Subgroups — C₂×#22577783 #31791205

Order 17536.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₄₈	C₂	1	4	yes	maximal
2	C₂×#17984088	C₂	1	2	yes	maximal
3	#22577783	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₄	C₂²	1	4	yes	—
6	C₂×Dic₅₄₈	—	2	16	no	—
7	C₂×Dic₅₄₈	C₂²	1	16	yes	—
8	#17984088	—	2	4	no	—
9	#17984088	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₃₇	—	2	4	no	—
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	4	no	—
15	C₂×Dic₂₇₄	—	2	16	no	—
16	C₂×Dic₂₇₄	C₂³	1	8	yes	—
17	C₂²×Dic₁₃₇	—	2	4	no	—
18	Dic₅₄₈	—	2	32	no	—
19	C₄⋊Dic₁₃₇	—	2	16	no	—
20	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	D₄	1	8	yes	—
23	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
24	C₄²×C₁₃₇	D₄	1	2	yes	—
25	C₄²×C₁₃₇	C₂³	1	2	yes	—
26	Dic₂₇₄	—	2	16	no	—
27	Dic₂₇₄	—	4	8	no	—
28	C₂×Dic₁₃₇	—	2	8	no	—
29	C₂×Dic₁₃₇	—	4	8	no	—
30	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	Q₁₆	1	8	yes	complemented
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
35	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
36	C₈×C₁₃₇	C₂×D₄	1	8	yes	—
37	C₂²×C₁₃₇	#127	1	2	yes	—
38	C₂²×C₁₃₇	C₂×Q₁₆	1	4	yes	—
39	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
40	C₄×C₁₃₇	#127	1	2	yes	—
41	C₄×C₁₃₇	C₂×Q₁₆	1	8	yes	complemented
42	C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
43	Dic₁₃₇	—	4	8	no	—
44	C₂×C₁₃₇	#494	1	4	yes	complemented
45	C₂×C₁₃₇	C₂×#127	1	1	yes	characteristic
46	C₂×C₁₃₇	C₂²×Q₁₆	1	2	yes	—
47	C₁₃₇	C₂×#494	1	1	yes	characteristic, complemented, Sylow
48	C₂×#494	—	137	1	no	maximal, Sylow
49	C₂²×Q₁₆	—	137	4	no	—
50	C₂×C₄⋊Q₈	—	137	2	no	—
51	#494	—	137	8	no	—
52	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
53	C₂²×Q₈	—	137	4	no	—
54	C₂×Q₁₆	—	274	16	no	—
55	C₂×Q₁₆	—	137	16	no	—
56	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
57	C₄⋊Q₈	—	274	4	no	—
58	C₄⋊Q₈	—	137	8	no	—
59	C₂×C₄⋊C₄	—	274	4	no	—
60	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₁₃₇	1	4	yes	—
62	C₂×Q₈	—	274	4	no	—
63	C₂×Q₈	—	274	16	no	—
64	C₂×Q₈	—	137	8	no	—
65	C₂²×C₄	—	274	4	no	—
66	C₂²×C₄	D₅₄₈	1	2	yes	—
67	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
68	Q₁₆	—	274	32	no	—
69	C₂×C₈	D₅₄₈	1	8	yes	—
70	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
71	C₄⋊C₄	—	274	16	no	—
72	C₄²	D₅₄₈	1	2	yes	—
73	C₄²	C₂²×D₁₃₇	1	2	yes	—
74	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
75	Q₈	—	274	16	no	—
76	Q₈	—	548	8	no	—
77	C₂×C₄	—	274	8	no	—
78	C₂×C₄	—	548	8	no	—
79	C₂×C₄	Dic₅₄₈	1	8	yes	complemented
80	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
81	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
82	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
83	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
84	C₈	C₂×D₅₄₈	1	8	yes	—
85	C₂²	#17984085	1	2	yes	—
86	C₂²	C₂×Dic₅₄₈	1	4	yes	—
87	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
88	C₄	—	548	8	no	—
89	C₄	#17984085	1	2	yes	—
90	C₄	C₂×Dic₅₄₈	1	8	yes	complemented
91	C₄	C₂²×D₅₄₈	1	2	yes	—
92	C₂	#22577783	1	4	yes	direct product factor
93	C₂	C₂×#17984085	1	1	yes	characteristic
94	C₂	C₂²×Dic₅₄₈	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×#494
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
3. #22577783	C₂	direct (N × H)
31. C₂×C₄×C₁₃₇	Q₁₆	semidirect
41. C₄×C₁₃₇	C₂×Q₁₆	semidirect
44. C₂×C₁₃₇	#494	semidirect
47. C₁₃₇	C₂×#494	semidirect
79. C₂×C₄	Dic₅₄₈	semidirect
90. C₄	C₂×Dic₅₄₈	semidirect
92. C₂	#22577783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₄₈	2
C₂×#17984088	2
#22577783	2
C₂×C₄×C₈×C₁₃₇	2
C₂×#494	137

Subgroups — C2×#22577783 #31791205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577783 #31791205