Finite Group Databasebeta

← #36354666

Subgroups — #36354666

Order 22144.

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

normal subgroups only

This group has 350 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#26012289C212yesmaximal, complemented
2#26011459C211yesmaximal, characteristic, complemented
3#26011458C212yesmaximal, complemented
4#26011249C211yesmaximal, characteristic, complemented
5#575×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7#19149710—22no—
8#19149709—22no—
9#19149705—22no—
10#19149704—22no—
11#19149699—24no—
12C2×#14583810C2212yes—
13C22×#13063436C2211yescharacteristic
14C22⋊#13063436—24no—
15#19149591—22no—
16C2×#41×C173C2212yes—
17#137×C173—22no—
18AGL(1,ℤ/8)×C173—22no—
19#135×C173—22no—
20C2×M16×C173C2211yescharacteristic
21#14583826—22no—
22#14583826—22no—
23C2×Dic346D411yescharacteristic
24C2×D692D411yescharacteristic
25C2×C22⋊D173—21no—
26C2×C4×D173—21no—
27#14583816—24no—
28#14583814—24no—
29#14583813—24no—
30#14583811—24no—
31#14583810—22no—
32#14583810—22no—
33C2×#13063436—21no—
34C2×#13063436—22no—
35C2×#13063436D414yes—
36#41×C173—22no—
37#41×C173—22no—
38#41×C173—22no—
39#41×C173—22no—
40C2×Q8×C173D412yes—
41C2×D4×C173—22no—
42C2×D4×C173D412yes—
43C22×C4×C173—22no—
44C22×C4×C173C2311yescharacteristic
45Q16×C173—22no—
46SD16×C173—24no—
47D8×C173—22no—
48M16×C173—22no—
49C2×C8×C173D412yes—
50C22×D173—21no—
51Dic346—22no—
52D692—22no—
53C22⋊D173—42no—
54C2×Dic173—21no—
55C4×D173—41no—
56C4×D173—41no—
57#13063436—24no—
58#13063436—41no—
59C23×C173—22no—
60C23×C173#4111yescharacteristic
61Q8×C173—22no—
62Q8×C173—22no—
63D4×C173—22no—
64D4×C173—22no—
65D4×C173—42no—
66D4×C173—42no—
67C2×C4×C173—22no—
68C2×C4×C173—42no—
69C2×C4×C173—42no—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D412yes—
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C8×C173—22no—
75C22×C173—41no—
76C22×C173—44no—
77C22×C173—42no—
78C22×C173#12111yescharacteristic
79C22×C173#12112yes—
80C2×D173—42no—
81C2×D173—41no—
82C4×C173—42no—
83C4×C173#12012yes—
84C4×C173#12011yescharacteristic
85C4×C173#12711yescharacteristic
86Dic173—41no—
87C2×C173—21no—
88C2×C173—22no—
89C2×C173—82no—
90C2×C173#39211yescharacteristic
91D173—81no—
92C173#201711yescharacteristic, complemented, Sylow
93#2017—1731nomaximal, Sylow
94#575—1732no—
95#575D17311yescharacteristic, complemented
96#408—1732no—
97#408—1731no—
98#338—1731no—
99C2×#41—1731no—
100C2×#41C2×D17312yes—
101#137—22no—
102#137—3462no—
103#137—3462no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—3462no—
106AGL(1,ℤ/8)—3462no—
107#135—22no—
108#135—3464no—
109C2×M16—1732no—
110C2×M16C2×D17311yescharacteristic
111C22×C8—1731no—
112C22⋊C8—3464no—
113C22⋊C8—3462no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—3462no—
119#41—3462no—
120C2×Q8—1731no—
121C2×Q8C22⋊D17312yes—
122C2×D4—22no—
123C2×D4—3461no—
124C2×D4—1731no—
125C2×D4C22⋊D17312yes—
126C22×C4—22no—
127C22×C4—3461no—
128C22×C4C22×D17311yescharacteristic
129Q16—22no—
130Q16—3464no—
131SD16—24no—
132SD16—3464no—
133SD16—3464no—
134D8—22no—
135D8—3464no—
136M16—22no—
137M16—3462no—
138M16—3462no—
139C2×C8—3462no—
140C2×C8—3461no—
141C2×C8—1734no—
142C2×C8C22⋊D17312yes—
143C23—22no—
144C23—3461no—
145C23#1458382511yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—3462no—
149D4—22no—
150D4—22no—
151D4—42no—
152D4—42no—
153D4—3462no—
154D4—6922no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—3461no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4D4×D17311yescharacteristic
162C2×C4D4×D17312yes—
163C2×C4C2×C22⋊D17312yes—
164C2×C4C2×C22⋊D17311yescharacteristic
165C8—22no—
166C8—3464no—
167C8—6921no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—6922no—
172C22—6921no—
173C22#1914965011yescharacteristic
174C22#1914965112yes—
175C4—42no—
176C4—6921no—
177C4#1914964712yes—
178C4#1914964811yescharacteristic
179C4#1914967511yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—13841no—
184C2#2601139911yescharacteristic
subgroupG/H
centerC4#19149675
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#575×C173C2
socleC2×C173#392
hypercenter#575D173
hypocenterC173#2017
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. #26012289C2semidirect
2. #26011459C2semidirect
3. #26011458C2semidirect
4. #26011249C2semidirect
5. #575×C173C2semidirect
92. C173#2017semidirect
95. #575D173semidirect

Maximal subgroups

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

subgroupindex
#260122892
#260114592
#260114582
#260112492
#575×C1732
#2017173