Finite Group Databasebeta

← #36352584

Subgroups — #36352584

Order 22144.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012049C211yesmaximal, characteristic
2#26011810C211yesmaximal, characteristic, complemented
3#26011807C211yesmaximal, characteristic, complemented
4#26011428C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C2×D4×D173—21no—
9#19149664—21no—
10#19149661—21no—
11#19149653C2211yescharacteristic
12#19149651—21no—
13#19149649—21no—
14#19149647—21no—
15C2×C4⋊Dic173—21no—
16C2×C4⋊Dic173C2211yescharacteristic, complemented
17C2×#14583803—21no—
18C2×#14583803C2211yescharacteristic
19#19149611—21no—
20#19149610—21no—
21#19149601—21no—
22#19149600—21no—
23C22⋊#13063436C2211yescharacteristic, complemented
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic, complemented
26#95×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×D692—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34D4×D173—42no—
35D4×D173—42no—
36C22×Dic173—21no—
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39C2×#13063436—21no—
40C4⋊Dic173—21no—
41C4⋊Dic173—21no—
42C4⋊Dic173—42no—
43C4⋊Dic173D411yescharacteristic
44#14583805—21no—
45C22⋊Dic173—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—42no—
51C2×D4×C173—21no—
52C2×D4×C173D411yescharacteristic
53C22×C4×C173—21no—
54C22×C4×C173C2311yescharacteristic
55C2×C8×C173—21no—
56C4⋊C4×C173D411yescharacteristic
57C22⋊C4×C173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—42no—
62C22×D173—41no—
63C22×D173—41no—
64C22×D173—81no—
65C22×D173—81no—
66C22×D173SD1612yes—
67D692—41no—
68D692—42no—
69C22⋊D173—42no—
70C22⋊D173—41no—
71C22⋊D173—41no—
72C22⋊D173—81no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—41no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C4×D173—42no—
80#13063436—41no—
81C23×C173—21no—
82C23×C173C2×D411yescharacteristic
83D4×C173—41no—
84D4×C173—41no—
85D4×C173—41no—
86C2×C4×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2×D411yescharacteristic
93C8×C173—41no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173#12011yescharacteristic
100C22×C173C2×SD1611yescharacteristic
101C22×C173AGL(1,ℤ/8)11yescharacteristic
102C2×D173—22no—
103C2×D173—41no—
104C2×D173—41no—
105C2×D173—42no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—41no—
109C2×D173—81no—
110C2×D173—84no—
111C4×C173—21no—
112C4×C173—21no—
113C4×C173—42no—
114C4×C173—41no—
115Dic173—41no—
116Dic173—41no—
117Dic173—81no—
118C2×C173—21no—
119C2×C173—21no—
120C2×C173—81no—
121C2×C173#45011yescharacteristic
122C2×C173#45311yescharacteristic
123C2×C173#45811yescharacteristic
124D173—42no—
125D173—81no—
126C173#161311yescharacteristic, complemented, Sylow
127#1613—1731nomaximal, Sylow
128#534—1731no—
129#481—1731no—
130#481—1731no—
131#399—1731no—
132#327—1731no—
133#327D17311yescharacteristic, complemented
134#323—1731no—
135C22×D4—3461no—
136#123—3461no—
137#123—3461no—
138#121—3461no—
139#121—1731no—
140#121C2×D17311yescharacteristic
141#120—3461no—
142#120—3461no—
143C2×C4⋊C4—3461no—
144C2×C4⋊C4—1731no—
145C2×C22⋊C4—3461no—
146C2×C22⋊C4—1731no—
147#106—3461no—
148#106—3461no—
149#102—3461no—
150#102—3461no—
151C22⋊C8—1731no—
152C22⋊C8C2×D17311yescharacteristic, complemented
153#95C2×D17311yescharacteristic, complemented
154C24—3461no—
155C24—1731no—
156C2×D4—21no—
157C2×D4—3461no—
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—3461no—
161C2×D4—1731no—
162C2×D4—6922no—
163C2×D4—6922no—
164C2×D4D69211yescharacteristic
165C22×C4—21no—
166C22×C4—3461no—
167C22×C4—3461no—
168C22×C4—1731no—
169C22×C4C22×D17311yescharacteristic
170C2×C8—21no—
171C2×C8—3461no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C4⋊C4—3461no—
175C4⋊C4—1731no—
176C4⋊C4—6922no—
177C4⋊C4D69211yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—6922no—
185C23—21no—
186C23—3461no—
187C23—3461no—
188C23—3461no—
189C23—13841no—
190C23—13841no—
191C23—1732no—
192C23—6921no—
193C23—6921no—
194C23—6922no—
195C23D4×D17311yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—13841no—
200D4—6921no—
201D4—6922no—
202D4—6922no—
203D4—6921no—
204D4—6921no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3461no—
211C2×C4—3461no—
212C2×C4—6922no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6921no—
217C2×C4D4×D17311yescharacteristic
218C2×C4C2×D69211yescharacteristic
219C8—41no—
220C8—6921no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—3462no—
227C22—13841no—
228C22—13844no—
229C22—6922no—
230C22—6921no—
231C22—6921no—
232C22—6921no—
233C22—6921no—
234C22—6921no—
235C22#1914964911yescharacteristic
236C22SD16×D17311yescharacteristic
237C22#1914970311yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—13841no—
243C4—6921no—
244C4—6921no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—13841no—
249C2—6922no—
250C2#2601164311yescharacteristic
251C2#2601166011yescharacteristic
252C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173#120
prime radical#26012049C2
hypercenter#327D173
hypocenterC173#1613
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26011810C2semidirect
3. #26011807C2semidirect
4. #26011428C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #327×C173C2semidirect
16. C2×C4⋊Dic173C22semidirect
23. C22⋊#13063436C22semidirect
25. C22⋊C8×C173C22semidirect
26. #95×C173C22semidirect
126. C173#1613semidirect
133. #327D173semidirect
152. C22⋊C8C2×D173semidirect
153. #95C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120492
#260118102
#260118072
#260114282
#260112242
#260112142
#327×C1732
#1613173