Finite Group Databasebeta

← #36354758

Subgroups — #36354758

Order 22144.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#19149655C211yesmaximal, characteristic
2C2×#19149611C211yesmaximal, characteristic, complemented
3C2×#19149603C211yesmaximal, characteristic, complemented
4C2×#19149591C211yesmaximal, characteristic
5#26011348C211yesmaximal, characteristic
6#26011253C211yesmaximal, characteristic, complemented
7C2×#103×C173C211yesmaximal, characteristic, complemented
8C22×C4×D173C2211yescharacteristic
9C22×#13063436C2211yescharacteristic
10#19149655—22no—
11#19149655—22no—
12C2×C4⋊Dic173C2211yescharacteristic
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583805—21no—
15C2×#14583803—21no—
16C2×#14583803—21no—
17#19149611—22no—
18#19149603—22no—
19#19149591—22no—
20#19149586—21no—
21C22×Q8×C173C2211yescharacteristic
22C22×C8×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic
24#103×C173—22no—
25C23×D173D411yescharacteristic
26C22×Dic173—21no—
27C22×Dic173—21no—
28C22×Dic173D411yescharacteristic
29C2×C4×D173—21no—
30C2×C4×D173—21no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—21no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C2×#13063436—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173—21no—
41C4⋊Dic173D412yes—
42#14583805—24no—
43#14583803—24no—
44#14583803—42no—
45C2×Q8×C173—21no—
46C2×Q8×C173—21no—
47C2×Q8×C173—21no—
48C2×Q8×C173—21no—
49C2×Q8×C173—21no—
50C2×Q8×C173D412yes—
51C22×C4×C173—21no—
52C22×C4×C173—21no—
53C22×C4×C173C2311yescharacteristic
54C2×C8×C173—21no—
55C2×C8×C173—21no—
56C2×C8×C173D412yes—
57C4⋊C4×C173—21no—
58C4⋊C4×C173—21no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173SD1612yes—
66C2×Dic173—21no—
67C2×Dic173—22no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C4×D173—41no—
76C4×D173—41no—
77C4×D173—41no—
78C4×D173—41no—
79#13063436—41no—
80C23×C173C2×D411yescharacteristic
81Q8×C173—22no—
82Q8×C173—22no—
83Q8×C173—41no—
84Q8×C173—41no—
85C2×C4×C173—22no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173#4111yescharacteristic
95C2×C4×C173#4111yescharacteristic
96C2×C4×C173#4111yescharacteristic
97C8×C173—22no—
98C22×C173#12011yescharacteristic
99C22×C173#12111yescharacteristic
100C22×C173#12311yescharacteristic
101C22×C173C2×SD1611yescharacteristic
102C22×C173#13511yescharacteristic
103C22×C173#13711yescharacteristic
104C22×C173#13711yescharacteristic
105C2×D173—22no—
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—41no—
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—41no—
112C4×C173—42no—
113C4×C173—81no—
114C4×C173#12111yescharacteristic
115C4×C173#12111yescharacteristic
116C4×C173#12311yescharacteristic
117C4×C173#12411yescharacteristic
118Dic173—41no—
119Dic173—42no—
120Dic173—81no—
121C2×C173#39411yescharacteristic
122C2×C173#44811yescharacteristic
123C2×C173#45211yescharacteristic
124C2×C173#46511yescharacteristic
125C2×C173#47011yescharacteristic
126C2×C173#48311yescharacteristic
127C2×C173#48511yescharacteristic
128D173—42no—
129C173#203511yescharacteristic, complemented, Sylow
130#2035—1731nomaximal, Sylow
131C2×C22⋊Q8—1731no—
132C2×#106—1731no—
133C2×#103—1731no—
134C2×#103D17311yescharacteristic, complemented
135C2×C22⋊C8—1731no—
136#380—1731no—
137#340—1731no—
138C22×Q8C2×D17311yescharacteristic
139C23×C4—1731no—
140C22×C8—1731no—
141C22×C8C2×D17311yescharacteristic
142C22⋊Q8—3462no—
143C22⋊Q8—3462no—
144C2×C4⋊C4—3461no—
145C2×C4⋊C4—1731no—
146C2×C4⋊C4—1731no—
147C2×C4⋊C4C2×D17311yescharacteristic
148C2×C22⋊C4—3461no—
149C2×C22⋊C4—3461no—
150#106—3462no—
151#103—22no—
152#103—3462no—
153C22⋊C8—3462no—
154#95—3461no—
155C24—1731no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8D69212yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—3461no—
165C22×C4—3461no—
166C22×C4—3461no—
167C22×C4—3461no—
168C22×C4—3461no—
169C22×C4—3461no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4—1731no—
173C22×C4C22×D17311yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—3461no—
177C2×C8—3461no—
178C2×C8—3461no—
179C2×C8C22⋊D17312yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—3464no—
183C4⋊C4—3461no—
184C4⋊C4—3461no—
185C4⋊C4—3461no—
186C4⋊C4—1732no—
187C22⋊C4—3464no—
188C22⋊C4—6922no—
189C23—3461no—
190C23—3461no—
191C23—3461no—
192C23—3461no—
193C23—3461no—
194C23—3461no—
195C23—1732no—
196C23D4×D17311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3462no—
208C2×C4—3461no—
209C2×C4—3461no—
210C2×C4—3461no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6921no—
217C2×C4—6921no—
218C2×C4—6921no—
219C2×C4—6921no—
220C2×C4D4×D17311yescharacteristic
221C2×C4C2×C22⋊D17311yescharacteristic
222C2×C4C2×D69211yescharacteristic
223C2×C4#1458382511yescharacteristic
224C2×C4#1458382511yescharacteristic
225C2×C4#1458382611yescharacteristic
226C8—22no—
227C8—6921no—
228C22—3462no—
229C22—3462no—
230C22—3462no—
231C22—6921no—
232C22—6921no—
233C22—6921no—
234C22—6921no—
235C22#1914964911yescharacteristic
236C22#1914965011yescharacteristic
237C22#1914966111yescharacteristic
238C22SD16×D17311yescharacteristic
239C22#1914970011yescharacteristic
240C22#1914971111yescharacteristic
241C22#1914971111yescharacteristic
242C4—42no—
243C4—81no—
244C4—13841no—
245C4—6922no—
246C4—6921no—
247C4#1914965111yescharacteristic
248C4#1914965311yescharacteristic
249C4#1914966311yescharacteristic
250C4#1914966511yescharacteristic
251C2—6922no—
252C2#2601140711yescharacteristic
253C2#2601163111yescharacteristic
254C2#2601165611yescharacteristic
255C2#2601171711yescharacteristic
256C2#2601174611yescharacteristic
257C2#2601182011yescharacteristic
258C2#2601183011yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#103×C173C2
socleC23×C173C2×D4
prime radicalC23×D173D4
hypercenterC2×#103D173
hypocenterC173#2035
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#19149611C2semidirect
3. C2×#19149603C2semidirect
6. #26011253C2semidirect
7. C2×#103×C173C2semidirect
129. C173#2035semidirect
134. C2×#103D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496552
C2×#191496112
C2×#191496032
C2×#191495912
#260113482
#260112532
C2×#103×C1732
#2035173