Finite Group Databasebeta

← #36354647

Subgroups — #36354647

Order 22144.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14583815C211yesmaximal, characteristic
2C22×#14583813C211yesmaximal, characteristic
3C2×#19149677C211yesmaximal, characteristic, complemented
4C2×#19149602C211yesmaximal, characteristic, complemented
5#26011392C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic, complemented
7C2×#103×C173C211yesmaximal, characteristic, complemented
8C22×Dic346C2211yescharacteristic
9C22×D692C2211yescharacteristic
10C2×#14583815—22no—
11C2×#14583815—22no—
12C2×#14583815—22no—
13C2×#14583813—22no—
14C2×#14583813—22no—
15C2×#14583813—22no—
16C22×#13063436C2211yescharacteristic
17#19149677—22no—
18#19149677—22no—
19C2×#14583805—22no—
20C2×#14583803—22no—
21C2×C4×Dic173C2211yescharacteristic
22#19149602—22no—
23C22×Q8×C173C2211yescharacteristic
24C22×C8×C173C2211yescharacteristic
25C2×C4⋊C4×C173C2211yescharacteristic
26#103×C173—22no—
27C23×D173—21no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—21no—
31C2×Dic346—21no—
32C2×Dic346—21no—
33C2×Dic346D412yes—
34C2×D692—21no—
35C2×D692—21no—
36C2×D692—21no—
37C2×D692—21no—
38C2×D692—21no—
39C2×D692D412yes—
40C22×Dic173—21no—
41C22×Dic173D412yes—
42#14583815—28no—
43#14583813—28no—
44C2×#13063436—21no—
45C2×#13063436—21no—
46C2×#13063436D412yes—
47#14583805—28no—
48#14583803—44no—
49C4×Dic173—21no—
50C4×Dic173—21no—
51C2×Q8×C173—21no—
52C2×Q8×C173—21no—
53C2×Q8×C173—21no—
54C2×Q8×C173—21no—
55C2×Q8×C173—21no—
56C2×Q8×C173D412yes—
57C22×C4×C173—21no—
58C22×C4×C173—21no—
59C22×C4×C173C2311yescharacteristic
60C2×C8×C173—21no—
61C2×C8×C173—21no—
62C2×C8×C173D412yes—
63C4⋊C4×C173—21no—
64C4⋊C4×C173—21no—
65C22×D173—22no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173—42no—
69C22×D173—42no—
70Dic346—22no—
71Dic346—22no—
72Dic346—41no—
73Dic346—41no—
74D692—22no—
75D692—22no—
76D692—41no—
77D692—41no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—22no—
81C2×Dic173—41no—
82C2×Dic173—41no—
83C2×Dic173SD1614yescomplemented
84#13063436—22no—
85C23×C173C2×D411yescharacteristic
86Q8×C173—22no—
87Q8×C173—22no—
88Q8×C173—41no—
89Q8×C173—41no—
90C2×C4×C173—22no—
91C2×C4×C173—41no—
92C2×C4×C173—41no—
93C2×C4×C173—41no—
94C2×C4×C173—42no—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173C2×D411yescharacteristic
99C2×C4×C173C2×D411yescharacteristic
100C2×C4×C173#4111yescharacteristic
101C8×C173—22no—
102C22×C173#12011yescharacteristic
103C22×C173#12111yescharacteristic
104C22×C173#12711yescharacteristic
105C22×C173C2×SD1612yes—
106C22×C173#13712yes—
107C2×D173—42no—
108C2×D173—44no—
109C2×D173—82no—
110C2×D173—81no—
111C2×D173—81no—
112C4×C173—42no—
113C4×C173—81no—
114C4×C173#12011yescharacteristic
115C4×C173#12011yescharacteristic
116C4×C173#12111yescharacteristic
117C4×C173#12111yescharacteristic
118Dic173—24no—
119Dic173—42no—
120C2×C173#39211yescharacteristic
121C2×C173#44812yes—
122C2×C173#46112yes—
123C2×C173#49211yescharacteristic
124C2×C173#49711yescharacteristic
125D173—82no—
126C173#201311yescharacteristic, complemented, Sylow
127#2013—1731nomaximal, Sylow
128C22×SD16—1731no—
129C22×SD16—1731no—
130C2×C4⋊Q8—1731no—
131C2×#103—1731no—
132C2×#103D17311yescharacteristic, complemented
133#390—1731no—
134#336—1731no—
135C22×Q8—1731no—
136C22×Q8C2×D17311yescharacteristic
137C22×D4—1731no—
138C2×SD16—3462no—
139C2×SD16—3462no—
140C2×SD16—3462no—
141C2×SD16—3462no—
142C2×SD16—3462no—
143C2×SD16—3462no—
144C22×C8—1731no—
145C22×C8C2×D17311yescharacteristic
146C4⋊Q8—3462no—
147C4⋊Q8—3462no—
148C2×C4⋊C4—3462no—
149C2×C4⋊C4C2×D17311yescharacteristic
150C2×C22⋊C4—3462no—
151C2×C42—1731no—
152#103—22no—
153#103—3462no—
154C24—3461no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—3461no—
161C2×Q8—3461no—
162C2×Q8—3461no—
163C2×Q8—3461no—
164C2×Q8—3461no—
165C2×Q8—1732no—
166C2×Q8D69212yes—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—1732no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—3461no—
176C22×C4—1732no—
177C22×C4C22×D17311yescharacteristic
178SD16—3468no—
179SD16—3468no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—3461no—
183C2×C8—3461no—
184C2×C8—1732no—
185C2×C8C22⋊D17312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3468no—
189C22⋊C4—6924no—
190C42—3461no—
191C42—3461no—
192C23—3462no—
193C23—6921no—
194C23—6921no—
195C23—6922no—
196C23—6922no—
197C23D4×D17311yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—3462no—
203Q8—3462no—
204Q8—6921no—
205Q8—6921no—
206D4—3462no—
207D4—3462no—
208D4—6921no—
209D4—6921no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—3462no—
216C2×C4—3462no—
217C2×C4—3462no—
218C2×C4—1734no—
219C2×C4—6921no—
220C2×C4—6921no—
221C2×C4D4×D17311yescharacteristic
222C2×C4D4×D17311yescharacteristic
223C2×C4D4×D17311yescharacteristic
224C2×C4C2×C22⋊D17311yescharacteristic
225C2×C4C2×D69211yescharacteristic
226C2×C4#1458382611yescharacteristic
227C8—22no—
228C8—3462no—
229C22—13841no—
230C22—13841no—
231C22—13842no—
232C22—6922no—
233C22—6924no—
234C22#1914964911yescharacteristic
235C22#1914965211yescharacteristic
236C22#1914967511yescharacteristic
237C22SD16×D17312yes—
238C22#1914971112yes—
239C4—42no—
240C4—81no—
241C4—3464no—
242C4—6922no—
243C4#1914964711yescharacteristic
244C4#1914964911yescharacteristic
245C4#1914965211yescharacteristic
246C4#1914965311yescharacteristic
247C2—13842no—
248C2#2601140011yescharacteristic
249C2#2601163112yes—
250C2#2601169512yes—
251C2#2601186211yescharacteristic
252C2#2601188411yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#103×C173C2
socleC23×C173C2×D4
prime radicalC22×D692C22
hypercenterC2×#103D173
hypocenterC173#2013
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#19149677C2semidirect
4. C2×#19149602C2semidirect
6. #26011245C2semidirect
7. C2×#103×C173C2semidirect
83. C2×Dic173SD16semidirect
126. C173#2013semidirect
132. C2×#103D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838152
C22×#145838132
C2×#191496772
C2×#191496022
#260113922
#260112452
C2×#103×C1732
#2013173