Subgroups — #46837944

Order 34688.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31773946	C₂	1	1	yes	maximal, characteristic
2	#31773667	C₂	1	2	yes	maximal
3	#31773662	C₂	1	2	yes	maximal
4	C₂₇₁⋊#338	C₂	1	1	yes	maximal, characteristic
5	#424×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864638	C₂²	1	2	yes	—
7	C₂×#16864638	C₂²	1	2	yes	—
8	#22576096	—	2	2	no	—
9	C₂×M₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
11	#117×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₇₁	—	2	2	no	—
13	#16864638	—	2	2	no	—
14	#16864638	—	2	2	no	—
15	#16864638	—	2	2	no	—
16	#16864638	—	2	2	no	—
17	C₂×C₂₇₁⋊C₈	—	2	2	no	—
18	C₂×C₂₇₁⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₇₁	—	2	2	no	—
21	C₂×C₈×C₂₇₁	—	2	1	no	—
22	C₂×C₈×C₂₇₁	—	2	2	no	—
23	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₇₁	—	2	1	no	—
25	C₂²⋊C₄×C₂₇₁	Q₈	1	2	yes	—
26	C₄²×C₂₇₁	—	2	1	no	—
27	C₂₇₁⋊C₈	—	4	2	no	—
28	C₂₇₁⋊C₈	—	4	2	no	—
29	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₂₇₁	—	2	2	no	—
31	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₇₁	#41	1	2	yes	—
33	C₂×C₄×C₂₇₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₂₇₁	#41	1	2	yes	—
35	C₈×C₂₇₁	—	2	2	no	—
36	C₈×C₂₇₁	—	4	1	no	—
37	C₂²×C₂₇₁	—	4	1	no	—
38	C₂²×C₂₇₁	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₂₇₁	#125	1	2	yes	—
40	C₄×C₂₇₁	—	4	2	no	—
41	C₄×C₂₇₁	#123	1	1	yes	characteristic
42	C₄×C₂₇₁	#124	1	1	yes	characteristic
43	C₄×C₂₇₁	#126	1	2	yes	—
44	C₂×C₂₇₁	—	2	1	no	—
45	C₂×C₂₇₁	—	2	2	no	—
46	C₂×C₂₇₁	#400	1	1	yes	characteristic
47	C₂₇₁	#2103	1	1	yes	characteristic, complemented, Sylow
48	#2103	—	271	1	no	maximal, Sylow
49	#430	—	271	1	no	—
50	#424	D₂₇₁	1	1	yes	characteristic
51	#344	—	271	2	no	—
52	#341	—	271	2	no	—
53	#338	—	271	1	no	—
54	C₂×M₁₆	—	271	2	no	—
55	C₂×M₁₆	—	271	2	no	—
56	C₂×M₁₆	C₂×D₂₇₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
58	#117	C₂×D₂₇₁	1	1	yes	characteristic
59	#108	—	542	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	542	2	no	—
64	M₁₆	—	542	2	no	—
65	M₁₆	—	542	2	no	—
66	M₁₆	—	542	2	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	542	2	no	—
70	C₂×C₈	—	542	2	no	—
71	C₂×C₈	C₂²⋊D₂₇₁	1	2	yes	—
72	C₄⋊C₄	—	2	1	no	—
73	C₂²⋊C₄	Dic₅₄₂	1	2	yes	—
74	C₄²	—	2	1	no	—
75	C₂³	C₂×Dic₅₄₂	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
78	C₂×C₄	#16864653	1	2	yes	—
79	C₂×C₄	#16864654	1	1	yes	characteristic
80	C₂×C₄	#16864655	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	4	1	no	—
83	C₈	—	1084	2	no	—
84	C₈	—	1084	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#22576139	1	1	yes	characteristic
87	C₂²	#22576151	1	2	yes	—
88	C₄	—	4	2	no	—
89	C₄	#22576141	1	1	yes	characteristic
90	C₄	#22576149	1	1	yes	characteristic
91	C₄	#22576155	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#31773836	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22576149
commutator, cosocle	C₂²×C₄×C₂₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	#424×C₂₇₁	C₂
socle	C₂×C₂₇₁	#400
prime radical	C₂³×C₂₇₁	C₂×Q₈
hypercenter	#424	D₂₇₁
hypocenter	C₂₇₁	#2103
LCS	G ▷ C₂²×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
47. C₂₇₁	#2103	semidirect

Maximal subgroups

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

subgroup	index
#31773946	2
#31773667	2
#31773662	2
C₂₇₁⋊#338	2
#424×C₂₇₁	2
#2103	271