Finite Group Databasebeta

← #36353512

Subgroups — #36353512

Order 22144.

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#26012327C211yesmaximal, characteristic
2#26011473C212yesmaximal, complemented
3#26011459C212yesmaximal, complemented
4#26011458C211yesmaximal, characteristic, complemented
5#413×C173C211yesmaximal, characteristic, complemented
6#19149727—21no—
7#19149726—22no—
8#19149725—22no—
9#19149724—21no—
10#19149724C414yes—
11C2×#14583826—22no—
12C2×#14583826C2212yes—
13#41×D173—24no—
14C2×#14583810C2211yescharacteristic, complemented
15C22×#13063436C2211yescharacteristic, complemented
16#19149597—22no—
17#19149595—22no—
18C22⋊#13063436—22no—
19#19149591—24no—
20C2×#41×C173C2211yescharacteristic
21C2×M16×C173C2212yescomplemented
22#101×C173—21no—
23#100×C173—21no—
24#14583827—24no—
25#14583826—24no—
26#14583826—24no—
27#14583826—24no—
28#14583826—22no—
29#14583826—22no—
30#14583826—22no—
31#14583826—22no—
32#14583825—24no—
33#14583825—28no—
34C2×Dic346—22no—
35C2×Dic346D412yes—
36Q8×D173—24no—
37C2×D692—22no—
38C2×D692D412yes—
39C2×C22⋊D173—24no—
40C2×C22⋊D173C2×C414yes—
41D4×D173—24no—
42D4×D173—28no—
43C2×C4×D173—24no—
44C2×C4×D173D414yes—
45#14583810—22no—
46C2×#13063436—22no—
47C2×#13063436—21no—
48C2×#13063436—21no—
49#41×C173—22no—
50#41×C173—22no—
51C2×Q8×C173D411yescharacteristic
52C2×D4×C173C2×C412yes—
53C2×D4×C173D411yescharacteristic
54C22×C4×C173D412yes—
55C22×C4×C173C2311yescharacteristic
56M16×C173—22no—
57M16×C173—22no—
58C2×C8×C173—22no—
59C22×D173—24no—
60C22×D173C22⋊C414yes—
61Dic346—24no—
62Dic346—22no—
63Dic346—22no—
64D692—24no—
65D692—22no—
66D692—22no—
67C22⋊D173—28no—
68C22⋊D173—24no—
69C22⋊D173—24no—
70C2×Dic173—24no—
71C2×Dic173C22⋊C414yes—
72C4×D173—28no—
73C4×D173—24no—
74C4×D173—24no—
75#13063436—41no—
76#13063436—41no—
77C23×C173C22⋊C412yes—
78C23×C173C22×C411yescharacteristic
79Q8×C173—22no—
80D4×C173—24no—
81D4×C173—22no—
82C2×C4×C173—22no—
83C2×C4×C173—22no—
84C2×C4×C173C22⋊C412yes—
85C2×C4×C173C2×D412yes—
86C2×C4×C173C2×D412yes—
87C2×C4×C173C2×D411yescharacteristic
88C2×C4×C173C2×D411yescharacteristic
89C8×C173—42no—
90C22×C173—22no—
91C22×C173—41no—
92C22×C173—42no—
93C22×C173C2×C22⋊C412yes—
94C22×C173C2×C22⋊C411yescharacteristic
95C2×D173—24no—
96C2×D173—44no—
97C2×D173—84no—
98C4×C173—22no—
99C4×C173#12012yes—
100C4×C173#12011yescharacteristic
101C4×C173#12011yescharacteristic
102Dic173—24no—
103C2×C173—22no—
104C2×C173—21no—
105C2×C173—42no—
106C2×C173#37911yescharacteristic
107D173—44no—
108C173#179111yescharacteristic, complemented, Sylow
109#1791—1731nomaximal, Sylow
110#585—1731no—
111#413—1732no—
112#413D17311yescharacteristic, complemented
113#408—1731no—
114#408—1732no—
1152−1+4—3462no—
1162−1+4—3461no—
117H(5,𝔽2)—3462no—
118H(5,𝔽2)—3461no—
119H(5,𝔽2)—1734no—
120C2×#41—3464no—
121C2×#41—3462no—
122C2×#41—1732no—
123C2×#41C2×D17311yescharacteristic
124C2×M16—1731no—
125C2×M16C2×D17312yescomplemented
126C22×C8—1731no—
127#101—21no—
128#101—3462no—
129#100—21no—
130#100—3462no—
131C22⋊C8—3464no—
132C22⋊C8—3462no—
133#41—22no—
134#41—22no—
135#41—3464no—
136#41—3464no—
137#41—3464no—
138#41—3464no—
139#41—3468no—
140#41—3462no—
141#41—3462no—
142#41—3462no—
143#41—3462no—
144#41—3464no—
145C2×Q8—3462no—
146C2×Q8—3464no—
147C2×Q8—1732no—
148C2×Q8C22⋊D17311yescharacteristic
149C2×D4—3462no—
150C2×D4—3464no—
151C2×D4—3468no—
152C2×D4—3464no—
153C2×D4—1734no—
154C2×D4—1732no—
155C2×D4C4×D17312yes—
156C2×D4C22⋊D17311yescharacteristic
157C22×C4—3464no—
158C22×C4—1734no—
159C22×C4D69212yes—
160C22×C4C22×D17311yescharacteristic
161M16—22no—
162M16—22no—
163M16—3462no—
164C2×C8—22no—
165C2×C8—3462no—
166C2×C8—3461no—
167C2×C8—3461no—
168C23—3464no—
169C23—1734no—
170C23#1458380312yes—
171C23C2×C4×D17311yescharacteristic
172Q8—22no—
173Q8—3464no—
174Q8—3462no—
175Q8—3462no—
176D4—24no—
177D4—22no—
178D4—3464no—
179D4—3468no—
180D4—3464no—
181D4—3464no—
182D4—3462no—
183D4—3462no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3468no—
187C2×C4—3464no—
188C2×C4—3464no—
189C2×C4—3464no—
190C2×C4—1734no—
191C2×C4#1458380312yes—
192C2×C4D4×D17312yes—
193C2×C4D4×D17312yes—
194C2×C4C2×C22⋊D17311yescharacteristic
195C2×C4C2×D69211yescharacteristic
196C8—42no—
197C8—6921no—
198C8—6921no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—3464no—
203C22—13844no—
204C22—6924no—
205C22C22⋊C4×D17312yes—
206C22C2×#1458380311yescharacteristic
207C4—22no—
208C4—3464no—
209C4#1914964712yes—
210C4#1914964911yescharacteristic
211C4#1914964911yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—6924no—
216C2#2601134611yescharacteristic
subgroupG/H
centerC4#19149649
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#413×C173C2
socleC2×C173#379
prime radical#26012327C2
cosocleC22×C4×C173C23
hypercenter#413D173
hypocenterC173#1791
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #26011473C2semidirect
3. #26011459C2semidirect
4. #26011458C2semidirect
5. #413×C173C2semidirect
14. C2×#14583810C22semidirect
15. C22×#13063436C22semidirect
21. C2×M16×C173C22semidirect
108. C173#1791semidirect
112. #413D173semidirect
125. C2×M16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260114732
#260114592
#260114582
#413×C1732
#1791173