Finite Group Databasebeta

← #36364638

Subgroups — #36364638

Order 22144.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012104C211yesmaximal, characteristic, complemented
2#26012059C211yesmaximal, characteristic
3#26011886C211yesmaximal, characteristic, complemented
4#26011864C211yesmaximal, characteristic, complemented
5#26011857C211yesmaximal, characteristic, complemented
6#26011716C212yesmaximal, complemented
7#26011690C212yesmaximal, complemented
8#26011656C212yesmaximal, complemented
9#26011643C212yesmaximal, complemented
10#26011550C211yesmaximal, characteristic, complemented
11#488×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14C2×#14583816C2212yescomplemented
15C2×#14583815C2212yes—
16C2×#14583815C2212yescomplemented
17C2×#14583814C2212yes—
18#19149679C2211yescharacteristic, complemented
19#19149676C2211yescharacteristic
20#19149675C2211yescharacteristic
21#19149674—22no—
22#19149670—22no—
23#19149668C2211yescharacteristic
24#19149659C2212yes—
25#19149652—22no—
26#19149649—22no—
27#19149644C2212yescomplemented
28#19149636C2211yescharacteristic
29#19149608C2212yescomplemented
30#19149599C2212yes—
31#19149599C2212yescomplemented
32#19149591C2212yes—
33#19149590C2211yescharacteristic, complemented
34#125×C173C2211yescharacteristic, complemented
35#124×C173C2211yescharacteristic
36#103×C173C2212yescomplemented
37#102×C173C2212yes—
38C4×C8×C173C2211yescharacteristic, complemented
39C23×D173—21no—
40#14583827—42no—
41C2×Dic346C2312yes—
42C2×D692—21no—
43C2×D692—22no—
44C2×D692C2311yescharacteristic
45C2×D692C2311yescharacteristic
46C2×C22⋊D173—21no—
47C2×C22⋊D173—22no—
48D4×D173—42no—
49C2×C4×D173—21no—
50C2×C4×D173C2311yescharacteristic
51#14583816—42no—
52#14583815—42no—
53#14583815—42no—
54#14583814—42no—
55C2×#13063436C2312yes—
56C4⋊Dic173—22no—
57C4⋊Dic173—22no—
58#14583805—22no—
59#14583805D412yes—
60#14583803—22no—
61#14583803—22no—
62#14583803D412yes—
63C4×Dic173—22no—
64C4×Dic173C2311yescharacteristic
65C2×Q8×C173C2311yescharacteristic
66C2×D4×C173C2311yescharacteristic
67C2×C8×C173C2312yes—
68C4⋊C4×C173—22no—
69C4⋊C4×C173C2312yes—
70C22⋊C4×C173—22no—
71C42×C173C2311yescharacteristic
72C22×D173—21no—
73C22×D173—21no—
74C22×D173—42no—
75C22×D173—82no—
76C22×D173C2×D411yescharacteristic
77Dic346—22no—
78Dic346—22no—
79D692—22no—
80D692—22no—
81D692—42no—
82D692—82no—
83C22⋊D173—42no—
84C22⋊D173—82no—
85C2×Dic173—22no—
86C2×Dic173C2×D411yescharacteristic
87C2×Dic173C2×D412yes—
88C4×D173—22no—
89C4×D173—42no—
90#13063436—22no—
91C23×C173—21no—
92Q8×C173—22no—
93D4×C173—22no—
94C2×C4×C173—21no—
95C2×C4×C173—22no—
96C2×C4×C173C2×D412yes—
97C2×C4×C173C2411yescharacteristic
98C8×C173—22no—
99C22×C173—41no—
100C22×C173—42no—
101C22×C173C22×D411yescharacteristic
102C2×D173—21no—
103C2×D173—22no—
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—42no—
107C2×D173—42no—
108C2×D173—84no—
109C4×C173—22no—
110C4×C173—41no—
111C4×C173—42no—
112C4×C173H(5,𝔽2)12yes—
113Dic173—21no—
114Dic173—22no—
115Dic173—42no—
116C2×C173—81no—
117C2×C173#53511yescharacteristic
118C2×C173#57712yes—
119D173—41no—
120D173—81no—
121D173—81no—
122C173#322211yescharacteristic, complemented, Sylow
123#3222—1731nomaximal, Sylow
124#541—1731no—
125#535—1731no—
126#497—1731no—
127#492—1731no—
128#490—1731no—
129#488D17311yescharacteristic, complemented
130#464—1732no—
131#460—1732no—
132#452—1732no—
133#450—1732no—
134#433—1731no—
135C2×#41—1731no—
136C22×D4—1731no—
137C2×Q16—1732no—
138C2×SD16—1732no—
139C2×SD16—1732no—
140C2×SD16—1732no—
141C4⋊Q8—1731no—
142#127—1731no—
143#127—1731no—
144#126—3462no—
145#125—3462no—
146#125C2×D17311yescharacteristic, complemented
147#124—1731no—
148#124C2×D17311yescharacteristic
149C22⋊Q8—1732no—
150#121—3462no—
151#120—3462no—
152C4×Q8—1732no—
153#117—1731no—
154C4⋊C8—1732no—
155#103C2×D17312yescomplemented
156#102—1732no—
157#102—1732no—
158#102C2×D17312yes—
159C22⋊C8—1732no—
160#97—1731no—
161C4×C8C2×D17311yescharacteristic, complemented
162C24—3461no—
163#41—6922no—
164C2×Q8—1732no—
165C2×Q8C22×D17311yescharacteristic
166C2×D4—3461no—
167C2×D4—3462no—
168C2×D4—3461no—
169C2×D4—3462no—
170C2×D4—1731no—
171C2×D4—1731no—
172C2×D4—6922no—
173C2×D4C22×D17311yescharacteristic
174C22×C4—3461no—
175C22×C4—1731no—
176Q16—6922no—
177SD16—6922no—
178SD16—6922no—
179SD16—6922no—
180C2×C8—1732no—
181C2×C8C22×D17312yes—
182C4⋊C4—22no—
183C4⋊C4—3462no—
184C4⋊C4—3462no—
185C4⋊C4—3462no—
186C4⋊C4—1732no—
187C4⋊C4C22×D17312yes—
188C22⋊C4—22no—
189C22⋊C4—3462no—
190C22⋊C4—3462no—
191C22⋊C4—1732no—
192C42—3462no—
193C42—1731no—
194C42C22×D17311yescharacteristic
195C23—21no—
196C23—3461no—
197C23—3461no—
198C23—13842no—
199C23—1731no—
200C23—6922no—
201Q8—22no—
202Q8—3462no—
203Q8—3462no—
204D4—22no—
205D4—3462no—
206D4—3462no—
207D4—13842no—
208D4—13842no—
209D4—6922no—
210D4—6922no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—3462no—
214C2×C4—3462no—
215C2×C4—1732no—
216C2×C4—1731no—
217C2×C4—6922no—
218C2×C4D4×D17312yes—
219C2×C4C23×D17311yescharacteristic
220C8—22no—
221C8—3462no—
222C22—42no—
223C22—41no—
224C22—3462no—
225C22—3461no—
226C22—13844no—
227C22—6922no—
228C22—6922no—
229C22—6921no—
230C22—6921no—
231C22C2×D4×D17311yescharacteristic
232C4—22no—
233C4—42no—
234C4—41no—
235C4—3462no—
236C4—3461no—
237C4—6922no—
238C4#1914972512yes—
239C2—81no—
240C2—13841no—
241C2—13841no—
242C2—6921no—
243C2#2601205911yescharacteristic
244C2#2601229812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
prime radical#26012059C2
hypercenter#488D173
hypocenterC173#3222
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26012104C2semidirect
3. #26011886C2semidirect
4. #26011864C2semidirect
5. #26011857C2semidirect
6. #26011716C2semidirect
7. #26011690C2semidirect
8. #26011656C2semidirect
9. #26011643C2semidirect
10. #26011550C2semidirect
11. #488×C173C2semidirect
14. C2×#14583816C22semidirect
16. C2×#14583815C22semidirect
18. #19149679C22semidirect
27. #19149644C22semidirect
29. #19149608C22semidirect
31. #19149599C22semidirect
33. #19149590C22semidirect
34. #125×C173C22semidirect
36. #103×C173C22semidirect
38. C4×C8×C173C22semidirect
122. C173#3222semidirect
129. #488D173semidirect
146. #125C2×D173semidirect
155. #103C2×D173semidirect
161. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260120592
#260118862
#260118642
#260118572
#260117162
#260116902
#260116562
#260116432
#260115502
#488×C1732
#3222173