Subgroups — #27170739

Order 12928.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265554	C₂	1	1	yes	maximal, characteristic
2	#20265525	C₂	1	4	yes	maximal, complemented
3	C₈×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈₀₈	—	2	2	no	—
6	#15692252	C₂²	1	2	yes	—
7	#15692175	—	2	8	no	—
8	C₄×#11717579	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₀₁	—	2	2	no	—
10	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×D₄₀₄	—	2	4	no	—
13	C₂×D₄₀₄	—	2	2	no	—
14	D₈₀₈	—	4	4	no	—
15	C₂×#11717579	D₄	1	4	yes	—
16	C₂×Q₈×C₁₀₁	—	2	2	no	—
17	Q₁₆×C₁₀₁	—	4	4	no	—
18	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₀₁	—	2	4	no	—
20	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₂²×D₁₀₁	—	4	2	no	—
22	Q₈×C₁₀₁	—	4	4	no	—
23	C₂×C₄×C₁₀₁	—	4	2	no	—
24	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
26	D₄₀₄	—	4	4	no	—
27	D₄₀₄	—	4	8	no	—
28	C₈×C₁₀₁	D₈	1	4	yes	—
29	#11717579	—	2	4	no	—
30	C₂²×C₁₀₁	#127	1	1	yes	characteristic
31	C₂×D₁₀₁	—	8	2	no	—
32	C₂×D₁₀₁	—	8	4	no	—
33	C₄×C₁₀₁	—	8	2	no	—
34	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
35	C₄×C₁₀₁	#135	1	4	yes	—
36	C₂×C₁₀₁	#493	1	1	yes	characteristic
37	C₂×C₁₀₁	#495	1	2	yes	—
38	D₁₀₁	—	16	2	no	—
39	#1709	—	101	1	no	maximal, Sylow
40	C₁₀₁	#1709	1	1	yes	characteristic, complemented, Sylow
41	#494	D₁₀₁	1	1	yes	characteristic, complemented
42	#493	—	101	1	no	—
43	#486	—	101	4	no	—
44	C₈²	—	101	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	202	2	no	—
47	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
48	#127	—	101	2	no	—
49	#102	—	202	8	no	—
50	C₄×C₈	—	101	2	no	—
51	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	202	4	no	—
54	C₂×D₄	—	202	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	404	4	no	—
57	C₂×C₈	—	101	4	no	—
58	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
61	C₂³	—	404	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	404	4	no	—
64	D₄	—	404	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
68	C₈	—	202	4	no	—
69	C₈	#13108059	1	4	yes	—
70	C₂²	—	808	2	no	—
71	C₂²	—	808	4	no	—
72	C₂²	#15692251	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#13108059	1	2	yes	—
75	C₄	#15692278	1	4	yes	—
76	C₂	—	1616	2	no	—
77	C₂	#20265553	1	1	yes	characteristic
78	C₂	#20265562	1	2	yes	—

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#494×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
prime radical	#20265554	C₂
hypercenter	#494	D₁₀₁
hypocenter	C₁₀₁	#1709
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #20265525	C₂	semidirect
3. C₈×#11717579	C₂	semidirect
4. #494×C₁₀₁	C₂	semidirect
40. C₁₀₁	#1709	semidirect
41. #494	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265554	2
#20265525	2
C₈×#11717579	2
#494×C₁₀₁	2
#1709	101