Finite Group Databasebeta

Subgroups — #35216403

Order 20864.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#19104678	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781747	C₂	1	1	yes	maximal, characteristic
4	#23781490	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781467	C₂	1	1	yes	maximal, characteristic
6	#23781440	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₃₀₄	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₆₃	—	2	1	no	—
10	#19104697	—	2	1	no	—
11	#19104678	C₄	1	4	yes	complemented
12	C₂×C₁₆₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19104669	C₂²	1	1	yes	characteristic
14	#19104659	—	2	1	no	—
15	#19104655	—	2	1	no	—
16	C₈×Dic₁₆₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
19	#107×C₁₆₃	C₂²	1	1	yes	characteristic
20	C₂×D₆₅₂	—	2	1	no	—
21	C₂×C₄×D₁₆₃	—	4	1	no	—
22	D₁₃₀₄	—	2	1	no	—
23	D₁₃₀₄	C₂×C₄	1	2	yes	—
24	C₂×C₁₆₃⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₁₆₃	—	2	1	no	—
26	C₄⋊Dic₁₆₃	—	4	1	no	—
27	(C₂×D₁₆₃)⋊C₄	—	4	1	no	—
28	C₄×Dic₁₆₃	—	4	1	no	—
29	C₄×Dic₁₆₃	D₄	1	1	yes	characteristic
30	C₁₆₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₁₆₃	—	2	1	no	—
32	Q₁₆×C₁₆₃	—	2	1	no	—
33	Q₁₆×C₁₆₃	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₆₃	—	2	1	no	—
36	C₁₆×C₁₆₃	—	2	1	no	—
37	C₂²×D₁₆₃	—	4	1	no	—
38	D₆₅₂	—	2	2	no	—
39	D₆₅₂	—	4	1	no	—
40	C₂×Dic₁₆₃	—	4	1	no	—
41	C₂×Dic₁₆₃	D₈	1	1	yes	characteristic
42	C₂×Dic₁₆₃	D₈	1	1	yes	characteristic
43	C₄×D₁₆₃	—	8	2	no	—
44	C₁₆₃⋊C₈	—	2	1	no	—
45	Q₈×C₁₆₃	—	2	2	no	—
46	Q₈×C₁₆₃	—	4	1	no	—
47	C₂×C₄×C₁₆₃	—	4	1	no	—
48	C₂×C₄×C₁₆₃	—	4	1	no	—
49	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₆₃	#41	1	1	yes	characteristic
52	C₂²×C₁₆₃	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₆₃	—	4	2	no	—
54	C₂×D₁₆₃	—	8	1	no	—
55	C₂×D₁₆₃	—	8	1	no	—
56	C₄×C₁₆₃	—	4	2	no	—
57	C₄×C₁₆₃	—	8	1	no	—
58	C₄×C₁₆₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₆₃	#135	1	1	yes	characteristic
60	Dic₁₆₃	—	2	1	no	—
61	Dic₁₆₃	—	2	1	no	—
62	Dic₁₆₃	—	8	1	no	—
63	C₂×C₁₆₃	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₆₃	#509	1	1	yes	characteristic
65	C₂×C₁₆₃	#510	1	1	yes	characteristic
66	D₁₆₃	—	8	2	no	—
67	C₁₆₃	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	163	1	no	maximal, Sylow
69	C₂×SD₃₂	—	163	1	no	—
70	C₄×Q₁₆	—	163	1	no	—
71	C₄×D₈	—	163	1	no	—
72	#366	—	163	1	no	—
73	#358	D₁₆₃	1	1	yes	characteristic, complemented
74	#357	—	163	1	no	—
75	#346	—	163	1	no	—
76	C₂×Q₁₆	C₂×D₁₆₃	1	1	yes	characteristic
77	C₂×D₈	—	163	1	no	—
78	C₄×Q₈	—	326	1	no	—
79	C₄×D₄	—	326	1	no	—
80	SD₃₂	—	163	4	no	—
81	C₂×C₁₆	—	163	1	no	—
82	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
83	#107	—	163	1	no	—
84	#107	C₂×D₁₆₃	1	1	yes	characteristic
85	#103	—	326	1	no	—
86	#102	—	326	1	no	—
87	C₄×C₈	—	163	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	326	1	no	—
90	C₂²×C₄	—	652	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₁₆₃	1	2	yes	complemented
93	D₈	—	326	1	no	—
94	D₈	—	163	2	no	—
95	C₂×C₈	—	163	1	no	—
96	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	326	1	no	—
99	C₄⋊C₄	—	652	1	no	—
100	C₂²⋊C₄	—	652	1	no	—
101	C₄²	—	652	1	no	—
102	C₄²	—	163	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	163	2	no	—
105	C₂³	—	652	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	326	2	no	—
109	D₄	—	652	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	1304	2	no	—
113	C₂×C₄	—	652	1	no	—
114	C₂×C₄	—	163	1	no	—
115	C₂×C₄	—	163	1	no	—
116	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
117	C₈	—	326	1	no	—
118	C₈	C₂×C₄×D₁₆₃	1	1	yes	characteristic
119	C₈	#14579698	1	1	yes	characteristic
120	C₂²	—	1304	1	no	—
121	C₂²	—	1304	1	no	—
122	C₂²	—	652	2	no	—
123	C₂²	D₈×D₁₆₃	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	1304	1	no	—
127	C₄	—	326	1	no	—
128	C₄	—	326	1	no	—
129	C₄	(C₂²⋊D₁₆₃)⋊C₄	1	1	yes	characteristic
130	C₄	#19104756	1	1	yes	characteristic
131	C₂	—	1304	2	no	—
132	C₂	#23781745	1	1	yes	characteristic
133	C₂	#23782102	1	1	yes	characteristic
134	C₂	#23782107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₆₃
commutator	C₈×C₁₆₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting	#358×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂×D₈
prime radical	C₂×D₁₃₀₄	C₂²
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	#358	D₁₆₃
hypocenter	C₁₆₃	#2175
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₆₃	C₂	semidirect
4. #23781490	C₂	semidirect
6. #23781440	C₂	semidirect
7. #358×C₁₆₃	C₂	semidirect
11. #19104678	C₄	semidirect
30. C₁₆₃⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₁₆₃	C₂×C₄	semidirect
67. C₁₆₃	#2175	semidirect
73. #358	D₁₆₃	semidirect
92. Q₁₆	C₄×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104678	2
Q₁₆×Dic₁₆₃	2
#23781747	2
#23781490	2
#23781467	2
#23781440	2
#358×C₁₆₃	2
#2175	163