Subgroups — #46834661

Order 34688.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31773950	C₂	1	2	yes	maximal
2	#31773619	C₂	1	2	yes	maximal, complemented
3	#31773618	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₂₇₁⋊C₈)	C₄	1	4	yes	—
7	C₂²⋊(C₂₇₁⋊C₈)	C₄	1	4	yes	—
8	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	—
9	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	—
10	C₂×M₁₆×C₂₇₁	C₄	1	2	yes	—
11	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
12	#97×C₂₇₁	C₄	1	2	yes	—
13	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₂₇₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₂×C₂₇₁⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₂₇₁	—	2	4	no	—
20	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
22	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
23	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
25	C₂₇₁⋊C₈	—	2	4	no	—
26	C₂₇₁⋊C₈	—	2	4	no	—
27	C₂³×C₂₇₁	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₂₇₁	—	2	2	no	—
29	C₂×C₄×C₂₇₁	—	2	4	no	—
30	C₂×C₄×C₂₇₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
32	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₇₁	—	2	2	no	—
35	C₈×C₂₇₁	M₁₆	1	4	yes	—
36	C₂²×C₂₇₁	—	2	1	no	—
37	C₂²×C₂₇₁	—	2	2	no	—
38	C₂²×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₂₇₁	—	2	1	no	—
40	C₄×C₂₇₁	—	2	2	no	—
41	C₄×C₂₇₁	C₂×M₁₆	1	2	yes	—
42	C₄×C₂₇₁	#131	1	4	yes	—
43	C₂×C₂₇₁	—	4	1	no	—
44	C₂×C₂₇₁	C₄×M₁₆	1	1	yes	characteristic
45	C₂×C₂₇₁	#405	1	2	yes	—
46	C₂₇₁	#1453	1	1	yes	characteristic, complemented, Sylow
47	#1453	—	271	1	no	maximal, Sylow
48	#431	—	271	2	no	—
49	C₄×M₁₆	D₂₇₁	1	1	yes	characteristic
50	#322	—	271	1	no	—
51	#322	—	271	2	no	—
52	C₈²	—	271	1	no	—
53	C₂×M₁₆	Dic₂₇₁	1	2	yes	—
54	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
55	C₄⋊C₈	—	271	4	no	—
56	C₂²⋊C₈	—	271	4	no	—
57	#97	Dic₂₇₁	1	2	yes	—
58	C₄×C₈	—	271	2	no	—
59	C₄×C₈	—	271	2	no	—
60	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
61	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
62	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
63	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	271	4	no	—
66	C₂×C₈	—	271	4	no	—
67	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
68	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
69	C₄²	C₄×D₂₇₁	1	2	yes	—
70	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
71	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
72	C₂³	C₄×Dic₂₇₁	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄×Dic₂₇₁	1	1	yes	characteristic
76	C₂×C₄	C₄×Dic₂₇₁	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
78	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	542	4	no	—
81	C₈	—	542	4	no	—
82	C₈	#16864638	1	4	yes	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄×Dic₂₇₁	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	2	no	—
88	C₄	C₂×#16864638	1	2	yes	—
89	C₄	#22576168	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₄×#16864638	1	1	yes	characteristic
92	C₂	#31773855	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂×C₄²
Fitting	C₄×M₁₆×C₂₇₁	C₂
prime radical	C₂³×C₂₇₁	C₄²
cosocle	C₄²×C₂₇₁	C₂³
hypercenter	C₄×M₁₆	D₂₇₁
hypocenter	C₂₇₁	#1453
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #31773619	C₂	semidirect
3. #31773618	C₂	semidirect
4. C₈×C₂₇₁⋊C₈	C₂	semidirect
46. C₂₇₁	#1453	semidirect

Maximal subgroups

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

subgroup	index
#31773950	2
#31773619	2
#31773618	2
C₈×C₂₇₁⋊C₈	2
C₄×M₁₆×C₂₇₁	2
#1453	271