Finite Group Databasebeta

← #36364635

Subgroups — #36364635

Order 22144.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012103C211yesmaximal, characteristic, complemented
2#26011886C211yesmaximal, characteristic, complemented
3#26011863C211yesmaximal, characteristic, complemented
4#26011855C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011710C212yesmaximal, complemented
7#26011688C212yesmaximal, complemented
8#26011653C212yesmaximal, complemented
9#26011642C212yesmaximal, complemented
10#26011552C211yesmaximal, characteristic, complemented
11#535×C173C211yesmaximal, characteristic
12C2×#14583816C2212yescomplemented
13C2×#14583814C2212yes—
14C2×#14583814C2212yes—
15C2×#14583814C2212yescomplemented
16#19149679C2211yescharacteristic, complemented
17#19149672—22no—
18#19149671C2211yescharacteristic, complemented
19#19149669—22no—
20#19149658C2212yes—
21C4×Dic346C2212yescomplemented
22C4⋊#13063436C2212yescomplemented
23#19149603C2212yescomplemented
24#19149600C2212yes—
25#19149600C2212yes—
26#19149600C2212yescomplemented
27C22⋊#13063436C2212yes—
28#19149590C2211yescharacteristic, complemented
29C4×#13063436C2211yescharacteristic, complemented
30C2×#41×C173C2211yescharacteristic
31C22×D4×C173C2211yescharacteristic
32#127×C173C2211yescharacteristic
33#127×C173C2211yescharacteristic
34#124×C173C2211yescharacteristic
35#124×C173C2211yescharacteristic
36#121×C173—22no—
37#120×C173—22no—
38#117×C173C2211yescharacteristic
39C2×Dic346C2312yes—
40#14583816—42no—
41#14583814—42no—
42#14583814—42no—
43#14583814—42no—
44C2×#13063436C2312yes—
45C2×#13063436C2312yes—
46C4⋊Dic173—22no—
47C4⋊Dic173C2312yes—
48#14583805—22no—
49#14583805—22no—
50#14583805—22no—
51C22⋊Dic173—22no—
52C4×Dic173—22no—
53C24×C173—21no—
54#41×C173—42no—
55C2×Q8×C173C2311yescharacteristic
56C2×D4×C173—21no—
57C2×D4×C173—21no—
58C2×D4×C173—22no—
59C2×D4×C173—22no—
60C2×D4×C173—42no—
61C2×D4×C173C2311yescharacteristic
62C2×D4×C173C2311yescharacteristic
63C2×D4×C173C2311yescharacteristic
64C22×C4×C173—21no—
65C22×C4×C173C2311yescharacteristic
66C4⋊C4×C173D412yes—
67C22⋊C4×C173—22no—
68C22⋊C4×C173—22no—
69C22⋊C4×C173D412yes—
70C42×C173C2311yescharacteristic
71C42×C173C2311yescharacteristic
72Dic346—22no—
73Dic346—22no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76#13063436—22no—
77#13063436—22no—
78C23×C173—21no—
79C23×C173—21no—
80C23×C173—21no—
81C23×C173—42no—
82C23×C173—82no—
83C23×C173C2×D411yescharacteristic
84Q8×C173—22no—
85D4×C173—22no—
86D4×C173—22no—
87D4×C173—22no—
88D4×C173—42no—
89D4×C173—42no—
90D4×C173—82no—
91D4×C173—82no—
92C2×C4×C173—22no—
93C2×C4×C173—21no—
94C2×C4×C173—42no—
95C2×C4×C173C2×D412yes—
96C2×C4×C173C2×D412yes—
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173C2411yescharacteristic
99C22×C173—21no—
100C22×C173—22no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—42no—
105C22×C173—42no—
106C22×C173—42no—
107C22×C173—84no—
108C22×C173C22×D411yescharacteristic
109C4×C173—21no—
110C4×C173—22no—
111C4×C173—22no—
112C4×C173—41no—
113C4×C173H(5,𝔽2)12yes—
114Dic173—42no—
115Dic173—42no—
116C2×C173—41no—
117C2×C173—81no—
118C2×C173—81no—
119C2×C173—81no—
120C2×C173#53511yescharacteristic
121C2×C173#57712yes—
122C173#322211yescharacteristic, complemented, Sylow
123#3222—1731nomaximal, Sylow
124#541—1731no—
125#535D17311yescharacteristic
126#497—1731no—
127#492—1731no—
128#490—1731no—
129#488—1731no—
130#464—1732no—
131#460—1732no—
132#452—1732no—
133#450—1732no—
134#433—1731no—
135C2×#41C2×D17311yescharacteristic
136C22×D4C2×D17311yescharacteristic
137C2×Q16—1732no—
138C2×SD16—1732no—
139C2×SD16—1732no—
140C2×SD16—1732no—
141C4⋊Q8—1731no—
142#127C2×D17311yescharacteristic
143#127C2×D17311yescharacteristic
144#126—3462no—
145#125—3462no—
146#125—1731no—
147#124C2×D17311yescharacteristic
148#124C2×D17311yescharacteristic
149C22⋊Q8—1732no—
150#121—22no—
151#120—22no—
152C4×Q8—1732no—
153#117C2×D17311yescharacteristic
154C4⋊C8—1732no—
155#103—1732no—
156#102—1732no—
157#102—1732no—
158#102—1732no—
159C22⋊C8—1732no—
160#97—1731no—
161C4×C8—1731no—
162C24—21no—
163#41—42no—
164C2×Q8—1732no—
165C2×Q8C22×D17311yescharacteristic
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4C22×D17311yescharacteristic
172C2×D4C22×D17311yescharacteristic
173C2×D4C22×D17311yescharacteristic
174C22×C4—21no—
175C22×C4C22×D17311yescharacteristic
176Q16—6922no—
177SD16—6922no—
178SD16—6922no—
179SD16—6922no—
180C2×C8—1732no—
181C2×C8—1732no—
182C4⋊C4—3462no—
183C4⋊C4—3462no—
184C4⋊C4—3462no—
185C4⋊C4—3462no—
186C4⋊C4—1732no—
187C4⋊C4C22⋊D17312yes—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—3462no—
191C22⋊C4C22⋊D17312yes—
192C42—3462no—
193C42C22×D17311yescharacteristic
194C42C22×D17311yescharacteristic
195C23—21no—
196C23—21no—
197C23—21no—
198C23—42no—
199C23—82no—
200C23C2×C22⋊D17311yescharacteristic
201Q8—22no—
202Q8—3462no—
203Q8—3462no—
204D4—22no—
205D4—22no—
206D4—22no—
207D4—42no—
208D4—42no—
209D4—82no—
210D4—82no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—3462no—
215C2×C4—3462no—
216C2×C4C2×C22⋊D17312yes—
217C2×C4C2×C22⋊D17312yes—
218C2×C4C2×C22⋊D17311yescharacteristic
219C2×C4C23×D17311yescharacteristic
220C8—3462no—
221C8—3462no—
222C22—22no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—84no—
231C22C22×C22⋊D17311yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—41no—
236C4—6922no—
237C4—6922no—
238C4#1914972412yes—
239C2—41no—
240C2—81no—
241C2—81no—
242C2—81no—
243C2#2601205811yescharacteristic
244C2#2601230212yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#535×C173C2
socleC22×C173C22×D4
hypercenter#535D173
hypocenterC173#3222
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26012103C2semidirect
2. #26011886C2semidirect
3. #26011863C2semidirect
4. #26011855C2semidirect
5. #26011845C2semidirect
6. #26011710C2semidirect
7. #26011688C2semidirect
8. #26011653C2semidirect
9. #26011642C2semidirect
10. #26011552C2semidirect
12. C2×#14583816C22semidirect
15. C2×#14583814C22semidirect
16. #19149679C22semidirect
18. #19149671C22semidirect
21. C4×Dic346C22semidirect
22. C4⋊#13063436C22semidirect
23. #19149603C22semidirect
26. #19149600C22semidirect
28. #19149590C22semidirect
29. C4×#13063436C22semidirect
122. C173#3222semidirect

Maximal subgroups

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

subgroupindex
#260121032
#260118862
#260118632
#260118552
#260118452
#260117102
#260116882
#260116532
#260116422
#260115522
#535×C1732
#3222173