Finite Group Databasebeta

← #35214485

Subgroups — #35214485

Order 20864.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782501C211yesmaximal, characteristic
2#23781647C212yesmaximal, complemented
3#23781633C212yesmaximal, complemented
4#23781632C211yesmaximal, characteristic, complemented
5#413×C163C211yesmaximal, characteristic, complemented
6#19104783—21no—
7#19104782—22no—
8#19104781—22no—
9#19104780—21no—
10#19104780C414yes—
11C2×#14579699—22no—
12C2×#14579699C2212yes—
13#41×D163—24no—
14C2×#14579683C2211yescharacteristic, complemented
15C22×C163⋊C8C2211yescharacteristic, complemented
16#19104653—22no—
17#19104651—22no—
18C22⋊(C163⋊C8)—22no—
19(C2×D163)⋊C8—24no—
20C2×#41×C163C2211yescharacteristic
21C2×M16×C163C2212yescomplemented
22#101×C163—21no—
23#100×C163—21no—
24#14579700—24no—
25#14579699—24no—
26#14579699—24no—
27#14579699—24no—
28#14579699—22no—
29#14579699—22no—
30#14579699—22no—
31#14579699—22no—
32#14579698—24no—
33#14579698—28no—
34C2×Dic326—22no—
35C2×Dic326D412yes—
36Q8×D163—24no—
37C2×D652—22no—
38C2×D652D412yes—
39C2×C22⋊D163—24no—
40C2×C22⋊D163C2×C414yes—
41D4×D163—24no—
42D4×D163—28no—
43C2×C4×D163—24no—
44C2×C4×D163D414yes—
45#14579683—22no—
46C2×C163⋊C8—22no—
47C2×C163⋊C8—21no—
48C2×C163⋊C8—21no—
49#41×C163—22no—
50#41×C163—22no—
51C2×Q8×C163D411yescharacteristic
52C2×D4×C163C2×C412yes—
53C2×D4×C163D411yescharacteristic
54C22×C4×C163D412yes—
55C22×C4×C163C2311yescharacteristic
56M16×C163—22no—
57M16×C163—22no—
58C2×C8×C163—22no—
59C22×D163—24no—
60C22×D163C22⋊C414yes—
61Dic326—24no—
62Dic326—22no—
63Dic326—22no—
64D652—24no—
65D652—22no—
66D652—22no—
67C22⋊D163—28no—
68C22⋊D163—24no—
69C22⋊D163—24no—
70C2×Dic163—24no—
71C2×Dic163C22⋊C414yes—
72C4×D163—28no—
73C4×D163—24no—
74C4×D163—24no—
75C163⋊C8—41no—
76C163⋊C8—41no—
77C23×C163C22⋊C412yes—
78C23×C163C22×C411yescharacteristic
79Q8×C163—22no—
80D4×C163—24no—
81D4×C163—22no—
82C2×C4×C163—22no—
83C2×C4×C163—22no—
84C2×C4×C163C22⋊C412yes—
85C2×C4×C163C2×D412yes—
86C2×C4×C163C2×D412yes—
87C2×C4×C163C2×D411yescharacteristic
88C2×C4×C163C2×D411yescharacteristic
89C8×C163—42no—
90C22×C163—22no—
91C22×C163—41no—
92C22×C163—42no—
93C22×C163C2×C22⋊C412yes—
94C22×C163C2×C22⋊C411yescharacteristic
95C2×D163—24no—
96C2×D163—44no—
97C2×D163—84no—
98C4×C163—22no—
99C4×C163#12012yes—
100C4×C163#12011yescharacteristic
101C4×C163#12011yescharacteristic
102Dic163—24no—
103C2×C163—22no—
104C2×C163—21no—
105C2×C163—42no—
106C2×C163#37911yescharacteristic
107D163—44no—
108C163#179111yescharacteristic, complemented, Sylow
109#1791—1631nomaximal, Sylow
110#585—1631no—
111#413—1632no—
112#413D16311yescharacteristic, complemented
113#408—1631no—
114#408—1632no—
1152−1+4—3262no—
1162−1+4—3261no—
117H(5,𝔽2)—3262no—
118H(5,𝔽2)—3261no—
119H(5,𝔽2)—1634no—
120C2×#41—3264no—
121C2×#41—3262no—
122C2×#41—1632no—
123C2×#41C2×D16311yescharacteristic
124C2×M16—1631no—
125C2×M16C2×D16312yescomplemented
126C22×C8—1631no—
127#101—21no—
128#101—3262no—
129#100—21no—
130#100—3262no—
131C22⋊C8—3264no—
132C22⋊C8—3262no—
133#41—22no—
134#41—22no—
135#41—3264no—
136#41—3264no—
137#41—3264no—
138#41—3264no—
139#41—3268no—
140#41—3264no—
141#41—3262no—
142#41—3262no—
143#41—3262no—
144#41—3262no—
145C2×Q8—3262no—
146C2×Q8—3264no—
147C2×Q8—1632no—
148C2×Q8C22⋊D16311yescharacteristic
149C2×D4—3262no—
150C2×D4—3264no—
151C2×D4—3268no—
152C2×D4—3264no—
153C2×D4—1634no—
154C2×D4—1632no—
155C2×D4C4×D16312yes—
156C2×D4C22⋊D16311yescharacteristic
157C22×C4—3264no—
158C22×C4—1634no—
159C22×C4D65212yes—
160C22×C4C22×D16311yescharacteristic
161M16—22no—
162M16—22no—
163M16—3262no—
164C2×C8—22no—
165C2×C8—3262no—
166C2×C8—3261no—
167C2×C8—3261no—
168C23—3264no—
169C23—1634no—
170C23(C2×D163)⋊C412yes—
171C23C2×C4×D16311yescharacteristic
172Q8—22no—
173Q8—3264no—
174Q8—3262no—
175Q8—3262no—
176D4—24no—
177D4—22no—
178D4—3264no—
179D4—3268no—
180D4—3264no—
181D4—3264no—
182D4—3262no—
183D4—3262no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3268no—
187C2×C4—3264no—
188C2×C4—3264no—
189C2×C4—3264no—
190C2×C4—1634no—
191C2×C4(C2×D163)⋊C412yes—
192C2×C4D4×D16312yes—
193C2×C4D4×D16312yes—
194C2×C4C2×C22⋊D16311yescharacteristic
195C2×C4C2×D65211yescharacteristic
196C8—42no—
197C8—6521no—
198C8—6521no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—13044no—
203C22—3264no—
204C22—6524no—
205C22C22⋊C4×D16312yes—
206C22C2×(C2×D163)⋊C411yescharacteristic
207C4—22no—
208C4—3264no—
209C4#1910470312yes—
210C4#1910470511yescharacteristic
211C4#1910470511yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—6524no—
216C2#2378152011yescharacteristic
subgroupG/H
centerC4#19104705
commutatorC23×C163C22×C4
FrattiniC22×C4C22×D163
Fitting#413×C163C2
socleC2×C163#379
prime radical#23782501C2
cosocleC22×C4×C163C23
hypercenter#413D163
hypocenterC163#1791
LCSG ▷ C23×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #23781647C2semidirect
3. #23781633C2semidirect
4. #23781632C2semidirect
5. #413×C163C2semidirect
14. C2×#14579683C22semidirect
15. C22×C163⋊C8C22semidirect
21. C2×M16×C163C22semidirect
108. C163#1791semidirect
112. #413D163semidirect
125. C2×M16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237825012
#237816472
#237816332
#237816322
#413×C1632
#1791163