Finite Group Databasebeta

← #36354667

Subgroups — #36354667

Order 22144.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012291C211yesmaximal, characteristic, complemented
2#26012289C212yesmaximal, complemented
3#26011457C212yesmaximal, complemented
4#26011249C211yesmaximal, characteristic, complemented
5#408×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7#19149712—22no—
8#19149710—22no—
9#19149709—22no—
10#19149707—22no—
11#19149705—22no—
12#19149704—22no—
13#19149701—22no—
14#19149699—24no—
15C2×#14583810C2212yes—
16#19149591—24no—
17C2×#41×C173C2211yescharacteristic
18C2×M16×C173C2211yescharacteristic
19C22×C8×C173C2211yescharacteristic
20C22⋊C8×C173—22no—
21#14583826—22no—
22#14583826—22no—
23#14583826—22no—
24#14583826—22no—
25C2×Dic346D412yes—
26C2×D692D412yes—
27C2×C22⋊D173—22no—
28C2×C4×D173—22no—
29Dic692—22no—
30#14583816—24no—
31#14583815—24no—
32#14583814—24no—
33#14583813—24no—
34D1384—22no—
35#14583811—24no—
36#14583810—22no—
37#14583810—22no—
38C2×#13063436D414yes—
39#41×C173—22no—
40#41×C173—22no—
41C2×Q8×C173D411yescharacteristic
42C2×D4×C173—21no—
43C2×D4×C173D411yescharacteristic
44C22×C4×C173—21no—
45C22×C4×C173C2311yescharacteristic
46M16×C173—22no—
47C2×C8×C173—22no—
48C2×C8×C173—21no—
49C2×C8×C173D412yes—
50C22×D173—22no—
51Dic346—22no—
52Dic346—22no—
53D692—22no—
54D692—22no—
55C22⋊D173—42no—
56C22⋊D173—42no—
57C2×Dic173—22no—
58C4×D173—42no—
59C4×D173—42no—
60#13063436—24no—
61C23×C173—21no—
62C23×C173#4111yescharacteristic
63Q8×C173—22no—
64D4×C173—22no—
65D4×C173—42no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173—41no—
69C2×C4×C173C2×D412yes—
70C2×C4×C173C2×D412yes—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—22no—
74C8×C173—41no—
75C22×C173—41no—
76C22×C173—42no—
77C22×C173—41no—
78C22×C173#12112yes—
79C22×C173#12111yescharacteristic
80C2×D173—44no—
81C2×D173—42no—
82C4×C173—41no—
83C4×C173#12011yescharacteristic
84C4×C173#12012yes—
85C4×C173#12711yescharacteristic
86Dic173—42no—
87C2×C173—22no—
88C2×C173—21no—
89C2×C173—81no—
90C2×C173#39211yescharacteristic
91D173—82no—
92C173#201711yescharacteristic, complemented, Sylow
93#2017—1731nomaximal, Sylow
94#575—1732no—
95#575—1731no—
96#408—1732no—
97#408D17311yescharacteristic, complemented
98#338—1731no—
99C2×#41—1732no—
100C2×#41C2×D17311yescharacteristic
101#137—3462no—
102#137—3462no—
103#137—3462no—
104AGL(1,ℤ/8)—3462no—
105AGL(1,ℤ/8)—3462no—
106AGL(1,ℤ/8)—3462no—
107#135—3464no—
108#135—3462no—
109C2×M16—1732no—
110C2×M16C2×D17311yescharacteristic
111C22×C8C2×D17311yescharacteristic
112C22⋊C8—22no—
113C22⋊C8—3464no—
114#41—22no—
115#41—22no—
116#41—3462no—
117#41—3462no—
118#41—3462no—
119#41—3462no—
120C2×Q8—1732no—
121C2×Q8D69211yescharacteristic
122C2×D4—21no—
123C2×D4—3462no—
124C2×D4—1732no—
125C2×D4D69211yescharacteristic
126C22×C4—21no—
127C22×C4—3462no—
128C22×C4C22×D17311yescharacteristic
129Q16—3462no—
130Q16—3464no—
131SD16—3464no—
132SD16—3464no—
133SD16—3464no—
134D8—3462no—
135D8—3464no—
136M16—22no—
137M16—3462no—
138M16—3462no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—1734no—
142C2×C8C22⋊D17312yes—
143C23—21no—
144C23—3462no—
145C23#1458382611yescharacteristic
146Q8—22no—
147Q8—3462no—
148Q8—3462no—
149D4—22no—
150D4—42no—
151D4—3462no—
152D4—3462no—
153D4—6922no—
154D4—6922no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—3462no—
159C2×C4—6922no—
160C2×C4—6922no—
161C2×C4D4×D17312yes—
162C2×C4D4×D17312yes—
163C2×C4C2×C22⋊D17311yescharacteristic
164C2×C4C2×D69211yescharacteristic
165C8—22no—
166C8—41no—
167C8—3464no—
168C22—41no—
169C22—42no—
170C22—41no—
171C22—6924no—
172C22—6922no—
173C22#1914965212yes—
174C22#1914965311yescharacteristic
175C4—41no—
176C4—6922no—
177C4#1914964711yescharacteristic
178C4#1914964912yes—
179C4#1914967511yescharacteristic
180C2—21no—
181C2—22no—
182C2—81no—
183C2—13842no—
184C2#2601140011yescharacteristic
subgroupG/H
centerC4#19149675
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#408×C173C2
socleC2×C173#392
hypercenter#408D173
hypocenterC173#2017
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #26012291C2semidirect
2. #26012289C2semidirect
3. #26011457C2semidirect
4. #26011249C2semidirect
5. #408×C173C2semidirect
92. C173#2017semidirect
97. #408D173semidirect

Maximal subgroups

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

subgroupindex
#260122912
#260122892
#260114572
#260112492
#408×C1732
#2017173