Finite Group Databasebeta

← #36364334

Subgroups — #36364334

Order 22144.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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#26011828C212yesmaximal, complemented
3#26011825C212yesmaximal, complemented
4#26011732C211yesmaximal, characteristic, complemented
5#26011730C212yesmaximal, complemented
6#26011648C212yesmaximal, complemented
7#26011645C212yesmaximal, complemented
8C2×#19149591C211yesmaximal, characteristic
9#467×C173C211yesmaximal, characteristic, complemented
10C2×Q8×D173C2212yes—
11C22×C4×D173C2211yescharacteristic
12C2×Dic692C2211yescharacteristic, complemented
13C2×#14583816C2212yescomplemented
14C22×#13063436C2211yescharacteristic
15#19149659C2212yes—
16#19149658C2212yes—
17#19149657—22no—
18#19149656—22no—
19#19149655C2212yes—
20C4⋊C4×D173—22no—
21C4⋊C4×D173C2212yes—
22C22⋊C4×D173—22no—
23#19149613C2211yescharacteristic, complemented
24#19149612C2212yescomplemented
25#19149604C2212yescomplemented
26#19149603C2212yescomplemented
27#19149602C2212yescomplemented
28#19149591C2212yes—
29#19149591C2212yes—
30C2×Q16×C173C2211yescharacteristic, complemented
31C22×C8×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#107×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—
49C2×#13063436—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—
62C2×C8×C173—41no—
63C2×C8×C173C2311yescharacteristic
64C2×C8×C173C2311yescharacteristic
65C4⋊C4×C173—22no—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173Q1614yescomplemented
71C22×D173C2×D412yes—
72Dic346—22no—
73Dic346—22no—
74Dic346—44no—
75C2×Dic173—22no—
76C2×Dic173—22no—
77C2×Dic173—41no—
78C2×Dic173C2×D412yes—
79C4×D173—22no—
80C4×D173—22no—
81C4×D173—42no—
82C4×D173—44no—
83C4×D173—44no—
84#13063436—22no—
85C23×C173C2×D411yescharacteristic
86Q8×C173—22no—
87Q8×C173—22no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2411yescharacteristic
94C8×C173—21no—
95C8×C173—21no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173C2×Q1612yes—
99C22×C173C22×D411yescharacteristic
100C2×D173—22no—
101C2×D173—22no—
102C2×D173—44no—
103C2×D173C2×Q1614yes—
104C4×C173—21no—
105C4×C173—42no—
106C4×C173—42no—
107C4×C173H(5,𝔽2)11yescharacteristic
108C4×C173H(5,𝔽2)11yescharacteristic
109Dic173—22no—
110Dic173—42no—
111Dic173—42no—
112C2×C173—22no—
113C2×C173#53411yescharacteristic
114C2×C173C22×Q1611yescharacteristic
115C2×C173#57511yescharacteristic
116D173—24no—
117C173#318911yescharacteristic, complemented, Sylow
118#3189—1731nomaximal, Sylow
119C2×C22⋊Q8—1732no—
120#484—1732no—
121#484—1732no—
122#467—1731no—
123#467—1732no—
124#467D17311yescharacteristic, complemented
125#451—1732no—
126#451—1732no—
127C2×C22⋊C8—1731no—
128C22×Q8—1732no—
129C23×C4—1731no—
130C2×Q16—1732no—
131C2×Q16—1731no—
132C2×Q16C2×D17311yescharacteristic, complemented
133C22×C8—1731no—
134C22×C8C2×D17311yescharacteristic
135C22⋊Q8—3462no—
136C22⋊Q8—3462no—
137C22⋊Q8—1732no—
138C22⋊Q8—1732no—
139C22⋊Q8—1732no—
140C22⋊Q8C2×D17312yes—
141C2×C4⋊C4—3462no—
142C2×C4⋊C4—1732no—
143C2×C22⋊C4—3462no—
144#107—1731no—
145#107—1732no—
146#107C2×D17311yescharacteristic, complemented
147#103—1732no—
148#103—1732no—
149#103—1732no—
150#103C2×D17312yescomplemented
151C22⋊C8—1732no—
152C22⋊C8—1732no—
153C24—1731no—
154C2×Q8—3462no—
155C2×Q8—1732no—
156C2×Q8—6922no—
157C2×Q8—6922no—
158C2×Q8C22×D17312yes—
159C22×C4—3462no—
160C22×C4—3462no—
161C22×C4—1732no—
162C22×C4—1731no—
163C22×C4—1732no—
164C22×C4—6921no—
165C22×C4—6921no—
166C22×C4C22×D17311yescharacteristic
167Q16—41no—
168Q16—6921no—
169Q16—6922no—
170C2×C8—41no—
171C2×C8—1732no—
172C2×C8—6921no—
173C2×C8C22×D17311yescharacteristic
174C2×C8C22×D17311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—3462no—
177C4⋊C4—3462no—
178C4⋊C4—3462no—
179C4⋊C4—3462no—
180C4⋊C4—1732no—
181C4⋊C4C22×D17312yes—
182C22⋊C4—22no—
183C22⋊C4—3462no—
184C22⋊C4—3462no—
185C22⋊C4—3462no—
186C23—1732no—
187C23—1734no—
188C23—6921no—
189C23—6921no—
190C23D4×D17311yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—3462no—
194Q8—3462no—
195Q8—6924no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—3462no—
201C2×C4—3462no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—1732no—
205C2×C4—6922no—
206C2×C4—6924no—
207C2×C4—6924no—
208C2×C4—6921no—
209C2×C4D4×D17311yescharacteristic
210C2×C4C23×D17311yescharacteristic
211C8—21no—
212C8—21no—
213C8—3462no—
214C22—21no—
215C22—21no—
216C22—3462no—
217C22—3462no—
218C22—1734no—
219C22—6924no—
220C22Q16×D17312yes—
221C22C2×D4×D17311yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—3462no—
226C4—6922no—
227C4—6922no—
228C4#1914972411yescharacteristic
229C4#1914972511yescharacteristic
230C2—22no—
231C2—3464no—
232C2#2601204911yescharacteristic
233C2C2×Q16×D17311yescharacteristic
234C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#467D173
hypocenterC173#3189
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
2. #26011828C2semidirect
3. #26011825C2semidirect
4. #26011732C2semidirect
5. #26011730C2semidirect
6. #26011648C2semidirect
7. #26011645C2semidirect
9. #467×C173C2semidirect
12. C2×Dic692C22semidirect
13. C2×#14583816C22semidirect
23. #19149613C22semidirect
24. #19149612C22semidirect
25. #19149604C22semidirect
26. #19149603C22semidirect
27. #19149602C22semidirect
30. C2×Q16×C173C22semidirect
33. #107×C173C22semidirect
34. #103×C173C22semidirect
70. C22×D173Q16semidirect
117. C173#3189semidirect
124. #467D173semidirect
132. C2×Q16C2×D173semidirect
146. #107C2×D173semidirect
150. #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
#260118282
#260118252
#260117322
#260117302
#260116482
#260116452
C2×#191495912
#467×C1732
#3189173