Finite Group Databasebeta

← #36364383

Subgroups — #36364383

Order 22144.

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

normal subgroups only

This group has 410 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
1C22⋊Q8×D173C212yesmaximal
2#26011823C212yesmaximal, complemented
3#26011820C212yesmaximal, complemented
4#26011749C211yesmaximal, characteristic, complemented
5#26011747C212yesmaximal, complemented
6#26011648C212yesmaximal, complemented
7#26011645C212yesmaximal, complemented
8#26011454C211yesmaximal, characteristic
9#470×C173C211yesmaximal, characteristic, complemented
10C2×Q8×D173C2212yes—
11C22×C4×D173C2211yescharacteristic
12C2×Dic692C2211yescharacteristic, complemented
13C2×#14583816C2212yescomplemented
14C2×#14583810C2211yescharacteristic
15#19149659C2212yes—
16#19149658C2212yes—
17#19149657—22no—
18#19149656—22no—
19#19149655C2212yes—
20C4⋊C4×D173—22no—
21C4⋊C4×D173C2212yes—
22C22⋊C4×D173—22no—
23#19149611C2211yescharacteristic, complemented
24#19149610C2212yescomplemented
25#19149604C2212yescomplemented
26#19149603C2212yescomplemented
27#19149602C2212yescomplemented
28#19149591C2212yes—
29#19149591C2212yes—
30C2×Q16×C173C2211yescharacteristic, complemented
31C2×M16×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#106×C173C2211yescharacteristic, complemented
34#103×C173C2212yescomplemented
35C23×D173D411yescharacteristic
36C2×Dic346—22no—
37C2×Dic346C2312yes—
38Q8×D173—42no—
39Q8×D173—42no—
40C22×Dic173D411yescharacteristic
41C2×C4×D173—22no—
42C2×C4×D173—22no—
43C2×C4×D173—41no—
44C2×C4×D173—41no—
45C2×C4×D173D412yes—
46C2×C4×D173C2312yes—
47Dic692—41no—
48#14583816—42no—
49#14583810—41no—
50C2×#13063436C2312yes—
51C4⋊Dic173—22no—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54#14583805—22no—
55#14583805—22no—
56C22⋊Dic173—22no—
57#14583803—22no—
58#14583803—22no—
59C2×Q8×C173C2312yes—
60C22×C4×C173C2311yescharacteristic
61Q16×C173—41no—
62M16×C173—41no—
63C2×C8×C173C2311yescharacteristic
64C2×C8×C173C2311yescharacteristic
65C4⋊C4×C173—22no—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22×D173—24no—
69C22×D173—41no—
70C22×D173C2×D412yes—
71Dic346—22no—
72Dic346—22no—
73Dic346—44no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76C2×Dic173—41no—
77C2×Dic173C2×D412yes—
78C4×D173—22no—
79C4×D173—22no—
80C4×D173—42no—
81C4×D173—44no—
82C4×D173—44no—
83#13063436—22no—
84C23×C173C2×D411yescharacteristic
85Q8×C173—22no—
86Q8×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2411yescharacteristic
93C8×C173—21no—
94C8×C173—21no—
95C22×C173—21no—
96C22×C173—22no—
97C22×C173C22×D411yescharacteristic
98C2×D173—22no—
99C2×D173—22no—
100C2×D173—44no—
101C2×D173#13714yes—
102C4×C173—21no—
103C4×C173—42no—
104C4×C173—42no—
105C4×C173H(5,𝔽2)11yescharacteristic
106C4×C173H(5,𝔽2)11yescharacteristic
107Dic173—22no—
108Dic173—42no—
109Dic173—42no—
110C2×C173—41no—
111C2×C173#53411yescharacteristic
112C2×C173C2×#13712yes—
113D173—24no—
114C173#319511yescharacteristic, complemented, Sylow
115#3195—1731nomaximal, Sylow
116C2×C22⋊Q8—1732no—
117#483—1732no—
118#483—1732no—
119#470—1731no—
120#470—1732no—
121#470D17311yescharacteristic, complemented
122#451—1732no—
123#451—1732no—
124#407—1731no—
125C22×Q8—1732no—
126C23×C4—1731no—
127C2×Q16—1732no—
128C2×Q16—1731no—
129C2×Q16C2×D17311yescharacteristic, complemented
130C2×M16—1731no—
131C2×M16C2×D17311yescharacteristic
132C22⋊Q8—3462no—
133C22⋊Q8—3462no—
134C22⋊Q8—1732no—
135C22⋊Q8—1732no—
136C22⋊Q8—1732no—
137C22⋊Q8C2×D17312yes—
138C2×C4⋊C4—3462no—
139C2×C4⋊C4—1732no—
140C2×C22⋊C4—3462no—
141#106—1731no—
142#106—1732no—
143#106C2×D17311yescharacteristic, complemented
144#103—1732no—
145#103—1732no—
146#103—1732no—
147#103C2×D17312yescomplemented
148C22⋊C8—1732no—
149C22⋊C8—1732no—
150C24—1731no—
151C2×Q8—3462no—
152C2×Q8—1732no—
153C2×Q8—6922no—
154C2×Q8—6922no—
155C2×Q8C22×D17312yes—
156C22×C4—3462no—
157C22×C4—3462no—
158C22×C4—1732no—
159C22×C4—1731no—
160C22×C4—1732no—
161C22×C4—6921no—
162C22×C4—6921no—
163C22×C4C22×D17311yescharacteristic
164Q16—41no—
165Q16—6921no—
166Q16—6922no—
167M16—41no—
168M16—6921no—
169C2×C8—1732no—
170C2×C8C22×D17311yescharacteristic
171C2×C8C22×D17311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—3462no—
174C4⋊C4—3462no—
175C4⋊C4—3462no—
176C4⋊C4—3462no—
177C4⋊C4—1732no—
178C4⋊C4C22×D17312yes—
179C22⋊C4—22no—
180C22⋊C4—3462no—
181C22⋊C4—3462no—
182C22⋊C4—3462no—
183C23—3464no—
184C23—1732no—
185C23—6921no—
186C23D4×D17311yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—3462no—
190Q8—3462no—
191Q8—6924no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3462no—
197C2×C4—3462no—
198C2×C4—3462no—
199C2×C4—3462no—
200C2×C4—1732no—
201C2×C4—6922no—
202C2×C4—6924no—
203C2×C4—6924no—
204C2×C4—6921no—
205C2×C4D4×D17311yescharacteristic
206C2×C4C23×D17311yescharacteristic
207C8—21no—
208C8—21no—
209C8—3462no—
210C22—22no—
211C22—21no—
212C22—3462no—
213C22—3462no—
214C22—1734no—
215C22—6924no—
216C22C2×D4×D17311yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—3462no—
221C4—6922no—
222C4—6922no—
223C4#1914972411yescharacteristic
224C4#1914972511yescharacteristic
225C2—41no—
226C2—3464no—
227C2#2601204911yescharacteristic
228C2#137×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#470D173
hypocenterC173#3195
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #26011823C2semidirect
3. #26011820C2semidirect
4. #26011749C2semidirect
5. #26011747C2semidirect
6. #26011648C2semidirect
7. #26011645C2semidirect
9. #470×C173C2semidirect
12. C2×Dic692C22semidirect
13. C2×#14583816C22semidirect
23. #19149611C22semidirect
24. #19149610C22semidirect
25. #19149604C22semidirect
26. #19149603C22semidirect
27. #19149602C22semidirect
30. C2×Q16×C173C22semidirect
33. #106×C173C22semidirect
34. #103×C173C22semidirect
114. C173#3195semidirect
121. #470D173semidirect
129. C2×Q16C2×D173semidirect
143. #106C2×D173semidirect
147. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
#260118232
#260118202
#260117492
#260117472
#260116482
#260116452
#260114542
#470×C1732
#3195173