Subgroups — #461×C₁₃₇ #22576996

Order 8768.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×C₁₃₇	C₂	1	2	yes	maximal
2	C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#103×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₁₃₇	—	2	1	no	—
9	SD₁₆×C₁₃₇	—	2	4	no	—
10	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄⋊C₄×C₁₃₇	—	2	1	no	—
12	C₄⋊C₄×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂³×C₁₃₇	—	2	1	no	—
17	Q₈×C₁₃₇	—	2	2	no	—
18	Q₈×C₁₃₇	—	2	2	no	—
19	D₄×C₁₃₇	—	2	1	no	—
20	D₄×C₁₃₇	D₄	1	2	yes	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	—	2	2	no	—
23	C₂×C₄×C₁₃₇	—	4	2	no	—
24	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₃₇	—	2	2	no	—
28	C₂²×C₁₃₇	—	2	2	no	—
29	C₂²×C₁₃₇	—	4	1	no	—
30	C₂²×C₁₃₇	—	4	1	no	—
31	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₃₇	—	2	1	no	—
33	C₄×C₁₃₇	—	4	2	no	—
34	C₄×C₁₃₇	—	4	1	no	—
35	C₄×C₁₃₇	SD₁₆	1	2	yes	complemented
36	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₁₃₇	#41	1	1	yes	characteristic
38	C₂×C₁₃₇	—	4	2	no	—
39	C₂×C₁₃₇	#121	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₃₇	#137	1	1	yes	characteristic
42	C₁₃₇	#461	1	1	yes	characteristic, direct product factor, Sylow
43	#461	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×SD₁₆	C₂×C₁₃₇	1	2	yes	—
45	C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
46	C₄×D₄	C₂×C₁₃₇	1	1	yes	characteristic
47	C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
48	#103	C₂×C₁₃₇	1	2	yes	complemented
49	C₂×Q₈	C₂²×C₁₃₇	1	2	yes	—
50	C₂×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	2	4	no	—
53	C₂×C₈	C₂²×C₁₃₇	1	2	yes	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	2	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₄×C₁₃₇	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	D₄×C₁₃₇	1	1	yes	characteristic
68	C₂×C₄	D₄×C₁₃₇	1	1	yes	characteristic
69	C₂×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	4	1	no	—
78	C₄	SD₁₆×C₁₃₇	1	2	yes	complemented
79	C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
80	C₄	#41×C₁₃₇	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#121×C₁₃₇	1	1	yes	characteristic
83	C₂	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
84	C₂	#137×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₃₇
prime radical	C₂×D₄×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄⋊Q₈×C₁₃₇	C₂	semidirect
4. C₄⋊C₈×C₁₃₇	C₂	semidirect
5. #103×C₁₃₇	C₂	semidirect
35. C₄×C₁₃₇	SD₁₆	semidirect
42. C₁₃₇	#461	direct (N × H)
43. #461	C₁₃₇	direct (N × H)
45. C₄⋊Q₈	C₂×C₁₃₇	semidirect
47. C₄⋊C₈	C₂×C₁₃₇	semidirect
48. #103	C₂×C₁₃₇	semidirect
78. C₄	SD₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×C₁₃₇	2
C₄⋊Q₈×C₁₃₇	2
C₄×D₄×C₁₃₇	2
C₄⋊C₈×C₁₃₇	2
#103×C₁₃₇	2
#461	137

Subgroups — #461×C137 #22576996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #461×C₁₃₇ #22576996