Subgroups — #27172890

Order 12928.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692195	C₂	1	1	yes	maximal, characteristic
2	#20265132	C₂	1	1	yes	maximal, characteristic
3	#20264995	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108058	C₄	1	1	yes	characteristic
6	#15692214	C₄	1	1	yes	characteristic
7	#15692195	—	2	2	no	—
8	C₂×#13108049	C₂²	1	2	yes	—
9	#15692165	C₄	1	2	yes	—
10	C₈×Dic₁₀₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₁	—	2	2	no	—
13	C₂×C₁₆×C₁₀₁	C₂²	1	2	yes	—
14	#13108058	—	2	2	no	—
15	C₂×#11717579	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₀₁	C₈	1	2	yes	—
17	C₂²⋊Dic₁₀₁	C₈	1	2	yes	—
18	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
19	#13108049	—	2	2	no	—
20	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
23	C₂×C₈×C₁₀₁	Q₈	1	2	yes	—
24	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₀₁	—	2	2	no	—
26	C₂³×C₁₀₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₁	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	—
32	C₈×C₁₀₁	C₄⋊C₄	1	2	yes	—
33	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
35	#11717579	—	2	2	no	—
36	C₂²×C₁₀₁	—	2	2	no	—
37	C₂²×C₁₀₁	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₀₁	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
42	Dic₁₀₁	—	2	4	no	—
43	C₂×C₁₀₁	—	2	2	no	—
44	C₂×C₁₀₁	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₀₁	#504	1	2	yes	—
46	#2151	—	101	1	no	maximal, Sylow
47	C₁₀₁	#2151	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	101	1	no	—
49	C₂²×C₁₆	D₁₀₁	1	1	yes	characteristic
50	#405	—	101	1	no	—
51	C₄⋊C₁₆	—	101	4	no	—
52	C₂×M₁₆	—	101	1	no	—
53	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
54	#117	—	101	1	no	—
55	M₃₂	—	202	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	101	2	no	—
58	C₂×C₁₆	C₂×D₁₀₁	1	2	yes	—
59	#97	—	101	2	no	—
60	C₄×C₈	—	101	2	no	—
61	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
62	M₁₆	—	202	2	no	—
63	C₂×C₈	—	101	2	no	—
64	C₂×C₈	Dic₂₀₂	1	2	yes	—
65	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
67	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
68	C₄⋊C₄	—	101	2	no	—
69	C₂²⋊C₄	—	101	2	no	—
70	C₄²	—	101	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	202	2	no	—
73	C₂³	C₈×D₁₀₁	1	1	yes	characteristic
74	C₂×C₄	—	101	4	no	—
75	C₂×C₄	#13108053	1	2	yes	—
76	C₂×C₄	C₈×D₁₀₁	1	1	yes	characteristic
77	C₂×C₄	#13108057	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
79	C₈	—	202	2	no	—
80	C₈	#13108053	1	2	yes	—
81	C₈	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
82	C₈	C₂×Dic₂₀₂	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#15692184	1	2	yes	—
85	C₂²	C₂×C₈×D₁₀₁	1	1	yes	characteristic
86	C₄	—	202	4	no	—
87	C₄	#15692184	1	2	yes	—
88	C₄	C₂×#13108053	1	1	yes	characteristic
89	C₄	C₂×#13108057	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#15692184	1	1	yes	characteristic
92	C₂	#20265593	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₂²×C₁₆×C₁₀₁	C₂
prime radical	C₂³×C₁₀₁	C₂×C₈
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	C₂²×C₁₆	D₁₀₁
hypocenter	C₁₀₁	#2151
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20264995	C₂	semidirect
47. C₁₀₁	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692195	2
#20265132	2
#20264995	2
C₂²×C₁₆×C₁₀₁	2
#2151	101