₄×Dic₁₃₇	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	C₄²×C₁₃₇	C₂²	1	2	yes	—
20	Dic₂₇₄	C₂×C₄	1	8	yes	—
21	Dic₂₇₄	C₂×C₄	1	8	yes	—
22	C₂×Dic₁₃₇	C₂×C₄	1	8	yes	—
23	C₂×Dic₁₃₇	C₂³	1	4	yes	—
24	C₂×Dic₁₃₇	C₂³	1	4	yes	—
25	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
28	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₁₃₇	—	2	2	no	—
31	C₂²×C₁₃₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
33	C₄×C₁₃₇	—	2	4	no	—
34	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
35	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
36	Dic₁₃₇	—	2	4	no	—
37	Dic₁₃₇	C₂²×C₄	1	8	yes	—
38	C₂×C₁₃₇	—	2	2	no	—
39	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
41	C₁₃₇	#519	1	1	yes	characteristic, complemented, Sylow
42	#519	—	137	1	no	maximal, Sylow
43	C₂²×Q₈	—	137	1	no	—
44	C₄×Q₈	—	137	4	no	—
45	C₄×Q₈	—	137	4	no	—
46	#117	—	137	1	no	—
47	#117	—	137	4	no	—
48	#117	D₁₃₇	1	1	yes	characteristic
49	C₂×Q₈	—	137	8	no	—
50	C₂×Q₈	—	137	2	no	—
51	C₂×Q₈	—	137	2	no	—
52	C₂²×C₄	—	137	2	no	—
53	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
54	C₄⋊C₄	—	137	8	no	—
55	C₄⋊C₄	—	137	2	no	—
56	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
57	C₂²⋊C₄	—	137	2	no	—
58	C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
59	C₄²	—	137	2	no	—
60	C₄²	—	137	8	no	—
61	C₄²	C₂×D₁₃₇	1	2	yes	—
62	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
63	Q₈	—	137	8	no	—
64	Q₈	—	137	8	no	—
65	C₂×C₄	—	137	4	no	—
66	C₂×C₄	—	137	8	no	—
67	C₂×C₄	—	137	4	no	—
68	C₂×C₄	C₄×D₁₃₇	1	4	yes	—
69	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
70	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₁₃₇	1	2	yes	—
74	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	274	4	no	—
77	C₄	—	137	8	no	—
78	C₄	C₂×C₄×D₁₃₇	1	2	yes	—
79	C₄	C₂×C₄×D₁₃₇	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
82	C₂	#17984135	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator	C₂×C₁₃₇	C₂³×C₄
Fitting	#117×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂⁴
hypercenter	#117	D₁₃₇
hypocenter	C₁₃₇	#519
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₂₇₄	C₂	semidirect
3. #17984053	C₂	semidirect
7. C₂×Dic₂₇₄	C₄	semidirect
41. C₁₃₇	#519	semidirect