Subgroups — #46835965

Order 34688.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774295	C₂	1	1	yes	maximal, characteristic, complemented
2	#31774270	C₂	1	1	yes	maximal, characteristic
3	#31774248	C₂	1	2	yes	maximal, complemented
4	#31774244	C₂	1	2	yes	maximal
5	C₈²×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864643	—	2	2	no	—
7	#22576160	C₂²	1	2	yes	—
8	#22576160	C₂²	1	1	yes	characteristic
9	#22576157	C₂²	1	1	yes	characteristic
10	#22576092	—	2	2	no	—
11	#22576085	—	2	4	no	—
12	#22576082	—	2	4	no	—
13	C₄×C₈×C₂₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₄₂	—	2	1	no	—
16	C₂×Dic₅₄₂	—	2	2	no	—
17	C₂×D₁₀₈₄	—	2	1	no	—
18	C₂×D₁₀₈₄	—	2	2	no	—
19	#16864643	—	4	4	no	—
20	C₄⋊Dic₂₇₁	—	2	2	no	—
21	C₄⋊Dic₂₇₁	—	2	2	no	—
22	C₄⋊Dic₂₇₁	—	2	2	no	—
23	C₂×C₈×C₂₇₁	D₄	1	4	yes	—
24	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
25	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
26	C₂²×D₂₇₁	—	4	1	no	—
27	Dic₅₄₂	—	4	2	no	—
28	Dic₅₄₂	—	4	4	no	—
29	D₁₀₈₄	—	4	2	no	—
30	D₁₀₈₄	—	4	4	no	—
31	C₂×Dic₂₇₁	—	4	1	no	—
32	C₂×Dic₂₇₁	—	4	2	no	—
33	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
35	C₈×C₂₇₁	—	2	4	no	—
36	C₈×C₂₇₁	SD₁₆	1	4	yes	—
37	C₂²×C₂₇₁	#127	1	1	yes	characteristic
38	C₂×D₂₇₁	—	8	1	no	—
39	C₂×D₂₇₁	—	8	2	no	—
40	C₄×C₂₇₁	C₂×SD₁₆	1	2	yes	—
41	C₄×C₂₇₁	#135	1	4	yes	—
42	Dic₂₇₁	—	8	1	no	—
43	Dic₂₇₁	—	8	2	no	—
44	C₂×C₂₇₁	#492	1	1	yes	characteristic
45	C₂×C₂₇₁	#495	1	2	yes	—
46	D₂₇₁	—	16	1	no	—
47	C₂₇₁	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	271	1	no	maximal, Sylow
49	#498	—	271	1	no	—
50	#492	—	271	1	no	—
51	#487	—	271	2	no	—
52	#486	—	271	2	no	—
53	C₈²	D₂₇₁	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	542	2	no	—
55	C₄⋊Q₈	—	271	2	no	—
56	C₄⋊Q₈	—	271	1	no	—
57	#127	—	271	1	no	—
58	#106	—	542	2	no	—
59	#103	—	542	4	no	—
60	#102	—	542	4	no	—
61	C₄×C₈	C₂×D₂₇₁	1	2	yes	—
62	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
63	C₂×Q₈	—	542	2	no	—
64	C₂×Q₈	—	542	1	no	—
65	C₂×D₄	—	542	2	no	—
66	C₂×D₄	—	542	1	no	—
67	SD₁₆	—	1084	4	no	—
68	C₂×C₈	D₁₀₈₄	1	4	yes	—
69	C₂×C₈	D₁₀₈₄	1	2	yes	—
70	C₄⋊C₄	—	542	2	no	—
71	C₄⋊C₄	—	542	2	no	—
72	C₄⋊C₄	—	542	2	no	—
73	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
74	C₂³	—	1084	1	no	—
75	Q₈	—	1084	2	no	—
76	Q₈	—	1084	4	no	—
77	D₄	—	1084	2	no	—
78	D₄	—	1084	4	no	—
79	C₂×C₄	—	1084	1	no	—
80	C₂×C₄	—	1084	2	no	—
81	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
82	C₂×C₄	C₂×D₁₀₈₄	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#16864643	1	4	yes	—
85	C₂²	—	2168	1	no	—
86	C₂²	—	2168	2	no	—
87	C₂²	#22576157	1	1	yes	characteristic
88	C₄	—	2168	1	no	—
89	C₄	—	2168	2	no	—
90	C₄	C₂×#16864643	1	2	yes	—
91	C₄	#22576182	1	4	yes	—
92	C₂	—	4336	1	no	—
93	C₂	#31774270	1	1	yes	characteristic
94	C₂	#31774281	1	2	yes	—

	subgroup	G/H
center	C₂²	#22576157
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	C₈²×C₂₇₁	C₂
socle	C₂²×C₂₇₁	#127
prime radical	#22576157	C₂²
hypercenter	C₈²	D₂₇₁
hypocenter	C₂₇₁	#1708
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #31774295	C₂	semidirect
3. #31774248	C₂	semidirect
5. C₈²×C₂₇₁	C₂	semidirect
47. C₂₇₁	#1708	semidirect
53. C₈²	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774295	2
#31774270	2
#31774248	2
#31774244	2
C₈²×C₂₇₁	2
#1708	271