Subgroups — #27186038

Order 12928.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266198	C₂	1	2	yes	maximal
2	#20265200	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11717581	—	2	2	no	—
4	C₂²⋊#11717581	C₄	1	4	yes	—
5	#15692310	—	2	2	no	—
6	C₂×#13108058	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	#15692184	—	2	2	no	—
9	C₂×M₁₆×C₁₀₁	C₄	1	1	yes	characteristic
10	C₂×#11717581	—	2	4	no	—
11	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
12	#13108058	—	4	1	no	—
13	C₂×#11717579	—	2	1	no	—
14	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
15	C₄×Dic₁₀₁	D₄	1	2	yes	—
16	C₄×Dic₁₀₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₀₁	—	4	1	no	—
19	C₂×C₈×C₁₀₁	—	2	1	no	—
20	C₂³×C₁₀₁	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₁₀₁	—	2	1	no	—
22	C₂×C₄×C₁₀₁	—	2	1	no	—
23	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×Dic₁₀₁	—	4	2	no	—
26	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
27	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
28	#11717581	—	4	4	no	—
29	C₈×C₁₀₁	—	4	1	no	—
30	#11717579	—	4	1	no	—
31	C₂²×C₁₀₁	—	2	1	no	—
32	C₂²×C₁₀₁	—	2	2	no	—
33	C₂²×C₁₀₁	#95	1	1	yes	characteristic
34	C₄×C₁₀₁	—	2	1	no	—
35	C₄×C₁₀₁	#100	1	1	yes	characteristic
36	C₄×C₁₀₁	#101	1	1	yes	characteristic
37	Dic₁₀₁	—	2	2	no	—
38	Dic₁₀₁	—	2	2	no	—
39	C₂×C₁₀₁	—	4	1	no	—
40	C₂×C₁₀₁	#343	1	1	yes	characteristic
41	C₂×C₁₀₁	#344	1	2	yes	—
42	#1307	—	101	1	no	maximal, Sylow
43	C₁₀₁	#1307	1	1	yes	characteristic, complemented, Sylow
44	#432	—	101	2	no	—
45	#423	—	101	1	no	—
46	C₂×M₁₆	—	101	1	no	—
47	C₂×M₁₆	#87938	1	1	yes	characteristic
48	C₂×C₄²	—	101	1	no	—
49	C₄⋊C₈	—	202	2	no	—
50	C₄⋊C₈	—	202	2	no	—
51	C₂²⋊C₈	—	101	4	no	—
52	#97	—	202	2	no	—
53	C₂²×C₄	—	101	2	no	—
54	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	M₁₆	—	404	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	202	4	no	—
59	C₂×C₈	—	202	1	no	—
60	C₄²	—	101	2	no	—
61	C₄²	—	101	1	no	—
62	C₄²	—	101	1	no	—
63	C₂³	C₄×#87938	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	101	2	no	—
67	C₂×C₄	—	101	2	no	—
68	C₂×C₄	—	404	2	no	—
69	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
70	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
71	C₈	—	4	1	no	—
72	C₈	—	404	4	no	—
73	C₈	—	404	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	202	2	no	—
79	C₄	—	202	2	no	—
80	C₄	#15692315	1	1	yes	characteristic
81	C₄	#15692316	1	1	yes	characteristic
82	C₂	—	4	1	no	—
83	C₂	#20266057	1	1	yes	characteristic
84	C₂	#20266060	1	2	yes	—

	subgroup	G/H
center	C₂²	(C₂×#87938)⋊C₄
commutator, prime radical	C₂³×C₁₀₁	C₄²
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂×M₁₆×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#95
cosocle	C₂×C₄×Dic₁₀₁	C₂²
hypercenter	C₂×M₁₆	#87938
hypocenter	C₁₀₁	#1307
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
43. C₁₀₁	#1307	semidirect

Maximal subgroups

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

subgroup	index
#20266198	2
#20265200	2
#1307	101