Finite Group Databasebeta

← #36363215

Subgroups — #36363215

Order 22144.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19149686C211yesmaximal, characteristic
3#26012062C211yesmaximal, characteristic
4#26011747C214yesmaximal, complemented
5#26011730C214yesmaximal, complemented
6#26011527C211yesmaximal, characteristic, complemented
7#26011500C211yesmaximal, characteristic, complemented
8#26011499C211yesmaximal, characteristic, complemented
9#536×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11C2×#14583816—24no—
12C2×#14583816C2212yes—
13C2×#14583816C2212yes—
14#19149686—24no—
15C2×#14583810C2212yes—
16C2×#14583810C2211yescharacteristic
17C22×#13063436C2212yes—
18C22×#13063436C2211yescharacteristic
19#19149677—21no—
20#19149666—21no—
21#19149660—22no—
22#19149658C2214yes—
23#19149638C2211yescharacteristic
24#19149612C2212yes—
25#19149610C2212yes—
26#19149603C2214yes—
27#19149602C2214yes—
28C2×#41×C173C2211yescharacteristic
29C22×Q8×C173C2211yescharacteristic
30C4⋊Q8×C173—21no—
31#124×C173—21no—
32#123×C173—22no—
33C22⋊Q8×C173C2214yes—
34#117×C173C2211yescharacteristic
35C2×Dic346—21no—
36C2×Dic346—22no—
37C2×Dic346—22no—
38C2×Dic346C2312yes—
39C22×Dic173—21no—
40#14583816—24no—
41#14583816—24no—
42#14583810—24no—
43#14583810—22no—
44C2×#13063436—24no—
45C2×#13063436D414yes—
46C2×#13063436C2312yes—
47C2×#13063436C2311yescharacteristic
48C2×#13063436C2311yescharacteristic
49C4⋊Dic173C2312yes—
50#14583805—24no—
51C22⋊Dic173—21no—
52C22⋊Dic173—24no—
53C4×Dic173—21no—
54#41×C173—22no—
55#41×C173—22no—
56C2×Q8×C173—21no—
57C2×Q8×C173—22no—
58C2×Q8×C173—22no—
59C2×Q8×C173D411yescharacteristic
60C2×Q8×C173C2312yes—
61C2×D4×C173D412yes—
62C2×D4×C173D411yescharacteristic
63C22×C4×C173—21no—
64C22×C4×C173C2312yes—
65C22×C4×C173C2311yescharacteristic
66C4⋊C4×C173—24no—
67C4⋊C4×C173C2312yes—
68C22⋊C4×C173—21no—
69C22⋊C4×C173—24no—
70C42×C173—21no—
71Dic346—22no—
72Dic346—22no—
73Dic346—42no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76C2×Dic173—42no—
77#13063436—22no—
78#13063436C2×D412yes—
79#13063436C2×D412yes—
80C23×C173C2×D412yes—
81C23×C173C2×D411yescharacteristic
82Q8×C173—22no—
83Q8×C173—22no—
84Q8×C173—22no—
85Q8×C173—42no—
86D4×C173—24no—
87D4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173—42no—
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2×D412yes—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173#4112yes—
97C2×C4×C173C2411yescharacteristic
98C22×C173—22no—
99C22×C173—22no—
100C22×C173—24no—
101C22×C173#12112yes—
102C22×C173C22×D411yescharacteristic
103C4×C173—22no—
104C4×C173—42no—
105C4×C173—42no—
106C4×C173#12112yes—
107C4×C173C22×D411yescharacteristic
108C4×C173C2×#4111yescharacteristic
109Dic173—42no—
110Dic173—42no—
111C2×C173—22no—
112C2×C173—42no—
113C2×C173C2×#12111yescharacteristic
114C2×C173#57812yes—
115C173#305911yescharacteristic, complemented, Sylow
116#3059—1731nomaximal, Sylow
117C22×Q16—1731no—
118C2×#131—1731no—
119#536—1731no—
120#536D17311yescharacteristic
121#470—1734no—
122#467—1734no—
123#427—1731no—
124#419—1731no—
125#419—1731no—
126C2×#41C2×D17311yescharacteristic
127C22×Q8—1731no—
128C22×Q8C2×D17311yescharacteristic
129C2×Q16—3464no—
130C2×Q16—1732no—
131C2×Q16—1732no—
132#131—3464no—
133C2×M16—1731no—
134C2×M16—1732no—
135C22×C8—1732no—
136C22×C8—1731no—
137C4⋊Q8—21no—
138C4⋊Q8—3461no—
139#124—21no—
140#124—3461no—
141#123—22no—
142#123—3462no—
143C22⋊Q8—1734no—
144C22⋊Q8C2×D17314yes—
145#117—1731no—
146#117C2×D17311yescharacteristic
147#107—1732no—
148#106—1732no—
149#103—1734no—
150#103—1734no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3461no—
157C2×Q8—3462no—
158C2×Q8—3462no—
159C2×Q8—1732no—
160C2×Q8C22⋊D17311yescharacteristic
161C2×Q8C22×D17312yes—
162C2×D4C22⋊D17312yes—
163C2×D4C22⋊D17311yescharacteristic
164C22×C4—21no—
165C22×C4—3461no—
166C22×C4C22×D17312yes—
167C22×C4C22×D17311yescharacteristic
168Q16—3464no—
169Q16—3464no—
170M16—3462no—
171M16—3464no—
172C2×C8—3464no—
173C2×C8—1732no—
174C2×C8—1734no—
175C2×C8—1731no—
176C2×C8—1731no—
177C4⋊C4—24no—
178C4⋊C4—3464no—
179C4⋊C4—1732no—
180C4⋊C4C22×D17312yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—3461no—
184C22⋊C4—3464no—
185C42—21no—
186C42—3461no—
187C23C2×C22⋊D17312yes—
188C23C2×C22⋊D17311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3462no—
194Q8—3462no—
195Q8—6922no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—6922no—
206C2×C4D4×D17312yes—
207C2×C4C2×C22⋊D17312yes—
208C2×C4C2×C22⋊D17311yescharacteristic
209C2×C4#1458382512yes—
210C2×C4C23×D17311yescharacteristic
211C8—3462no—
212C8—1732no—
213C8—1732no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1914965112yes—
218C22C22×C22⋊D17311yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—6922no—
223C4—6922no—
224C4#1914965112yes—
225C4C2×D4×D17311yescharacteristic
226C4C2×#1458382511yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1914965111yescharacteristic
230C2#2601230612yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×C173C22
hypercenter#536D173
hypocenterC173#3059
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#14583816C2semidirect
4. #26011747C2semidirect
5. #26011730C2semidirect
6. #26011527C2semidirect
7. #26011500C2semidirect
8. #26011499C2semidirect
115. C173#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#145838162
C2×#191496862
#260120622
#260117472
#260117302
#260115272
#260115002
#260114992
#536×C1732
#3059173