Finite Group Databasebeta

← #36364630

Subgroups — #36364630

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#26012102C211yesmaximal, characteristic, complemented
2#26012059C211yesmaximal, characteristic
3#26011865C211yesmaximal, characteristic, complemented
4#26011856C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011715C212yesmaximal, complemented
7#26011689C212yesmaximal, complemented
8#26011652C212yesmaximal, complemented
9#26011640C212yesmaximal, complemented
10#26011553C211yesmaximal, characteristic, complemented
11#497×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14C2×#14583813C2212yes—
15C2×#14583813C2212yescomplemented
16#19149676C2211yescharacteristic
17#19149675C2211yescharacteristic
18#19149674—22no—
19#19149671C2211yescharacteristic, complemented
20#19149669—22no—
21#19149668C2211yescharacteristic
22#19149655C2212yes—
23#19149652—22no—
24#19149649—22no—
25Q8×Dic173C2212yescomplemented
26#19149636C2211yescharacteristic
27#19149608C2212yescomplemented
28#19149603C2212yescomplemented
29#19149601C2212yes—
30#19149601C2212yescomplemented
31#19149600C2212yes—
32#19149591C2212yes—
33C4×#13063436C2211yescharacteristic, complemented
34C2×Q16×C173C2212yescomplemented
35C2×SD16×C173C2212yes—
36C4⋊Q8×C173C2211yescharacteristic, complemented
37#124×C173C2211yescharacteristic
38#97×C173C2211yescharacteristic, complemented
39C23×D173—21no—
40#14583827—42no—
41C2×D692—21no—
42C2×D692—22no—
43C2×D692C2311yescharacteristic
44C2×D692C2311yescharacteristic
45C2×C22⋊D173—21no—
46C2×C22⋊D173—22no—
47D4×D173—42no—
48C2×C4×D173—21no—
49C2×C4×D173C2311yescharacteristic
50#14583813—42no—
51#14583813—42no—
52C2×#13063436C2312yes—
53C4⋊Dic173—22no—
54C4⋊Dic173C2312yes—
55#14583805—22no—
56#14583805—22no—
57#14583805D412yes—
58#14583803—22no—
59#14583803—22no—
60#14583803D412yes—
61C4×Dic173—22no—
62C4×Dic173C2311yescharacteristic
63C2×Q8×C173C2312yes—
64C2×Q8×C173C2311yescharacteristic
65C2×D4×C173C2311yescharacteristic
66Q16×C173—42no—
67SD16×C173—42no—
68C2×C8×C173C2312yes—
69C4⋊C4×C173—22no—
70C22⋊C4×C173—22no—
71C42×C173C2311yescharacteristic
72C22×D173—21no—
73C22×D173—21no—
74C22×D173—42no—
75C22×D173—82no—
76C22×D173C2×D411yescharacteristic
77D692—22no—
78D692—22no—
79D692—42no—
80D692—82no—
81C22⋊D173—42no—
82C22⋊D173—82no—
83C2×Dic173—22no—
84C2×Dic173C2×D411yescharacteristic
85C2×Dic173C2×D412yes—
86C4×D173—22no—
87C4×D173—42no—
88#13063436—22no—
89C23×C173—21no—
90Q8×C173—22no—
91Q8×C173—22no—
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—42no—
111C4×C173—41no—
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#497D17311yescharacteristic, complemented
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×#41—1731no—
136C22×D4—1731no—
137C2×Q16C2×D17312yescomplemented
138C2×SD16—1732no—
139C2×SD16—1732no—
140C2×SD16C2×D17312yes—
141C4⋊Q8C2×D17311yescharacteristic, complemented
142#127—1731no—
143#127—1731no—
144#126—3462no—
145#125—3462no—
146#125—1731no—
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#103—1732no—
156#102—1732no—
157#102—1732no—
158#102—1732no—
159C22⋊C8—1732no—
160#97C2×D17311yescharacteristic, complemented
161C4×C8—1731no—
162C24—3461no—
163#41—6922no—
164C2×Q8C22×D17311yescharacteristic
165C2×Q8C22×D17312yes—
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—42no—
177SD16—42no—
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⋊C4—1732no—
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—22no—
203Q8—22no—
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—41no—
234C4—42no—
235C4—3462no—
236C4—3461no—
237C4—6922no—
238C4#1914972412yes—
239C2—81no—
240C2—13841no—
241C2—13841no—
242C2—6921no—
243C2#2601205411yescharacteristic
244C2#2601230012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
prime radical#26012059C2
hypercenter#497D173
hypocenterC173#3222
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26012102C2semidirect
3. #26011865C2semidirect
4. #26011856C2semidirect
5. #26011845C2semidirect
6. #26011715C2semidirect
7. #26011689C2semidirect
8. #26011652C2semidirect
9. #26011640C2semidirect
10. #26011553C2semidirect
11. #497×C173C2semidirect
15. C2×#14583813C22semidirect
19. #19149671C22semidirect
25. Q8×Dic173C22semidirect
27. #19149608C22semidirect
28. #19149603C22semidirect
30. #19149601C22semidirect
33. C4×#13063436C22semidirect
34. C2×Q16×C173C22semidirect
36. C4⋊Q8×C173C22semidirect
38. #97×C173C22semidirect
122. C173#3222semidirect
126. #497D173semidirect
137. C2×Q16C2×D173semidirect
141. C4⋊Q8C2×D173semidirect
160. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121022
#260120592
#260118652
#260118562
#260118452
#260117152
#260116892
#260116522
#260116402
#260115532
#497×C1732
#3222173