Finite Group Databasebeta

← #36364606

Subgroups — #36364606

Order 22144.

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

normal subgroups only

This group has 420 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#26012095C211yesmaximal, characteristic, complemented
2#26012054C211yesmaximal, characteristic, complemented
3#26011878C211yesmaximal, characteristic, complemented
4#26011855C211yesmaximal, characteristic, complemented
5#26011846C211yesmaximal, characteristic, complemented
6#26011704C212yesmaximal, complemented
7#26011684C212yesmaximal, complemented
8#26011633C212yesmaximal, complemented
9#26011623C212yesmaximal, complemented
10#26011550C211yesmaximal, characteristic, complemented
11#493×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14C2×#14583814C2212yescomplemented
15C2×#14583811C2212yes—
16#19149675C2211yescharacteristic
17#19149671C2211yescharacteristic, complemented
18#19149667C2211yescharacteristic
19#19149666C2211yescharacteristic
20#19149660—22no—
21#19149651—22no—
22#19149651—22no—
23#19149650C2212yes—
24#19149647—22no—
25D4×Dic173C2212yescomplemented
26#19149636C2211yescharacteristic
27#19149608C2212yescomplemented
28#19149604C2212yescomplemented
29#19149601C2212yes—
30#19149600C2212yes—
31#19149600C2212yescomplemented
32#19149591C2212yes—
33#19149590C2211yescharacteristic, complemented
34C2×D8×C173C2212yes—
35C2×D8×C173C2212yescomplemented
36#127×C173C2211yescharacteristic
37#127×C173C2211yescharacteristic, complemented
38C4×C8×C173C2211yescharacteristic, complemented
39C23×D173—21no—
40#14583825—42no—
41C2×Dic346C2311yescharacteristic
42C2×D692C2311yescharacteristic
43C2×C22⋊D173—21no—
44C2×C22⋊D173—22no—
45C2×C22⋊D173—21no—
46C2×C22⋊D173—22no—
47D4×D173—42no—
48C22×Dic173—21no—
49C22×Dic173—22no—
50C2×C4×D173C2311yescharacteristic
51#14583814—42no—
52#14583811—42no—
53C2×#13063436C2312yes—
54C4⋊Dic173C2312yes—
55#14583805—22no—
56#14583805D412yes—
57C22⋊Dic173—22no—
58C22⋊Dic173—22no—
59C22⋊Dic173—22no—
60#14583803—22no—
61#14583803D412yes—
62C4×Dic173C2311yescharacteristic
63C2×D4×C173—22no—
64C2×D4×C173—22no—
65C2×D4×C173C2311yescharacteristic
66C2×D4×C173C2312yes—
67C2×D4×C173C2311yescharacteristic
68D8×C173—42no—
69D8×C173—42no—
70C2×C8×C173C2312yes—
71C42×C173C2311yescharacteristic
72C22×D173—21no—
73C22×D173—42no—
74C22×D173—82no—
75C22×D173C2×D411yescharacteristic
76Dic346—22no—
77D692—22no—
78C22⋊D173—42no—
79C22⋊D173—42no—
80C22⋊D173—82no—
81C22⋊D173—82no—
82C2×Dic173—21no—
83C2×Dic173—22no—
84C2×Dic173—42no—
85C2×Dic173—82no—
86C2×Dic173C2×D411yescharacteristic
87C2×Dic173C2×D412yes—
88C4×D173—22no—
89#13063436—22no—
90C23×C173—21no—
91C23×C173—22no—
92C23×C173—21no—
93D4×C173—22no—
94D4×C173—22no—
95D4×C173—22no—
96D4×C173—22no—
97D4×C173—82no—
98D4×C173—82no—
99C2×C4×C173C2×D412yes—
100C2×C4×C173C2411yescharacteristic
101C8×C173—22no—
102C22×C173—42no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173—42no—
106C22×C173—42no—
107C22×C173—42no—
108C22×C173—42no—
109C22×C173C22×D411yescharacteristic
110C2×D173—21no—
111C2×D173—22no—
112C2×D173—41no—
113C2×D173—42no—
114C2×D173—84no—
115C4×C173—22no—
116C4×C173H(5,𝔽2)12yes—
117Dic173—21no—
118Dic173—22no—
119Dic173—41no—
120Dic173—42no—
121C2×C173—81no—
122C2×C173—82no—
123C2×C173—81no—
124C2×C173#53511yescharacteristic
125C2×C173#57612yes—
126D173—41no—
127D173—81no—
128C173#322011yescharacteristic, complemented, Sylow
129#3220—1731nomaximal, Sylow
130#540—1731no—
131#535—1731no—
132#496—1731no—
133#493D17311yescharacteristic, complemented
134#490—1731no—
135#488—1731no—
136#463—1732no—
137#459—1732no—
138#449—1732no—
139#447—1732no—
140#433—1731no—
141C2×#41—1731no—
142C22×D4—1731no—
143C2×SD16—1732no—
144C2×D8—1732no—
145C2×D8C2×D17312yes—
146C2×D8C2×D17312yescomplemented
147#127—1731no—
148#127C2×D17311yescharacteristic
149#127C2×D17311yescharacteristic, complemented
150#125—1731no—
151#124—1731no—
152#124—1731no—
153#123—3462no—
154#121—3462no—
155#121—3462no—
156#121—1732no—
157#120—3462no—
158C4×D4—1732no—
159#117—1731no—
160C4⋊C8—1732no—
161#103—1732no—
162#102—1732no—
163#102—1732no—
164#102—1732no—
165C22⋊C8—1732no—
166#97—1731no—
167C4×C8C2×D17311yescharacteristic, complemented
168C24—3461no—
169#41—6922no—
170C2×Q8—1731no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—3462no—
174C2×D4—3461no—
175C2×D4—3462no—
176C2×D4—3461no—
177C2×D4—1731no—
178C2×D4—6922no—
179C2×D4C22×D17311yescharacteristic
180C2×D4C22×D17311yescharacteristic
181C2×D4C22×D17312yes—
182C22×C4—3462no—
183C22×C4—3461no—
184C22×C4—1731no—
185SD16—6922no—
186D8—42no—
187D8—42no—
188D8—6922no—
189C2×C8—1732no—
190C2×C8C22×D17312yes—
191C4⋊C4—3462no—
192C4⋊C4—1732no—
193C4⋊C4—1732no—
194C22⋊C4—3462no—
195C22⋊C4—3462no—
196C22⋊C4—3462no—
197C22⋊C4—3462no—
198C22⋊C4—1732no—
199C42—1731no—
200C42C22×D17311yescharacteristic
201C23—22no—
202C23—21no—
203C23—21no—
204C23—3461no—
205C23—13842no—
206C23—1731no—
207C23—6922no—
208Q8—3462no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—22no—
213D4—82no—
214D4—82no—
215D4—3462no—
216D4—13842no—
217D4—13842no—
218D4—6922no—
219D4—6922no—
220C2×C4—3462no—
221C2×C4—3461no—
222C2×C4—3462no—
223C2×C4—13842no—
224C2×C4—1732no—
225C2×C4—1731no—
226C2×C4—6922no—
227C2×C4D4×D17312yes—
228C2×C4C23×D17311yescharacteristic
229C8—22no—
230C8—3462no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—3462no—
239C22—3461no—
240C22—13844no—
241C22—6922no—
242C22—6921no—
243C22C2×D4×D17311yescharacteristic
244C4—22no—
245C4—3462no—
246C4—3461no—
247C4—6922no—
248C4—6921no—
249C4#1914972412yes—
250C2—82no—
251C2—81no—
252C2—81no—
253C2—13841no—
254C2—6921no—
255C2#2601205411yescharacteristic
256C2#2601229212yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#493×C173C2
socleC22×C173C22×D4
hypercenter#493D173
hypocenterC173#3220
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #26012095C2semidirect
2. #26012054C2semidirect
3. #26011878C2semidirect
4. #26011855C2semidirect
5. #26011846C2semidirect
6. #26011704C2semidirect
7. #26011684C2semidirect
8. #26011633C2semidirect
9. #26011623C2semidirect
10. #26011550C2semidirect
11. #493×C173C2semidirect
14. C2×#14583814C22semidirect
17. #19149671C22semidirect
25. D4×Dic173C22semidirect
27. #19149608C22semidirect
28. #19149604C22semidirect
31. #19149600C22semidirect
33. #19149590C22semidirect
35. C2×D8×C173C22semidirect
37. #127×C173C22semidirect
38. C4×C8×C173C22semidirect
128. C173#3220semidirect
133. #493D173semidirect
146. C2×D8C2×D173semidirect
149. #127C2×D173semidirect
167. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120952
#260120542
#260118782
#260118552
#260118462
#260117042
#260116842
#260116332
#260116232
#260115502
#493×C1732
#3220173