Subgroups — #46837969

Order 34688.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22576100	C₂	1	2	yes	maximal
2	#31773675	C₂	1	4	yes	maximal, complemented
3	#405×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#22576100	C₄	1	4	yes	complemented
5	#22576100	C₂²	1	4	yes	—
6	C₂×C₂₇₁⋊C₁₆	C₄	1	4	yes	complemented
7	C₂×M₁₆×C₂₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
9	#117×C₂₇₁	C₄	1	1	yes	characteristic
10	#97×C₂₇₁	C₂²	1	2	yes	—
11	C₄×C₈×C₂₇₁	C₄	1	2	yes	—
12	C₂₇₁⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
14	M₁₆×C₂₇₁	—	2	2	no	—
15	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₇₁	—	2	1	no	—
21	C₂²⋊C₄×C₂₇₁	—	2	1	no	—
22	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
23	C₂³×C₂₇₁	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₂₇₁	—	2	2	no	—
25	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
26	C₂×C₄×C₂₇₁	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₂₇₁	M₁₆	1	2	yes	—
28	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₂₇₁	—	2	2	no	—
30	C₈×C₂₇₁	C₄²	1	1	yes	characteristic
31	C₈×C₂₇₁	C₄²	1	1	yes	characteristic
32	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
33	C₂²×C₂₇₁	—	4	1	no	—
34	C₂²×C₂₇₁	#97	1	2	yes	—
35	C₂²×C₂₇₁	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₂₇₁	—	4	2	no	—
37	C₄×C₂₇₁	#97	1	2	yes	—
38	C₄×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₂₇₁	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₂₇₁	—	2	1	no	—
41	C₂×C₂₇₁	—	2	2	no	—
42	C₂×C₂₇₁	C₂×#97	1	1	yes	characteristic
43	C₂₇₁	#2110	1	1	yes	characteristic, complemented, Sylow
44	#2110	—	271	1	no	maximal, Sylow
45	C₂×M₃₂	—	271	2	no	—
46	#405	D₂₇₁	1	1	yes	characteristic
47	#347	—	271	4	no	—
48	C₂×M₁₆	Dic₂₇₁	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
50	#117	Dic₂₇₁	1	1	yes	characteristic
51	M₃₂	—	271	4	no	—
52	M₃₂	—	271	4	no	—
53	C₂×C₁₆	—	271	4	no	—
54	#97	C₂×D₂₇₁	1	2	yes	—
55	C₄×C₈	Dic₂₇₁	1	2	yes	—
56	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
59	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
60	C₂×C₈	C₂×Dic₂₇₁	1	2	yes	—
61	C₂×C₈	C₂×Dic₂₇₁	1	1	yes	characteristic
62	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×Dic₂₇₁	1	2	yes	—
66	C₁₆	—	271	8	no	—
67	C₂³	#16864638	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄×Dic₂₇₁	1	2	yes	—
70	C₂×C₄	#16864638	1	2	yes	—
71	C₂×C₄	#16864638	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₄×Dic₂₇₁	1	1	yes	characteristic
75	C₈	C₄×Dic₂₇₁	1	1	yes	characteristic
76	C₈	C₂×C₄×D₂₇₁	1	2	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	#22576071	1	2	yes	—
79	C₂²	C₂×#16864638	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#22576071	1	2	yes	—
82	C₄	C₂×C₄×Dic₂₇₁	1	1	yes	characteristic
83	C₄	C₂×#16864638	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#22576071	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₂₇₁
commutator	C₄×C₂₇₁	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	#405×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×#97
prime radical	C₂³×C₂₇₁	M₁₆
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	#405	D₂₇₁
hypocenter	C₂₇₁	#2110
LCS	G ▷ C₄×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #31773675	C₂	semidirect
4. #22576100	C₄	semidirect
6. C₂×C₂₇₁⋊C₁₆	C₄	semidirect
43. C₂₇₁	#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22576100	2
#31773675	2
#405×C₂₇₁	2
#2110	271