Finite Group Databasebeta

← #36354610

Subgroups — #36354610

Order 22144.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19149709C212yesmaximal, complemented
2C2×#19149606C212yesmaximal, complemented
3#26011396C211yesmaximal, characteristic, complemented
4#26011260C211yesmaximal, characteristic, complemented
5#537×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7#19149709—24no—
8#19149709—24no—
9C2×#14583816—22no—
10C2×#14583814—22no—
11C2×#14583810C2212yes—
12C2×C4×Dic173C2211yescharacteristic
13#19149606—24no—
14#19149586—22no—
15C2×#41×C173C2212yes—
16C22⋊Q8×C173—22no—
17#121×C173—22no—
18#121×C173—22no—
19C2×C4⋊C4×C173C2211yescharacteristic
20C2×Dic346—22no—
21C2×Dic346—22no—
22C2×Dic346—21no—
23C2×Dic346D412yescomplemented
24C22×Dic173—21no—
25C22×Dic173—21no—
26#14583816—28no—
27#14583814—28no—
28#14583810—22no—
29#14583810D414yescomplemented
30C2×#13063436—22no—
31C4×Dic173—22no—
32#41×C173—22no—
33#41×C173—22no—
34#41×C173—22no—
35#41×C173—22no—
36C2×Q8×C173D412yes—
37C2×D4×C173—22no—
38C2×D4×C173—22no—
39C2×D4×C173D412yes—
40C22×C4×C173—22no—
41C22×C4×C173D412yes—
42C22×C4×C173C2311yescharacteristic
43C4⋊C4×C173—22no—
44C4⋊C4×C173—22no—
45C22⋊C4×C173—24no—
46Dic346—24no—
47Dic346—42no—
48C2×Dic173—22no—
49C2×Dic173—22no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C2×Dic173—41no—
53C2×Dic173—41no—
54#13063436—24no—
55C23×C173—22no—
56C23×C173C2×D411yescharacteristic
57Q8×C173—22no—
58Q8×C173—22no—
59D4×C173—22no—
60D4×C173—22no—
61D4×C173—42no—
62D4×C173—42no—
63D4×C173—82no—
64C2×C4×C173—22no—
65C2×C4×C173—22no—
66C2×C4×C173—42no—
67C2×C4×C173—42no—
68C2×C4×C173—42no—
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D412yes—
71C2×C4×C173C2×D412yes—
72C2×C4×C173#4111yescharacteristic
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—42no—
78C22×C173#12011yescharacteristic
79C22×C173#12111yescharacteristic
80C22×C173#12711yescharacteristic
81C4×C173—42no—
82C4×C173—42no—
83C4×C173#12012yes—
84C4×C173#12112yes—
85Dic173—42no—
86Dic173—42no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173—82no—
90C2×C173#39211yescharacteristic
91C2×C173#45511yescharacteristic
92C2×C173#45611yescharacteristic
93C173#200711yescharacteristic, complemented, Sylow
94#2007—1731nomaximal, Sylow
95C2×#137—1732no—
96#537D17311yescharacteristic
97C2×#104—1732no—
98#391—1731no—
99#343—1731no—
100C2×#41C2×D17312yes—
101C22×Q8—1731no—
102#137—3464no—
103#137—3464no—
104C2×Q16—3462no—
105C2×SD16—3462no—
106C2×M16—1732no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D17311yescharacteristic
111C2×C42—1731no—
112#104—3464no—
113#95—3462no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—3461no—
119C2×Q8—3462no—
120C2×Q8—3462no—
121C2×Q8—1732no—
122C2×Q8C22⋊D17312yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D17312yes—
126C22×C4—22no—
127C22×C4—3461no—
128C22×C4—3461no—
129C22×C4C22⋊D17312yes—
130C22×C4C22×D17311yescharacteristic
131Q16—3468no—
132SD16—3468no—
133M16—3462no—
134M16—1734no—
135C2×C8—3462no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—3462no—
140C23—22no—
141C23C2×C22⋊D17311yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—3464no—
145Q8—6922no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—3462no—
157C2×C4—3462no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4—6921no—
162C2×C4D4×D17311yescharacteristic
163C2×C4D4×D17312yes—
164C2×C4C2×C22⋊D17312yes—
165C2×C4#1458382511yescharacteristic
166C8—3464no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1914964811yescharacteristic
173C22#1914965011yescharacteristic
174C22#1914967511yescharacteristic
175C4—42no—
176C4—42no—
177C4—6922no—
178C4—6922no—
179C4#1914964712yes—
180C4#1914965112yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2601139911yescharacteristic
185C2#2601166911yescharacteristic
186C2#2601167411yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting, prime radical#537×C173C2
socleC22×C173#120
hypercenter#537D173
hypocenterC173#2007
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#19149709C2semidirect
2. C2×#19149606C2semidirect
3. #26011396C2semidirect
4. #26011260C2semidirect
23. C2×Dic346D4semidirect
29. #14583810D4semidirect
93. C173#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
C2×#191496062
#260113962
#260112602
#537×C1732
#2007173