Subgroups — C₂×#22577801 #31791288

Order 17536.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984088	C₂	1	2	yes	maximal
2	#22577801	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984020	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₄	—	2	2	no	—
6	#17984088	—	2	4	no	—
7	#17984088	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
10	#17984020	C₂²	1	16	yes	—
11	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
14	C₂×Dic₂₇₄	—	2	8	no	—
15	C₂×Dic₂₇₄	—	4	8	no	—
16	C₂²×Dic₁₃₇	—	2	4	no	—
17	C₄⋊Dic₁₃₇	—	2	8	no	—
18	C₄⋊Dic₁₃₇	—	2	4	no	—
19	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
20	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	Q₈	1	8	yes	complemented
23	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
24	C₄²×C₁₃₇	D₄	1	2	yes	—
25	C₄²×C₁₃₇	C₂³	1	2	yes	—
26	Dic₂₇₄	—	4	16	no	—
27	C₂×Dic₁₃₇	—	2	8	no	—
28	C₂×Dic₁₃₇	—	4	8	no	—
29	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	SD₁₆	1	8	yes	—
31	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
35	C₈×C₁₃₇	C₂×Q₈	1	8	yes	complemented
36	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
37	C₂²×C₁₃₇	C₂×SD₁₆	1	4	yes	—
38	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
40	C₄×C₁₃₇	C₂×SD₁₆	1	8	yes	—
41	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
42	Dic₁₃₇	—	4	8	no	—
43	C₂×C₁₃₇	#498	1	4	yes	complemented
44	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₃₇	C₂²×SD₁₆	1	2	yes	—
46	C₁₃₇	C₂×#498	1	1	yes	characteristic, complemented, Sylow
47	C₂×#498	—	137	1	no	maximal, Sylow
48	C₂×C₄⋊Q₈	—	137	2	no	—
49	#498	—	137	8	no	—
50	C₂×#106	—	137	4	no	—
51	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
52	C₂²×Q₈	—	274	2	no	—
53	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
54	C₄⋊Q₈	—	274	4	no	—
55	C₄⋊Q₈	—	137	8	no	—
56	C₂×C₄⋊C₄	—	274	2	no	—
57	C₂×C₄⋊C₄	—	137	4	no	—
58	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
59	#106	—	137	16	no	—
60	C₄×C₈	C₂×D₁₃₇	1	4	yes	—
61	C₂×Q₈	—	274	8	no	—
62	C₂×Q₈	—	548	8	no	—
63	C₂²×C₄	—	274	4	no	—
64	C₂²×C₄	D₅₄₈	1	2	yes	—
65	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
66	C₂×C₈	Dic₂₇₄	1	8	yes	complemented
67	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
68	C₄⋊C₄	—	274	8	no	—
69	C₄⋊C₄	—	274	4	no	—
70	C₄⋊C₄	—	137	8	no	—
71	C₄²	D₅₄₈	1	2	yes	—
72	C₄²	C₂²×D₁₃₇	1	2	yes	—
73	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
74	Q₈	—	548	16	no	—
75	C₂×C₄	—	274	8	no	—
76	C₂×C₄	—	548	8	no	—
77	C₂×C₄	#14555499	1	8	yes	—
78	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
79	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
80	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
81	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
82	C₈	C₂×Dic₂₇₄	1	8	yes	complemented
83	C₂²	#17984088	1	2	yes	—
84	C₂²	C₂×#14555499	1	4	yes	—
85	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
86	C₄	—	548	8	no	—
87	C₄	#17984088	1	2	yes	—
88	C₄	C₂×#14555499	1	8	yes	—
89	C₄	C₂²×Dic₂₇₄	1	2	yes	—
90	C₂	#22577801	1	4	yes	direct product factor
91	C₂	C₂×#17984088	1	1	yes	characteristic
92	C₂	C₂²×#14555499	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₄₈
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#498
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22577801	C₂	direct (N × H)
22. C₂×C₈×C₁₃₇	Q₈	semidirect
35. C₈×C₁₃₇	C₂×Q₈	semidirect
43. C₂×C₁₃₇	#498	semidirect
46. C₁₃₇	C₂×#498	semidirect
66. C₂×C₈	Dic₂₇₄	semidirect
82. C₈	C₂×Dic₂₇₄	semidirect
90. C₂	#22577801	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984088	2
#22577801	2
C₂×#17984020	2
C₂×C₄×C₈×C₁₃₇	2
C₂×#498	137

Subgroups — C2×#22577801 #31791288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577801 #31791288