Finite Group Databasebeta

← #36354010

Subgroups — #36354010

Order 22144.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#26012003C211yesmaximal, characteristic
2C2×#19149636C211yesmaximal, characteristic, complemented
3C2×#19149607C211yesmaximal, characteristic, complemented
4C2×#19149605C211yesmaximal, characteristic, complemented
5#26011454C211yesmaximal, characteristic, complemented
6#26011250C211yesmaximal, characteristic, complemented
7#339×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C22×C4×D173C2211yescharacteristic
10C2×#14583810C2211yescharacteristic
11#19149663—21no—
12#19149659C412yescomplemented
13#19149654C412yescomplemented
14#19149649—21no—
15C4×D692—22no—
16#19149636—22no—
17#19149636—22no—
18C2×#14583805—21no—
19C2×#14583803—21no—
20C2×C4×Dic173C2211yescharacteristic
21#19149607—22no—
22#19149605—22no—
23#19149591—22no—
24C2×M16×C173C2211yescharacteristic
25#117×C173C2211yescharacteristic
26C2×C42×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583826—21no—
29#14583826—21no—
30#14583826—21no—
31#14583826—21no—
32C2×Dic346C2×C411yescharacteristic
33C2×D692—22no—
34C2×D692C2×C411yescharacteristic
35C2×C22⋊D173—21no—
36C22×Dic173—21no—
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40C2×C4×D173—21no—
41C2×C4×D173—21no—
42C2×C4×D173—21no—
43C2×C4×D173C2×C412yes—
44C2×C4×D173D412yes—
45#14583810—21no—
46#14583810—21no—
47C2×#13063436—21no—
48C4⋊Dic173—22no—
49#14583805—24no—
50#14583803—24no—
51#14583803—22no—
52#14583803—22no—
53C4×Dic173—21no—
54C4×Dic173D412yescomplemented
55C22×C4×C173—21no—
56C22×C4×C173C2311yescharacteristic
57M16×C173—21no—
58M16×C173—21no—
59C2×C8×C173—21no—
60C4⋊C4×C173C2×C412yes—
61C22⋊C4×C173—21no—
62C42×C173—21no—
63C42×C173—21no—
64C42×C173D412yescomplemented
65C22×D173—21no—
66C22×D173—21no—
67C22×D173—21no—
68C22×D173—42no—
69C22×D173C22⋊C412yes—
70Dic346—21no—
71Dic346—21no—
72D692—21no—
73D692—21no—
74D692—44no—
75C22⋊D173—41no—
76C22⋊D173—41no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—22no—
81C2×Dic173—42no—
82C2×Dic173C22⋊C412yes—
83C4×D173—22no—
84C4×D173—22no—
85C4×D173—22no—
86C4×D173—22no—
87C4×D173—41no—
88C4×D173—41no—
89#13063436—41no—
90C23×C173C2×D411yescharacteristic
91C2×C4×C173—22no—
92C2×C4×C173—22no—
93C2×C4×C173—42no—
94C2×C4×C173C22×C411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173#4111yescharacteristic
99C2×C4×C173#4111yescharacteristic
100C8×C173—41no—
101C22×C173—22no—
102C22×C173C2×C22⋊C411yescharacteristic
103C22×C173#12011yescharacteristic
104C22×C173#12311yescharacteristic
105C2×D173—22no—
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—41no—
111C2×D173—42no—
112C4×C173—42no—
113C4×C173—42no—
114C4×C173C4×D411yescharacteristic
115C4×C173C4×D411yescharacteristic
116C4×C173#12111yescharacteristic
117C4×C173#12111yescharacteristic
118Dic173—22no—
119Dic173—42no—
120Dic173—41no—
121C2×C173—21no—
122C2×C173—21no—
123C2×C173#38611yescharacteristic
124C2×C173#42112yes—
125D173—42no—
126D173—81no—
127C173#189711yescharacteristic, complemented, Sylow
128#1897—1731nomaximal, Sylow
129#525—1731no—
130C2×#117—1731no—
131C2×#104—1731no—
132C2×#104—1731no—
133#407—1731no—
134#339—1731no—
135#339D17311yescharacteristic, complemented
136C2×#41—1731no—
137C23×C4—1731no—
138C2×M16—1731no—
139C2×M16C2×D17311yescharacteristic
140#123—3461no—
141C22⋊Q8—1732no—
142#121—1732no—
143#120—3461no—
144C4×D4—3462no—
145#117—3462no—
146#117—3462no—
147#117C2×D17311yescharacteristic
148C2×C4⋊C4—3461no—
149C2×C22⋊C4—3461no—
150C2×C42—1731no—
151C2×C42C2×D17311yescharacteristic
152#104—3462no—
153#104—3462no—
154C22⋊C8—3462no—
155C24—1731no—
156#41—3461no—
157#41—3461no—
158#41—3461no—
159#41—3461no—
160C2×Q8—1731no—
161C2×D4—3462no—
162C2×D4—3461no—
163C2×D4—1731no—
164C22×C4—21no—
165C22×C4—3461no—
166C22×C4—3461no—
167C22×C4—3461no—
168C22×C4—3461no—
169C22×C4—3461no—
170C22×C4—3461no—
171C22×C4—1731no—
172C22×C4—1732no—
173C22×C4—1732no—
174C22×C4C22×D17311yescharacteristic
175M16—21no—
176M16—21no—
177M16—3461no—
178M16—3461no—
179C2×C8—21no—
180C2×C8—3461no—
181C4⋊C4—3464no—
182C4⋊C4—3462no—
183C4⋊C4C4×D17312yes—
184C22⋊C4—21no—
185C22⋊C4—3464no—
186C22⋊C4—3462no—
187C22⋊C4—3462no—
188C42—21no—
189C42—21no—
190C42—3461no—
191C42—1732no—
192C42D69212yescomplemented
193C23—3461no—
194C23—3461no—
195C23—3461no—
196C23—1732no—
197C23—6922no—
198C23D4×D17311yescharacteristic
199Q8—3461no—
200Q8—3461no—
201D4—3461no—
202D4—3461no—
203D4—6924no—
204D4—6921no—
205D4—6921no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3461no—
210C2×C4—3461no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—3462no—
214C2×C4—3462no—
215C2×C4—3461no—
216C2×C4—3462no—
217C2×C4—1732no—
218C2×C4—6922no—
219C2×C4—6921no—
220C2×C4—6921no—
221C2×C4C2×C4×D17311yescharacteristic
222C2×C4D4×D17311yescharacteristic
223C2×C4D4×D17311yescharacteristic
224C2×C4C2×D69211yescharacteristic
225C2×C4#1458382611yescharacteristic
226C2×C4#1458382711yescharacteristic
227C8—41no—
228C8—6921no—
229C22—22no—
230C22—3462no—
231C22—3462no—
232C22—3462no—
233C22—6922no—
234C22—6922no—
235C22—6922no—
236C22—6921no—
237C22C22⋊C4×D17311yescharacteristic
238C22#1914964911yescharacteristic
239C22#1914966411yescharacteristic
240C4—42no—
241C4—42no—
242C4—3462no—
243C4—6922no—
244C4—6921no—
245C4#1914964111yescharacteristic
246C4C4×D69211yescharacteristic
247C4#1914965211yescharacteristic
248C4#1914965411yescharacteristic
249C2—21no—
250C2—21no—
251C2—13841no—
252C2—6922no—
253C2#2601136911yescharacteristic
254C2#2601150612yes—
subgroupG/H
centerC2×C4D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#339×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012003C2
cosocleC22×C4×C173C23
hypercenter#339D173
hypocenterC173#1897
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#19149636C2semidirect
3. C2×#19149607C2semidirect
4. C2×#19149605C2semidirect
5. #26011454C2semidirect
6. #26011250C2semidirect
7. #339×C173C2semidirect
12. #19149659C4semidirect
13. #19149654C4semidirect
54. C4×Dic173D4semidirect
64. C42×C173D4semidirect
127. C173#1897semidirect
135. #339D173semidirect
192. C42D692semidirect

Maximal subgroups

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

subgroupindex
#260120032
C2×#191496362
C2×#191496072
C2×#191496052
#260114542
#260112502
#339×C1732
#1897173