Finite Group Databasebeta

← #36363217

Subgroups — #36363217

Order 22144.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14583816C211yesmaximal, characteristic, complemented
2C2×#19149687C211yesmaximal, characteristic
3#26012067C211yesmaximal, characteristic
4#26012065C211yesmaximal, characteristic
5#26011748C214yesmaximal, complemented
6#26011731C214yesmaximal, complemented
7#26011527C211yesmaximal, characteristic, complemented
8#26011501C211yesmaximal, characteristic, complemented
9#419×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C22×Dic346C2211yescharacteristic
12C2×#14583816—24no—
13C2×#14583816C2212yes—
14C2×#14583816C2212yes—
15#19149687—24no—
16C2×#14583809C2212yes—
17C22×#13063436C2211yescharacteristic
18C2×C8×D173C2212yes—
19#19149679—21no—
20#19149677—21no—
21#19149668—21no—
22#19149667—21no—
23#19149663—22no—
24#19149662—22no—
25#19149659C2214yes—
26#19149655C2214yes—
27#19149638C2211yescharacteristic
28#19149612C2212yes—
29#19149610C2212yes—
30#19149604C2214yes—
31C22×Q8×C173C2211yescharacteristic
32C2×M16×C173C2211yescharacteristic
33#117×C173C2211yescharacteristic
34#103×C173C2214yes—
35#14583826—22no—
36#14583826—22no—
37C2×Dic346—21no—
38C2×Dic346—22no—
39C2×Dic346—22no—
40C2×Dic346D411yescharacteristic
41C2×Dic346C2312yes—
42C2×D692D411yescharacteristic
43C2×C22⋊D173D412yes—
44C22×Dic173—21no—
45C2×C4×D173C2312yes—
46#14583816—24no—
47#14583816—24no—
48#14583809—24no—
49C2×#13063436D414yes—
50C2×#13063436C2311yescharacteristic
51C2×#13063436C2311yescharacteristic
52C8×D173—24no—
53C4⋊Dic173—24no—
54C4⋊Dic173C2312yes—
55#14583805—24no—
56C22⋊Dic173—21no—
57#14583803—24no—
58#14583803—24no—
59C4×Dic173—21no—
60C2×Q8×C173—21no—
61C2×Q8×C173—22no—
62C2×Q8×C173—22no—
63C2×Q8×C173C2312yes—
64C22×C4×C173—21no—
65C22×C4×C173C2311yescharacteristic
66M16×C173—22no—
67C2×C8×C173C2312yes—
68C4⋊C4×C173C2312yes—
69C22⋊C4×C173—21no—
70C42×C173—21no—
71C22×D173C2×D412yes—
72Dic346—22no—
73Dic346—22no—
74Dic346—22no—
75Dic346—42no—
76D692—22no—
77C22⋊D173—24no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—42no—
81C2×Dic173C2×D412yes—
82C4×D173—22no—
83C4×D173—22no—
84#13063436C2×D412yes—
85#13063436C2×D412yes—
86C23×C173C2×D411yescharacteristic
87Q8×C173—22no—
88Q8×C173—22no—
89Q8×C173—42no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173—42no—
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2×D412yes—
95C2×C4×C173#4112yes—
96C2×C4×C173C2411yescharacteristic
97C8×C173—22no—
98C22×C173—22no—
99C22×C173#12112yes—
100C22×C173C22×D411yescharacteristic
101C2×D173—22no—
102C2×D173—24no—
103C4×C173—42no—
104C4×C173—42no—
105C4×C173#12112yes—
106C4×C173C22×D411yescharacteristic
107C4×C173C2×#4111yescharacteristic
108Dic173—22no—
109Dic173—42no—
110Dic173—42no—
111C2×C173—22no—
112C2×C173C2×#12111yescharacteristic
113C2×C173#57812yes—
114D173—42no—
115C173#305911yescharacteristic, complemented, Sylow
116#3059—1731nomaximal, Sylow
117C22×Q16—1731no—
118C2×#131—1731no—
119#536—1731no—
120#536—1731no—
121#470—1734no—
122#467—1734no—
123#427—1731no—
124#419—1731no—
125#419D17311yescharacteristic, complemented
126C2×#41—1731no—
127C22×Q8—1731no—
128C22×Q8C2×D17311yescharacteristic
129C2×Q16—3464no—
130C2×Q16—1732no—
131C2×Q16—1732no—
132#131—3464no—
133C2×M16—1732no—
134C2×M16C2×D17311yescharacteristic
135C22×C8—1732no—
136C22×C8—1731no—
137C4⋊Q8—3461no—
138C4⋊Q8—3461no—
139#124—3461no—
140#124—3461no—
141#123—3462no—
142#123—3462no—
143C22⋊Q8—1734no—
144C22⋊Q8—1734no—
145#117—1731no—
146#117C2×D17311yescharacteristic
147#107—1732no—
148#106—1732no—
149#103—1734no—
150#103C2×D17314yes—
151#41—3462no—
152#41—3462no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3461no—
157C2×Q8—3462no—
158C2×Q8—3462no—
159C2×Q8—1732no—
160C2×Q8—1731no—
161C2×Q8C22×D17312yes—
162C2×D4—1732no—
163C2×D4—1731no—
164C22×C4—21no—
165C22×C4—3461no—
166C22×C4—1732no—
167C22×C4C22×D17311yescharacteristic
168Q16—3464no—
169Q16—3464no—
170M16—22no—
171M16—3464no—
172C2×C8—3464no—
173C2×C8—1734no—
174C2×C8—1731no—
175C2×C8—1731no—
176C2×C8C22×D17312yes—
177C4⋊C4—3464no—
178C4⋊C4—3464no—
179C4⋊C4—1732no—
180C4⋊C4C22×D17312yes—
181C22⋊C4—21no—
182C22⋊C4—3464no—
183C22⋊C4—3464no—
184C22⋊C4—3461no—
185C42—21no—
186C42—3461no—
187C23—1732no—
188C23D4×D17311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3462no—
193Q8—3462no—
194Q8—3462no—
195Q8—6922no—
196D4—3464no—
197D4—3462no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—3462no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—1732no—
206C2×C4—6922no—
207C2×C4D4×D17311yescharacteristic
208C2×C4D4×D17312yes—
209C2×C4#1458382612yes—
210C2×C4C23×D17311yescharacteristic
211C8—22no—
212C8—1732no—
213C8—1732no—
214C22—22no—
215C22—3462no—
216C22—3464no—
217C22#1914965212yes—
218C22C2×D4×D17311yescharacteristic
219C4—42no—
220C4—42no—
221C4—3462no—
222C4—6922no—
223C4—6922no—
224C4#1914965212yes—
225C4C2×D4×D17311yescharacteristic
226C4C2×#1458382611yescharacteristic
227C2—22no—
228C2—6922no—
229C2C2×#1914965211yescharacteristic
230C2#2601230712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#419×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#419D173
hypocenterC173#3059
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14583816C2semidirect
5. #26011748C2semidirect
6. #26011731C2semidirect
7. #26011527C2semidirect
8. #26011501C2semidirect
9. #419×C173C2semidirect
115. C173#3059semidirect
125. #419D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838162
C2×#191496872
#260120672
#260120652
#260117482
#260117312
#260115272
#260115012
#419×C1732
#3059173