Finite Group Databasebeta

← #36353994

Subgroups — #36353994

Order 22144.

360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#14583810C211yesmaximal, characteristic
2C2×#19149658C211yesmaximal, characteristic
3C2×#19149602C212yesmaximal, complemented
4#26011253C212yesmaximal, complemented
5#380×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7C2×#14583810—22no—
8C2×#14583810—24no—
9C22×#13063436C2212yes—
10#19149658—22no—
11#19149658C414yescomplemented
12C2×C4⋊Dic173C2211yescharacteristic
13C2×#14583805—21no—
14C2×C22⋊Dic173—21no—
15#19149602—24no—
16C23×C4×C173C2211yescharacteristic
17C2×C4⋊C4×C173C2212yes—
18C2×C22⋊C4×C173—21no—
19#95×C173—21no—
20C2×Dic346—21no—
21C2×Dic346—22no—
22C2×Dic346—22no—
23C2×Dic346C2×C412yes—
24C22×Dic173—21no—
25C22×Dic173—21no—
26#14583810—28no—
27C2×#13063436—22no—
28C2×#13063436—22no—
29C2×#13063436D414yes—
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C412yes—
32#14583805—24no—
33C22⋊Dic173—24no—
34C24×C173D411yescharacteristic
35C22×C4×C173—22no—
36C22×C4×C173—22no—
37C22×C4×C173—22no—
38C22×C4×C173C2×C412yes—
39C22×C4×C173D411yescharacteristic
40C22×C4×C173D412yes—
41C22×C4×C173C2311yescharacteristic
42C4⋊C4×C173—22no—
43C4⋊C4×C173—22no—
44C22⋊C4×C173—42no—
45Dic346—24no—
46Dic346—41no—
47Dic346—41no—
48C2×Dic173—22no—
49C2×Dic173—22no—
50C2×Dic173—42no—
51C2×Dic173—41no—
52C2×Dic173—41no—
53#13063436—24no—
54C23×C173—22no—
55C23×C173—21no—
56C23×C173—22no—
57C23×C173—21no—
58C23×C173C22⋊C412yes—
59C23×C173C2×D411yescharacteristic
60C2×C4×C173—22no—
61C2×C4×C173—24no—
62C2×C4×C173—22no—
63C2×C4×C173—22no—
64C2×C4×C173—42no—
65C2×C4×C173—42no—
66C2×C4×C173—42no—
67C2×C4×C173C22⋊C412yes—
68C2×C4×C173C22×C411yescharacteristic
69C2×C4×C173C2×D412yes—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173#4112yes—
72C22×C173—22no—
73C22×C173—22no—
74C22×C173—22no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173C2×C22⋊C411yescharacteristic
78C22×C173#12112yes—
79C22×C173AGL(1,ℤ/8)11yescharacteristic
80C22×C173#13711yescharacteristic
81C22×C173#13712yes—
82C4×C173—22no—
83C4×C173—82no—
84C4×C173C4×D412yes—
85C4×C173#12011yescharacteristic
86C4×C173#12311yescharacteristic
87Dic173—42no—
88Dic173—42no—
89C2×C173—42no—
90C2×C173#38611yescharacteristic
91C2×C173#41711yescharacteristic
92C2×C173#41911yescharacteristic
93C2×C173#46812yes—
94C2×C173#47012yes—
95C173#189411yescharacteristic, complemented, Sylow
96#1894—1731nomaximal, Sylow
97C22×M16—1731no—
98C2×C22⋊Q8—1731no—
99C2×#103—1732no—
100#380D17311yescharacteristic
101#340—1732no—
102C22×Q8—1731no—
103C23×C4C2×D17311yescharacteristic
104C2×M16—3462no—
105C2×M16—3464no—
106C22×C8—1732no—
107C22⋊Q8—3462no—
108C22⋊Q8—1734no—
109C2×C4⋊C4—3461no—
110C2×C4⋊C4—1731no—
111C2×C4⋊C4C2×D17312yes—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—3461no—
114#103—3464no—
115#95—21no—
116C24C22⋊D17311yescharacteristic
117C2×Q8—3461no—
118C2×Q8—3462no—
119C2×Q8—3462no—
120C2×Q8—1732no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—3461no—
125C22×C4—3461no—
126C22×C4C4×D17312yes—
127C22×C4C22⋊D17311yescharacteristic
128C22×C4D69212yes—
129C22×C4C22×D17311yescharacteristic
130M16—3468no—
131C2×C8—3462no—
132C2×C8—3462no—
133C2×C8—1734no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—3461no—
137C4⋊C4—3464no—
138C4⋊C4—1732no—
139C22⋊C4—42no—
140C22⋊C4—3464no—
141C23—22no—
142C23—21no—
143C23—22no—
144C23—21no—
145C23#1458380312yes—
146C23C2×C22⋊D17311yescharacteristic
147Q8—3464no—
148Q8—6921no—
149Q8—6921no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—6922no—
160C2×C4—6921no—
161C2×C4—6921no—
162C2×C4#1458380312yes—
163C2×C4C2×C4×D17311yescharacteristic
164C2×C4D4×D17312yes—
165C2×C4C2×D69211yescharacteristic
166C2×C4#1458382512yes—
167C8—3464no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×#1458380311yescharacteristic
174C22#1914965112yes—
175C22#1914970511yescharacteristic
176C22#1914970912yes—
177C22#1914971011yescharacteristic
178C4—22no—
179C4—82no—
180C4—6922no—
181C4—6922no—
182C4#1914963912yes—
183C4#1914964911yescharacteristic
184C4#1914966311yescharacteristic
185C2—42no—
186C2#2601136811yescharacteristic
187C2#2601149011yescharacteristic
188C2#2601149911yescharacteristic
189C2#2601173512yes—
190C2#2601174712yes—
subgroupG/H
centerC23C2×C22⋊D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#380×C173C2
socleC23×C173C2×D4
prime radicalC24×C173D4
cosocleC22×C4×C173C23
hypercenter#380D173
hypocenterC173#1894
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#19149602C2semidirect
4. #26011253C2semidirect
11. #19149658C4semidirect
95. C173#1894semidirect

Maximal subgroups

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

subgroupindex
C22×#145838102
C2×#191496582
C2×#191496022
#260112532
#380×C1732
#1894173