Finite Group Databasebeta

← #36354627

Subgroups — #36354627

Order 22144.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#26011998C212yesmaximal, complemented
2#26011508C212yesmaximal, complemented
3#26011454C211yesmaximal, characteristic, complemented
4#26011248C211yesmaximal, characteristic, complemented
5#575×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C2211yescharacteristic
7C2×#14583810C2211yescharacteristic
8#19149660—22no—
9#19149655—22no—
10#19149650—22no—
11#19149647—22no—
12C4×C22⋊D173—24no—
13#19149638C2212yes—
14#19149606—24no—
15#19149591—22no—
16C2×#41×C173C2212yes—
17#137×C173—22no—
18AGL(1,ℤ/8)×C173—22no—
19#135×C173—22no—
20C2×M16×C173C2211yescharacteristic
21C23×D173D411yescharacteristic
22C2×C22⋊D173—24no—
23C22×Dic173D411yescharacteristic
24C2×C4×D173—21no—
25C2×C4×D173—22no—
26C2×C4×D173—22no—
27C2×C4×D173—21no—
28#14583810—22no—
29C2×#13063436—21no—
30C4⋊Dic173—22no—
31#14583805—24no—
32C22⋊Dic173—24no—
33C22⋊Dic173—22no—
34#14583803—24no—
35C4×Dic173D414yescomplemented
36#41×C173—22no—
37#41×C173—22no—
38#41×C173—22no—
39#41×C173—22no—
40C2×Q8×C173D412yescomplemented
41C2×D4×C173—22no—
42C2×D4×C173D412yescomplemented
43C22×C4×C173—22no—
44C22×C4×C173C2311yescharacteristic
45Q16×C173—22no—
46SD16×C173—24no—
47D8×C173—22no—
48M16×C173—22no—
49C2×C8×C173D412yescomplemented
50C22×D173—22no—
51C22×D173—21no—
52C22×D173—42no—
53C22⋊D173—48no—
54C2×Dic173—22no—
55C2×Dic173—24no—
56C2×Dic173—21no—
57C4×D173—41no—
58C4×D173—42no—
59C4×D173—41no—
60#13063436—41no—
61C23×C173—22no—
62C23×C173C2×D411yescharacteristic
63Q8×C173—22no—
64Q8×C173—22no—
65D4×C173—22no—
66D4×C173—22no—
67D4×C173—42no—
68D4×C173—42no—
69C2×C4×C173—22no—
70C2×C4×C173—42no—
71C2×C4×C173—42no—
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D412yes—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173#4111yescharacteristic
76C8×C173—22no—
77C22×C173—41no—
78C22×C173—44no—
79C22×C173—42no—
80C22×C173#12012yes—
81C22×C173#12111yescharacteristic
82C2×D173—24no—
83C2×D173—22no—
84C2×D173—42no—
85C2×D173—41no—
86C2×D173—41no—
87C4×C173—42no—
88C4×C173#12011yescharacteristic
89C4×C173#12112yes—
90C4×C173#12711yescharacteristic
91Dic173—44no—
92Dic173—41no—
93C2×C173—22no—
94C2×C173—21no—
95C2×C173—82no—
96C2×C173#39211yescharacteristic
97D173—42no—
98C173#201011yescharacteristic, complemented, Sylow
99#2010—1731nomaximal, Sylow
100#575D17311yescharacteristic, complemented
101#525—1732no—
102#421—1732no—
103#407—1731no—
104H(3,ℤ/4)—1731no—
105C2×#41C2×D17312yes—
106C23×C4—1731no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—1731no—
111C2×M16C2×D17311yescharacteristic
112#123—3462no—
113C22⋊Q8—3462no—
114#121—3462no—
115#120—3462no—
116C4×D4—3464no—
117#117—1732no—
118#104—3464no—
119C22⋊C8—3462no—
120C24—1731no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D17312yescomplemented
126C2×D4—22no—
127C2×D4—3464no—
128C2×D4C22⋊D17312yescomplemented
129C22×C4—22no—
130C22×C4—3462no—
131C22×C4—3461no—
132C22×C4—3462no—
133C22×C4—3461no—
134C22×C4—1731no—
135C22×C4C22×D17311yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—3462no—
141C2×C8—3461no—
142C2×C8C22⋊D17312yescomplemented
143C4⋊C4—3464no—
144C4⋊C4—3462no—
145C22⋊C4—3464no—
146C22⋊C4—3464no—
147C22⋊C4—3462no—
148C42—1734no—
149C23—22no—
150C23—3462no—
151C23—3461no—
152C23—6922no—
153C23D4×D17311yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—6928no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—3464no—
165C2×C4—3462no—
166C2×C4—3461no—
167C2×C4—6921no—
168C2×C4—6922no—
169C2×C4—6921no—
170C2×C4D4×D17312yes—
171C2×C4C2×C22⋊D17312yes—
172C2×C4C2×C22⋊D17311yescharacteristic
173C2×C4#1458382511yescharacteristic
174C8—22no—
175C8—6921no—
176C22—41no—
177C22—44no—
178C22—42no—
179C22—3464no—
180C22—3462no—
181C22—6922no—
182C22—6921no—
183C22—6921no—
184C22#1914964712yes—
185C22#1914965011yescharacteristic
186C4—42no—
187C4—6924no—
188C4—6921no—
189C4#1914964811yescharacteristic
190C4#1914965112yes—
191C4#1914967511yescharacteristic
192C2—22no—
193C2—21no—
194C2—82no—
195C2—6922no—
196C2#2601139911yescharacteristic
subgroupG/H
centerC4#19149675
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#575×C173C2
socleC2×C173#392
hypercenter#575D173
hypocenterC173#2010
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #26011998C2semidirect
2. #26011508C2semidirect
3. #26011454C2semidirect
4. #26011248C2semidirect
5. #575×C173C2semidirect
35. C4×Dic173D4semidirect
40. C2×Q8×C173D4semidirect
42. C2×D4×C173D4semidirect
49. C2×C8×C173D4semidirect
98. C173#2010semidirect
100. #575D173semidirect
125. C2×Q8C22⋊D173semidirect
128. C2×D4C22⋊D173semidirect
142. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
#260115082
#260114542
#260112482
#575×C1732
#2010173