Finite Group Databasebeta

← #36364357

Subgroups — #36364357

Order 22144.

424 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 113 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#26012004C212yesmaximal, complemented
2#26011831C212yesmaximal, complemented
3#26011818C212yesmaximal, complemented
4#26011727C211yesmaximal, characteristic, complemented
5#26011722C212yesmaximal, complemented
6#26011652C212yesmaximal, complemented
7#26011639C212yesmaximal, complemented
8C2×#19149591C211yesmaximal, characteristic, complemented
9#467×C173C211yesmaximal, characteristic, complemented
10C2×#14583827C2212yes—
11C22×C4×D173C2211yescharacteristic
12C2×#14583813C2212yescomplemented
13C2×D1384C2211yescharacteristic, complemented
14C22×#13063436C2211yescharacteristic
15#19149664—22no—
16#19149655C2212yes—
17#19149654C2212yes—
18#19149653C2212yes—
19#19149649—22no—
20#19149641—22no—
21#19149639—22no—
22#19149637C2212yes—
23#19149613C2211yescharacteristic, complemented
24#19149610C2212yescomplemented
25#19149603C2212yescomplemented
26#19149601C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yes—
29#19149591C2212yes—
30C2×Q16×C173C2211yescharacteristic, complemented
31C22×C8×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#107×C173C2211yescharacteristic, complemented
34#103×C173C2212yescomplemented
35C23×D173D411yescharacteristic
36#14583827—42no—
37#14583827—42no—
38C2×D692—22no—
39C2×D692—22no—
40C2×D692C2312yes—
41C2×C22⋊D173—22no—
42C22×Dic173D411yescharacteristic
43C2×C4×D173—22no—
44C2×C4×D173—41no—
45C2×C4×D173—41no—
46C2×C4×D173D412yes—
47C2×C4×D173C2312yes—
48#14583813—42no—
49D1384—41no—
50C2×#13063436—41no—
51C2×#13063436C2312yes—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54#14583803—22no—
55#14583803—22no—
56#14583803—22no—
57#14583803—22no—
58C4×Dic173—22no—
59C2×Q8×C173C2312yes—
60C22×C4×C173C2311yescharacteristic
61Q16×C173—41no—
62C2×C8×C173—41no—
63C2×C8×C173C2311yescharacteristic
64C2×C8×C173C2311yescharacteristic
65C4⋊C4×C173—22no—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22×D173—22no—
69C22×D173—22no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173C2×D412yes—
73D692—22no—
74D692—22no—
75D692—44no—
76D692—44no—
77C22⋊D173—44no—
78C2×Dic173—22no—
79C2×Dic173—41no—
80C2×Dic173C2×D412yes—
81C4×D173—22no—
82C4×D173—22no—
83C4×D173—42no—
84C4×D173—44no—
85#13063436—22no—
86C23×C173C2×D411yescharacteristic
87Q8×C173—22no—
88Q8×C173—22no—
89C2×C4×C173—21no—
90C2×C4×C173—21no—
91C2×C4×C173—22no—
92C2×C4×C173—22no—
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2411yescharacteristic
95C8×C173—21no—
96C8×C173—21no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173#13512yes—
100C22×C173C22×D411yescharacteristic
101C2×D173—22no—
102C2×D173—22no—
103C2×D173—22no—
104C2×D173—44no—
105C2×D173—42no—
106C2×D173—42no—
107C2×D173—42no—
108C4×C173—21no—
109C4×C173—42no—
110C4×C173—42no—
111C4×C173H(5,𝔽2)11yescharacteristic
112C4×C173H(5,𝔽2)11yescharacteristic
113Dic173—22no—
114Dic173—42no—
115C2×C173—22no—
116C2×C173#53411yescharacteristic
117C2×C173C2×#13511yescharacteristic
118C2×C173#57511yescharacteristic
119D173—42no—
120D173—82no—
121C173#319111yescharacteristic, complemented, Sylow
122#3191—1731nomaximal, Sylow
123#525—1732no—
124#485—1732no—
125#482—1732no—
126#467D17311yescharacteristic, complemented
127#466—1731no—
128#465—1732no—
129#452—1732no—
130#449—1732no—
131C2×C22⋊C8—1731no—
132C2×#41—1732no—
133C23×C4—1731no—
134C2×Q16C2×D17311yescharacteristic, complemented
135C2×SD16—1732no—
136C2×D8—1731no—
137C22×C8—1731no—
138C22×C8C2×D17311yescharacteristic
139#123—3462no—
140C22⋊Q8—1732no—
141C22⋊Q8C2×D17312yes—
142#121—1732no—
143#121—1732no—
144#120—3462no—
145C4×D4—3462no—
146C4×D4—3462no—
147#117—1732no—
148#107—1731no—
149#107C2×D17311yescharacteristic, complemented
150#106—1732no—
151#103—1732no—
152#103C2×D17312yescomplemented
153#102—1732no—
154#102—1732no—
155C22⋊C8—1732no—
156C22⋊C8—1732no—
157C24—1731no—
158#41—6922no—
159#41—6922no—
160C2×Q8C22×D17312yes—
161C2×D4—3462no—
162C2×D4—3462no—
163C2×D4—3462no—
164C2×D4—1732no—
165C22×C4—3462no—
166C22×C4—1732no—
167C22×C4—1731no—
168C22×C4—1732no—
169C22×C4—6921no—
170C22×C4—6921no—
171C22×C4C22×D17311yescharacteristic
172Q16—41no—
173SD16—6922no—
174D8—6921no—
175C2×C8—41no—
176C2×C8—1732no—
177C2×C8—6921no—
178C2×C8C22×D17311yescharacteristic
179C2×C8C22×D17311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—3462no—
182C4⋊C4—1732no—
183C4⋊C4C22×D17312yes—
184C22⋊C4—22no—
185C22⋊C4—3462no—
186C22⋊C4—3462no—
187C22⋊C4—3462no—
188C22⋊C4—3462no—
189C42—3462no—
190C23—3462no—
191C23—3462no—
192C23—1732no—
193C23—6921no—
194C23—6921no—
195C23D4×D17311yescharacteristic
196Q8—22no—
197Q8—22no—
198D4—3462no—
199D4—3462no—
200D4—6924no—
201D4—6924no—
202D4—6924no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—3462no—
208C2×C4—3462no—
209C2×C4—3462no—
210C2×C4—1732no—
211C2×C4—6922no—
212C2×C4—6924no—
213C2×C4—6921no—
214C2×C4D4×D17311yescharacteristic
215C2×C4C23×D17311yescharacteristic
216C8—21no—
217C8—21no—
218C8—3462no—
219C22—21no—
220C22—21no—
221C22—3462no—
222C22—3462no—
223C22—3462no—
224C22—6924no—
225C22—6922no—
226C22—6922no—
227C22—6922no—
228C22#1914970212yes—
229C22C2×D4×D17311yescharacteristic
230C4—21no—
231C4—42no—
232C4—42no—
233C4—3462no—
234C4—6922no—
235C4#1914972411yescharacteristic
236C4#1914972511yescharacteristic
237C2—22no—
238C2—13842no—
239C2—6922no—
240C2#2601204911yescharacteristic
241C2C2×#1914970211yescharacteristic
242C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
hypercenter#467D173
hypocenterC173#3191
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26012004C2semidirect
2. #26011831C2semidirect
3. #26011818C2semidirect
4. #26011727C2semidirect
5. #26011722C2semidirect
6. #26011652C2semidirect
7. #26011639C2semidirect
8. C2×#19149591C2semidirect
9. #467×C173C2semidirect
12. C2×#14583813C22semidirect
13. C2×D1384C22semidirect
23. #19149613C22semidirect
24. #19149610C22semidirect
25. #19149603C22semidirect
26. #19149601C22semidirect
27. #19149599C22semidirect
30. C2×Q16×C173C22semidirect
33. #107×C173C22semidirect
34. #103×C173C22semidirect
121. C173#3191semidirect
126. #467D173semidirect
134. C2×Q16C2×D173semidirect
149. #107C2×D173semidirect
152. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
#260118312
#260118182
#260117272
#260117222
#260116522
#260116392
C2×#191495912
#467×C1732
#3191173