Finite Group Databasebeta

← #36363093

Subgroups — #36363093

Order 22144.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19149710C211yesmaximal, characteristic
2C22×#14583816C211yesmaximal, characteristic, complemented
3#26012067C211yesmaximal, characteristic
4#26011713C214yesmaximal, complemented
5#26011701C214yesmaximal, complemented
6#26011518C211yesmaximal, characteristic, complemented
7#26011499C211yesmaximal, characteristic, complemented
8#26011485C211yesmaximal, characteristic
9#519×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C22×Dic346C2211yescharacteristic
12#19149710—24no—
13C2×#14583816—24no—
14C2×#14583816C2212yes—
15C2×#14583816C2214yes—
16C2×#14583813C2212yes—
17C2×#14583810C2211yescharacteristic
18C22×#13063436C2211yescharacteristic
19#19149679C2212yes—
20#19149667C2212yes—
21#19149663—22no—
22#19149659—22no—
23C4⋊#13063436C2214yes—
24#19149602C2212yes—
25#19149602C2214yes—
26#19149599C2212yes—
27C22×Q8×C173C2211yescharacteristic
28C4×Q8×C173—22no—
29C4×Q8×C173C2214yes—
30#117×C173—22no—
31#117×C173C2211yescharacteristic
32#117×C173C2211yescharacteristic
33#14583826—22no—
34#14583826—22no—
35C2×Dic346—21no—
36C2×Dic346—22no—
37C2×Dic346—22no—
38C2×Dic346C2312yes—
39C2×Dic346C2311yescharacteristic
40C2×D692C2311yescharacteristic
41C2×C22⋊D173—21no—
42C22×Dic173—21no—
43C2×C4×D173—21no—
44#14583816—28no—
45#14583816—24no—
46#14583813—24no—
47#14583810—22no—
48C2×#13063436—22no—
49C2×#13063436C2312yes—
50C2×#13063436C2312yes—
51C4⋊Dic173—24no—
52#14583803—24no—
53C2×Q8×C173—21no—
54C2×Q8×C173—22no—
55C2×Q8×C173D414yes—
56C2×Q8×C173C2312yes—
57C22×C4×C173—21no—
58C22×C4×C173C2311yescharacteristic
59C4⋊C4×C173—24no—
60C4⋊C4×C173D412yes—
61C4⋊C4×C173C2312yes—
62C22⋊C4×C173—21no—
63C22⋊C4×C173D412yes—
64C42×C173—24no—
65C42×C173—21no—
66C42×C173C2312yes—
67C22×D173—21no—
68Dic346—22no—
69Dic346—21no—
70Dic346—22no—
71Dic346—21no—
72Dic346—42no—
73D692—21no—
74D692—21no—
75C22⋊D173—42no—
76C2×Dic173—21no—
77C2×Dic173—22no—
78C2×Dic173—42no—
79C4×D173—42no—
80#13063436—22no—
81#13063436—22no—
82C23×C173C2×D411yescharacteristic
83Q8×C173—24no—
84Q8×C173—22no—
85Q8×C173C2×D414yes—
86C2×C4×C173—24no—
87C2×C4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173C2×D414yes—
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D412yes—
92C2×C4×C173#4112yes—
93C2×C4×C173C2411yescharacteristic
94C22×C173—22no—
95C22×C173#12112yes—
96C22×C173C22×D411yescharacteristic
97C2×D173—41no—
98C2×D173—42no—
99C4×C173—24no—
100C4×C173—24no—
101C4×C173—42no—
102C4×C173#12112yes—
103C4×C173C22×D411yescharacteristic
104C4×C173C2×#4111yescharacteristic
105Dic173—41no—
106Dic173—42no—
107C2×C173—22no—
108C2×C173C2×#12111yescharacteristic
109C2×C173#57812yes—
110D173—81no—
111C173#304411yescharacteristic, complemented, Sylow
112#3044—1731nomaximal, Sylow
113C2×#137—1731no—
114C22×Q16—1731no—
115#536—1731no—
116#519D17311yescharacteristic, complemented
117#464—1734no—
118#462—1734no—
119#424—1731no—
120#419—1731no—
121#416—1731no—
122C2×#41—1731no—
123C22×Q8—1731no—
124C22×Q8C2×D17311yescharacteristic
125#137—3464no—
126C2×Q16—3464no—
127C2×Q16—1734no—
128C2×Q16—1732no—
129C2×SD16—1732no—
130C2×M16—1731no—
131C22×C8—1731no—
132C4⋊Q8—1732no—
133#124—1732no—
134#123—3462no—
135C22⋊Q8—3462no—
136C4×Q8—22no—
137C4×Q8C2×D17314yes—
138#117—22no—
139#117C2×D17311yescharacteristic
140#117C2×D17311yescharacteristic
141C4⋊C8—1734no—
142#103—1732no—
143#103—1734no—
144#102—1732no—
145#41—3462no—
146#41—3462no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3461no—
150C2×Q8—3462no—
151C2×Q8—3462no—
152C2×Q8—1732no—
153C2×Q8—1731no—
154C2×Q8D69214yes—
155C2×Q8C22×D17312yes—
156C2×D4—3461no—
157C2×D4—1731no—
158C22×C4—21no—
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4C22×D17311yescharacteristic
162Q16—3468no—
163Q16—3464no—
164SD16—3464no—
165M16—3462no—
166C2×C8—3462no—
167C2×C8—1732no—
168C2×C8—1732no—
169C4⋊C4—24no—
170C4⋊C4—3464no—
171C4⋊C4C22⋊D17312yes—
172C4⋊C4C22×D17312yes—
173C22⋊C4—21no—
174C22⋊C4—3464no—
175C22⋊C4C22⋊D17312yes—
176C42—24no—
177C42—21no—
178C42C22×D17312yes—
179C23—3461no—
180C23C2×C22⋊D17311yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—3462no—
184Q8—3461no—
185Q8—3462no—
186Q8—3461no—
187Q8—6922no—
188Q8C2×D69214yes—
189D4—3461no—
190D4—3461no—
191D4—6922no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3462no—
196C2×C4—3461no—
197C2×C4—6922no—
198C2×C4—6922no—
199C2×C4C2×C22⋊D17314yes—
200C2×C4C2×C22⋊D17311yescharacteristic
201C2×C4C2×D69212yes—
202C2×C4#1458382612yes—
203C2×C4C23×D17311yescharacteristic
204C8—3462no—
205C8—3462no—
206C22—22no—
207C22—6921no—
208C22—6922no—
209C22#1914965312yes—
210C22C22×C22⋊D17311yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—6921no—
215C4—6922no—
216C4#1914965312yes—
217C4C22×D69211yescharacteristic
218C4C2×#1458382611yescharacteristic
219C2—22no—
220C2—13841no—
221C2C2×#1914965311yescharacteristic
222C2#2601230612yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#519×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#519D173
hypocenterC173#3044
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#14583816C2semidirect
4. #26011713C2semidirect
5. #26011701C2semidirect
6. #26011518C2semidirect
7. #26011499C2semidirect
9. #519×C173C2semidirect
111. C173#3044semidirect
116. #519D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C22×#145838162
#260120672
#260117132
#260117012
#260115182
#260114992
#260114852
#519×C1732
#3044173