Subgroups — #31774292

Order 17344.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864644	C₂	1	2	yes	maximal, complemented
2	C₂×#16864642	C₂	1	2	yes	maximal
3	#22576160	C₂	1	1	yes	maximal, characteristic, complemented
4	#22576071	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₂	C₂²	1	2	yes	—
7	#16864644	—	2	4	no	—
8	#16864642	—	2	4	no	—
9	C₂×C₂₇₁⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₂₇₁	—	2	2	no	—
11	C₂×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
14	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
15	Dic₅₄₂	—	2	2	no	—
16	Dic₅₄₂	—	2	2	no	—
17	C₂×Dic₂₇₁	—	2	2	no	—
18	C₂₇₁⋊C₈	D₄	1	4	yes	—
19	C₂³×C₂₇₁	—	2	1	no	—
20	Q₈×C₂₇₁	—	2	2	no	—
21	D₄×C₂₇₁	—	2	2	no	—
22	C₂×C₄×C₂₇₁	—	2	1	no	—
23	C₂×C₄×C₂₇₁	D₄	1	2	yes	—
24	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₇₁	—	4	1	no	—
26	C₂²×C₂₇₁	—	4	2	no	—
27	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₇₁	—	2	2	no	—
29	C₄×C₂₇₁	—	4	1	no	—
30	C₄×C₂₇₁	C₂×D₄	1	2	yes	—
31	Dic₂₇₁	—	4	2	no	—
32	C₂×C₂₇₁	—	8	1	no	—
33	C₂×C₂₇₁	#127	1	1	yes	characteristic
34	C₂×C₂₇₁	#137	1	2	yes	—
35	C₂₇₁	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	271	1	no	maximal, Sylow
37	C₂×Q₁₆	—	271	2	no	—
38	C₂×SD₁₆	—	271	2	no	—
39	C₄⋊Q₈	—	271	1	no	—
40	#124	D₂₇₁	1	1	yes	characteristic
41	#97	—	271	1	no	—
42	C₂×Q₈	—	271	2	no	—
43	C₂×Q₈	C₂×D₂₇₁	1	1	yes	characteristic
44	C₂×D₄	C₂×D₂₇₁	1	1	yes	characteristic
45	Q₁₆	—	542	4	no	—
46	SD₁₆	—	542	4	no	—
47	C₂×C₈	—	271	2	no	—
48	C₄⋊C₄	—	542	2	no	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	542	2	no	—
54	Q₈	—	542	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	542	2	no	—
58	C₂×C₄	C₂²⋊D₂₇₁	1	2	yes	—
59	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
60	C₈	—	271	4	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	1084	2	no	—
67	C₄	D₄×D₂₇₁	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#22576156	1	1	yes	characteristic
70	C₂	#22576190	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#124×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₄
prime radical	C₂×D₄×C₂₇₁	C₂²
hypercenter	#124	D₂₇₁
hypocenter	C₂₇₁	#497
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#16864644	C₂	semidirect
3. #22576160	C₂	semidirect
4. #22576071	C₂	semidirect
35. C₂₇₁	#497	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864644	2
C₂×#16864642	2
#22576160	2
#22576071	2
#124×C₂₇₁	2
#497	271