Finite Group Databasebeta

Subgroups — #27173029

Order 12928.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265615	C₂	1	1	yes	maximal, characteristic
2	#20265298	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265288	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264989	C₂	1	1	yes	maximal, characteristic, complemented
5	#20264986	C₂	1	1	yes	maximal, characteristic, complemented
6	#20264954	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#15692277	C₂²	1	1	yes	characteristic
9	C₂×D₈₀₈	C₂²	1	1	yes	characteristic
10	#15692263	—	2	1	no	—
11	#15692217	—	2	1	no	—
12	#15692200	C₄	1	2	yes	complemented
13	D₁₆₁₆	C₄	1	2	yes	complemented
14	#15692195	C₂²	1	1	yes	characteristic
15	#15692181	—	2	1	no	—
16	#15692175	—	2	1	no	—
17	#15692165	C₂²	1	1	yes	characteristic
18	M₃₂×C₁₀₁	C₂²	1	1	yes	characteristic
19	#108×C₁₀₁	C₂²	1	1	yes	characteristic
20	#106×C₁₀₁	C₂²	1	1	yes	characteristic
21	#13108074	—	2	1	no	—
22	C₂×D₄₀₄	—	2	1	no	—
23	C₂×C₄×D₁₀₁	—	4	1	no	—
24	Dic₄₀₄	C₂×C₄	1	1	yes	characteristic, complemented
25	#13108063	—	2	1	no	—
26	D₈₀₈	—	2	1	no	—
27	D₈₀₈	C₂×C₄	1	2	yes	—
28	D₈₀₈	C₂×C₄	1	1	yes	characteristic, complemented
29	#13108057	—	4	1	no	—
30	C₂×#11717579	D₄	1	1	yes	characteristic
31	C₈×D₁₀₁	—	4	1	no	—
32	#13108051	—	4	1	no	—
33	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
34	#13108049	—	2	1	no	—
35	M₁₆×C₁₀₁	—	2	1	no	—
36	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₀₁	—	2	1	no	—
38	C₁₆×C₁₀₁	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₀₁	—	4	1	no	—
40	C₂×C₄×C₁₀₁	—	4	1	no	—
41	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₁₀₁	—	4	1	no	—
43	C₂×Dic₁₀₁	—	2	1	no	—
44	D₄₀₄	—	2	2	no	—
45	D₄₀₄	—	2	1	no	—
46	D₄₀₄	—	4	1	no	—
47	C₄×D₁₀₁	—	4	1	no	—
48	C₄×D₁₀₁	—	8	2	no	—
49	Dic₂₀₂	—	2	1	no	—
50	C₈×C₁₀₁	—	4	1	no	—
51	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₀₁	#41	1	1	yes	characteristic
53	#11717579	—	2	1	no	—
54	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₀₁	—	4	1	no	—
56	C₂×D₁₀₁	—	4	2	no	—
57	C₂×D₁₀₁	—	8	1	no	—
58	C₂×D₁₀₁	—	8	1	no	—
59	C₄×C₁₀₁	—	8	1	no	—
60	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₀₁	—	4	1	no	—
63	Dic₁₀₁	—	4	1	no	—
64	C₂×C₁₀₁	—	2	1	no	—
65	C₂×C₁₀₁	#442	1	1	yes	characteristic
66	D₁₀₁	—	8	1	no	—
67	D₁₀₁	—	8	2	no	—
68	AGL(1,ℤ/16)	—	101	1	no	maximal, Sylow
69	C₁₀₁	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	—	101	1	no	—
71	#444	—	101	1	no	—
72	#442	—	101	1	no	—
73	#365	D₁₀₁	1	1	yes	characteristic, complemented
74	#361	—	101	1	no	—
75	#360	—	101	1	no	—
76	#347	—	101	1	no	—
77	#135	—	101	1	no	—
78	C₂×D₈	—	101	1	no	—
79	#131	—	202	1	no	—
80	C₄×D₄	—	202	1	no	—
81	SD₃₂	—	101	2	no	—
82	D₁₆	—	101	2	no	—
83	M₃₂	—	101	1	no	—
84	M₃₂	C₂×D₁₀₁	1	1	yes	characteristic
85	#108	C₂×D₁₀₁	1	1	yes	characteristic
86	#106	C₂×D₁₀₁	1	1	yes	characteristic
87	#104	—	202	1	no	—
88	#102	—	202	1	no	—
89	#97	—	101	1	no	—
90	#41	—	202	1	no	—
91	C₂×D₄	—	202	1	no	—
92	C₂²×C₄	—	404	1	no	—
93	Q₁₆	—	101	1	no	—
94	SD₁₆	—	202	1	no	—
95	D₈	—	101	2	no	—
96	D₈	—	101	1	no	—
97	D₈	—	202	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	404	1	no	—
100	C₂×C₈	—	101	1	no	—
101	C₂×C₈	—	404	1	no	—
102	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	404	1	no	—
105	C₄²	—	101	1	no	—
106	C₁₆	—	202	1	no	—
107	C₁₆	C₄×D₁₀₁	1	2	yes	complemented
108	C₂³	—	404	1	no	—
109	Q₈	—	202	1	no	—
110	D₄	—	404	1	no	—
111	D₄	—	404	1	no	—
112	D₄	—	202	2	no	—
113	D₄	—	202	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	808	2	no	—
116	C₂×C₄	—	404	1	no	—
117	C₂×C₄	—	202	1	no	—
118	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	202	1	no	—
121	C₈	C₂×C₄×D₁₀₁	1	1	yes	characteristic
122	C₈	#13108075	1	1	yes	characteristic
123	C₂²	—	808	1	no	—
124	C₂²	—	808	1	no	—
125	C₂²	—	404	1	no	—
126	C₂²	—	404	2	no	—
127	C₂²	#15692282	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	404	1	no	—
130	C₄	—	404	1	no	—
131	C₄	#15692217	1	1	yes	characteristic
132	C₄	#15692282	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	808	2	no	—
135	C₂	—	808	1	no	—
136	C₂	#20265288	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20265288
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#365×C₁₀₁	C₂
socle	C₂×C₁₀₁	#442
prime radical	#20265615	C₂
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	#365	D₁₀₁
hypocenter	C₁₀₁	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
2. #20265298	C₂	semidirect
3. #20265288	C₂	semidirect
4. #20264989	C₂	semidirect
5. #20264986	C₂	semidirect
6. #20264954	C₂	semidirect
7. #365×C₁₀₁	C₂	semidirect
12. #15692200	C₄	semidirect
13. D₁₆₁₆	C₄	semidirect
24. Dic₄₀₄	C₂×C₄	semidirect
28. D₈₀₈	C₂×C₄	semidirect
38. C₁₆×C₁₀₁	C₂×C₄	semidirect
69. C₁₀₁	AGL(1,ℤ/16)	semidirect
73. #365	D₁₀₁	semidirect
107. C₁₆	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265615	2
#20265298	2
#20265288	2
#20264989	2
#20264986	2
#20264954	2
#365×C₁₀₁	2
AGL(1,ℤ/16)	101