Finite Group Databasebeta

Subgroups — #35216436

Order 20864.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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	#23782103	C₂	1	1	yes	maximal, characteristic
2	#23781788	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781778	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781488	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781479	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781476	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19104757	C₂²	1	1	yes	characteristic
9	C₂×D₁₃₀₄	C₂²	1	1	yes	characteristic
10	#19104741	—	2	1	no	—
11	C₄×D₆₅₂	—	2	1	no	—
12	#19104680	C₄	1	2	yes	complemented
13	D₂₆₀₈	C₄	1	2	yes	complemented
14	#19104671	C₂²	1	1	yes	characteristic
15	#19104667	C₂²	1	1	yes	characteristic
16	#19104663	—	2	1	no	—
17	#19104657	—	2	1	no	—
18	M₃₂×C₁₆₃	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₆₃	C₂²	1	1	yes	characteristic
20	#97×C₁₆₃	C₂²	1	1	yes	characteristic
21	#14579699	—	2	1	no	—
22	C₂×D₆₅₂	—	2	1	no	—
23	C₂×C₄×D₁₆₃	—	4	1	no	—
24	Dic₆₅₂	C₂×C₄	1	1	yes	characteristic, complemented
25	#14579688	—	2	1	no	—
26	D₁₃₀₄	—	2	1	no	—
27	D₁₃₀₄	C₂×C₄	1	2	yes	—
28	D₁₃₀₄	C₂×C₄	1	1	yes	characteristic, complemented
29	#14579683	—	2	1	no	—
30	#14579682	—	4	1	no	—
31	C₈×D₁₆₃	—	4	1	no	—
32	C₄⋊Dic₁₆₃	—	2	1	no	—
33	(C₂×D₁₆₃)⋊C₄	—	4	1	no	—
34	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
36	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
37	C₁₆×C₁₆₃	—	2	1	no	—
38	C₁₆×C₁₆₃	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₆₃	—	4	1	no	—
40	Dic₃₂₆	—	2	1	no	—
41	D₆₅₂	—	2	2	no	—
42	D₆₅₂	—	2	1	no	—
43	D₆₅₂	—	4	1	no	—
44	C₂²⋊D₁₆₃	—	4	1	no	—
45	C₂×Dic₁₆₃	—	4	1	no	—
46	C₄×D₁₆₃	—	4	1	no	—
47	C₄×D₁₆₃	—	8	2	no	—
48	C₁₆₃⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₆₃	—	2	1	no	—
50	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₆₃	—	2	1	no	—
52	C₈×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₆₃	#41	1	1	yes	characteristic
54	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₆₃	—	4	1	no	—
56	C₂×D₁₆₃	—	4	2	no	—
57	C₂×D₁₆₃	—	8	1	no	—
58	C₂×D₁₆₃	—	8	1	no	—
59	C₄×C₁₆₃	—	4	1	no	—
60	C₄×C₁₆₃	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₆₃	—	4	1	no	—
63	Dic₁₆₃	—	8	1	no	—
64	C₂×C₁₆₃	—	2	1	no	—
65	C₂×C₁₆₃	#442	1	1	yes	characteristic
66	D₁₆₃	—	8	1	no	—
67	D₁₆₃	—	8	2	no	—
68	C₁₆₃	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	163	1	no	maximal, Sylow
70	#509	—	163	1	no	—
71	#444	—	163	1	no	—
72	#442	—	163	1	no	—
73	#365	—	163	1	no	—
74	#361	—	163	1	no	—
75	#360	—	163	1	no	—
76	#347	D₁₆₃	1	1	yes	characteristic, complemented
77	#135	—	163	1	no	—
78	C₂×D₈	—	163	1	no	—
79	#131	—	326	1	no	—
80	C₄×D₄	—	326	1	no	—
81	SD₃₂	—	163	2	no	—
82	D₁₆	—	163	2	no	—
83	M₃₂	C₂×D₁₆₃	1	1	yes	characteristic
84	M₃₂	C₂×D₁₆₃	1	1	yes	characteristic
85	#108	—	163	1	no	—
86	#106	—	163	1	no	—
87	#104	—	326	1	no	—
88	#102	—	326	1	no	—
89	#97	C₂×D₁₆₃	1	1	yes	characteristic
90	#41	—	326	1	no	—
91	C₂×D₄	—	326	1	no	—
92	C₂²×C₄	—	652	1	no	—
93	Q₁₆	—	163	1	no	—
94	SD₁₆	—	326	1	no	—
95	D₈	—	326	1	no	—
96	D₈	—	163	1	no	—
97	D₈	—	163	2	no	—
98	M₁₆	—	326	1	no	—
99	M₁₆	—	652	1	no	—
100	C₂×C₈	—	652	1	no	—
101	C₂×C₈	D₆₅₂	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
103	C₄⋊C₄	—	326	1	no	—
104	C₂²⋊C₄	—	652	1	no	—
105	C₄²	D₆₅₂	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₁₆₃	1	2	yes	complemented
108	C₂³	—	652	1	no	—
109	Q₈	—	326	1	no	—
110	D₄	—	326	2	no	—
111	D₄	—	326	1	no	—
112	D₄	—	652	1	no	—
113	D₄	—	652	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	1304	2	no	—
116	C₂×C₄	—	652	1	no	—
117	C₂×C₄	—	652	1	no	—
118	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	652	1	no	—
121	C₈	C₂×C₄×D₁₆₃	1	1	yes	characteristic
122	C₈	#14579699	1	1	yes	characteristic
123	C₂²	—	1304	1	no	—
124	C₂²	—	1304	1	no	—
125	C₂²	—	652	1	no	—
126	C₂²	—	652	2	no	—
127	C₂²	#19104763	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	1304	1	no	—
130	C₄	—	652	1	no	—
131	C₄	C₄×D₆₅₂	1	1	yes	characteristic
132	C₄	#19104763	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1304	2	no	—
135	C₂	—	1304	1	no	—
136	C₂	#23781778	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781778
commutator	C₈×C₁₆₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting	#347×C₁₆₃	C₂
socle	C₂×C₁₆₃	#442
prime radical	#23782103	C₂
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	#347	D₁₆₃
hypocenter	C₁₆₃	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #23781788	C₂	semidirect
3. #23781778	C₂	semidirect
4. #23781488	C₂	semidirect
5. #23781479	C₂	semidirect
6. #23781476	C₂	semidirect
7. #347×C₁₆₃	C₂	semidirect
12. #19104680	C₄	semidirect
13. D₂₆₀₈	C₄	semidirect
24. Dic₆₅₂	C₂×C₄	semidirect
28. D₁₃₀₄	C₂×C₄	semidirect
38. C₁₆×C₁₆₃	C₂×C₄	semidirect
68. C₁₆₃	AGL(1,ℤ/16)	semidirect
76. #347	D₁₆₃	semidirect
107. C₁₆	C₄×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782103	2
#23781788	2
#23781778	2
#23781488	2
#23781479	2
#23781476	2
#347×C₁₆₃	2
AGL(1,ℤ/16)	163